numerical assessment of material models for coated fabrics in...

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Permission for use of content

The author gives permission to make this master dissertation available for consultation and to copy parts of this master dissertation for personal use. In the case of any other use, the limitations of the copyright have to be respected, in particular with regard to the obligation to state expressly the source when quoting results from this master dissertation. Kristine Vandenboer, 4 juni 2012

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Toelating tot bruikleen

De auteur geeft de toelating deze masterproef voor consultatie beschikbaar te stellen en delen van de masterproef te kopiëren voor persoonlijk gebruik. Elk ander gebruik valt onder de beperkingen van het auteursrecht, in het bijzonder met betrekking tot de verplichting de bron uitdrukkelijk te vermelden bij het aanhalen van resultaten uit deze masterproef. Kristine Vandenboer, 4 juni 2011

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Acknowledgements

The completion of this thesis has only been possible with the help of several people. Firstly, I would like to thank prof. dr. ir. Wim Van Paepegem and prof. dr. ir.-arch. Marijke Mollaert for giving me the opportunity to work on this fascinating subject. I am especially appreciative to prof. Van Paepegem for always being supportive. The help of my supervisor dr. ir. Ali Rezaei is acknowledged, for providing the necessary guidance throughout my thesis and the useful introduction to Abaqus. Furthermore, I am grateful to several people of the Free University of Brussels for providing extensive experimental data: Paolo Topalli, dr. ir. Lars de Laet and prof. dr. ir. Danny Van Hemelrijck. I also want to thank dr. ir. Ives De Baere, ir. Nicolas Lammens and ir. Klaas Allaer who helped me with the microscopical examination of the tent fabric. Of course, the support of my parents has been essential throughout my studies and master thesis. Finally, I am very grateful to my boyfriend Mathias for believing in me and encouraging me in the moments I needed it the most with his patience, love and endless support.

Kristine Vandenboer, 4 juni 2012

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Numerical assessment of material models for

coated fabrics in foldable tent structures

by

Kristine Vandenboer

Masterproef ingediend tot het behalen van de academische graad van Master in de ingenieurswetenschappen: Bouwkunde

Promotoren: prof. dr. ir. Wim Van Paepegem, prof. dr. Ir. Marijke Mollaert

Begeleider: dr. ir. Ali Rezaei

Vakgroep Toegepaste materiaalwetenschappen Voorzitter: prof. dr. ir. Joris Degrieck

Faculteit Ingenieurswetenschappen en Architectuur Universiteit Gent

Academiejaar 2011–2012

Summary

At present days, the design of tent structures is based on highly simplified material models. The

aim of this thesis concerns the numerical assessment of material models for the structural

behavior of a coated fabric. To this end, the material behavior of a coated fabric is studied with

the use of both uniaxial and biaxial experiments. To obtain representative and reproducible

experimental results, an appropriate geometry of the biaxial test sample is of major importance.

For this reason, a numerical study is performed to the influence of (i) specimen size, (ii) presence

of arm slits, (iii) number of arm slits and (iv) location of arm slits, for the stress field in the central

part of the cruciform sample. Based on the obtained knowledge of the experimentally recorded

stress-strain curves, the suitability of a material model for simulating the behavior of a coated

fabric true-to-nature is judged. Several built-in material models in the finite element method

software Abaqus are examined and discussed. Starting from a very simple linear elastic material

model, we evolve to several test data based fabric material models and finally end up with the

anisotropic Hill plasticity material model.

Keywords

Coated fabric, uniaxial and biaxial experiments, Finite element method (FEM), anisotropic Hill

plasticity

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Numerical assessment of material models for coated fabrics in foldable tent structures

Kristine Vandenboer

Supervisors: dr. ir. Ali Rezaei, prof. dr. ir. Marijke Mollaert, prof. dr. ir. Wim Van Paepegem

Abstract- At present, the design of tent structures is largely

based on highly simplified material models, leading to inaccurate results. The aim of the present thesis concerns the numerical assessment of the structural behavior of a coated fabric. Several material models in the finite element method (FEM) package Abaqus are investigated, discussed and judged.

Both uniaxial and biaxial tensile tests have been performed and analyzed in order to accurately survey the material behavior of a coated fabric. With the gained knowledge, a numerical FEM model is set up, in which various material models have been defined. The numerical computations reveal that both the test data based FABRIC model and the Hill plasticity model are appropriate for the modeling and simulation of a coated fabric.

Keywords- Coated fabric, uniaxial and biaxial experiments, Finite Element Method (FEM), anisotropic plasticity

I. INTRODUCTION

Besides the classic permanent applications, tent structures currently gain a lot of interest as temporary shelters in disaster areas or festival zones in the form of foldable structures. Current popular design methods are largely based on highly simplified material models, leading to inaccurate and misleading results, while others are so advanced and complex, demanding a unrealistic computational time for the simulation of a complete tent structure. This study strives for the golden mean, i.e. a representative material model in combination with an acceptable computation time. The proposed material models are judged on the basis of the results of the performed uniaxial and biaxial experiments.

II. MATERIAL BEHAVIOR

The coated fabric consists of woven polyester fibers covered with a polyvinylchloride (PVC) coating. It is a flexible material, having a completely different structural behavior compared to conventional rigid materials. Due to the particular weaving method, the fibers in warp direction are initially straight while the fibers in fill direction are curved. Applying an external load to the fabric results in a reallocation of the fibers, and consequently large strains. The level of fiber reallocation depends on both the load ratio and the load history [1, 2].

A. Uniaxial test

Contrary to the warp direction, large initial strains are observed in fill direction during a uniaxial test (orange arrow in figure 1 (b)), which are related to the straightening of the fibers. When decreasing the load to a pre-stress value of 2.5 MPa, the fibers in fill direction stay straight. When further reducing the applied load, the fibers partially curve back. Plasticity phenomena are observed when exceeding the yield

stress (≈ 14 MPa) in both warp and fill direction (blue arrow in figure 1 (a-b)).

Figure 1: Recorded stress-strain curve for a uniaxial test with different load cycles: warp direction (a) and fill direction (b).

B. Biaxial test

The structural behavior of a coated fabric, when applying a biaxial stress state, is investigated on a cruciform sample. The sample consists of a central part and four arms to which slits are applied. A numerical study has been performed to the influence of the geometry of the cruciform sample on the stress distribution in the central part. It is revealed that the geometrical shape of the slit tips as well as the size of the cruciform sample have a negligible influence. The number of slits and their spatial location on the other hand have a substantial influence: each slit in the arms induces a stress peak in the central part of the sample. A larger number of slits leads to smaller stress variations and a higher average stress value, which is advantageous for the correct and unambiguous interpretation of an experiment. However, the insertion of the slits is a time-consuming process, for which the neccessary expertise is required. The results of a biaxial test with each time 3 cycles of the sequencing load ratios 1:1, 2:1, 1:2, 1:0, 0:1 and 1:1 with a maximum stress of 24.1 MPa, are presented in figure 2.

Figure 2: Recorded stress-strain curve for a biaxial test with different load ratios en a maximum stress value of 24.1 MPa: warp direction

(a) and fill direction (b).

For each load ratio, a different slope in the stress-strain diagram can be observed. When passing on to the next load ratio, a small increase or decrease of the permanent strain is observed. It can be stated that the stress-strain behavior is stabilized after 3 identical loading cycles [2]. The cycles with load ratio 1:1 are applied at the beginning as well as at the end

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of the biaxial experiment. The recorded permanent strains at the end of the respective cycles differ from each other, which indicates that repeating a cycle cannot fully remove the influence of the load history.

III. MATERIAL MODELS

C. Linear elastic model

Many software packages are based on a linear elastic material model [3]. However a coated fabric exhibits both non-linear effects and permanent strain (figure 3). Hence, the quest for more advanced simulation techniques imparts a significant challenge in current research.

Figure 3: Numerical calculation based on a linear elastic material model (FEM) and recorded stress-strain curve for a uniaxial test:

warp direction (a) and fill direction (b).

D. Test data based FABRIC model

The FABRIC model based on experimental data is able to combine non-linearity, orthotropy and permanent strain. By implementing the experimentally observed large initial strain in fill direction, which is understood in terms of the straightening of the initially curved fill fibers, the numerical model completely accounts for this effect in a uniaxial test (figure 4 (a-b)).

Figure 4: Numerical calculation based on a test data based FABRIC model (FEM) and recorded stress-strain curve for warp direction and

fill direction: uniaxial test (a) and biaxial test (b).

Since the FABRIC model consists of a phenomenological description, the influence of the load ratio is not taken into account. This causes the strain to be overestimated for a biaxial test in the FEM model (figure 4 (c-d)). An even more important drawback concerns the missing of a coupling mechanism between the warp direction and the fill direction in the FABRIC model. This incorrect feature is reflected in the insensitivity of the slopes with respect to the load ratio in the numerically computed stress-strain curves (figure 4 (c-d)).

E. Combined orthotropic elastic-plastic Hill model

It can be expected that an elastic-plastic model provides a more realistic description for the coated fabric. Though, a major disadvantage concerns the artificial implementation of the reallocation of the fibers in the yield behavior. Furthermore, the anisotropy of the yield behavior is defined by means of a single parameter. The latter obviously limits the correct implementation of the stress-strain curve to only one of the orthotropic directions. The uniaxial and biaxial results, in which the elastic-plastic behavior is harmonized to the fill direction, are presented in figure 5. This is clearly reflected by the inferior results obtained in warp direction.

Figure 5: Numerical calculation based on the Hill model (FEM) and

recorded stress-strain curve for warp direction and fill direction: uniaxial test (a-b) and biaxial test (c-d).

IV. CONCLUSION

Both the test data based FABRIC model and the combined orthotropic elastic-plastic Hill model are investigated. It was found that both models are more appropriate to model the structural behavior of a coated fabric, compared to the widespread linear elastic material model. Both material models take into account non-linear effects as well as plasticity phenomena. The results are in reasonable agreement with the experimental uniaxial and biaxial results. Though, caution is needed: in this study, the material models were applied to small-scale samples. Further research is required to grasp the material behavior of a complete foldable tent structure.

V. REFERENCES

[1] Bridgens, B.N., P.D. Gosling, and M.J.S. Birchall, Membrane material behavior: concepts, practice and developments, in The structural engineer. 2003. p. 6.

[2] Galliot, C. and R.H. Luchsinger, Determination of the response of coated fabrics under biaxial stress: Comparison between different test procedures, in International conference on textile composites and inflatable structures: Strucural membranes, E. Onate, B. Kroplin, and K.-U. Bletzinger, Editors. 2011. p. 12.

[3] Uhlemann, J., et al., Effects on elastic constants of technical membranes applying the evaluation methods of MSAJ/M-02-1995, in International conference on Textile Composites and Inflatable Structures: Structural membranes, O. E., K. B., and B. K.-U., Editors. 2011. p. 12.

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Numerieke studie naar materiaalmodellen voor gecoate weefsels van opvouwbare tentstructuren

Kristine Vandenboer

Begeleiders: dr. ir. Ali Rezaei, prof. dr. ir. Marijke Mollaert, prof. dr. ir. Wim Van Paepegem

Abstract- Het ontwerp van tentconstructies gebeurt momenteel met behulp van softwarepakketten die gebaseerd zijn op sterk vereenvoudigde materiaalmodellen, waardoor zeer inaccurate resultaten worden bekomen. Het doel van deze thesis is de geschiktheid van verschillende materiaalmodellen in Abaqus te onderzoeken en te beoordelen voor een gecoat weefsel.

Uniaxiale en biaxiale trektesten zijn uitgevoerd en geanalyseerd om het materiaalgedrag van een gecoat weefsel zo accuraat mogelijk in kaart te brengen. Met behulp van de verworven kennis uit de experimenten, is een numeriek model opgesteld. De numerieke berekeningen onthullen dat zowel het FABRIC model, met als input werkelijke testgegevens, als het Hill plasticiteitsmodel geschikt zijn om het materiaalgedrag van een gecoat weefsel realistisch te modelleren en simuleren.

Trefwoorden- Gecoat weefsel, uniaxiale en biaxiale experimenten, eindige elementen methode (EEM), anisotrope plasticiteit

I. INLEIDING

Tentconstructies worden, behalve voor permanente toepassingen, steeds vaker ingezet als tijdelijke schuilplaatsen in rampgebieden en op festivalweides onder de vorm van opvouwbare structuren. De bestaande ontwerpmethoden zijn echter gebaseerd op uiterst vereenvoudigde materiaalmodellen die leiden tot grote onnauwkeurigheden en zelfs tot problemen tijdens de montage van tentstructuren. Andere materiaal-modellen zijn daarentegen zo geavanceerd en complex dat deze gepaard gaan met extreem hoge rekentijden, en bijgevolg onbruikbaar zijn bij het ontwerp van een volledige tentconstructie. Deze studie bewandelt de gulden middenweg, namelijk de combinatie van een accuraat materiaalmodel en een acceptabele berekeningstijd. De voorgestelde materiaal-modellen worden beoordeeld op basis van de resultaten van de uitgevoerde uni- en biaxiaaltesten.

II. MATERIAALGEDRAG

Het gecoat weefsel bestaat uit gewoven polyestervezels bedekt met een polyvinylchloride (PVC) coating. Het is een soepel materiaal waarvan het structureel gedrag sterk verschilt van alledaagse starre materialen. De weefmethode zorgt ervoor dat de vezels in scheringrichting aanvankelijk recht zijn, terwijl de vezels in inslagrichting gekromd zijn. Wanneer het weefsel onderworpen wordt aan een uni- of biaxiale trek-belasting, zullen de vezels zich herschikken, namelijk strekken of krommen, wat resulteert in grote vervormingen. De mate van herschikking van de vezels is afhankelijk van zowel de belastingverhouding als de belastinggeschiedenis [1, 2].

A. Uniaxiaaltest

In tegenstelling tot de scheringrichting, leidt een uniaxiaal-test in inslagrichting tot grote initiële rekken (oranje pijl in figuur 1 (b)), die veroorzaakt worden door het strekken van de

vezels. Bij het verminderen van de belasting tot 2.5 MPa blijven de inslagvezels gestrekt, bij lagere spanningen krom-men deze gedeeltelijk terug. Na het bereiken van de vloeigrens (≈ 14 MPa) treden plasticiteitsverschijnselen op in zowel schering- als inslagrichting (blauwe pijl in figuur 1(a-b)).

Figuur 1: Spanning-rek diagram opgemeten tijdens een uniaxiaaltest met verschillende belastings- en ontlastingscycli: scheringrichting (a)

en inslagrichting (b)

B. Biaxiaaltest

Het gedrag van het gecoat weefsel onder een biaxiale spanningstoestand wordt onderzocht met behulp van een kruisvormig proefstuk, dat bestaat uit een centraal deel en vier armen die verdeeld worden door snedes. Er is een numerieke studie uitgevoerd naar de invloed van de geometrie van het proefstuk op de spanningsverdeling in het centrale deel. Hierruit blijkt dat zowel de vorm van de snedeuiteindes als de grootte van het proefstuk een verwaarloosbare invloed hebben. Het aantal snedes en hun locatie daarentegen beinvloeden wel degelijk de spannings-toestand: elke snede in de arm induceert een spanningspiek in het centrale gedeelte. Een groter aantal snedes leidt tot kleinere lokale piekwaarden en een hogere gemiddelde spanningswaarde, wat voordelig is voor een correcte en eenduidige interpretatie van een experiment. Langs de andere kant is het aanbrengen van snedes een zeer tijdrovend proces, waarvoor de nodige expertise vereist is. In figuur 2 zijn de resultaten van een biaxiaal test met telkens 3 cycli van de opeenvolgende belastingverhoudingen 1:1, 2:1, 1:2, 1:0, 0:1 en 1:1 getoond, waarin de maximale aangelegde spanning 24.1 MPa bedraagt.

Figuur 2: Spanning-rek diagram opgemeten tijdens een biaxiaaltest met verschillende belastingverhoudingen en een maximale spanning

van 24.1 MPa: scheringrichting (a) en inslagrichting (b).

Met elke belastingverhouding stemt een verschillende helling van het spanning-rek diagram overeen. Bij de

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overgang naar een volgende belastingsverhouding, wordt een kleine toe- of afname van de permanente rek waargenomen. Men kan stellen dat na 3 identieke cycli, het spanning-rek gedrag zich stabiliseert [2]. De belasting-verhouding 1:1 wordt zowel aan het begin als aan het einde van het experiment aangelegd. De permanente rek opgemeten aan het eind van deze cycli is verchillend, waaruit blijkt dat het herhalen van eenzelfde cyclus de invloed van de belastinggeschiedenis niet volledig kan wegnemen.

III. MATERIAALMODELLERING

C. Lineair elastisch model

Veel softwaremodellen zijn gebaseerd op een lineair elastisch materiaalmodel [3]. Een gecoat weefsel vertoont echter zowel niet-lineair gedrag als permanente rek, wat de noodzaak voor meer geavanceerde materiaalmodellen uitdrukt (zie figuur 3).

Figuur 3: Berekening op basis van een lineair elastisch model (EEM)

en experimenteel spanning-rek diagram voor een uniaxiaaltest: scheringrichting (a) en inslagrichting (b).

D. FABRIC model gebaseerd op experimentele data

Het FABRIC model, gebaseerd op experimentele data, is in staat niet-lineariteit, orthotropie en permanente rek te combi-neren. Door de experimenteel waargenomen grote initiële rek in inslagrichting, ten gevolge van het strekken van de initieel gekromde vezels te implementeren, wordt dit effect in een uniaxiaaltest voorspeld door het EEM model (figuur 4 (a-b)).

Figuur 4: Berekening op basis van een FABRIC model met als input experimentele data (EEM) en opgemeten spanning-rek diagram voor schering- en inslagrichting: uniaxiaaltest (a-b) en biaxiaaltest (c-d).

Aangezien het een fenomenologische implementatie betreft, wordt de invloed van de belastingverhouding echter niet in acht genomen waardoor de rek, in het geval van een biaxiaaltest, overschat door het EEM model (figuur 4 (c-d)). Belangrijker is het ontbreken van een koppeling tussen schering- en inslagrichting in het FABRIC model, waardoor de verschillende hellingen voor verschillende belasting-verhoudingen niet tot uiting komen in het berekende spanning-rekdiagram (figuur 4 (c-d)).

E. Gecombineerd orthotroop elastisch-plastisch Hill model

Men kan verwachten dat een elastisch-plastisch model een meer realistische beschrijving biedt voor het gecoate weefsel. Het grote nadeel echter betreft de nodige implementatie van de vezelherschikking op een artificiële manier in het vloeigedrag. Daarenboven wordt de anisotropie van het Hill plasticiteitsmodel bepaald door slechts één factor. Dit beperkt een correcte implementatie van de spanning-rekdiagrammen tot één van beide orthotrope richtingen. Figuur 5 toont uniaxiale en biaxiale resultaten waarin het elastisch-plastisch gedrag is afgestemd op de inslagrichting. Dit wordt duidelijk gereflecteerd in de inferieure resultaten in de scheringrichting.

Figuur 5: Berekening op basis van het Hill model (EEM) en

experimenteel spanning-rek diagram voor schering- en inslagrichting: uniaxiaaltest (a-b) en biaxiaaltest (c-d).

IV. CONCLUSIE

Zowel het FABRIC model, gebaseerd op experimentele data, als het gecombineerd anisotroop elastisch-plastisch Hill model zijn veel performanter voor het modelleren van het materiaal gedrag van een gecoat weefsel, in vergelijking met het vaak gebruikte lineair elastisch model. Beide materiaal-modellen brengen zowel niet-lineaire effecten als plastische fenomenen in rekening, in goede overeenkomst met de experimentele resultaten. Voorzichtigheid is echter geboden: de materiaalmodellen werden toegepast op kleinschalige proefstukken, verder onderzoek is vereist naar het materiaalgedrag van een volledige opvouwbare tentstructuur.

V. REFERENTIES

[1] Bridgens, B.N., P.D. Gosling, and M.J.S. Birchall, Membrane material behavior: concepts, practice and developments, in The structural engineer. 2003. p. 6.

[2] Galliot, C. and R.H. Luchsinger, Determination of the response of coated fabrics under biaxial stress: Comparison between different test procedures, in International conference on textile composites and inflatable structures: Strucural membranes, E. Onate, B. Kroplin, and K.-U. Bletzinger, Editors. 2011. p. 12.

[3] Uhlemann, J., et al., Effects on elastic constants of technical membranes applying the evaluation methods of MSAJ/M-02-1995, in International conference on Textile Composites and Inflatable Structures: Structural membranes, O. E., K. B., and B. K.-U., Editors. 2011. p. 12.

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Contents

Permission for use of content ..................................................................................................................... i

Toelating tot bruikleen ................................................................................................................................ ii

Acknowledgements ..................................................................................................................................... iii

Overview ...................................................................................................................................................... iv

Extended abstract: English version ........................................................................................................... v

Extended abstract: Nederlandstalige versie ............................................................................................ vii

Contents........................................................................................................................................................ ix

Chapter 1. Introduction ......................................................................................................................... 1

1.1. Objectives ..................................................................................................................................... 1

1.2. History ........................................................................................................................................... 2

1.3. Coated fabric material and production techniques ................................................................. 2

1.4. Material used in this master thesis ............................................................................................ 4

1.4.1. Data sheet ............................................................................................................................. 4

1.4.2. Microscopic examination ................................................................................................... 5

1.5. Form and physical behavior [5, 6] ............................................................................................. 9

1.5.1. Tensioned surfaces ............................................................................................................ 10

1.4.3. Air supported structures ................................................................................................... 14

1.4.4. Air inflated structures ....................................................................................................... 15

1.4.5. Hybrid structures ............................................................................................................... 15

1.6. Quality characteristics of tensile structures ............................................................................ 16

1.6.1. Lightweight ......................................................................................................................... 16

1.6.2. Transportability ................................................................................................................. 16

1.6.3. Translucency ...................................................................................................................... 16

1.6.4. Flexibility ............................................................................................................................ 17

1.6.5. Sculptural ............................................................................................................................ 18

1.6.6. Safety ................................................................................................................................... 18

1.6.7. Weather protection ........................................................................................................... 18

1.6.8. Convertibility and adaptability ......................................................................................... 19

1.7. Internal environment ................................................................................................................ 19

1.7.1. Thermal environment ....................................................................................................... 19

1.7.2. Lighting environment ....................................................................................................... 20

1.7.3. Acoustical environment.................................................................................................... 21

Contents

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Chapter 2. Behavior and available modeling methods for coated fabrics .................................... 22

2.1. Introduction ............................................................................................................................... 22

2.2. Behavior of coated membranes ............................................................................................... 22

2.2.1. Nonlinear orthotropic behavior ...................................................................................... 22

2.2.2. Crimp interchange ............................................................................................................. 22

2.2.3. Load ratio dependency ..................................................................................................... 23

2.2.4. Cycle repetition .................................................................................................................. 24

2.2.5. Load history ....................................................................................................................... 25

2.2.6. Hysteresis ............................................................................................................................ 26

2.2.7. Strain rate ............................................................................................................................ 26

2.2.8. Pre-stress ............................................................................................................................ 27

2.2.9. Tensile strength ................................................................................................................. 27

2.2.10. Influence of temperature .................................................................................................. 28

2.3. Modeling the membrane behavior .......................................................................................... 29

2.3.1. Linear elastic orthotropic plane stress model ................................................................ 29

2.3.2. Cable networks .................................................................................................................. 32

2.3.3. Piecewise linear elastic orthotropic plane stress model ............................................... 33

2.3.4. Nonlinear elastic material model, taking the influence of the load ratio into account

33

2.3.5. Response surfaces with division in quadrilaterals ......................................................... 35

2.3.6. Response surfaces without division in quadrilaterals ................................................... 35

2.3.7. Day’s method ..................................................................................................................... 36

2.3.8. Micro-mechanical models ................................................................................................ 36

2.4. Reciprocal relationship.............................................................................................................. 37

2.5. Quantifying and understanding the biaxial behavior of different membrane types ........ 38

2.6. Conclusions ................................................................................................................................ 39

Chapter 3. Uniaxial tensile tests .......................................................................................................... 41

3.1. Introduction ............................................................................................................................... 41

3.2. Experimental setup .................................................................................................................... 41

3.3. Experimental test results .......................................................................................................... 42

3.3.1. Loading until failure .......................................................................................................... 42

3.3.2. Load cycle repetition ......................................................................................................... 44

3.4. FEM model of a uniaxial tensile test ...................................................................................... 57

3.5. Conclusions ................................................................................................................................ 59

Contents

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Chapter 4. Biaxial tensile tests ............................................................................................................. 60

4.1. Introduction ............................................................................................................................... 60

4.2. Experimental setup .................................................................................................................... 60

4.2.1. Biaxial extension of cruciform samples ......................................................................... 60

4.2.2. Bubble inflation tests ........................................................................................................ 61

4.2.3. Biaxial experimental tests for this project ...................................................................... 62

4.3. Numerical study of the geometry of the cruciform sample ................................................ 62

4.3.1. Principles and assumptions for the FEM simulation ................................................... 63

4.3.2. Shape of the slit tips .......................................................................................................... 67

4.3.3. Size of the sample.............................................................................................................. 70

4.3.4. Number of slits .................................................................................................................. 72

4.3.5. Location of slits ................................................................................................................. 76

4.4. Experimental test results .......................................................................................................... 79

4.5. Conclusions ................................................................................................................................ 83

Chapter 5. Modeling the material behavior of a coated fabric ....................................................... 84

5.1. Introduction ............................................................................................................................... 84

5.2. Selection criteria ......................................................................................................................... 84

5.3. Orthotropic linear elastic material behavior .......................................................................... 84

5.4. Orthotropic multi-linear material behavior ........................................................................... 86

5.5. Isotropic hyper elastic material behavior with permanent set ............................................ 88

5.6. Anisotropic hyper elastic material behavior .......................................................................... 90

5.6.1. Generalized Fung potential.............................................................................................. 90

5.6.2. Holzapfel-Gasser-Ogden potential ................................................................................. 91

5.7. Test data based fabric material behavior ................................................................................ 93

5.7.1. Uniaxial test data based FABRIC model ....................................................................... 95

5.7.2. Biaxial test data based FABRIC model .......................................................................... 99

5.7.3. Conclusion........................................................................................................................ 100

5.8. Combined orthotropic elastic-plastic Hill material model ................................................. 101

5.8.1. Hardening law .................................................................................................................. 103

5.8.2. Lankford ratios ................................................................................................................ 106

5.8.3. Hill’s plasticity excluding crimp interchange ............................................................... 109

5.8.4. Hill’s plasticity including crimp interchange ................................................................ 113

5.9. Overall strain field ................................................................................................................... 118

5.9.1. Strain in warp direction .................................................................................................. 118

Contents

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5.9.2. Strain in fill direction ...................................................................................................... 121

5.10. Conclusion ............................................................................................................................ 123

Chapter 6. Conclusions and recommendations.............................................................................. 125

Appendices ................................................................................................................................................ 128

Appendix A .......................................................................................................................................... 128

Appendix B ........................................................................................................................................... 153

Appendix C ........................................................................................................................................... 156

Bibliography .............................................................................................................................................. 159

List of figures ............................................................................................................................................ 162

List of tables .............................................................................................................................................. 171

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List of acronyms

FEM Finite element method

FWO Fonds voor wetenschappelijk onderzoek

PVC Polyvinylchloride

PTFE Polytetrafluorethylene

UV Ultra violet

UTS Ultimate tensile strength

MSAJ Testing method for elastic constants of membrane materials (Japanese standard)

DIC Digital image correlation

RP Reference point

VUB Free University of Brussels

MeMC Mechanics of material and constructions

i.e. Id est

e.g. Exempli gratia

etc. Etcetera

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List of symbols

Ew, Ef [Pa] Young’s modulus in warp, fill direction

Gwf [Pa] Shear modulus in the plane of the fabric

νwf, νfw [-] Poisson ratio

σw, σf [Pa] Stress in warp, fill direction

σwf [Pa] Shear stress in the plane of the fabric

εw, εf [-] Engineering strain in warp, fill direction

εwf [-] Half of the engineering shear strain

t [m] thickness

nx, ny [Pa] Tension in x-direction, y-direction

R, r [m] Radius of curvature

nij [N/m] Force density

Pij [N] Force in a segment

lij [m] Segment length

Fij [N] Force in a node

Xi, Xj, Yi, Yj, Zi, Zj [N] Force component in a node

Sg [N/m] Geometric stiffness matrix

RF [N] Reaction force

A [m2] Area

pi [Pa] Internal pressure

D [m] Diameter

d [m] Distance

γt [-] High temperature reduction factor

f1, f2 Tensile strength at 70°C, 23°C

Sε, Sσ [Pa, -] Strain error, stress error

γw, γf [-] Normalized load ratio in warp, fill direction

U [Pa] Energy potential

List of symbols

xv

x, y, z [m] Cartesian coordinates

S11, S22, S33 [Pa] Young’s modulus in warp, fill, thickness direction

S12, S13, S23 [Pa] Shear modulus in the principal planes of the fabric

E11, E22, E12 [-] Engineering strain in warp, fill, shear direction

F, Fe, Fp [-] Deformation gradient, elastic and plastic part of deformation gradient

J, Jel, Jth [-] Total volume ratio, elastic volume ratio, thermal volume ratio

bijkl [-] symmetric fourth order tensor of anisotropic material constants

�� [1/Pa] Initial deviatoric elasticity tensor

K0 [Pa] Bulk modulus

ψ� [Pa] Isochoric deformation potential

I�̅ [-] First deviatoric strain invariant

N [-] Number of families of fibers

� [-] Level of dispersion in the fiber directions

ρθ� [-] Orientation density function

E�� [-] Strain-like quantity

�� [Pa] Equivalent stress

σy0, τy0 [Pa] Reference yield stress, reference yield shear stress

ε�� [-] Plastic strain

Rij [-] Yield stress ratio

λ [-] Stretch ratio

C(p) [-] Cost function

χ��, χ�� [-] experimental, numerical response

rα [-] Lankford ratio in the direction α rotated to the reference direction

r̅ [-] Coefficient of normal anisotropy

∆r [-] Degree of planar anisotropy

1

Chapter 1.Introduction After a short statement of the objectives of the present master thesis, the origin, the development

as well as the usage of tent structures throughout the history of mankind is described. This is

followed by a basic description about the constituents of a coated fabric as well as the production

techniques. Emphasis is put on the connection between the production method and the

structural behavior of a coated fabric, i.e. large mutual influences are present. A microscopic

examination of the coated fabric is done with the aim to obtain additional information about the

structure of the fibers and the yarns. After this, several characteristics about the form and

physical behavior of tent structures are covered, followed by an overview of the actual

improvements of tent structures compared to more classical structures. Finally, a short note is

given about the internal atmosphere of a tent structure, e.g. temperature control, acoustical

performance, etc.

1.1. Objectives The aim of this master thesis is to check the feasibility to model and simulate a tent structure

true-to-nature by means of the finite element method (FEM) software Abaqus. The design of

tent structures is nowadays mainly based on experience. At present, tent structures are

increasingly used for a variety of critical applications, while their size expands exponentially.

Several software packages were developed during the last decades, though most of them are

found on largely simplified material models and consequently cannot capture the correct

behavior of coated fabrics. On the other hand, packages were developed which use more

advanced modeling techniques. The main drawback of these packages concerns the extremely

high computation time making the simulation of a complete tent structure unrealistic. Hence a

package is needed which strikes the golden mean, i.e. an acceptable material model and a

reasonable computation time.

The FEM software Abaqus has already proven its suitability to model and simulate rigid

structures and materials. Here, the ability of modeling and simulating a coated fabric, used for

modern tent structures, will be explored and discussed. This master thesis is part of a ‘Fonds

voor Wetenschappelijk onderzoek’ (FWO) project, starting in 2012, which aims to model

transformable tent structures, including the folding and unfolding procedures. The focus of this

master thesis is limited to the modeling of the material behavior under various loading states.

However, the fact that the material model will probably be used for simulating large foldable tent

structures is always kept in mind. A variety of built-in material models in Abaqus is studied, their

appropriateness for modeling and simulating a coated fabric is examined and will be discussed in

detail. Hence the possibility of a user defined subroutine, to simulate the material behavior of a

tent structure, is not considered. In order to be able to select an appropriate material model, a

profound knowledge about the material behavior of a tent fabric is required. Therefore, a big

share of this master thesis is devoted to the understanding of the structural behavior of the

coated fabric.

Chapter 1. Introduction

2

1.2. History The history of tent structures goes back to ancient times. Nomads used them as mobile homes,

because of the comfort to break them down and rebuild them later and also because other

building materials were rare. Those initial tent structures were very simple: a skeleton made of

bones or wood draped with an animal skin.

Figure 1.1: Indian teepee.

Tents developed over the years and the animal skin was replaced by textiles, which were woven

from natural fibers like goat or sheep wool. When people started to tension them, large spans

could be achieved. Because of the tensioning, the fabric needed reinforcements and the

connections between membrane and skeleton were improved. Since then, tent structures could

be employed for several simple additional applications, e.g. shade structures, army tents, circus

tents, ceremony tents, sailing ships, etc.

Recently, the use of tent structures in advanced structural applications gained interest, due to the

improvement of the properties of the fabric material. Nowadays, textiles are characterized by a

light weight to high strength ratio, enabling large spans. Their size ranges from that of ordinary

bivouac tents, just big enough for one person, to large permanent buildings capable of seating

thousands of people. Furthermore, modern tent structures are easy to clean and have a high life

span, which makes them attractive as permanent roof structures. Besides permanent objectives,

large foldable tent structures are increasingly used as temporary housing for medical and

coordination teams during natural disasters because of their compact transportation and quick

establishment (Figure 1.2). Also at concerts and festivals, they are often used as temporary

shelters for the rain (Figure 1.3).

Figure 1.2: Emergency tent used by the Red Cross [1].

Figure 1.3: Festival tent [2].

1.3. Coated fabric material and production techniques The manufacture of textiles is one of the oldest human technologies. To make a textile, the first

requirement is a source of fibers from which a yarn can be made, primarily by spinning. Spinning


3

is the twisting together of drawn out strands of fibers. Fibers can be both natural and man-made.

Natural fibers come either from animals (e.g. sheep, goat, rabbit), minerals (e.g. asbestos) or

plants (e.g. cotton, flex, sisal). Many processes are needed before a clean even staple is obtained.

With the exception of silk, the fibers are short (several centimeters) and have a rough surface that

enables it to bond with similar staples. Artificial fibers on the other hand are made by extruding a

polymer trough a spinneret into a medium where it hardens. These fibers are lengthy but can be

batched and cut in order to process them like a natural fiber. Some examples of commonly used

fibers in tent structures are: polyester fibers, polyamide fibers, aramid fibers, polyethylene fibers,

glass fibers and steel wire. Besides its use in tent structures, textiles are further utilized in clothing,

furnishing, window shades, baskets, parachutes, sails, industrial geotextiles and composite

materials.

The yarn is processed by weaving, knitting, crocheting, knotting or pressing, which turns it into a

textile. Weaving is the common manufacturing method for tent textiles, in which a set of warp

fibers are interlaced with a set of weft or fill fibers. The warp fibers are held in tension on a loom

(Figure 1.4), while the weft or fill fibers are inserted over and under the warp fibers [3].

Figure 1.4: Loom to weave a fabric.

Because of this weaving technique, the warp fibers are more or less straight, while the fill fibers

are curved (Figure 1.5).

Figure 1.5: Woven fabric with straight warp fibers and curved fill fibers [4].

The warp and fill directions are the two orthogonal principal directions of the textile. The way

the warp and fill fibers interlace with each other is called the weave. The majority of woven

products are created with the following basic weaves: (i) plain weave, (ii) satin weave or (iii) twill

weave. In a plain weave (Figure 1.6 (a)), each fill yarn crosses the warp yarns by going over one,

then under the next, and so on. The next fill thread goes over the warp yarns, that its neighbor


4

went under and vice versa. It is the most basic of three fundamental types, because of its strength

and hardwearing. Basket weave is a variation of plain weave in which two or more fibers are

bundled and then woven as if it were one. The satin weave (Figure 1.6 (b)) is characterized by

four or more fill yarns floating over a warp yarn and vice versa. The light reflecting is not

scattered as much by the fibers, which have fewer tucks, causing a lustrous or ‘silky’ appearance.

Twill weaving (Figure 1.6 (c)) is characterized by a pattern of diagonal ribs. This is done by

passing the fill yarns over one or more warp yarns and then under two or more warp yarns and

so on, with an offset between the different rows.

Figure 1.6: Plain weave (a), Satin weave (b) and twill weave (c).

A coating is applied, which covers the outsides of the fabric and fills the holes between the fibers.

This coating protects the textile for chemical attack and UV-light, and makes it possibly water-

and airtight, anti-stain, antistatic, abrasion resistant, and printable. The choice of coating is of

critical importance because of its influence on the density, suppleness, weldability, light

transmission, cleanability and fire behavior of the fabric. Coatings that are used most in the

application of tent structures are polyvinylchloride (PVC) and polytetrafluorethylene (PTFE).

PVC is flexible, but has a restricted ultraviolet (UV)-resistance. It is mostly used for polyester

fabrics. PTFE is less flexible, but has a good resistance against several chemicals, solvents,

moisture, UV-rays, high temperatures and is dirt repellent. The PTFE coating is often used for

fabrics made of glass fibers. The nature of (i) the fibers, (ii) the matrix, (iii) the weaving method,

(iv) the coating and (v) the deposition method of the coating, all influence the material behavior

of the membrane.

1.4. Material used in this master thesis

1.4.1. Data sheet The coated fabric used in this master thesis is T2107 from the company SIOEN, and is shown in

Figure 1.7.


5

Figure 1.7: The coated fabric T2107 that is used in this master thesis.

The fibers are made of polyester (polyethylene terephthalate) and are woven with a plain weave

pattern. The fibers have a mass density of 1100 dtex (1100 grams per 10000 meters). In warp

direction the density of reinforcement is 12 yarns/cm, the one in fill direction is 13 yarns/cm.

The type of coating is a plasticized PVC, to which flame retardant additives (antimony trioxide)

and fungicide are added. The outside of the tent structure is exposed to the natural elements, e.g.

rain, sunshine, UV-rays, etc., and is therefore protected with an extra coating layer. For this

reason the outside of the material has a smooth surface (Figure 1.8 (a)) while the inside has a

typical surface profile (Figure 1.8 (b)).

Figure 1.8: Photo of the surface of the coated fabric of the outer surface (a) and the inner surface (b).

The coated fabric has a thickness t = 0.83 mm and a surface mass density of 1050 g/m2. The

company SIOEN has provided some indicating values for the strength of the material: the

breaking strength in both warp and fill direction is 96.4 MPa, the temperature at which the

material may be exposed, is in the range of -30°C to 70°C.

1.4.2. Microscopic examination In this subsection, a microscopical examination of the coated fabric is done. The aim is to

improve the knowledge about the structure of the woven fibers and the coating. From the data

sheet it is not clear whether the coating forms two separate layers, one on top and one at the

bottom of the woven fabric, or if the coating is present between the fiber bundles as well as

inside the fiber bundles.


6

Figure 1.9 and Figure 1.10 show magnifications of samples respectively cut out along the warp

and the fill direction. The applied magnification is x200.

Figure 1.9: Microscopic image (x200) of the coated fabric, cut out along the warp direction.

Figure 1.10: Microscopic image (x200) of the coated fabric, cut out along the fill direction.

The straight warp fibers (Figure 1.9) and the curved fill fibers (Figure 1.10) are clearly

distinguishable. Individual fibers perpendicular to the sheet as well as some loose fibers parallel to

the sheet can be observed in the fiber bundles. One also clearly identifies the presence of coating

between the fiber bundles. However, because of the low contrast pictures, it is not clear if coating

is present inside the fiber bundles, i.e. whether the fibers are wet or dry.

Therefore, the samples are embedded and afterwards polished. The samples are cut along warp

and fill direction and embedded in Epofix hardener, to which a fluorescent yellow pigment has

been added. If the fiber bundles are dry, the Epofix will penetrate them during hardening. A

curing cycle of one day at room temperature and 1 hour in the oven at 60°C is considered.

Afterwards, the samples are polished with different abrasive papers, having a sequential

roughness of 180, 320, 800, 1200 and 4000. The first polishing session removes burr and major

imperfections, while the last session takes off very small irregularities. The embedded samples are

shown in Figure 1.11.


7

Figure 1.11: Photos of the embedded samples in Epofix with a fluorescent yellow color.

The microscopic pictures of the polished samples cut out along the warp and fill direction are

shown in Figure 1.12 and Figure 1.13 respectively.

Figure 1.12: : Microscopic image of the coated fabric, cut out along the warp direction after embedding and polishing

with magnification x55 (a) and x200 (b).

Figure 1.13: Microscopic image of the coated fabric, cut out along the fill direction after embedding and polishing with

magnification x55 (a) and x200 (b).


8

It is seen that the inside of a fiber bundle has a fluorescent yellow color, indicating that the

embedding material is absorbed between the fibers. However, this does not necessarily reveal that

the fiber bundles were dry before the treatment: after further polishing them by hand, the

intensity of the fluorescent yellow color decreases, as can be seen in Figure 1.14 and Figure 1.15.

Figure 1.14: Microscopic image of the coated fabric, cut out along the warp direction after embedding, polishing and

further polishing by hand with magnification x55 (a) and x200 (b).

Figure 1.15: Microscopic image of the coated fabric, cut out along the fill direction after embedding, polishing and

further polishing by hand with magnification x55 (a) and x200 (b).

These results could indicate that the origin of the yellow color of the fibers is not to be found in

the absorption of the embedding material during hardening, but rather in the pollution of the top

of the fibers with particles of the embedding material during the mechanical polishing.

In order to make sure that the yellow shine is due to the absorption of the embedding material

into the fiber bundles, another technique is used in which the embedded material is not polished.

Instead of polishing, the top layer is removed by means of a cut-off machine. This way, the top

of the sample cannot be polluted with particles of the embedding material. The microscopic


9

pictures of the samples cut out along the warp and fill direction are shown in Figure 1.16 and

Figure 1.17 respectively.

Figure 1.16: Microscopic image of the coated fabric, cut out along the warp direction after embedding and cutting by

means of a cut-off machine with magnification x55 (a) and x200 (b).

Figure 1.17: Microscopic image of the coated fabric, cut out along the fill direction after embedding and cutting by

means of a cut-off machine with magnification x55 (a) and x200 (b).

There is still a clear yellow shine observed, indicating that the Epofix with fluorescent color was

absorbed by the fiber bundles during curing, rather than a pollution of Epofix on top of the

sample due to polishing. The fact that the fiber bundles can absorb the embedding material,

indicates that no coating was present at the inside of the fiber bundles before embedding, thus

the fibers of the coated fabric material are dry.

1.5. Form and physical behavior [5, 6] The form and physical behavior of fabric structures are very different from those of conventional

rigid structures. From an engineering point of view, tent structures are thin membranes, with a

constant thickness, able to support the imposed loads by their surface shape and pre-tensioning.


10

Four different types of membrane structures are known: (i) tensioned surfaces, (ii) air supported

structures, (iii) air inflated structures and (iv) hybrid systems. The three last ones represent the

pneumatic structures, which are characterized by an overpressure in the created cavity of the

membrane. Each of the different types will be discussed in the following subsections.

1.5.1. Tensioned surfaces Membranes are tensioned by compression and/or bending elements, such as masts, compression

rings, arches and beams. Because membranes only resist tension, the basic shape must be an

anticlastic surface, which means that the surface has both a positive and a negative curvature in

each point (figure 18). A set of ‘arching’ tensile elements (A) works in opposite way to a similar

set of ‘hanging’ elements (B).

Figure 1.18: Anticlastic surface consisting of arching (A) and hanging (B) elements [5].

The need for an anticlastic surface can be demonstrated by considering the membrane equation

(neglecting the weight of both membrane and cables):

�� + � � = 0 (1.1)

with �� = tension in x-direction (MPa), � = tension in y-direction (MPa), �� = radius of

curvature in x-direction (m) and � = radius of curvature in y-direction (m). Both the pre-

stresses �� and � are positive because of the inherent tensioned state of the membrane. Hence

the principal curvatures 1 ��$ and 1 � $ must have an opposite sign in every material point to

satisfy equation (1.1). This means that the Gaussian index %1 ��$ ∗ 1 � $ ' is negative, which

characterizes an anticlastic surfaces.

Physically, the arching set (A) and the hanging set (B) represent the orthogonal directions of the

membrane, i.e. warp and fill. This configuration provides the possibility to pre-stress the

membrane without significantly changing the overall shape of the surface. The hanging and

arching tensile elements each carry a different kind of load: downward forces are carried by the

fibers with the hanging curvature, while upward forces are carried by the fibers with the arching

curvature. The designer of the construction should make sure that the pre-stress is high enough

in order to always maintain some tension in the membrane when external loads are applied. If


11

this condition is not satisfied, wrinkles will occur in the membrane, which reduce the bearing

strength of the structure into a great extent. Some examples of anticlastic shapes are shown in

Figure 1.19.

Figure 1.19: Saddle structure (a), high point structure [2] (b), ridge and valley structure [5] (c) and hypar structure [5]

(d).

Unlike conventional stiff materials, it is impossible to impose the geometry of the membrane’s

surface. The geometry is rather defined by the ‘internal equilibrium of pre-stresses’ within the

boundary systems of the support. Under certain assumptions [6] this internal equilibrium is given

by the ‘minimal surface’ geometry of the considered system. This is an ideal state of the textile’s

geometry and can be understood by the physical analogy of the soap film because of its structural

resemblance to a membrane (only resist tension). When two ends of a wire are connected to each

other and dipped in a bath of soapy water, a soap film is formed which spans the minimal surface

for the given boundaries (Figure 1.20 and Figure 1.21). Joseph A. F. Plateau, after whom the

former building of the faculty ‘Engineering sciences and architecture’ of our university is named,

studied these phenomena. The mathematical problem of the existence of a minimal surface with

a given boundary is named for him as well [7].

Figure 1.20: Home-made soap film models, representing the minimal surface for the chosen boundaries.


12

Figure 1.21: Soap film model [8] (a) and a realized membrane structure with the same shape [9] (b).

Of all possible surfaces, for a given boundary, the minimal surface is the one with minimal

energy, i.e. the one in which the membrane is most relaxed. Using the membrane equation, the

minimal surface is achieved by the condition �)* = − �),. In this case the membrane forces �� and � are identical and a uniform tensile stress state exists in the membrane.

Sometimes several minimal surfaces can exist for given boundary conditions (degeneration). The

most common example involves the case of two parallel rings as a boundary. The possible

minimal surfaces are (i) a hyperboloid (Figure 1.22(a)) and (ii) two disks (Figure 1.22(b)). This

degeneration implies that a minimal surface exists locally.

Figure 1.22: Minimal surfaces for two parallel rings: hyperboloid (a) and two parallel rings (b) [10].

It is important to note that there are physical limitations to the boundary conditions in the

analogy of the soap films. For example, you cannot draw the two parallel rings infinitely apart

because the soap film will collapse at a certain distance d between the two rings. Day determined

the limiting ratios of ring diameters D to distance d in order to have a stable minimal surface [11].

Change of the boundary, leads to a change of the minimal surface. Boundary conditions in real

tent structures are the disposition of all elements that connect and provide support to the

membrane, e.g. masts, arches, ridge and valley cables, beams, etc. Another kind of frequently

used boundary conditions in real structures involve pre-stressed cables which support the

membrane’s surface. These pre-stressed cables can be implemented in the soap film by threads

lying within the film. In the example of the two parallel rings, a set of threads can be running

between the upper and lower ring. Instead of representing pre-stressed cables, these threads can

also represent a non-uniform and varying stress field within a slightly altered geometry with larger

pre-stress forces. Hence, the surface shape is the outcome of both the chosen boundary

conditions and the chosen pre-stress ratios within the boundary conditions. Obtaining a certain

shape of the surface is an iterative process, called 'form finding', in which the heights and spacing

of supports and pre-stress ratios are optimized. The choice of the boundary conditions is primary

to the resulting surface shape. The choice of the pre-stress ratios then modifies this shape. A


13

‘form finding method’ that is often used for tent structures is the ‘force density method’ [6],

which has the great advantage that every equilibrium state of a general network configuration is

obtained by solving a linear system of equations. Disadvantageous in this method is the

unrealistic representation of the membrane as a cable network (Figure 1.23).

Figure 1.23: Membrane structure simplified as a cable network.

The segments can either be rigid or elastic and are mutually connected by hinges. The method is

characterized by using the ‘force densities’ which are the forces Pij (N) in the segments divided by

the segment lengths lij (m), i.e. n�. = /01�01 as design parameters. By pre-defining these ‘force

densities’, the equations of the displacement method can be solved to determine the equilibrium

shape. If the segment ij is subjected to a pre-stress P�., the nodes are subjected to the forces:

34567 = 899:99;<5=5>5<6=6>6?99

@99A =

899999:99999;−B6 − B5C56−D6 − D5C56−E6 − E5C56B6 − B5C56D6 − D5C56E6 − E5C56 ?9

9999@99999A

F56 (1.2)

For every free node k in the cable network, the forces XH, YH and ZH are equal to zero. The

unknown reactions in the fixed points (the anchor points) as well as the coordinates x�, x., y�, y., z� and z. of the equilibrium shape are determined by solving the linear equations of the

displacement method:


14

899:99;<5=5>5<6=6>6?99

@99A =

899999:99999;−B6 − B5C56−D6 − D5C56−E6 − E5C56B6 − B5C56D6 − D5C56E6 − E5C56 ?9

9999@99999A

F56 =�56 NOOOOP1 0 00 1 00 0 1

−1 0 00 −1 00 0 −1−1 0 00 −1 00 0 −11 0 00 1 00 0 1 QRR

RRS89:9;B5D5E5B6D6E6?9@

9A (1.3)

in which

TUVW56 =�56 NOOOOP1 0 00 1 00 0 1

−1 0 00 −1 00 0 −1−1 0 00 −1 00 0 −11 0 00 1 00 0 1 QRR

RRS (1.4)

TSYW�. is called the geometric stiffness matrix. Equation (1.3)can be written as

ZUV� UV[UV\ UV]^ _B�B[` = _ 0�4` (1.5)

With ax�b the unknown coordinates vector of the free nodes, ax[b the known coordinates vector

of the anchor nodes and aRFb the unknown reaction forces in the anchor nodes. Solving

equation (1.5) gives the equilibrium shape and the reaction forces.

1.4.3. Air supported structures The first type of pneumatic structures are literally structures supported by air. The form active

membrane floats over the enclosed air, having a sufficient internal pressure in order to resist

external loads. An example of an air supported structure is the Tokyo dome (Figure 1.24).

Figure 1.24: Tokyo dome ‘Big egg’ football stadium [12].


15

One of the design aspects is the choice of the profile, usually it is cylindrical or dome shaped.

Comparing a hemisphere with radius r (m) and a partial sphere with bigger radius R (m) covering

the same area A = πr[ with the same internal pressure pi (Pa), gives higher membrane forces in

the partial sphere, than in the hemisphere, respectively �0∗g[ and

�0∗h[ . From this, a high profile

seems preferable, however there are some other aspects that should be considered. A membrane

with a high profile encloses a significantly higher construction volume and this leads to greater

requirements for the mechanical equipment that provides thermal comfort. In addition there is

an increased possibility for a membrane, having a high profile, to be subject to wind pressures

causing indentations. The building skin is never completely airtight, thus a continuous air support

is needed.

1.4.4. Air inflated structures The bearing elements of air inflated structures are closed membrane elements that are put under

very high pressure and are called ‘pneumatics’. The stiffness of a pneumatic is strongly dependent

on the applied internal overpressure, which must ensure that under the action of external forces

the membrane forces are always positive (tension) and do not exceed the tensile strength. For

this, high strength materials are required. Furthermore, special attention must be paid to the

airtightness because usually it is impossible to provide continuous air supply. Examples of air

inflated structures are tubular beams, columns, arches and walls (Figure 1.25). Tensairities (Figure

1.26) are air inflated structures to which stiffeners or cables are added in order to gain mechanical

stiffness, while the mass is kept small.

Figure 1.25: Modern teahouse 2007 (Frankfurt) [13].

Figure 1.26: Tensairity used as bearing element of a

bridge [14].

1.4.5. Hybrid structures Hybrid structures are a combination of air supported and air inflated structures or a

combinations of traditional ‘hard’ bearing systems and pneumatic structures. Double walled air

domes (Figure 1.27) are an example of the first type. The advantages of these structures are the

improved insulation performance and the fact that a deflection of the outer skin is not visible

within the building volume.


16

Figure 1.27: Fuji group pavilion, expo 1970 Osaka [15].

1.6. Quality characteristics of tensile structures Tensioned membranes have very special qualities, mainly as a result of their unique form and

the nature of membrane materials. For these reasons they invite architectural solutions that

conventional building systems cannot offer.

1.6.1. Lightweight Tensile structures are much lighter than conventional structures because their structural stability

results from their pre-stressed shape rather than the mass of the material. Thus large spans can

be obtained with only little material use.

1.6.2. Transportability Mobile and temporary tensile structures are a synthesis of the nomad tent and the permanent

settlement. The lightness and flexibility of the material permits these structures to be carried

and deployed easily and quickly. When not in use, the structures occupy a relatively limited

volume. These are valuable characteristics for temporary housing and mobile structures, and are

essential in the case of catastrophes and emergencies where shelters are required for a great

number of people in a short time.

1.6.3. Translucency Translucency can provide the necessary amount of daylight, which plays an important role in

the building’s energy efficiency. The translucency also offers great aesthetic possibilities for

architectural design, through the use of natural and artificial daylight. At night the membrane’s

translucency transforms the structure into a sculpture of light. The translucency of engineering

fabrics depends on the type of fiber as well as on the coating. The translucency of typical

membranes varies between 10% and 40%. To the human eye such roofs appear bright and

translucent even when the light transmission is only a few percent, giving the appearance of a

lightweight roof structure. The Eden project is an example in which natural daylight was a

major design criterion (Figure 1.28).


17

Figure 1.28: The Eden project [16].

A special application of membrane structures is found in its use as a façade for an ordinary

building. The membrane acts as a second skin for the building, hence preventing the inside heat

to escape during cold days on the one hand and offering a shade structure during warm days on

the other hand. No other building material matches these criteria better than foils or

membranes because of their thin, light and transparent/translucent properties. The training

center of the Bavarian mountain rescue (Figure 1.29) is one of the first buildings equipped with

an ETFE façade. Further note that the second skin can be located at the inside of the building

as well. An old industrial steel and glass shed was insulated thermally in that way (Figure 1.30).

Figure 1.29: ETFE Facade as second skin for the Training center for the Bavarian mountain rescue

(Bad Tolz, Germany) [17].

Figure 1.30: Eco membrane at the inside of the

Deckelhalle as thermal insulation (Munich, Germany) [18].

1.6.4. Flexibility Tensioned membranes are not rigid and do not achieve their strength by using heavy and thick

materials, but because the external forces influence the form. The geometry of the membrane

responds to the applied loading by deforming, and so doing developing a more efficient form for


18

bearing the loads. This principle is illustrated for a cable in Figure 1.31. In this way, much force

can be absorbed, using relatively low weight. This is in clear contrast with concrete or steel

structures.

Figure 1.31: Deformation of a cable to a more efficient structure due to the load [6].

1.6.5. Sculptural A tensile membrane’s unique curved shape has a very strong sculptural quality. The tensioned

membrane is in equilibrium with itself and it is possible to create large buildings which seem to

be floating effortless. The unique shape of tensioned membrane structures present a natural

beauty that creates interesting landmarks. The sculptural quality is sensational, both outside and

inside the building and give architects the possibility of creating dramatic and complex spaces.

The play of natural and artificial lighting using both the translucency and the reflectivity of the

membrane materials transforms the structure into a sculpture of light.

Figure 1.32: Hovering archives: temporary art project

(Hamburg, Germany) [19].

Figure 1.33:Green void: a temporary construction in the Sydney custom house (Sydney, Australia) [20].

1.6.6. Safety Lightweight tensioned structures are safe structures and are particularly stable in horizontal loads

like earthquakes. In an unforeseen case of collapse, tent structures tend to be less dangerous than

conventional building systems because of their much lower mass.

1.6.7. Weather protection One of the main functions of a roof is to provide shelter and comfort in weather conditions such

as sun, rain, wind and snow. Membrane materials with a porous weave can be used for shading,

they bring diffuse light into the space within and stimulate natural ventilation. This can keep the

surface temperatures at the level of the ambient air and avoid downward radiation of heat. For

protection from rain and snow, the form should allow fast and easy drainage of water and snow.

The particular forms of tensile structures make it possible to avoid ponding of rain water and

large accumulations of snow. To keep rainwater out, the membrane material and its joints have to


19

be watertight. The Venezuelan pavilion (Figure 1.34) is a convertible structure, which acts as a

shading structure when opened, and provides rain and wind protection when closed.

Figure 1.34: Venezuelan pavilion [21].

1.6.8. Convertibility and adaptability Convertible structures are adaptive systems which change according to necessity. They are

flexible and adaptable in their spatial arrangement response to climatic variations. This design

approach leads automatically to the possibility of energy saving through the control of natural

light and internal temperature. Figure 1.35 and Figure 1.36 show the convertible roof structures

of respectively a swimming pool and a football stadium.

Figure 1.35: Convertible cover of a swimming pool in

Seville, Spain [22].

Figure 1.36: Retractable roof of the Toyota stadium

(Nagoya, Japan) [23].

1.7. Internal environment For a long time, thermal, acoustic and lighting performance of tent structures were of minor

importance However, as the application of membrane structures has extended to more complex

and permanent buildings, the need for thermal, acoustic and lighting comfort received

increasingly interest. Due to their extreme lightness and translucency, membrane enclosures

should be considered as filters rather than barriers to external conditions. The development of

material technology made it possible to improve the comfort conditions within the spaces they

enclose.

1.7.1. Thermal environment Tensile membrane structures tend to create internal environments that depend on the properties

of the textile skin as well as on the topology of the space they enclose. A tensile membrane skin is

extremely thin and lightweight, so it provides little or no thermal buffering to the interior.

Therefore a usual bivouac tent feels extremely hot when in the hot sun and cools down very fast

at night.


20

To improve the environmental behavior of the membrane enclosure, the number of layers

constituting the building skin can be increased. The use of a multiple skin does not only reduce

the heat transfer through the building skin, also solar gains and daylight transmission can be

controlled, acoustical characteristics are improved, the risk of condensation is reduced and it

provides additional fire protection. A double layer membrane construction exists of an outer

membrane and an inner membrane, separated by an air space of variable thickness. The inner

layer can be made out of a much lighter material than the outer skin, since it does not have to

withstand the pre-stress and external loadings, meanwhile it is chosen in accordance to the

requirements of fire resistance and permeability to water. The airspace reduces the heat transfer

by convection occurring between the outer membrane and the enclosed space. This technique

was amongst others used in the Eden project (Figure 1.28).

Insulated membrane structures contain a layer of low-density insulating material sandwiched

between the external structural membrane skin and internal membrane layers. Available insulating

materials range from simple foam coatings to fiber mats and self-contained air cell films. The

insulating layer can significantly reduce the heat transfer across the thickness of the membrane

construction allowing a tighter control of the thermal conditions inside the enclosure.

Solar energy can penetrate a space that is covered by a tensile membrane structure through direct

gains, namely transmitted solar radiation and indirect gains, i.e. the absorbed solar radiation. The

coating of most architectural membrane materials offer high reflection to solar radiation and this

can be exploited for effective shading of the covered space. Unpigmented coatings, having a

small solar absorption, should be preferred if shading is the main function of the structure. The

cooling effect provided by shading should be enhanced by promoting natural ventilation. If

shelter from rain is not a functional requirement, the use of open mesh fabrics can further

improve the cooling effect by allowing wind driven air flow to circulate through the fabric.

1.7.2. Lighting environment The translucency of architectural fabric skin is often seen as the most valuable asset of tensile

membrane structures. Despite the great adaptability of the human eye to brightness level, the

strong difference in brightness between the translucent ceiling and the other internal surfaces of

the building can cause problems. This can be avoided by providing direct lighting (Figure 1.37)

inside the building via glazed surfaces, which improve contrast whilst restoring the balance

between the brightness of the ceiling and the other internal surfaces.


21

Figure 1.37: Integration of glazed areas in the supporting steelwork to introduce direct illumination in the enclosure,

APPP church (Maassluis, the Netherlands) [24].

Daytime artificial lighting is usually not necessary in a space covered by a membrane. Nighttime

artificial lighting is often achieved by lighting the underside of the membrane roof as this reflects

a large proportion of the incident light back into the internal space. This creates a diffuse internal

lighting environment similar to daytime. The light transmitted through the membrane produces a

warm glowing effect when viewed from the outside (Figure 1.38).

Figure 1.38: The water cube, Bejing [25].

1.7.3. Acoustical environment The lightweight nature of membrane materials brings particular problems to the acoustical

designers, whereby the ‘blocking’ of sound using mass, as used traditionally, contradicts the

concept of membrane structures. Up to now, little success has been obtained with single skin

membrane materials as preventing noise infiltration or leakage is concerned. In general, the

transmission loss of normal double-layer membrane structures is only between 5 and 10 dB, in

fact offering very little protection from external sound sources. Beside external noise, internal

noise leakage and skin drumming are factors that should be taken into consideration. Skin

drumming is the repercussion of the patterns and intensity of rain and hail on the noise levels

experienced within the space.

22

Chapter 2.Behavior and available modeling methods for coated fabrics

2.1. Introduction The aim of this chapter is to gain knowledge about the material behavior of coated fabrics in

general on the one hand and to explore the possibilities for implementing a coated fabric in a

FEM model on the other hand. This chapter starts with an overview of the material behavior of

coated fabrics, which is largely based on a literature review. In the second part, several methods

for modeling the behavior, which are described in literature, are shortly introduced and discussed.

The chapter ends with an overview of the relevance of the reciprocal relation, together with a

representation of the material behavior in strain-strain space.

2.2. Behavior of coated membranes A coated membrane is flexible, cannot support pressure [26] and has no bending stiffness, hence

it only resists tension [27, 28]. Compared to homogeneous materials, the behavior of coated

fabrics when applying tension is more complicated. Crimp interchange, load ratio and load

history are some parameters that considerably influence the behavior of these materials. The

most important characteristics of coated fabrics are explained in the following subsections.

2.2.1. Nonlinear orthotropic behavior A membrane can be considered as a composite because of the combination of fibers and coating.

The stiffness of the fibers is considerably higher than the stiffness of the coating. Both the fibers

and the coating show nonlinear behavior at high tensile strengths, attributable to plastic behavior.

Moreover, the woven structure of the fibers causes an additional nonlinearity, called ‘crimp

interchange’ (see subsection 2.2.2). The resulting material consequently behaves nonlinear [28,

29].

The anisotropic nature of the weave, combined with the isotropy of the coating, results in a

membrane which behaves orthotropic [4].

2.2.2. Crimp interchange The composition of straight warp fibers and curved fill fibers makes the membrane behave in a

special way when forces are applied. Figure 2.1 (a) shows the stress-strain diagram for a PTFE

glass fiber textile recorded during a uniaxial test (see section 3.1) in the fill direction.

Chapter 2. Behavior and available modeling methods for coated fabrics

23

Figure 2.1: Load-strain curve for uniaxial test in fill (a) and warp (b) direction [6].

The curved fill fibers will first straighten, before they stretch (Figure 2.2), which results in a fairly

large apparent initial strain in the fill direction. This process is called crimp interchange [4, 30-33].

After crimp interchange, the stress-strain curve becomes steeper, revealing the actual stiffness of

the fabric. For a uniaxial tensile test in warp direction (Figure 2.1 (b)), no apparent strain is

observed and the stiffness of the material remains approximately constant during the experiment.

This is because the warp yarns were already straightened at the beginning of the experiment,

hence the real stress-strain behavior of straight yarns is observed.

Figure 2.2: Crimp interchange.

2.2.3. Load ratio dependency The stress-strain curves depend on the load ratio (or stress ratio) in warp and fill direction. When

stresses are applied in both directions (biaxial test, see section 3.2), both fiber directions try to

straighten. Because this cannot happen independently, the resulting geometry of the fabric is a

function of the ratio of the stresses applied in both directions [4, 27, 28, 30, 34, 35]. Hence, a

tensed fabric has a special kind of Poisson-effect: the force applied in one direction influences the


24

deformation in the other direction. When the stress-strain curve obtained with the biaxial test

with stress ratio 1:1 for a PTFE-glass fiber fabric (Figure 2.3) is compared to the one obtained

with the uniaxial test in fill direction for the same material (Figure 2.1), large differences are

observed. It is seen in Figure 2.3 that the initial strain in the fill direction for the biaxial test is

smaller than the initial strain corresponding to the uniaxial test. This is natural since the warp

fibers are tensioned as well and therefore want to stay straight. This prevents the fill fibers from

straightening and thus less apparent strains is observed.

Figure 2.3: Stress-strain behavior for a biaxial test with load ratio 1:1 on a PTFE-glass fiber membrane. Warp and fill

yarns show a different behavior [4].

Similarly, the strains in warp direction are smaller compared to the uniaxial test in warp direction

because of the tension in fill direction.

In Figure 2.4 (a) a biaxial test is shown for a PTFE –glass fiber membrane which was performed

with load ratio 1:5. Negative strains appear for the warp direction because of the higher tension

in the fill direction: the originally curved fill fibers will straighten, and consequently the warp

fibers have to curve and thus contract. Figure 2.4 (b) shows the stress-strain behavior for a biaxial

test with load ratio 5:1. Because a higher stress is applied to the originally straight warp yarns,

they stay straight and stretch. The fill yarns exhibit almost no strains since they retain their

original curved shape.

Figure 2.4: Stress - strain curves for a biaxial test with load ratio 1:5 (a) and 5:1 (b) on a PTFE-glass fiber membrane [4].

2.2.4. Cycle repetition When a loading cycle is repeated several times, the response of the coated fabric is different for

each cycle [35]. Figure 2.5 shows the stress-strain curve which is obtained if 5 cycles of a biaxial

test with load ratio 1:1 are performed on a PVC-polyester membrane. Each cycle, the stiffness

changes and the permanent strain diminishes. This process is called the removal of residual strain


25

and can be understood as follows: the fill yarns, which are curved around the warp yarns at the

beginning of the experiment, get more straightened during each load cycle. So each cycle the

crimp of the fill fibers is smaller than that of the previous cycle and consequently the apparent

strain decreases. This causes the stress-strain curve to become steeper. Generally, after 3 to 5

cycles this process stabilizes, as can be observed in Figure 2.5. Note that the residual strain is of

minor importance for the warp direction because these fibers are already relatively straight at the

beginning of the experiment.

Figure 2.5: Influence of cycle repetition for a 1:1 load ratio for a PVC-polyester membrane with a maximum stress of

12 kN/m [35].

2.2.5. Load history The load history plays an important role in the material behavior of membranes. The previously

applied loading determines the configuration of the fibers: a previously applied load in warp

direction has straightened the warp yarns, while a previously applied load in fill direction has

straightened the fill yarns. This configuration of the fibers is called the state of the material or the

crimp in the fibers. This state of the material can be categorized as one of the following:

(i) straight warp fibers and curved fill fibers, resulting in a stiffer behavior in warp direction than

in fill direction (ii) curved warp fibers and straight fill fibers, resulting in a stiffer behavior in fill

direction than in warp direction or (iii) something in between (i) and (ii). For this reason, the load

history plays an important role [4, 35]. Figure 2.6 shows the loading cycles, for a biaxial test with

load ratio 1:1, each time anteceded by another load history. Figure 2.7 shows the obtained stress-

strain curves. It is observed that the influence of the load history is significant during the first

loading cycles (Figure 2.7(a)). An increase of the load ratio of the previous loading cycles leads to

a decrease of the total strain (apparent and real strain). After five cycles, the influence of the load

history is diminished, due to the repetition of load cycles (Figure 2.7(b)).


26

Figure 2.6: Loading cycles, used to investigate the influence of load history [35].

Figure 2.7: Influence of the load history in warp direction for a load ratio 1:1 for a PVC coated polyester fabric [35].

2.2.6. Hysteresis When both loading and unloading cycle are performed, hysteresis is observed. In Figure 2.5 the

loading cycle with a load ratio 1:1 was repeated 5 times on a PVC coated polyester fabric. This

resulted in 5 elliptically shaped hysteresis loops in the stress-strain curve. This hysteresis can be

attributed to the loss of energy due to friction between fibers mutually on the one hand and

between fibers and coating on the other hand [4].

2.2.7. Strain rate Figure 2.8 shows the tensile uniaxial response for different loading times. It is seen that the strain

rate has a certain influence on the material behavior but practically, this dependency is so small

that it can be neglected [35].


27

Figure 2.8: Influence of the loading rate measured with uniaxial tensile tests in warp and fill direction [35].

2.2.8. Pre-stress The pre-stress is the stress level that is maintained in the membrane when no external loads are

applied. Since it is a long term loading, it results in creep of the material. Higher pre-stress leads

to less residual strains, and thus results in a stiffer material behavior during the first loading cycles

[4]. In Figure 2.9, results are shown for two virginal samples which were maintained under pre-

stress during 6 hours, at respectively 4% and 1.3% of the ultimate tensile strength (UTS) and

afterwards loaded with load ratio 1:1.

Figure 2.9: Influence of initial pre-stress level for a 1:1 loading in warp direction (a) and in fill direction (b) [35].

Figure 2.9 reveals that the warp yarns are not influenced by the initial pre-stress, while the

opposite is observed for the fill yarns. The fill yarns respond initially stiffer when applying a

higher pre-stress level [35].

2.2.9. Tensile strength The coating material does not contribute to the strength of a coated fabric, hence the strength is

exclusively determined by the strength of the yarns. Though, a correction factor has to be taken

into account because of two major reasons: (i) the yarns experience a reduction in strength during

both the weaving and coating process and (ii) the deflection of a yarn when crossing the

perpendicular yarns. Due to the latter, the tensile strength in warp direction can differ from that

in fill direction, even though the material of the yarns is the same [5].


28

Furthermore, the tensile strength depends on the presence of tears in the membrane. The

resistance against propagation of tears, which is called the tear strength, should be sufficiently

large in order to avoid pre-existing small flaws to propagate when applying the design loads. The

tear strength of a membrane depends on both the nature of the fibers and the coating. The

resistance for tear propagation is higher in case of a multi-axial stress state than for a uniaxial

stress state [36, 37].

Except for the tears, the connections and seams can reduce the tensile strength of the structure

significantly. Moreover they are subjected to creep, so that the rupture strength is dependent on

the duration of loading and temperature.

Due to many uncertainties in the design of tensile structures, the maximum stresses allowed in

the membrane are around 25% of the tensile strength [34].

2.2.10. Influence of temperature Temperature effects are usually found to be less significant for fabric structures when compared

to rigid constructions [5].

Figure 2.10 shows experimentally measured stress-strain curves for a PTFE coated glass fiber

fabric tested at temperatures ranging from -20°C up to 70°C, which are representable

temperatures for real tensile structures. It is clear that no temperature dependency of the

mechanical properties is observed [38].

Figure 2.10: Stress-strain behavior under different temperatures for a PTFE coated glass fiber fabric material in warp

direction (a) and fill direction (b) [38].

The influence of temperature on the tensile strength on the other hand depends on both the

fibers and the weaving method. A low temperature reduces crimp deformation and increases the

resistance against fiber pullout. Besides, fibers become more brittle and the creep rate decreases.

Therefore, the tensile strength increases while the strain at failure decreases, for decreasing

temperature. On the contrary, at a high temperature the fibers become soft while the creep rate

increases. Therefore, the tensile strength decreases while the strain at failure increases for

increasing temperature. This decrease can be expressed with the high temperature reduction factor

γt (-), which is defined as:

ij =��[


29

where f1 (MPa), f2 (MPa) is the tensile strength at 70°C, respectively at 23°C. The high

temperature reduction factor for PTFE coated fabric is 1.01 in warp direction and 1.03 in fill

direction, indicating that the effect can be neglected for PTFE coated glass fiber fabrics.

Note that in the previous only the influence of temperature on the coated fabric itself was

discussed. When the seams of the structure are exposed to high temperatures, their creep rate

increases and thus changes the behavior and tensile strength of the construction.

2.3. Modeling the membrane behavior Because of the nonlinear characteristics of the stress-strain behavior of coated fabrics, it is

difficult to establish a single function which adequately represents the membrane response. In the

following subsections some of the developed material models from literature are presented and

discussed.

The most common assumptions in frequently used material models involve orthotropic behavior

and a plane stress state. The assumption of orthotropy is justified because typical fabrics exist of

yarns, which are arranged along two orthogonal directions [39]. The second assumption is easily

understood by taking into account the small thickness of the fabric (usually smaller than 1 mm).

This induces that stress components along the direction of the thickness can be neglected, hence

resulting in a plane stress state.

2.3.1. Linear elastic orthotropic plane stress model One of the simplest approaches is considering the membrane material as a linear elastic

orthogonal anisotropic two-dimensional plane-stress structure. Five independent elastic constants

[39], namely two Young’s moduli EkandE�, a shear modulus Gk� and two Poisson’s ratio νk� and ν�k (see section 2.4), are determined from the warp and fill stress-strain curves at a load ratio

and stress magnitude typical for the structure [4]. These values remain constant throughout the

structural analysis [30, 40].

p qrqs2qrsu = NOOOOOP 1vr −wsrvs 0−wrsvr 1vs 00 0 1xrsQR

RRRRSp �r�s�rsu (2.1)

Because of the assumption of plane stress, the out of plane stresses are equal to 0. This method is

numerically efficient, but it does not capture the material behavior in a proper way. Another

disadvantage is that four independent parameters are obtained from the experiments, i.e. two

Young’s moduli and two Poisson’s ratios. The compliance matrix is forced to be symmetrical by

averaging the Poisson’s ratios, but this results in a loss of accuracy.

The Japanese standard MSAJ/M-02-1995 [33, 34] uses this method, taking into account following

reciprocal relationship in order to obtain a symmetric stiffness matrix:

vrvs = wsrwrs (2.2)


30

The set of fictitious elastic constants, which consist of the Young’s moduli Ek and E� and the

Poisson’s ratio νk� (the shear modulus Gk� is not considered) is determined from the load-strain-

paths in a double step correlation analysis. The experimentally determined stress-strain curves are

substituted by a straight line, with a slope that satisfies the assumed linear elastic plane stress

behavior. The set of four fictitious constants has to satisfy the experimental loading paths for all

five load ratios as optimal as possible. The optimum set of parameters is determined by the least

square method. In this method, the sum of the squared errors for a certain subject interval [a, b]

between a continuous function f(x) and an approximation equation y(x) is minimized:

U = yz�B� − DB�{[|B}~ → �� (2.3)

The errors can either be defined as the vertical differences (load errors S�, see Figure 2.11(a)) or

the horizontal differences (strain errors S�, see Figure 2.11(b))

Figure 2.11: Load errors (a) and strain errors (b) [34].

Hence, one can minimize either the load term or the strain term: S� → min or S� → min. The

basis of the routine is the calculation of regression lines using the least squares method. A

regression line in a load-strain-diagram follows the linear equation

� = � ∗ q + � (2.4) With m the slope and b the intersection point of the regression line with the load-axis at zero

strain.

In the first step, the routine optimizes the regression lines of the load-strain-paths for all

experimentally evaluated load ratios (Figure 2.12). Usually, the regression line has another

intersection point b with the load-axis at zero strain than the test data path itself. To determine

the stiffness, the intersection point is not important but the slope, thus the intersection point of

the regression line may be switched into the intersection point of the test data path for the plots

(Figure 2.12).


31

Figure 2.12: Optimized regression line and switched regression line [34].

In the second step, all possible combinations of the three fictitious elastic constants within limit

values are generated.

In the third step, the strain values of the fictitious load-strain lines are calculated for one arbitrary

load level at each load ratio according to

qr = �rvr ∗ � −wrs ∗ �svs ∗ � qs = �svs ∗ � −wsr ∗ �rvr ∗ � (2.5)

Knowing the strain values enables the evaluation of the slope of the fictitious load-strain lines.

The sum of the squared strain errors over all n test data points and m load-strain-paths,

considered in the determination of constants, is calculated using:

U� =��q5�~�� − q5��[�5��

�6�� (2.6)

With ε�� = �0��1�1 being the strain value for a fictitious load-strain line j for each existent test

data point i of the related load-strain-path j. It is assumed that the minimum value of S� provides

the optimum set of elastic constants. For a Glass/PTFE material the procedure was evaluated. In

Figure 2.13 the calculated and experimental strains are plotted against the ‘leading membrane

force’, which is the larger one for each load ratio.


32

Figure 2.13: Experimental and calculated (according to MSAJ standard) stress strain curves for different load ratios

[34].

Taking into account the linearity, good agreement is observed for load ratios 1:1, 2:1 and 1:2 but

not for load ratios 5:1 and 1:5 for a load smaller than 30 kN/m.

The method is easily applicable, though the assumptions lead to an unrealistic simplification of

the real material behavior. The material model does not take into account the hysteresis, the

crimp interchange, the influence of load history, etc. This implies that the determined fictitious

constants largely depend upon the underlying determination options.

2.3.2. Cable networks A method quiet often used to design membranes for architectural applications is the ‘form

finding’ method. The material behavior is assumed to be orthotropic linear elastic. The difference

with the previous model is the fact that each strip of the membrane is approximated by a cable

and the whole curved membrane surface by a cable network. The ‘form finding’ method was

described in subsection 1.3.1. By means of this method, the equilibrium shape and the forces in

the cables are determined, starting from the ‘force densities’ and the anchor points. External

forces acting on the membrane are taken into account by substituting the zeros of the reaction

force components of the free nodes by the components of the external load. This method is

implemented in several software packages (e.g. EASY). In this method, the calculation times are

short because of the linear character of the equations. It is easy to understand that this model

does not give accurate results. On the one hand a membrane is a continuous surface rather than a

discontinuous cable network, thus the deformation by shearing [41, 42] is not determined


33

correctly. On the other hand the nonlinearity and permanent deformation of the material are not

taken into account.

2.3.3. Piecewise linear elastic orthotropic plane stress model

In this material model, stresses and strains are replaced by small increments Δσ and Δε in order

to linearize an interval of the nonlinear stress-strain curve [4, 40]. For example, the three moduli

(Ek, E�andνk�) are determined between an assumed pre-stress and an upper value. To simulate

different environmental loads, incremental loading is used, both positive and negative. This

model takes into account the nonlinearity of the stress-strain curves by approximating them with

a piecewise linear curve. Because of the linearization. the numerical computations are very simple

and time inexpensive. However hysteresis, residual strains, influence of load ratio and load

history are disregarded, which makes this model inappropriate to rigorously model a tent fabric.

2.3.4. Nonlinear elastic material model, taking the influence of the load ratio

into account The model is based upon three main assumptions:

• The material behavior is linear elastic with plane stress orthotropy for a given load ratio:

Z∆qr∆qs ^ = �1vrir�−wsrvsis�

−wrsvrir�1vsis� � ZΔ�rΔ�s ^ (2.7)

• The Poisson’s ratio νk� is independent of the normalized load ratio’s and the reciprocal

relation �� = �� is fulfilled and thus:

Z∆qr∆qs ^ = �1vrir�−wrsvrir�

−wrsvrir�1vsis� � ZΔ�rΔ�s ^ (2.8)

• The Young’s moduli Ek and E� are formulated as a linear function of the normalized load

ratios γk and γ� [30]:

vrir� = vr�:� + Δvr %ir − 1√2'

vs£is¤ = vs�:� + Δvs %is − 1√2'

(2.9)

where γk and γ� are the normalized load ratios in warp and fill direction and they are

used instead of the common load ratio σk: σ� [30]. They are defined as:

ir = �r¥�r[ +�s[

is = �s¥�r[ +�s[ (2.10)


34

Ek�:� and E��:� are the reference values for warp and fill Young’s moduli given for the 1:1 load

ratio (γk = �√[ respectively γ� = �√[), ΔEk and ΔE� represent the variation of warp and fill Young’s moduli for the complete range

of load ratios (0 ≤ γk ≤ 1 respectively 0 ≤ γ� ≤ 1), while νk� is the in-plane Poisson’s ratio.

The material model has five parameters: Ek�:�, E��:�, ΔEk, ΔE� and νk� which are obtained by a

least square fit in which the deviation of the experimental and modeled strains is minimized. The

modeled strains are calculated by equations (2.7), (2.9) and (2.10) for all applied stresses and

loads, while the experimental strains are measured directly. This material model can then be

integrated in finite element software by programming a user defined material routine (called

‘usermat’). At every time step t and every integration point i, the stresses σ�§, strains ε.§ and strain

increments Δε.§ are passed to the usermat, which then updates the stresses in order to obtain σ�§¨©§. In Figure 2.14 the results of the previous model are compared to the experimental measurements

for different load ratios. Note that an extra FEA model is included in the figure, namely an

orthotropic S FEA model, which is not described here. Further details about the latter can be

found in [30].

Figure 2.14: Experimental and calculated (according to a nonlinear elastic material model, taking the load ratio into

account) stress strain curves for a PTFE coated glass fiber fabric for different load ratio and a maximum load of 12kN/m [30].

Only a small difference is observed between the FEA - Model and the experimental results. Even

for different load ratios, the membrane behavior is accurately predicted. However, crimp

interchange, hysteresis and load history are not taken into account. Yet it is possible making the

stiffness constants dependent on the load ratio. Implementation in finite element methods is

more laborious and time-consuming than the previous methods.


35

2.3.5. Response surfaces with division in quadrilaterals Response surfaces are a visualization of the stress-strain behavior by plotting the orthogonal

stresses (σk, σ�) and strains (εk, ε�) in the σk, σ�, εk and σk, σ�, ε� coordinate systems, for

several stress ratios. The surface is divided into quadrilaterals, in which a plane stress linear

orthotropic material model is assumed. The elastic constants are determined in each quadrilateral

[4, 30, 43]. In this way, the elastic constants and thus the compliance matrix are established using

a multi-step linear approximation. The size of the quadrilaterals is critical to ensure the accurate

capturing of the fabric behavior. The use of response surfaces is particularly interesting for

modeling non-linear behavior, dependent on the load ratio, which is the case for membranes.

However, high amounts of experimental data are required and a large number of parameters

needs to be calculated, leading to large computational times.

2.3.6. Response surfaces without division in quadrilaterals Another approach is the use of stress-stress-strain response surfaces without dividing the surface

in quadrilaterals in which the test data are fitted to a plane stress material model. This method has

advantages since the plane-stress assumption is not appropriate for describing the behavior of a

coated fabric [4, 33]. For this, numerous stress states are explored giving a much wider

population of the data space. A response surface can be fitted to these scattered data points

(Figure 2.15), and a direct correlation between stresses and strains can be used for structural

analysis.

Figure 2.15: Response surfaces: experimental data point in stress-stress-strain space (a) and fitted surface (b).

Direct correlation is used between pairs of warp and fill strains to obtain stresses, thus surface

gradients are not required. A ‘look-up’ table of warp and fill stresses and strains replaces the

elastic constants in the analysis. Consequently, the surface does not need to be defined by a

fitting procedure, nor should it be differentiable. The difference in loading and unloading

behavior gives data points on two surfaces which define the upper and lower bounds of the

fabric response. In this modeling approach, the influence of residual strain, which depends on the

pre-stress level and load history, is not included. With this method, a good correlation between

the model and the experiments can be obtained. However, a large quantity of test data is required

in order to obtain an accurate model and the use in FEM analysis is difficult.


36

2.3.7. Day’s method Day’s method is based on the representation of the nonlinear stress-strain behavior in soil

mechanics. In this representation, the mean and the difference of the principal strains are related.

Because the membrane material can be treated as orthotropic, the principal stresses lie in warp

and fill directions. In this way, the shear stresses can be treated separately. This behavior is

described by following set of equations [4, 40, 43, 44]:

��~� =�� +� 2 q��~� =q� +q 2 �ª5ss�«�� =� −��2 qª5ss�«�� =q −q�2

(2.11)

Where σ� and σ¬ are the principal stresses, and ε� and ε¬ are the corresponding principal strains. σ�� and σ®��h�� are respectively the mean and the difference of the principal stresses. q��~� and qª5ss�«�� are the mean and the difference of the principal strains. The relations

between the mean and the difference are given by

��~� =��q��~�� + �[qª5ss�«�� ª5ss�«�� =�\q��~�� + �]qª5ss�«�� (2.12)

in which f� to f] are functions to be determined. Shear stress and strain are related by an

independent linear function f°:

�� =�°q� � (2.13)

Contrary to the universally adopted plane stress approach, this method attempts to encapsulate

the data for three different stress ratios in two equations. However, up to now it is unclear

whether this method can be applied to a wide range of stress states. In addition, questions arise

about the reliability of the used equations when interpolating the tested stress ratios [4].

2.3.8. Micro-mechanical models Micro-mechanical models derive the fabric behavior from a more fundamental basis, i.e. the

microstructure of the material [29, 30, 45]. They have already been applied successfully for a

variety of materials and they emphasize the great influence of the local weave geometry as well as

local mechanisms on the global material behavior. A micromechanical material model of the

yarns and the coating enables to include (i) crimp interchange, (ii) yarn and coating extension and

(iii) friction between the warp and fill yarns [30]. Generally, the micromechanical model is limited

to a unit cell which represents the complete fabric structure. Such a unit cell is then implemented

in a FEM simulation. With the use of periodic boundary conditions, one is able to simulate the

mechanical behavior of the material. These models can represent the mechanical response of the

fabric up to some extent. However the complexity leads to large computation times. The

requirement of a large number of parameters makes it a very laborious method [30].


37

2.4. Reciprocal relationship The plane stress assumption leads to a number of inconsistencies for membrane materials. For

example, linear elastic orthotropic materials subjected to biaxial tests satisfy the reciprocal

relationship

wrsvr = wsrvs (2.14)

This relation is invalid for membranes [4]. In [46] following values were obtained for some typical

membranes:

PTFE/glass fiber: �� = 2157 and

�� = 296 → �� ≠ �� PVC/Polyester:

�� = 2242and �� = 395 → �� ≠ ��

The maximum load of the biaxial tests was 25 % of the UTS. Residual strain has been removed

from the test results and the mean value of the loading and unloading curves has been calculated.

It was proposed that the values adhere more closely to an inverse of the reciprocal relationship ¸¹º»º = ¸º¹»¹ . However, for this inverse reciprocal relation to hold, a constant C needs to be

introduced:

wrsvs = ¼ ∗ wsrvr (2.15)

Values of C are reasonably consistent for a certain fabric material:

PTFE/glass: mean value of C = 1.40 and standard deviation = 0.11

PVC/polyester: mean value of C = 1.51 and standard deviation = 0.33

If a similar constant C' is introduced in the reciprocal relationship (2.14), the values of C' are

much more variable:

PTFE/glass: mean value of C’ = 7.32 and standard deviation = 2.82

PVC/polyester: mean value of C’ = 5.70 and standard deviation = 2.54

Hence based upon this, it can be concluded that the inverse reciprocal relationship is more

appropriate for coated woven fabrics. Though, the need for the additional constant C to match

the inverse reciprocal relation violates the conservation of energy E. Consider a unit square of

material and two loading conditions:

a: σ� and σ� in the warp and fill directions respectively

b: σ� and σ® in the warp and fill directions respectively

If ‘a’ is followed by ‘b’, the strain energy U can be calculated as:

½ =12�~q~ +�}q}� +12��q� +�ªqª� +�~q� +�}qª� (2.16)


38

If on the other hand ‘b’ is followed by ‘a’, the strain energy can be calculated as:

½ =12�~q~ +�}q}� +12��q� +�ªqª� +��q~ +�ªq}� (2.17)

The principle of conservation of energy implies that the strain energy U is independent of the

loading path and loading sequence. This results in:

�~q� +�}qª� = ��q~ +�ªq}� (2.18)

Using the strain-stress relationships εk = �»¹ ∗ σk − ¸º¹»º ∗ σ� and ε� = − ¸¹º»º ∗ σk + �»º ∗ σ� one

obtains following equation:

¾ ∗ �~ ∗ �� + ¿ ∗ �~ ∗ �ª + ¼ ∗ �} ∗ �� + À ∗ �} ∗ �ª= ¾ ∗ �~ ∗ �� + ¿ ∗ �} ∗ �� + ¼ ∗ �~ ∗ �ª + À ∗ �} ∗ �ª

(2.19)

or

−ν�kE� ∗ �~ ∗ �ª − νk�E� ∗ �} ∗ �� = −ν�kE� ∗ �} ∗ �� − νk�E� ∗ �~ ∗ �ª (2.20)

Which can only be satisfied for ¸º¹»º = ¸¹º»º , which is the reciprocal relation. The latter is correct in

the context of a homogeneous material. But, as already stated, this is highly inaccurate for a

coated woven fabric. This inaccuracy largely arises from the interaction of warp and fill yarns, in

combination with the behavior of the twisted yarn structure. This effect is further augmented by

the fact that the fabric is composed of two different materials, dominating the fabric response at

different load levels (essentially the coating at low loads, the yarns at high loads). Any lack of

conservation of energy is due to frictional effects at crossovers, inelastic yarn crushing and

inelastic coating extension [4, 46].

2.5. Quantifying and understanding the biaxial behavior of

different membrane types In this section a tool for quantifying and understanding the biaxial behavior of membranes is

presented. The test regime for determining the behavior of coated fabrics is stress controlled,

which is appropriate if the range of stresses is known in advance. However, a FEM calculates

displacements from which warp and fill strains are determined. Therefore it is useful to have an

idea about the strain values that occur in a tent structure. This can be achieved by plotting the

test data in the strain-strain space. In Figure 2.16 the experimental data points are plotted in the

strain-strain space for both a PVC coated polyester fabric and a PTFE coated glass fiber fabric

[4].


39

Figure 2.16: Experimental data points in plotted strain-strain space showing the bounds of the feasible membrane

response for a PVC coated polyester fabric (a) and for a PTFE coated glass fiber fabric (b) [4].

The plotted data points are extracted from tests with stresses ranging from zero through pre-

stress up to 20 or 25% of the UTS. Hence, the population of the strain-strain space indicates the

bounds of the feasible membrane response. Figure 2.16(b) shows that the PTFE-glass fiber fabric

has a very discrete response envelope. The behavior is dominated by crimp interchange with little

extension of the stiff glass fiber yarns. In contrast, the polyester yarns are more easily extensible

and thus giving a greater range of possible strain states (Figure 2.16 (a)).

2.6. Conclusions The behavior of coated fabrics (both PVC coated polyester fabrics and PTFE coated glass fiber

fabrics) is discussed in detail, a relevant literature overview is given. It is clear that there are some

major differences compared to the behavior of rigid materials. Coated fabrics can be considered

as a composite, which consist of orthogonal fibers and a coating. In general, these fabrics are

characterized by severe nonlinear orthotropic behavior. The characteristic structure of the woven

fibers causes a complicated interaction between the warp and fill direction, which is called crimp

interchange. The fill yarns of a virginal coated fabric are initially curved, while the warp yarns are

straight. Hence depending on the applied load, the yarns can be reallocated. It is easy to

understand that this reallocation is highly dependent upon the applied load ratio and the load

history. By repeating the load cycle three to five times, the load history is effectively removed.

The introduction of this load cycle repetition induces a hysteresis effect.

Several authors have tried to capture this complex behavior with the use of a FEM software

package. The most popular methods assume a linear elastic orthotropic material behavior, which is a

huge simplification for the complex material behavior of coated fabrics. A variant of this method

encloses the form finding method, which is based on cable networks and the associated equilibrium

state. More realistic models assume piecewise linear elastic behavior and are therefore capable to

approximately model the nonlinear response of coated fabrics. However, permanent

deformation, load ratio dependency, load history dependency and hysteresis are not taken into

account. Therefore, more advanced models have been introduced. Response surfaces with division in

quadrilaterals capture both the nonlinear behavior and load ratio dependency of a fabric. In that

method, the response surfaces are divided in quadrilaterals, in which a plane stress linear elastic


40

material behavior is assumed. Another method also uses the response surfaces but without division in

quadrilaterals. They are based on a look-up table which consists of many data points obtained by

experiments exploring a wide range of the stress-stress-strain behavior of the coated fabric.

Another way of modeling a coated fabric is found in Day’s method. This method relates the means

and the differences of stresses and strains. Permanent strain, hysteresis and load history

dependence are not included and the reliability when interpolating between different load ratios is

not yet proven. The most advanced models are the so-called micromechanical models, which model

the fabric on a microscopic scale, i.e. the yarns, the coating as well as their interactions are taken

into account. The main disadvantage of these models is found in the extremely high

computational time, making them inappropriate for the simulation of a complete tent structure.

41

Chapter 3.Uniaxial tensile tests

3.1. Introduction In order to develop an appropriate FEM model of a coated fabric, a profound knowledge of both

the material properties and the structural behavior of the coated fabric is required. Since these

characteristics are highly dependent on the type of material, it is essential to experimentally

investigate the material of interest. In general, a coated fabric can be characterized by the

application of a uniaxial stress state as well as a biaxial stress state. After a description of the

experimental setup for the uniaxial tensile test, the post-processed experimental results are

presented and discussed. In the penultimate section, the principles for modeling a uniaxial tensile

test in Abaqus are described. This chapter ends with several relevant conclusions.

3.2. Experimental setup Uniaxial tensile tests are performed by a standard tensile testing machine. The membrane samples

have a rectangular shape and are cut out along the direction of interest. By applying load to the

ends, the sample is subjected to a state of uniaxial tension (Figure 3.1). The strains are

continuously measured as a function of the applied load during the experiment.

Figure 3.1: Photo of a uniaxial experiment on a membrane [47].

The main advantage of the uniaxial test is its simplicity to perform an experiment: (i) easy to set-

up, (ii) easy to post-process (iii) short experimental time and (iv) small amount of required

material. As a result, several material parameters can be investigated in a relative short time [48].

Generally two types of experiments are carried out. In the first type, the load is continuously

increased until the UTS is reached at which failure occurs. The second type is ‘load cycle

repetition’, in which a loading and unloading cycle is repeated several times. The load varies

between a pre-load and a maximum load. In this way, both the initial material behavior (when

Chapter 3. Uniaxial tensile tests

42

crimp is still present) and stabilized behavior are determined. Usually, three to five identical load

cycles are necessary to exclude the effects of crimp. The uniaxial experiments can be either load

or displacement controlled.

Because the loading rate dependency is quite moderate, the membrane behavior does not need to

be investigated for different loading rates [35]. However, to improve the consistency of the

experimental results, it is important to keep the loading rate more or less constant during the

experiment.

Due to the specific characteristics of a membrane, uniaxial test data do not provide sufficient

information for developing an accurate material model [49], therefore a more advanced test

procedure, i.e. the biaxial tensile test, has to be taken into account (see chapter 4).

For this project, the experimental tensile tests are carried out at the ‘Free University of Brussels'

(VUB) at the department 'Mechanics of Materials and constructions’ (MeMC) by Paolo Topalli

[50]. In this thesis the results are further processed in order to obtain useful information, which is

discussed in detail in the following section.

During the experiments, strains are determined by means of ‘Digital image correlation’ (DIC).

This is an experimental technique which offers the possibility to determine displacement and

deformation fields at the surface of objects, based on a comparison between images taken at

different load steps. The DIC software processes and visualizes the gathered data in order to

obtain an impression of the distribution of strains in the measured object. Two cameras are used

in order to measure both in-plane and out-of-plane displacements [51]. The strains are measured

in the central part of the cruciform sample.

The applied loads and the elongations are recorded by the loading devices of the uniaxial testing

device. The stresses themselves at the center of the specimens are not measured. Instead, they are

calculated by dividing the applied load by the cross section (thickness multiplied by width). For

this, the stresses shown in all following curves are not the true stresses present at the center of

the sample. However, for uniaxial tensile tests, the differences are very small.

3.3. Experimental test results

3.3.1. Loading until failure

In the first step of experiments, three different samples were tested up to failure: (i) samples

along the warp directions, (ii) samples along the fill directions and (iii) shear samples rotated 45°

relative to warp and fill direction (Figure 3.2). These tests provide the ultimate tensile strength of

the material for the three cases.


43

Figure 3.2: Schematic drawing of the uniaxial test in warp direction (a), in fill direction (b) and in shear (c).

The tested samples have a thickness of 0.83 mm, a width of 50 mm and a length of 200 mm. The

experiments are displacement controlled with a speed of 100 mm/min. Both the load and the

displacement are recorded by the loading device. The experiments were performed at ambient

temperature.

3.3.1.1. Warp

Figure 3.3 shows the recorded load – elongation curves for six identical tensile tests in warp

direction.

Figure 3.3: Uniaxial tests up to failure in warp direction.

The load varies from zero up to the UTS. For the six tests, the mean of the UTS is 4672 N with a

standard deviation of 145 N. The minimum and maximum UTS are respectively 4392 N and

4784 N.

3.3.1.2. Fill

The load – elongation curves for five tests in fill direction up to failure are shown in Figure 3.4.


44

Figure 3.4: Uniaxial tests up to failure in fill direction.

The mean of the UTS is 4109 N with a standard deviation of 157 N. The minimum and

maximum UTS are respectively 3997 N and 4360 N. The UTS in warp direction differs from the

UTS in fill direction (further information in paragraph 2.1.9). For the further experiments, the

UTS is assumed to be 4000 N for both warp and fill direction. With a thickness of 0.83 mm and a

width of 50 mm for the sample, a load of 4000 N corresponds to a stress of 96.4 MPa.

3.3.1.3. Shear

The results of six uniaxial tensile tests on samples rotated over 45° with warp direction are shown

in Figure 3.5.

Figure 3.5: Uniaxial tests up to failure in shear.

The mean of the UTS is 2352 N with a standard deviation of 39 N. The UTS at 45° corresponds

to 53 MPa and is considerably smaller compared to the UTS value in warp and fill direction. This

is simply understood taking into account that the load is not applied along the fibers.

3.3.2. Load cycle repetition

In this master thesis, the Japanese standard MSAJ [34] is followed, which prescribes the pre-load

and the maximum applied load to be respectively 2.5 % and 25 % of the UTS. For the present

material, this results in the following standards:

(i) Warp and fill direction:

��~� = 25% ∗ ½ÂU = 25% ∗ 96.4ÄFÅ = 24.1ÄFÅ


45

��«�Æj«�ÆÆ = 2.5% ∗ ½ÂU = 2.5% ∗ 96.4ÄFÅ = 2.41ÄFÅ

(ii) 45° direction:

��~� = 25% ∗ ½ÂU = 25% ∗ 53ÄFÅ = 13.3ÄFÅ

��«�Æj«�ÆÆ = 2.5% ∗ ½ÂU = 2.5% ∗ 53ÄFÅ = 1.33ÄFÅ

The tested samples have a width of 50 mm and a length of 200 mm. The experiments are

displacement controlled. The experiment is conducted at ambient temperature.

3.3.2.1. Warp

The evolution of the load as a function of time is shown in Figure 3.6. The load increases from

zero to pre-stress in 500 seconds, thereafter the load varies five times between pre-stress and the

maximum load.

Figure 3.6: Load as a function of time for a uniaxial cycle repetition test in warp direction.

The elongation of the sample is recorded during the experiment and shown in Figure 3.7. After

the first cycle a large permanent elongation is observed. During the following load cycles, this

permanent elongation continues to increases but the process stabilizes. The total permanent

elongation after five cycles amounts 4.33 mm.

Figure 3.7: Elongation as a function of time for a uniaxial cycle repetition test in warp direction.

Figure 3.8 gives the strains in warp and fill direction as a function of time measured by DIC. As

could be expected, the strain in warp direction is positive and similar to the elongation curve. The

strain in fill direction is negative due to the Poisson effect.


46

Figure 3.8: Strain (in warp and fill direction) as a function of time for a uniaxial cycle repetition test in warp direction.

In Figure 3.9 the resulting stress-strain curve is shown. This curve clearly shows the presence of

nonlinearity, permanent strain and hysteresis effect, which is characteristic for the investigated

material. From the sequencing hysteresis loops, the stabilizing behavior can be clearly observed.

Since it is a uniaxial test in warp direction, there is no influence observed of crimp interchange.

Figure 3.9: Stress (warp) as a function of strain (warp) for a uniaxial cycle repetition test in warp direction.

In chapter 2, a stress-strain curve from literature (shown in Figure 3.10 (a)) for a uniaxial test in

warp direction on a glass fiber ETFE foil was discussed. The behavior was almost linear

elastic.Figure 3.10 (b) shows the corresponding stress-strain curve (only the first loading cycle)

for a uniaxial test on the PVC-polyester fabric derived from Figure 3.9.


47

Figure 3.10: Stress (warp) as a function of strain (warp) of the first loading for a uniaxial tensile test in warp direction

for a glass fiber ETFE foil (a) and for the PVC-polyester fabric (b).

Major differences are observed between the two stress-strain curves for the different materials.

For the ETFE–glass fabric, the material stiffness is constant, while it starts decreasing for the

PVC–polyester fabric at a stress of approximately 14 MPa. The dissimilarity of the stress-strain

curves is probably caused by the occurrence of plasticity phenomena in the case of the PVC-

polyester fabric: the yield stress for PVC-polyester is reached during the uniaxial test, while this is

not the case for PTFE-glass. This will be discussed further in detail in paragraph and paragraph

3.3.2.6.

From the slope of the unloading curve (Figure 3.9) the Young’s modulus in warp direction can be

estimated:

vr =∆�Ç��È~ª5�V∆qÇ��È~ª5�V ≈ 23.91ÄFÅ − 2.47ÄFÅ0.039203 − 0.020222 = 1.130xFÅ (3.1)

In Figure 3.11, the Poisson ratio νk� =− �º�¹ as a function of time is presented. It is seen that the

value of the Poisson ratio is variable during the experiment. This is probably due to the crimp

interchange between the warp and fill yarns during loading and unloading. Note that the Poisson

ratio becomes negative during preloading, which is easily understood by the observation that the

strain in both warp and fill direction increases, although it is expected to decrease in fill direction.

This is presumably due to an inaccuracy of the experiment: the reference image for DIC was

taken at the moment the sample was not yet preloaded in the tensile machine. For this reason,

the sample was not flat, but was rather hanging under the influence of gravity. Only after

applying the preload, this effect was eliminated and the sample became flat. However, the strains

are calculated with respect to the reference image taken from the hanging sample, and are

therefore not completely correct. This can explain the deceptive negative Poisson ratio.


48

Figure 3.11: Poisson ratio νwf as a function of time for a uniaxial cycle repetition test in warp direction.

3.3.2.2. Fill

Similar to the previous subsection, the next figure shows the applied load in fill direction as a

function of time.

Figure 3.12: Load as a function of time for a uniaxial cycle repetition test in fill direction.

The recorded elongation as a function of time is shown in Figure 3.13. The permanent elongation

is large for the first cycle and stabilizes during the next cycles. The total permanent elongation at

the end of the experiment is 17.36 mm. This is four times higher than the permanent elongation

in warp direction for a uniaxial test in warp direction.

Figure 3.13: Elongation as a function of time for a uniaxial cycle repetition test in fill direction.

The following curve shows the measured strains in fill and warp direction as a function of time.

The strain in fill direction is positive and similar to the elongation. The strain in warp direction is


49

negative due to both the Poisson effect and crimp interchange, because the warp yarns were

stretched at the beginning of the experiment and thus forced to bend during the experiment.

Figure 3.14: Strain in fill and warp direction as a function of time for a uniaxial cycle repetition test in fill direction

Figure 3.15 shows the stress-strain graph for the uniaxial test in fill direction. Again nonlinearity,

hysteresis and permanent strain are observed.

Figure 3.15: Stress (fill) as a function of strain (fill) for a uniaxial cycle repetition test in fill direction.

As in the previous paragraph, comparison of the first cycle of the stress-strain curve for an

ETFE-glass fabric (Figure 3.16 (a)) with the PVC polyester fabric (Figure 3.16 (b)) reveals major

differences.

Figure 3.16: Stress (fill) as a function of strain (fill) of the first loading for a uniaxial tensile test in fill direction for a

glass fiber ETFE foil (a) and for the PVC-polyester fabric (b).


50

At the beginning of the experiment a small stiffness is observed for both materials due to the

straightening of the initially curved fill fibers. Once the fill fibers are straight, the stiffness

increases indicating that the fibers are stretched. For the PTFE-glass fabric, this stiffness remains

constant during the remainder of the experiment, while for the PVC-polyester fabric the stiffness

decreases at a stress which is more or less equal to 14 MPa. As for the warp direction, this

behavior can be probably ascribed to the occurrence of plasticity phenomena (see paragraph

3.3.2.7).

From the slope of the unloading curve, the Young’s modulus in fill direction is estimated as:

vs =∆�Ç��È~ª5�V∆qÇ��È~ª5�V ≈ 23.64ÄFÅ − 2.44ÄFÅ0.1083 − 0.0826 = 0.825xFÅ (3.2)

Figure 3.17 shows the Poisson ratio ν�k =− �¹�º as a function of time. Similar to the results of

Figure 3.11, the value of the Poisson ratio varies during the experiment. During preloading, the

Poisson ratio is negative because the strain in both warp and fill direction increases with

increasing load. As explained in paragraph 3.3.2.1 it is plausible that the measured strains are not

completely reliable, because the reference image was taken at a moment in which the sample was

not yet preloaded. After preloading, the Poisson ratio becomes positive and large variations are

observed.

Figure 3.17: Poisson ratio νfw as a function of time for a uniaxial cycle repetition test in fill direction.

3.3.2.3. Shear

In the shear tests, six loading and unloading cycles were applied with a maximum load of 550 N.

The applied load as a function of time is shown in Figure 3.18, and the corresponding elongation

as a function of time is presented in Figure 3.19.


51

Figure 3.18: Load as a function of time for a uniaxial cycle repetition test in shear.

Figure 3.19: Elongation as a function of time for a uniaxial cycle repetition test in shear.

Because there are no fibers along the loading direction (Figure 3.2), the stiffness is much lower

compared to the stiffness in warp and fill direction. The coating of the sample gets stretched,

while the fibers carry only a very small load, causing an angular rotation between the warp and fill

fibers. This is schematically shown in Figure 3.20.

Figure 3.20: Deformation of the sample during a uniaxial tensile test in shear.

The total permanent deformation at the end of the experiment equals 46.22 mm. Compared to

the uniaxial test in warp and fill direction, this value is more than 10 times, respectively 2 times

higher.

The measured strains are plotted in Figure 3.21 as a function of time. The strain in loading

direction is similar to the elongation in loading direction. The strain perpendicular to the loading

direction is negative and in absolute value higher than the strain parallel to the loading direction.


52

Figure 3.21: Strain as a function of time for a uniaxial cycle repetition test in shear.

The stress-strain curve (shown in Figure 3.22) resembles the corresponding curves for uniaxial

tests in warp and fill direction, except for the higher permanent strain and the larger zone of low

stiffness. From the slope of the unloading curve the shear modulus can be estimated:

xrs =∆�Ç��È~ª5�V∆qÇ��È~ª5�V ≈ 13.05ÄFÅ − 2.44ÄFÅ0.28016 − 0.24481 = 0.3xFÅ (3.3)

Figure 3.22: Stress as a function of strain for a uniaxial cycle repetition test in shear.

3.3.2.4. Warp (loading up to ½ of 25 % UTS)

Even though the Japanese standard prescribes a maximum load of 25 % UTS, it is interesting to

investigate the response of the coated fabric for a maximum load which is only half of that. In

the following figures, the corresponding load and strain values as a function of time are plotted,

as well as the stress-strain curve.


53

Figure 3.23: Load (up to 500 N) as a function of time for a uniaxial cycle repetition test in warp direction.

Figure 3.24: Strain (warp and fill) as a function of time for a uniaxial cycle repetition test in warp direction up to a load

of 500 N.

Figure 3.25: Stress – strain curve for a uniaxial cycle repetition test in warp direction up to a load of 500 N.

The curves representing load and strain as a function of time are similar to the corresponding

curves for a maximum load of 1000 N. For the stress-strain curve on the other hand, clear

differences are observed: for a maximum load of 500 N, there is almost no permanent strain nor

hysteresis. The absence of permanent strain is most probably because the yield stress is not

crossed, so no plasticity phenomena are induced. The fact that no hysteresis is observed for this

experiment strongly suggests that this effect is mainly caused or triggered by the occurrence of

plasticity.


54

3.3.2.5. Fill (loading up to ½ of 25 % UTS)

For the same reasons as explained in paragraph 3.3.2.4, uniaxial tests are performed in fill

direction with a maximum load equal to ½ of 25 % of the UTS. In the following figures, the

corresponding load and strain values as a function of time are plotted, as well as the stress-strain

curve.

Figure 3.26: Load (up to 500 N) as a function of time for a uniaxial cycle repetition test in fill direction.

Figure 3.27: Strain (warp and fill) as a function of time for a uniaxial cycle repetition test in fill direction up to a load of

500 N.

Figure 3.28: Stress – strain curve for a uniaxial cycle repetition test in fill direction up to a load of 500 N.

The stress-strain curve in fill direction (Figure 3.28) is not similar to the previous one (Figure

3.15). Both curves show initially a low stiffness, followed by an increased stiffness. The test with

a maximum stress of 12 MPa however, does not reach the yield point, hence no plastic behavior

is induced. However, due to the straightening of the fill yarns at the beginning of the experiment,


55

there is a considerable permanent strain. It becomes obvious from these results that the

permanent deformation caused by crimp interchange should not be confused with plasticity.

3.3.2.6. Warp, increasing load level

The previous paragraphs suggest that the coated fabric shows plastic behavior when the stresses

surpass approximately 14 MPa. In order to confirm this supposition, additional uniaxial

experiments were performed, in order to explore the material behavior under several loading

cycles having different stress levels. Figure 3.29 shows the applied loads as a function of time,

starting with a loading cycle up to 200 N. Each next cycle, the maximum load is increasing by

200 N, until a maximum load of 1400 N is reached. In this way, some loading and unloading

cycles beneath the yield point are included, as well as some loading and unloading cycles above

the previously defined maximum load of 25% UTS.

Figure 3.29: Load as a function of time for a uniaxial test in warp direction with increasing load.

The corresponding strains in both warp and fill direction versus the time are plotted in Figure

3.30. During the first three cycles, no permanent strain is observed in warp direction, nor in fill

direction. The observed absence of permanent strain indicates that the applied stress does not

reach the yield point. During the fourth cycle however, the maximum applied load amounts

800 N, which corresponds to a stress of 19.3 MPa. This stress value overruns the value of the

yield stress, which is around 14MPa. For this reason, permanent strain can be observed. In the

subsequent cycles, the load is successively increased by 200 N, which consequently results in an

increasing permanent strain.

Figure 3.30: Strain (warp and fill) as a function of time for a uniaxial test in warp direction with increasing load.


56

In Figure 3.31, the stress-strain curve is shown. The material behaves almost linear elastically

during the first three cycles. Once the stress surpasses 14 MPa, the slope of the loading curve

becomes less steep and permanent deformation is observed. This strongly indicates the

occurrence of plasticity phenomena in the material. The same conclusion can be drawn about the

hysteresis effect as was already done in paragraph 3.3.2.4: practically no hysteresis effect is

observed during the linear elastic regime. Beyond the yield stress the hysteresis loops grow for

increasing load, which means that the amount of energy loss increases for each subsequent cycle.

Figure 3.31: Stress-strain curve for a uniaxial test in warp direction with increasing load.

3.3.2.7. Fill, increasing load level

The same kind of experiment, as discussed in paragraph 4.5.2.7, was performed in fill direction.

The applied load as a function of time and the corresponding measured strains are shown in

Figure 3.32 and Figure 3.33 respectively.

Figure 3.32: Load as a function of time for a uniaxial test in fill direction with increasing load.


57

Figure 3.33: Strain (warp and fill) as a function of time for a uniaxial test in fill direction with increasing load.

In contrast to Figure 3.30, permanent strain is observed from the first loading cycle on. The

origin of this is the occurrence of crimp interchange: the originally curved fill fibers become

straight during loading, and do not go back to their initially curved configuration during

unloading. Figure 3.34 shows the stress-strain curve for this experiment. The graph shows the

sequence of a low stiffness, followed by a high stiffness and again a low stiffness. Even at the

first cycle, permanent strain is observed due to the straightening of the fill fibers. At later cycles,

it is seen that the plasticity contributes to the permanent strain.

Figure 3.34: Stress-strain curve for a uniaxial test in fill direction with increasing load.

Hysteresis already occurs during the first cycles, in contrast to Figure 3.31, where it only starts

after the yield point. In case of a uniaxial fill test, the hysteresis in the first cycles represents the

loss of energy because of friction between (i) the yarns and the coating and (ii) the yarns mutually

while reorganizing during crimp interchange. Another interesting aspect is the shape of the

hysteresis loop. While it is almost elliptical in case of the load cycle repetition tests with constant

maximum load (Figure 3.15 and Figure 3.28), it has a kind of peak going to the origin for the

experiment with increasing maximum load (Figure 3.34). This discrepancy is simply due to the

fact that for the latter, the sample is unloaded each cycle until zero stress, instead of pre-stress.

The peak of the hysteresis indicates that unloading until 0 MPa causes the fill fibers to become

more curved again, as in the original configuration.

3.4. FEM model of a uniaxial tensile test In this master thesis, following convention regarding the coordinate system is used: the X-axis or

1-direction corresponds to the warp direction, while the Y-axis or 2-direction corresponds to the


58

fill direction. The Z-axis or 3-direction is of minor importance for a plane membrane structure,

and thus is omitted Both coordinate systems are shown in Figure 3.35. For instance, S11

represents the stress in warp direction, while S12 represents the shear stress in the plane of the

membrane.

The geometry is adopted from the experimental samples, i.e. a width of 50 mm, a length of

200 mm and a thickness of 0.83 mm. Considering the symmetry of the experimental setup, only a

quarter of the geometry is simulated by adding symmetry boundary conditions. This reduction

significantly increases the computational efficiency. This is schematically shown in Figure 3.35.

Figure 3.35: Geometry and loading of the uniaxial sample (a) and a quarter of the geometry with symmetry boundary

conditions (b).

In the FEM model the load is applied to a reference point (RP) which is connected to the sample

by means of a coupling constraint. Due to the coupling, the ends of the strips follow the

displacement of the RP, i.e. in the loading direction. This is schematically illustrated in Figure

3.36.

Figure 3.36: Coupling between the boundary nodes of the sample and the RP in the loading direction. The load is

applied in the reference point.

Since the uniaxial test approximately corresponds to a two-dimensional problem, the

displacements in z-direction are set to zero. Membrane elements (M3D4) are used for the finite

element calculations. The membrane elements carry membrane forces, but have no bending or

transverse shear stiffness. Hence the only nonzero stress components are those parallel to the

middle surface, resulting in a state of plane stress [52]. Results obtained with this numerical

model will be discussed in detail in chapter 5 for a variety of material models.


59

3.5. Conclusions Uniaxial testing is a very simple and effective technique to examine the material behavior of a

coated fabric. The obtained stress-strain curves provide good knowledge about the behavior of

the T2107 material, at least for the case of uniaxial tension.

The stress-strain curve in warp direction is characterized by an initial elastic behavior, followed by

plasticity phenomena. The yield stress approximately corresponds to 14 MPa. If the maximum

stress exceeds the yield stress, a large permanent deformation is observed after the first loading

and unloading cycle. This permanent strain increases a little during the next loading and

unloading cycles but stabilizes after 3-5 cycles. In case the yield stress is not reached, neither

permanent strain nor hysteresis is observed. This strongly suggests that the hysteresis effect in

warp direction is triggered by plasticity phenomena of the coated fabric.

The stress-strain curve in the fill direction mainly differs from the one in warp direction during

the first loading and unloading cycle. In fill direction, the coated fabric initially responds with a

very low stiffness. It is concluded that this stage is mainly governed by the occurrence of crimp

interchange: the curved fill yarns straighten because of the applied tension force, leading to large

permanent strains. After this initial stage, the coated fabric is actually stretched, resulting in a

more stiff linear elastic behavior. As was the case for the warp direction, both hysteresis effects

and plasticity phenomena are observed when exceeding a stress value of 14MPa. Nevertheless,

even when loaded beneath 14MPa, a permanent strain is observed. This can be attributed to the

fact that the straightening of the fill yarns is not completely reversible during unloading. More

evidence of this behavior is found by decreasing the load below the preload level. A sudden

change in strain is observed, indicating that the fibers become partially curved again. In contrast

to the warp direction, hysteresis is observed even if the yield point is not crossed. Therefore,

hysteresis in fill direction can be attributed to both plasticity phenomena and crimp interchange.

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Chapter 4.Biaxial tensile tests

4.1. Introduction In addition to uniaxial tensile tests, biaxial tensile tests are performed in order to determine the

membrane material behavior. Moreover, because of the interaction between the orthogonal warp

yarns and fill yarns, biaxial tests are essential to fully capture the material behavior of a membrane

[3]. The present chapter starts with a description of the experimental setup, followed by an

examination of the influence of the cruciform geometry. After this, the experimental results are

discussed and the chapter ends with some conclusions.

4.2. Experimental setup

4.2.1. Biaxial extension of cruciform samples

Cross-shaped samples, which are cut out along the fiber directions of the membrane, are tested in

a biaxial testing machine (Figure 4.1). This method is universally accepted and widely applied for

determining the material behavior of architectural membranes.

Figure 4.1: Experimental setup for a biaxial test [35].

The cruciform sample consists of (i) a central part in which the stress combinations are obtained

and (ii) four arms to which the external load is applied. The arms consist of different strips

divided by slits. The presence of the slits forces the applied stresses to run through the central

part instead of around it, what results in a higher stress introduction and a more uniform stress

distribution at the central part. The corners between the arms are rounded to prevent stress

concentrations. The influence of geometry and the size of the sample as well as the influence of

the slits are examined into more detail in subsections 4.3.3, 4.3.4 and 4.3.5.

Biaxial tests are load controlled and in analogy to the uniaxial cycle repetition tests, the strains in

the central part are measured continuously. For this study, the biaxial tests are performed with

Chapter 4. Biaxial tensile tests

61

repetition of load cycles, to explore both the initial and the stabilized behavior of the coated

fabric, at limited load levels (well below the UTS level of the fabric to prevent failure). Since three

to five identical load cycles are necessary to exclude the presence of residual strain, a possible

load history looks like the one shown in Figure 4.2.

Figure 4.2: Possible load history (MSAJ test protocol) [35].

Some authors [4, 30] investigated the material behavior at a much wider range of different load

ratios, in order to explore the limits of the fabric response. An example of such kind of extreme

loading is shown in Figure 4.3 in a warp-fill stress space.

Figure 4.3: Radial load paths in the warp-fill stress space [4].

The number of radial arms and data points can vary depending on practical considerations, time

constraints and the level of detail required. Further testing can be carried out at stress states that

are important for the particular design.

As is the case for the uniaxial tensile tests, it is recommended to keep the loading rate more or

less constant during the experiment in order to improve the consistency of the experiments.

4.2.2. Bubble inflation tests

Flat circular membrane samples are clamped in a bubble inflation test device between a plate and

a ring (Figure 4.4) [48].


62

Figure 4.4: Bubble inflation test device for performing biaxial tests [48].

Air is injected between the aluminum plate and the foil, resulting in a spherical deformation of

the foil. The deformations are measured on the pole of the bubble, where a 1:1 load ratio is

obtained. The stresses at the pole are calculated by the pressure vessel theory. It is noted that the

bubble inflation test is not used for determining the material behavior of architectural

membranes because the loading is limited to a 1:1 ratio. Nevertheless this method is suitable for

the investigation of the material failure under biaxial stress, because the failure always occurs in

the pole region.

4.2.3. Biaxial experimental tests for this project

The experimental biaxial tests are, like the uniaxial tests, carried out at the VUB at the department

MeMC by Paolo Topalli [50]. The processed results are presented and discussed in detail in

section 4.4. The strains at the central part are determined using DIC technique. The applied load

and the elongation are recorded at the arms of the biaxial testing device. The stresses are not

evaluated at the center of the specimens, but they are determined by dividing the applied load by

the cross section. For this exact reason, the stress-strain curves in this thesis do not show the true

stress at the center of the sample, but the applied stress at the arms. Though it has to be

mentioned that for biaxial tests the difference can be considerable and consequently this

difference should be taken into account in the numerical computations, by consistently evaluating

the applied stresses instead of the computed stresses.

4.3. Numerical study of the geometry of the cruciform

sample Biaxial tests are carried out on cruciform samples with slits in the arms. However, the exact

geometry, i.e. the size of the sample and the amount and locations of the slits are variable. The

aim of this section is to select an optimal geometry. Basically, there are two important conditions

which increase the reliability of the measurement in a biaxial test. On the one hand, there must be

a sufficiently large area in the central part with a uniform stress and strain distribution in order to

obtain good DIC results. On the other hand, the stress value in the center must correspond as

good as possible to the applied stress. The latter concerns the fact that stresses applied to the


63

arms of the cruciform sample do not run fully through the central part. Hence this results in a

lower stress state in the central part. The presence of slits in the arms diminishes this effect [52].

Apart from this, it is also important to avoid both unnecessary consumption of material and

time-consuming preparations, in order to facilitate the experiments.

After a description of the used methods and assumptions of the FEM model of the biaxial test,

the influence of the shape of the tips of the slits is studied. After that, the influence of the size of

the cruciform sample and the number and locations of the slits are investigated.

4.3.1. Principles and assumptions for the FEM simulation

The convention regarding the coordinate system is adopted from section 3.4: the X-axis or 1-

direction corresponds to the warp direction, while the Y-axis or 2-direction corresponds to the

fill direction (see Figure 4.5). The Z-axis or 3-direction can be neglected for the present

investigation.

Considering the symmetry of the cruciform shape, only a quarter of the geometry is simulated by

adding symmetry boundary conditions as shown in Figure 4.5, in analogy to the FEM of the

uniaxial test.

Figure 4.5: Geometry of the cruciform sample used in experimental biaxial tests (a) and a quarter of the sample with boundary conditions and loading (b).

In the experiments, each arm of the cruciform sample is loaded by means of a single clamping

device. For this reason, the ends of the strips cannot move independently. In the FEM model,

this is simulated by coupling the displacements of the ends of the strips to one RP. In analogy to

the FEM model of the uniaxial test, the external load is applied in this RP. This is schematically

illustrated in Figure 4.6.

Figure 4.6: Loading the ends of the arms by coupling the boundary nodes to a reference point in which the load is applied.


64

The two reference points are located as shown in Figure 4.7. Because of the symmetry of the

geometry, their only degree of freedom is in the direction of the applied load (along the arms).

Figure 4.7: Geometry of the finite element simulation, including symmetry boundary conditions and reference points.

For simplicity a linear elastic orthotropic material behavior is assumed in this section. Once a

good understanding is obtained about the simulation of the biaxial test, more advanced material

models will be investigated (chapter 5). The company EMPA performed biaxial tests on samples,

made of a similar material as the one used for this master thesis, and linearized the stress-strain

curves in order to determine the elasticity constants. The obtained elasticity constants and some

other characteristics are given in Table 4.1.

Table 4.1: Elasticity constants and other material constants for the membrane material.

The size of the mesh determines the accuracy of the results on the one hand and the

computation time on the other hand. Smaller elements result in a higher accuracy of the

calculations, but increases the computation time. Hence, a sufficient mesh has to be defined in

order to do accurate calculations. Typically, several element sizes are used: large elements in the

arms, while smaller ones dictate the zone of interest, i.e. the central part of the cruciform sample.

Further mesh refinement is done in zones where a geometrical discontinuity is encountered, i.e.

the slit tips. Membrane elements (M3D4) with a thickness of 0.83 mm are used. To verify

whether a correct mesh size is used, a mesh convergence check is performed: 7 possible meshes

with a different size of mesh element are compared. As geometry, a cruciform sample with a

central part with an area of 100 mm x 100 mm and arms with a length of 100 mm are used. A

load ratio 1:1 is modeled by applying an equal load of 2080 N in both warp and fill direction. The

applied load corresponds to a stress of 25 MPa. Figure 4.8 shows the evolution of both the S11


65

and S22 value in the center of the cruciform sample as a function of the number of mesh

elements. It can be seen that the stress values increase for an increasing number of mesh

elements, which obviously corresponds to a decreasing size of the mesh elements. One can see

that the computed stress values converge to 24.847 MPa. It can be stated that the 5th mesh is

completely stabilized, indicating that its element size is reasonable to provide accurate results.

Figure 4.8: Value of the stresses S11 (a) and S22 (b) at the central point of the cruciform sample for 7 meshes with a different number of mesh elements.

In the above convergence study, only the central point of the cruciform sample was considered.

In order to have a better overview of the evolution of the stresses in the complete central part for

an increasing number of elements, the stress S11 is plotted along three different paths in the

central part. The horizontal, vertical and diagonal paths all start at the central point of the

cruciform geometry (see Figure 4.9).

Figure 4.9: Horizontal, vertical and diagonal path in central part along which the stresses and strains are compared.

In Figure 4.10 the stress S11 is shown along these paths for the 7 different meshes. It is clear that

mesh 1 and mesh 2 are too rough to obtain accurate results. Mesh 5, 6 and 7 largely result in the

same value for stress component S11 along all the paths. For this reason, the same conclusion

can be drawn as from Figure 4.8: the refinement level of mesh 5 is adequate for the biaxial

simulation. It is furthermore noted that mesh 5 does not cause large calculation times.


66

Figure 4.10: Stress S11 along the diagonal path (a), horizontal path (b) and vertical path (c) for 7 different meshes.

The partitioning of the sample for mesh 5 is shown in Figure 4.11. This mesh was designed for a

cruciform geometry with four slits in each arm. Note that, compared to the element size in the

arms, the elements have smaller dimensions in the zone of interest, i.e. the central part of the

cruciform sample. When changing the geometry of the sample or the assigned element type, the

previous convergence study has to be repeated.


67

Figure 4.11: Mesh used in the FEM for a sample with four slits in each arm.

4.3.2. Shape of the slit tips

In this subsection, the influence of the physical shape of the slits on the stress distribution in the

central part is investigated. In the experimental setup, the slits in the arms are incisions within the

material and therefore have a limited physical width. For this reason, a very small width is

attributed to the slits in the FEM model. Though, this provides several possibilities for the shape

of the tips of the slits. Three different kinds of shapes are considered: (i) rectangular tips,

(ii) round tips and (iii) peak tips (Figure 4.12).

Figure 4.12: Three different shapes of the tips of the slits: rectangular, round, peak.

The same geometry and loading conditions are used as described in subsection 4.3.1. Following

contour plots for the stresses S11, S22 and S12 are obtained by means of the FEM for the three

different shapes. The stresses S33, S13 and S23 are equal to zero since the assumption of plane

stress is inherent to the definition of membrane elements. Considering the linear elastic material

model, the plots of the strains are similar to the plots of the stresses.


68

Figure 4.13: Contour plots of stresses S11 in biaxial test sample under 1:1 load ratio (25 MPa) for orthotropic linear elastic material behavior: rectangular tips (a), round tips (b), peak tips (c) and legend (Pascal) (d)

Figure 4.14: Contour plots of stresses S22 in biaxial test sample under 1:1 load ratio (25 MPa) for orthotropic linear elastic material behavior: rectangular tips (a), round tips (b), peak tips (c) and legend (Pascal) (d)

Figure 4.15: Contour plots of stresses S12 in biaxial test sample under 1:1 load ratio (25 MPa) for orthotropic linear elastic material behavior: rectangular tips (a), round tips (b), peak tips (c) and legend (Pascal) (d).

It is observed that the contourplots for the different slit tips look very similar. To make the stress

differences more visible, the stresses are plotted on a chosen ‘path’ in the cruciform sample. Since

we are interested in the stress introduction and stress distribution in the central part, the

comparison between the various slit ends is limited to the stress results in the central part.

Besides, the sharp corners at the slit ends give rise to singularities in the stress concentration for

decreasing mesh size, so the comparison of the stresses at those places is pointless. A meaningful

comparison of the three possibilities is made by plotting the stresses S11, S22 and S12 on a

diagonal path in the central part of the sample, starting at the center and ending at the rounded

corners (Figure 4.9). The plots along the diagonal path for different slit tip geometries are shown

in Figure 4.16, Figure 4.17 and Figure 4.18.


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Figure 4.16: Stress curves S11 along diagonal path for rectangular tips, round tips and peak tips


Figure 4.18: Stress curves S12 along diagonal path for rectangular tips, round tips and peak tips.

Previous graphs make clear that the difference in stress, for the different geometries of the slit

ends, are very small in the central part of the cruciform sample. Figure 4.19, Figure 4.20 and

Figure 4.21 show a magnification of the previous graphs. Still, hardly any difference is observed.

Note that the shear stress S12 does not contribute significantly to the total stress state in the

central part of the sample.

Figure 4.19: Stress curves S11 along diagonal path for rectangular tips, round tips and peak tips (magnification)


70

Figure 4.20: Stress curves S22 along diagonal path for rectangular tips, round tips and peak tips (magnification)

Figure 4.21: Stress curves S12 along diagonal path for rectangular tips, round tips and peak tips (magnification).

Based on this, it is concluded that the shape of the slit ends does not affect the stress distribution

in the central part of the sample. From now on, round tips are used because this enables applying

easily a mesh refinement in that area.

4.3.3. Size of the sample

In literature [31, 33], cruciform test samples with a central part of size 200 mm x 200 mm up to

700 mm x700 mm are commonly used. In order to reduce the material consumption for the

experiments, it is investigated whether such big samples are necessary, or not. Since the stresses

in the experiments are determined by means of DIC, a small central region with uniform stress

distributions is sufficient. In the following, two cruciform samples with different size (Figure

4.22) are simulated and discussed with respect to each other.

Figure 4.22: Comparison size of geometry of two cruciform samples: small geometry (a) and big geometry (b).

Similar to the previous subsection, the stresses of the two samples are compared at the diagonal

path in the central area. Figure 4.23, Figure 4.24 and Figure 4.25 show respectively S11, S22 and

S12 in function of the normalized distance along the path for the two geometries.


71

Figure 4.23: Stress curves S11 along diagonal path for the big geometry and the small geometry (a) and magnification (b).




72

On a normalized scale, both graphs look very similar, especially for a normalized distance d <

0.5, which is the zone of interest. The distance along the path ranges from 0 mm to 369 mm for

the big geometry, while it ranges from 0 mm to 73 mm for the small geometry. However, the

uniform area in the center is still sufficiently large to perform a correct DIC analysis. From this it

can be concluded that working with a smaller test sample, for example with a central part of 100mmx100mm does not cause any addition problems.

The first experimental biaxial tests were performed with samples having a central part of 100mmx100mm. Though, because of some practical aspects during the experiments (the

stability), the sample size was later enlarged up to a central part of 200mmx200mm [50].

4.3.4. Number of slits

As stated above, the slits provide a higher stress introduction and a more uniform stress

distribution in the central part of the cruciform sample [27]. Since the slits must be placed very

precisely for accurate experiments, they require a lot of preparatory work. Hence, the question

arises: how many slits are necessary or can the slits be omitted without loss of accuracy? Nine

samples with different numbers of slits are modeled and the stress distributions in the central

parts are compared. For this study, the strips of a sample have an equal physical width. The

investigated geometries are shown in Figure 4.26 having respectively 0, 1, 2, 3, 4, 5, 6, 9 and 19

slits in each arm.

Figure 4.26: Geometry of samples with different number of slits studied in order to determine the influence of the number of slits on the stress state in the central part.

For all samples, the same boundary conditions and loads are applied. The stresses S11, S22 and

S12 respectively are plotted in Figure 4.27, Figure 4.28 and Figure 4.29 as a function of the

normalized distance along the diagonal path.


73

Figure 4.27: Stresses S11 along diagonal path for geometries with different numbers of slits (a) and magnification (b).



74


Figure 4.27 (a), Figure 4.28(a) and Figure 4.29(a) indicate that the differences in stress distribution

along the diagonal path are rather small. However, the magnifications of the graphs (Figure

4.27 (b), Figure 4.28 (b) and Figure 4.29 (b)) show different stress distributions along the diagonal

path for the different geometries, all characterized by oscillations. For example, if Figure 4.27 (b)

is studied more thoroughly, it becomes clear that each positive oscillation in the stress curve

corresponds to a slit in the arms. This is illustrated in Figure 4.30 in case of 1, 2 and 3 slits. Note

that the graph is mirrored with respect to the origin (in contrast to the previously shown graphs).

Figure 4.30: Correspondence between location of slits in the geometry and oscillations of the stresses S11 in the central part.


75

The location of the oscillations of the stresses S11 in the central part of the sample correspond to

the location of the slits in the sample. Similar conclusions can be drawn when considering the

stress S22. This means that a slit in the arm of the sample causes increased stresses at the

corresponding location on the diagonal path in the central part. This implies that no oscillations

in the stresses S11 and S22 are observed for the sample without slits. From Figure 4.27 (b) it can

be derived that the amplitude of the oscillations decreases when the number of slits increases, so

the requirement related to the uniform stress distribution is better fulfilled by samples with many

slits (apart from samples without slits). However, this is only one of the two requirements for a

good biaxial test. The second requirement, namely introducing the highest possible stresses into

the central part is also dependent on the number of slits in the arms. In Figure 4.31 the mean

values of the stresses S11 and S22 for all modeled samples between 0 and 0.8 of the normalized

distance along the diagonal path are compared.

Figure 4.31: Mean values of stresses S11 over a range of 80 % of the diagonal path for geometries with a different number of slits.

An increasing number of slits provides higher stresses S11 in the central part and thus a better

stress introduction. In Figure 4.32 the standard deviations of stresses S11 over the same range are

shown for all samples. Ignoring the sample without slits, a decreasing trend for increasing

number of slits can be observed. Hence, increasing the number of slits leads to a smaller standard

deviation of the stresses and thus to a more uniform stress distribution. The same conclusion has

already been drawn based on the stress curves along the diagonal path.

In order to obtain both a uniform stress distribution and a high stress introduction, samples with

a high number of slits are most appropriate. Of course, this causes more work during the

preparations of an experimental test.

Figure 4.32: Standard deviation of stresses S11 over a range of 80 % of the diagonal path for geometries with a different number of slits.


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4.3.5. Location of slits

In this subsection, the influence of the location of the slits is studied. It is assumed that two slits

are applied in each arm of the cruciform sample. The six different geometries that are compared

to each other are shown in Figure 4.33.

Figure 4.33: Configurations with different locations of two slits.

Again a comparison is made by plotting the stresses S11, S22 and S12 on a diagonal path in the

central part (Figure 4.34, Figure 4.35 and Figure 4.36).


77

Figure 4.34: Stresses S11 along diagonal path for configurations with different location of slits (a) and magnification (b).


6

5

1 4 3 2

6 5

1 4 3 2


78


As could be expected, all configurations have similar stress distributions in the central area (see

Figure 4.34 (a), Figure 4.35 (a) and Figure 4.36 (a)). Magnifications of the results reveal

oscillations in the stress curves, whose locations correspond to the locations of the slits in the

arms (see Figure 4.34 (b), Figure 4.35 (b) and Figure 4.36 (b)). Apart from this, the amplitude of

each oscillation also depends on the location of the corresponding slit. Figure 4.34 (b) and Figure

4.35 (b) show that the amplitude of the oscillation is smallest for configuration 2. In following

figures, both the mean value and standard deviation are compared for the different

configurations.

Figure 4.37: Mean values of stresses S11 over a range of 80 % of the diagonal path for geometries with different location of slits.


79

Figure 4.38: Standard deviations of stresses S11 over a range of 80 % of the diagonal path for geometries with different locations of slits.

Figure 4.37 shows that the mean value of S11 over 80 % of the diagonal path increases steadily

from configuration 1 to configuration 6. Though note that the differences are rather small., hence

it is justified to ignore the dependency of the stresses in the central part on the location of the

slits. According to Figure 4.38, the standard deviation has a comparable value for all the

investigated configurations.

From this analysis, it can be concluded that a higher number of slits provides more accurate

stress-strain results. On the other hand, an increasing number of slits drastically increases the

time to prepare a sample. It was decided [50] that the improved stress introduction in the central

part of the sample does not outweigh the large preparation time, and thus the experiments have

been performed on cruciform samples with no slits. The difference in stress value is taken into

account by plotting the ‘applied stress’ as a function of the measured strain for both

experimentally and numerically obtained stress-strain curves.

4.4. Experimental test results The exact geometry of the biaxial test sample is shown in Figure 4.39.

Figure 4.39: Exact geometry of the biaxial test sample used for the experiments.

The used load ratios in sequence for the biaxial tests are 1:1, 2:1, 1:2, 1:0, 0:1, according to the

Japanese standard MSAJ [34]. Each load ratio is repeated 3 times so that both the initial behavior

and the stabilized behavior can be determined. Since the width of each arm is 200 mm, the

maximum applied load must be 4000 N in order to achieve a maximum stress of 24.1 MPa.

Figure 4.40 shows the applied load cycle in both warp and fill direction versus the time. The

different load ratios are easily identified in this graph.


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Figure 4.40: Load as a function of time for a biaxial test with successive load ratios 1:1, 2:1, 1:2, 1:0 and 0:1.

The strains in warp and fill direction, measured by DIC, are shown in Figure 4.41. Note that the

strain in fill direction is greater than the strain in warp direction, which is due to the crimp

interchange in fill direction. The permanent strain after three load cycles with load ratio 1:1 is in

warp and fill direction respectively 0.0144 and 0.0583.

Figure 4.41: Strain (warp and fill) as a function of time for a biaxial test with successive load ratios 1:1, 2:1, 1:2, 1:0 and 0:1.

Figure 4.42 and Figure 4.43 show the stress–strain curves in respectively warp and fill direction.

All load ratios are plotted in the same curve, but can be distinguished by the use of a different

color.


81

Figure 4.42: ‘Applied stress’-strain curve in warp direction for a biaxial test with successive load ratios 1:1, 2:1, 1:2, 1:0 and 0:1 and a maximum stress of 24.1 MPa.

Figure 4.43: ’Applied stress’-strain curve in fill direction for a biaxial test with successive load ratios 1:1, 2:1, 1:2, 1:0 and 0:1 and a maximum stress of 24.1 MPa.

The main differences between the stress-strain curves in warp and fill direction appear during the

first loading cycle, i.e. the cycle where the largest permanent strain occurs. After this, both curves

look very similar, at least when taking into account the load ratios. The slopes of the stress-strain

curves change for each load ratio, probably because of the presence of both the Poisson effect

and crimp interchange.

The behavior of the tested material during the first loading cycle with a load ratio of 1:1 shows

both agreements and disagreements with the uniaxial tests. In fill direction, the sequence of low

stiffness, high stiffness, low stiffness is not observed unambiguously for the biaxial test, while it

was in case of the uniaxial test. The reason for this can be understood by taking into account that

the fill fibers are no longer free to straighten because of the applied stress in warp direction.

Though, some agreements between the uniaxial and biaxial experiment are found: (i) the overall

behavior in fill direction is less stiff than in the warp direction, (ii) the permanent strain in fill

direction is remarkably higher when compared to the results in warp direction and (iii) an elastic

regime followed by a plastic regime is observed for both warp and fill direction, in which a yield

stress of approximately 14 MPa can be determined.


82

The transition of load ratio 1:1 to 2:1, as well as 1:2 to 1:0, results in a small increase of the

permanent strain in warp direction, and consequently a small decrease in fill direction, probably

because the warp yarns become more straightened. The reverse is true for the transition of load

ratio 2:1 to 1:2, as well as 1:0 to 0:1.

Figure 4.44 and Figure 4.45 show the stress-strain curves in respectively warp and fill direction

for a biaxial test, similar to the previous ones, but in which 3 additional cycles at load ratio 1:1

were performed at the end. This means the experiment was both started (green curve) and ended

(blue curve) with 3 cycles at load ratio 1:1.

Figure 4.44: Stress - strain curve in warp direction for a biaxial test with successive load ratios 1:1, 2:1, 1:2, 1:0, 0:1 and 1:1.

Figure 4.45: Stress - strain curve in fill direction for a biaxial test with successive load ratios 1:1, 2:1, 1:2, 1:0, 0:1 and 1:1.

As could be expected, the green and the blue curves show different behavior in their first cycle,

because of the different load history present in the material. At the third cycle however, the

influence of the load history should be of minor importance and thus the same behavior should

be observed. The slope and the shape are indeed the same for the green and blue curve, but one

clearly observes that, compared to the green curve, the blue curve is shifted to the right over a

certain distance. The reason for this is that the fabric was subjected to plastic deformation during


83

the cycles in between the cycles corresponding to the green and the blue curve. The induced

plastic deformation causes a permanent strain which cannot be removed by simply repeating the

cycles with load ratio 1:1 three or more times. As a consequence, the effect of load history did

not disappear completely, at least for the investigated system.

4.5. Conclusions In this chapter, the influence of the geometry of the cruciform sample was studied by FEM. For

simplicity, a linear elastic material model was assumed. Three different shapes of the slit tips, i.e.

round, rectangular and peak, were studied. It can be concluded that the shape of the slits does

not have a significant influence on the stress distributions in the central part of the cruciform

sample. For the remainder, round tips are applied, mainly because of the straightforward mesh

partitioning associated with them. Furthermore, two cruciform samples with a different size were

investigated, i.e. a central part of 100 mm x 100 mm and a central part of 500 mm x 500 mm.

Both samples provide a sufficiently large area in the central part with a uniform stress

distribution, which is important to comfort the DIC measurement. After this, an investigation

was performed to the influence of the presence of slits for the stress in the central part of the

sample. It was found that the presence of slits in the arms causes stress oscillations at the

corresponding locations in the central part. The amplitudes of the oscillations become smaller

when increasing the number of slits. For this reason, more slits provide a more uniform stress

distribution. One exception of the latter rule is observed, namely the case in which the sample

has no slits. Furthermore, it was observed that a higher number of slits results in a higher mean

stress value in the central part. Hence from this analysis, it can be concluded that a higher

number of slits provides more accurate stress-strain results. On the other hand, an increasing

number of slits drastically increases the time to prepare a sample. It was decided that the

improved stress introduction in the central part of the sample does not outweigh the large

preparation time, and thus the experiments have been performed on cruciform samples with no

slits. The difference in stress value is taken into account by plotting the ‘applied stress’ as a

function of the measured strain for both experimentally and numerically obtained stress-strain

curves.

Once the geometry of the biaxial sample was fixed, experiments with different load ratios were

performed. The stress-strain curves in warp and fill directions show a different behavior during

the first loading cycle, as could be expected on the basis of chapter 2. However, the differences

are smaller compared to the results of the uniaxial tests. The transition of one load ratio to

another causes a small increase or decrease of the permanent strain. Apart from that, a different

slope of the stress-strain curves can be observed for different load ratios. It was assumed that

repeating each load cycle 3 times could exclude the influence of load history. However, a biaxial

experiment starting and ending with a load ratio 1:1 was carried out and different total permanent

strains were measured. This indicates that the load history introduces, besides the reversible

crimp interchange, plasticity phenomena which are inherently irreversible. For this reason,

different stress-strain curves are expected when repeating the biaxial test with a different

sequence of the same load ratios. Taking this into account, more experimental biaxial tests must

be performed in order to characterize the material behavior completely.

84

Chapter 5.Modeling the material behavior of a coated fabric

5.1. Introduction The quest for a material model, which represents the material behavior of the coated fabric in full-

scale foldable tent structures true-to-nature, is put forward. Given the fact that the stresses in

foldable tent structures are limited to 25% UTS [34], the stress range of interest lies between

0 MPa and 24.1 MPa for the investigated T2107 material. No attention is paid to the rupture and

failure of a coated fabric.

It is not the purpose of this master thesis to write a user-subroutine, nor to implement a micro

scale model. It is rather examined whether one of the built-in material model of Abaqus is capable

to describe the material behavior of the coated fabric in a proper way. Starting with simple linear

elasticity principles, we finally evolve to the Hill plasticity material model which accounts for the

nonlinear anisotropic nature of the tent material.

5.2. Selection criteria The suitability of a material model is evaluated by verifying following criteria:

(1) Orthotropic behavior

(2) Nonlinear behavior

(3) Orthotropic permanent deformation

It is far from easy to find a material model that satisfies all 3 criteria. One could argue that a 4th

criterion involves the hysteresis effect, which is due to the friction between fill and warp yarns and

plasticity phenomena during loading and unloading. Though, the hysteresis has only a small effect

on the material behavior and thus it is acceptable to neglect.

The material models that appear to be suitable based on the above selection criteria, are examined

and discussed in a more profound way by modeling both uniaxial and biaxial tensile tests. The

numerically applied loading conditions are modeled identical to the loading conditions of which

experimental data are available: (i) a uniaxial tensile test with 5 identical load cycles with a

maximum stress of 24.1 MPa in both warp and fill direction, (ii) a uniaxial test with 8 different load

cycles, in which the stress increases with increments of 4.82 MPa from 4.82 MPa to 33.73 MPa in

both warp and fill direction and (iii) a biaxial test with sequencing load ratios 1:1, 2:1, 1:2, 1:0 and

0:1 having a maximum stress of 24.1 MPa. The material models are evaluated by comparing the

calculated stresses and strains with the measured stresses and strains in the corresponding

experiment.

5.3. Orthotropic linear elastic material behavior As already described in chapter 2, orthotropic linear elastic material models are often used to

design tent structures [4, 30, 33, 34, 39, 40]. Although these models do not provide accurate

Chapter 5. Modeling the material behavior of a coated fabric

85

results, they have become very popular, mainly because of their simplicity. Orthotropic linear

elastic material behavior can be easily assigned in Abaqus\Standard, Abaqus\Explicit and

Abaqus\CAE [53].

An orthotropic linear elastic material model requires the definition of several material constants.

The Young’s moduli in warp and fill direction as well as the shear modulus are determined from

the uniaxial experiments in chapter 3. From the same experiments, Poisson ratios were calculated

according to νk� =− �º0ËË�¹ÌÍÎ ≈ 0.04 and ν�k =− �¹ÌÍÎ�º0ËË ≈ 0.25. However, the reciprocal

relationship ¸¹º»¹ = ¸º¹»º is not valid for coated fabrics (see section 2.4), which results in an anti-

symmetrical stiffness matrix. This is in contradiction with the assumption of a symmetrical

stiffness matrix, defined by only one Poisson ratio, in the linear elastic material model. Here we

premise νk� as Poisson ratio, its appropriateness will be checked later in the present section by

means of a combined numerical-experimental approach.

A. Uniaxial tests

The material constants are assigned to the FEM model which corresponds to the uniaxial tensile

test with load cycle repetition having a maximum load level of 24.1 MPa. In Figure 5.1 the

numerically computed stress-strain curves are compared to the corresponding experimentally

determined stress-strain curves for both warp and fill direction. It is clear that the numerical results

do not correspond to the experiments However, if the permanent strain from the first loading-

unloading cycle is excluded from the experimental results, the linear elastic model does 'describe'

the stiffness of the coated fabric in a uniaxial tensile state in a more or less appropriate way. It is

easy to understand that the permanent strain of the first cycles has an important influence on the

overall behavior of a tent structure and therefore cannot be ignored. Moreover, a pure uniaxial

loading and unloading state rarely occurs in a tent structure.

Figure 5.1: Calculated and experimentally measured stress-strain curves for a uniaxial tensile test with load cycle repetition having a maximum stress level of 24.1 MPa: warp direction (a) and fill direction (b). The calculations have

been performed by means of an orthotropic linear elastic material model.

In Figure 5.2, a comparison between the calculated and the experimentally measured strain in the

direction perpendicular to the loading direction is shown for the last loading and unloading cycle

of the experiment, i.e. when crimp interchange and plasticity phenomena are of no importance.

For the uniaxial test in warp direction, the calculated strain in fill direction is higher than the

corresponding experimental strain, indicating that the used Poisson ratio is too high. For the

uniaxial test in fill direction, the opposite conclusion can be drawn, namely that the assigned


86

Poisson ratio is too small. These results indicate that one single Poisson ratio cannot describe the

interaction effect between warp and fill direction in a proper way. The effect of the Poisson ratio

will be further investigated in the discussion concerning the biaxial tests.

Figure 5.2: Calculated and measured strain in the direction opposite to the loading direction for a uniaxial test in warp direction (a) and in fill direction (b) as a function of the normalized time during the last loading and unloading cycle.

B. Biaxial tests

In Figure 5.3 the numerically calculated stress-strain curves of the biaxial tensile test are shown

together with the corresponding experimental curves. The same material model as for the uniaxial

tests is assumed, i.e. orthotropic linear elastic. Similar observations can be made as in the uniaxial

case: the permanent strain is not modeled by the linear elastic material model and the numerically

calculated slopes of the stress-strain curves for different load ratios correspond very well to the

slopes of the experimental stress-strain curves. Hence more advanced material models are needed

to capture the permanent strain. An example of a generated input file of a linear elastic model is

shown in appendix A1.

Figure 5.3: Calculated and experimentally measured stress-strain curves for a biaxial test with 5 identical cycles having a maximum stress of 24.1 MPa. The material model is an orthotropic linear elastic material model, the Poisson ratio

corresponds to 0.04.

5.4. Orthotropic multi-linear material behavior The previous section clearly indicates that the orthotropic linear elastic material model is not

appropriate for modeling a coated membrane true-to-nature. However, the shape of the

experimental stress-strain curves for a uniaxial test in both warp and fill direction (see Figure 3.9

and Figure 3.15) reveals that a multi-linear material law could be used to describe the material

behavior. The stress-strain curve in warp direction increases almost linearly up to the yield point.


87

At this point the slope changes, while the curve stays almost linear until the maximum load is

reached. For the unloading curve, a more complex progress is observed, though it is an acceptable

approximation to linearize the unloading curve by simply connecting the points of maximum and

minimum stress. The stress-strain curve corresponding to the fill direction can be linearized in the

same way. At the beginning however, an extra linear part with a small slope is added. With an

orthotropic linear elastic material behavior, the correct slopes of each part of the stress-strain

curves can be assigned for both warp and fill direction simultaneously. These properties are shown

in Table 5.1.

Table 5.1: Young’s moduli in warp and fill direction for all linearized parts of the stress-strain curve

In order to make the consecutive linear behavior of the coated fabric possible in Abaqus, a tool for

transferring results between different Abaqus analyses is used. Abaqus provides the capability to

import a deformed mesh and its associated material state at the end of a calculation, into a new

model in which it is used as a starting condition (in Abaqus\Standard, Abaqus\Explicit and

Abaqus\CAE) [54]. This feature can be used to model each linear part of the stress-strain curve in

an individual model, in which each model has its own elasticity constants, and stitch the different

models to each other. In the first model, the load is increased up to approximately 5 MPa. The

state of the sample at the end of this simulation is imported into a second model, having different

linear elastic properties. A new calculation is performed in which the load increases from 5 MPa to

15 MPa. At the end of this stage, the results are imported into a third model and so on. The

numerical procedure stops when the sample gets unloaded to 0 MPa. The stress-strain curves

calculated by the FEM model are plotted in Figure 5.4, together with the corresponding

experimentally recorded stress-strain curves for both warp and fill direction. Good agreement is

observed between the FEM results and the experimental results. It is furthermore noted that the

resemblance can be improved by subdividing the stress-strain curves into more linearized parts. In

that way, even the hysteresis loops can be included. An example of a generated input file of a

multilinear model is shown in appendix A2.

Figure 5.4: Calculated and experimentally measured stress-strain curves for a uniaxial test in warp direction (a) and in fill direction (b). The material model is an orthotropic multilinear material model.


88

This method seems to be very appropriate for modeling the behavior of a coated fabric. However,

there are some practical problems. For a complete tent structure which is subjected to both

preload and external loads, e.g. wind load and snow load, it is not known in advance when exactly

the membrane will be subjected to a stress for which the models must be changed. Moreover, this

moment can differ for each point of the tent structure. For this reason the present modeling

method is inapplicable to model a complete tent structure. Hence, there is no point to extend the

current analysis to biaxial tests. A better, more sophisticated material model is needed.

5.5. Isotropic hyper elastic material behavior with

permanent set The isotropic hyper elastic material model is valid for isotropic, nonlinear materials that exhibit an

elastic response, even at large strains [55]. In Abaqus, isotropic hyper elasticity can be combined

with permanent deformations which are called permanent set in rubberlike materials [56]. The

combination of the nonlinear hyper elasticity and the permanent deformations fulfills several

criteria for appropriately modeling the behavior of a coated fabric. It is available in

Abaqus\Standard, Abaqus\Explicit and Abaqus\CAE [55, 56].

Hyper elastic materials are described in terms of a strain energy potential U, which defines the

strain energy stored in the material per unit of reference volume as a function of the strain. Several

strain energy potentials are available in Abaqus: Arruda-Boyce, Marlow, Mooney-Rivley, neo-

Hookean, Ogden, Polynomial, reduced polynomial, Yeoh and Van der Waals. The material

coefficients of the hyper elastic models can be calibrated from experimental stress-strain data

which are obtained by means of 4 simple tests: (i) uniaxial test, (ii) equibiaxial test, (iii) planar test

and (iv) volumetric compression test. Based on these experimental data, Abaqus computes the

hyper elastic material parameters through a least-squares-fit procedure in which the relative error in

stress is minimized. For this application, the Yeoh strain energy potential is employed, which is

defined as:

½ = ¼��Ï�̅ − 3� +¼[�Ï�̅ − 3�[ +¼\�Ï�̅ − 3�\ + 1À� Ð�� − 1�[

+ 1À[ Ð�� − 1�] + 1À] Ð�� − 1�Ñ

(5.1)

where U is the strain energy per unit of reference volume, Ci0 and Di are material parameters, Ð�� is the elastic volume ratio and I�̅ is the first deviatoric strain invariant. This relation is investigated

and discussed into more detail.

The elastic volume ratio J�� relates the total volume ratio J to the thermal volume ratio Jth:

Ð�� = ÐÐjÓ = detÖ�1 + q�jÓ�1 + q[jÓ�1 + q\jÓ� (5.2)

with ε�§×, ε[§× and ε\§× the principal thermal expansion strains. For this master thesis J§× = 1

since temperature differences are not taken into account. Furthermore, the degree of

compressibility is not crucial for applications where the material is not extremely confined,

therefore the assumption that the material is fully incompressible is justified: the volume of the

material cannot change or J = 1.This implies that J�� = 1 and therefore the three last terms of


89

equation (5.1) are equal to zero. This results in the following relation for the Yeah strain energy

potential:

½ =¼��Ï�̅ − 3� +¼[�Ï�̅ − 3�[ +¼\�Ï�̅ − 3�\ (5.3)

The first deviatoric strain invariant Ï�̅ is defined as:

Ï�̅ = Ø̅�[ +Ø̅[[ +Ø̅\[ (5.4)

with Ø̅5 = Ð��/\Ø5 the deviatoric stretches. Because of the assumption of an incompressible

material, or equivalently J=1, the expression for the first deviatoric strain invariant I reduces to:

Ï�̅ =Ø�[ + Ø[[ + Ø\[ (5.5)

Permanent set is a feature to model the permanent strain which is observed for certain materials

undergoing a loading and unloading cycle. It is captured by isotropic Mises plasticity, combined

with a flow rule. In the context of finite elastic strains associated with the hyper elastic material

behavior, the plasticity is modeled using a multiplicative split of the deformation gradient into

elastic and plastic components:

Ö = Ö� ∙ Ö� (5.6)

where Û� is the elastic part of the deformation gradient (representing the hyper elastic behavior)

and Û� is the plastic part of the deformation gradient.

A uniaxial tensile test with increasing load is simulated in Abaqus with the use of a material model

which consists of both a permanent set law and a Yeoh hyperelastic behavior. The experimental

stress-strain data of the loading curve of a uniaxial test in warp direction have been applied as test

data to the numerical material model. The material coefficients of the Yeoh hyperelastic model, i.e.

C10, C20 and C30, were calibrated to these test data by Abaqus. The numerically computed stress-

strain curves are shown in Figure 5.5. The isotropic hyper elastic material model with permanent

set enables to define the loading curve of the coated fabric in warp direction in a very precise way.

However, because of the major differences of the stress-strain behavior between warp and fill

direction, the inherent isotropy prevents the model to be useful for simulating a coated fabric. In

addition, the unloading behavior cannot be predicted in a proper way because the definition of the

multiplicative split for modeling permanent deformations entails a similar shape for both the

loading and unloading curve. The latter is in clear contrast with the loading and unloading behavior

observed in the experiments. An example of a generated input file of the isotropic hyper elastic

model with permanent set can be found in appendix A3.


90

Figure 5.5: Calculated and experimentally measured stress-strain curves for a uniaxial test with increasing load in warp direction (a) and fill direction (b). The calculations have been performed by means of a test data based Yeoh hyper

elasticity model fitted to the warp direction combined with a permanent set law.

5.6. Anisotropic hyper elastic material behavior An anisotropic hyper elastic material model provides a general capability for modeling materials

that exhibit both anisotropic and nonlinear elastic behavior and it is valid up to large strains [57].

Because of the orthotropy, the nonlinear behavior and the finite strain possibilities, it seems a very

promising material model to describe the first loading cycle of a uniaxial or biaxial test. The

material model is supported by Abaqus\Standard, Abaqus\Explicit [57].

In Abaqus, two strain energy potentials are available: the generalized Fung form and the potential

proposed by Holzapfel, Gasser and Ogden. Both potentials are adequate for modeling orthotropic

hyper elastic material behavior. However, whereas Fung’s form is purely phenomenological, the

Holzapfel-Gasser-Ogden form is micromechanically based [58]. Both forms are described and

discussed a bit more into detail.

5.6.1. Generalized Fung potential

The generalized Fung strain energy potential has the following form:

½ = Ü2expÞ� − 1� + 1ÀßÐ��[ − 12 − ln Ð��á (5.7)

where U is the strain energy function per reference volume, c and D are material parameters, Q is a

dimensionless quantity and Jel is the elastic volume ratio. These quantities are discussed more in

detail.

The first term on the right hand side of (5.7) is the distortional or isochoric part [59], in which Q is

defined as

Þ = qâ̅ ∶ ä ∶ qâ̅ =q5̅6â �56å�qå̅�â (5.8)

where b�.H� is a dimensionless symmetric fourth-order tensor of anisotropic material constants [59].

The number of independent constants depends on the level of anisotropy of the material. Based

on the symmetry of εçè there are 21 constants for the general anisotropic case and 9 constants for

the orthotropic case. The εç�.è are the components of the modified green tensor [60]. The factor

c>0 is a material parameter in units of stress, and is dependent on the initial deviatoric elasticity

tensor �� [57, 59]:


91

Ü = äé�� (5.9)

The second term on the right hand side of (5.7) is the dilatational or volumetric part, in which D is

a material parameter which depends on the bulk modulus ê�:

À = 2ê� (5.10) J�� is the same elastic volume ratio as was defined in equation (5.2), and thus equals unity.

Therefore, the volumetric term becomes zero and consequently the Fung strain energy potential

reduces to:

½ = Ü2expÞ� − 1� (5.11)

It is clear that this strain energy potential inherently provides an exponential relationship

(e.g. Figure 5.6) between the stress and the strain [61, 62]. It is exactly this exponential curve that

impedes Fung hyper elasticity to be appropriate for modeling the behavior of a tent membrane.

Indeed, compare the stress-strain curve obtained with Fung hyper elasticity to the experimentally

determined stress strain curve of a coated fabric (Figure 5.6). These curves can never get matched

to each other, no matter which constants have been defined in the numerical model. An example

of a generated input file of the anisotropic hyper elastic material mode, in which the generalized

Fung potential is used, is shown in appendix A4.

Figure 5.6: Calculated and experimentally measured stress-strain curves for a uniaxial test with increasing load in warp direction (a) and fill direction (b). The calculations have been performed by means of a hyper elastic Fung model with a

scaling factor.

5.6.2. Holzapfel-Gasser-Ogden potential

The form of the strain energy potential is given by:

½ = ¼��Ï�̅ − 3� + ë�2ë[ � ìBízë[⟨vçï⟩[{ − 1ñï�� + 1ÀßÐ��[ − 12 − ln Ð��á (5.12)

Where U is the strain energy potential per unit of reference volume. Analogue to the general Fung

form, the third term on the right hand side of (5.12) which describes the volumetric part of the

energy potential can be set to 0. The first two terms of (5.17) describe the isochoric deformation

behavior ψ� , which is represented by a superposition of the isotropic potential ψ� �òó and the

anisotropic potential ôç~�5ÆÈ [63, 64].


92

ôç = ôç5ÆÈ +ôç~�5ÆÈ (5.13)

The isotropic contribution ψ� �òó is given by:

ôç5ÆÈ =¼��Ï�̅ − 3� (5.14) and mainly governs the initial stiffness represented by the elasticity of the non-fibrous substances.

The factor C10 is a material parameter and depends on the initial deviatoric elasticity tensor ��; I�̅

is the first deviatoric strain invariant.

The anisotropic contribution ôç~�5ÆÈ is given by:

ôç~�5ÆÈ = ë�2ë[ �ìBízë[⟨vçï⟩[{ − 1�ñï�� (5.15)

This part governs the much higher stiffness at large strains induced by the fibers [58]. k1>0 and

k2>0 describe the mechanical properties of the fibers and are respectively a stress-like and a

dimensionless material parameter [65, 66]. N corresponds to the number of families of fibers

(N≤3). All families of fibers have the same mechanical properties, i.e. the same set (k1, k2) [64]. The

model assumes that the directions of the fibers within each family are dispersed with rotational

symmetry about a mean preferred direction. The parameter � (0 ≤ � ≤ 1/3) describes the level

of dispersion in the fiber directions. If ρθ� is the orientation density function that represents the

normalized number of fibers with orientations in the range zΘ, Θ + dΘ{ with respect to the mean

direction, the parameter � can be written as

� = 14y ö÷�ø��\÷|÷ù� (5.16)

When κ = 0, the fibers are perfectly aligned, while in the other limit, when κ = �\, they are

randomly distributed. In the latter case, the material behaves mechanically isotropic. It is assumed

that all families of fibers have the same dispersion.

The strain-like quantity E�� is given by

vçï ≝ �Ï�̅ − 3� +1 − 3��£Ï]̅ïï� − 1¤ (5.17)

and characterizes the deformation of the family of fibers with mean direction ü�. I]̅�� are

pseudo-invariants of Cç and ü�. For perfectly aligned fibersκ = 0� the strain-like quantity

becomes I]̅�� − 1, while for randomly distributed fibers κ = �\� it is I�̅ − 3. A fundamental

hypothesis of the model is that the fibers can only support tension, because they would buckle

under the smallest compressive load [64]. This condition is enforced by the term ⟨E��⟩ in equation

(5.15), where the operator ⟨∙⟩ stands for the Macauley bracket and is defined as ⟨x⟩ = �[ |x| + x�, thus the anisotropic contribution in the strain energy function appears only when the strain of the

fibers is positive or, equivalently, when ⟨E��⟩ > 0.

The presence of both an isotropic term, which represents the behavior of the coating, and an

anisotropic term, which describes the influence of the fibers, indicates that the Holzapfel-Gasser-

Ogden form may be well suited for modeling a coated fabric in a realistic way. However, the

behavior of the fibers is limited to an exponential relationship between the stress and the strain (as


93

for the Fung form). For this reason, it is impossible to reproduce the experimentally recorded

stress-strain curves, which have a non-exponential course, by means of an anisotropic hyper elastic

material model using the Holzapfel-Gasser-Ogden potential.

Hence, the above discussed hyperelastic models seem to be qualified to describe the material

behavior of a coated fabric partly. The isotropic hyper elastic model has the big advantage that the

actual experimental test data can be imported right away. But it cannot describe the orthotropic

nature of a coated fabric. The anisotropic hyper elastic model on the other hand does take into

account orthotropy, but makes use of an implemented strain energy potential. The available

potentials, namely the Fung potential and Holzapfel-Gasser-Ogden potential, have been discussed

and analyzed. It was found that both potentials cannot capture the material behavior of a coated

fabric, mainly because of the presence of an exponential factor in the strain energy expression. It is

clear that a combination between the isotropic and anisotropic hyper elastic models would lead to

a representative material model of a coated fabric. The possibility to combine them has been

checked and investigated, but it was found to be impossible in Abaqus 6.11. Hence our view is

expanded to other material models, which rely on completely different fundamental principles,

starting with the fabric material model in the next section.

5.7. Test data based fabric material behavior The nonlinear anisotropic behavior of woven fabrics can be modeled in Abaqus/Explicit by means

of a test data-based fabric material behavior [67]. It is a phenomenological model that captures the

mechanical response of a woven fabric. The state of the fabric is described in terms of (i) the

nominal direct strains, ε1 and ε2, in the fabric plane along the warp and the fill directions

respectively, and (ii) the angle ô�[ between the two fiber directions. The material orthogonal basis

is illustrated in Figure 5.7 for both the reference (E1 and E2) and the deformed configuration (e1

and e2), together with the fiber local directions in the reference (N1 and N2) and the deformed

configuration (n1 and n2).

Figure 5.7: Material orthogonal basis and yarn local directions for the reference configuration (a) and the deformed configuration (b) [67]

For typical coated fabrics used in tent structures, the warp and fill fibers are orthogonal in the

reference configuration, i.e. ψ�[� = π/2. The engineering nominal shear strain γ12 is defined as the

change in angle between the two fiber directions: γ�[ =ψ�[ −ψ�[� . Based on the nominal strains

along the fiber directions ε1 and ε2 in the deformed configuration as well as the applied test data,

the corresponding nominal stresses T11, T22 and T12 are calculated. Finally, the relationship between

the Cauchy stress σ and the nominal stress T is given by [67]:


94

Ð� = Ø�Â�� +Ø[Â[[�[�[ +Â�[ÜøÜô�[��[ + �[��− Â�[Ü��ô�[�� + �[�[� (5.18)

While the Poisson ratio accounts for the coupling between the different directions in real rigid

homogeneous materials, a coated fabric is in addition to the Poisson ratio governed by crimp

interchange of the fibers. The fabric material model based on test data however, assumes that the

responses along the fill and the warp directions are independent of each other and secondly that

the shear response is independent of the direct response along the yarns. Hence, each component-

wise fabric stress response depends only on the fabric strain in that component. For this reason,

the overall behavior of the fabric consists of three independent component-wise responses: (i) the

direct response along the fill yarn to the nominal strain in the fill yarn, (ii) the direct response along

the warp yarn to the nominal strain in the warp yarn and (iii) the shear response to the change in

angle between the warp and fill yarns. The accuracy of the inherent independency of the test data

based fabric material model will be checked later in the present section by comparing the

numerically computed stress-strain curves to the experimentally obtained stress-strain curves.

Within each component, i.e. the warp and fill direction, test data must be provided, which define

the response of the fabric. The test data can be specified separately for the loading and unloading

data. For this reason it is possible to define models that exhibit permanent deformation upon

unloading. The loading behavior is defined by specifying the fabric stress as a nonlinear function of

the fabric strain in table format. The unloading behavior controls the amount of energy dissipated

as well as the amount of permanent deformation. The unloading behavior can be specified in one

of the following ways: (i) an analytical unloading curve having an exponential or quadratic shape,

(ii) a unloading curve which is interpolated from multiple user-specified unloading curves or (iii) a

user-specified unloading curve which is shifted to the point of maximum stress (see Figure 5.8). In

Figures 3.31 and 3.34, it was shown that all unloading curves are more or less parallel and have

approximately the same shape for different maximum applied stresses in both warp and fill

direction. For this reason, the third way of defining the unloading behavior is employed by

entering the fabric stress as a function of the fabric strain in table format in a similar way as was

done for the loading behavior.

Figure 5.8: The actual unloading curve (BCD) is obtained by shifting the user-specified unloading curve horizontally [67]

The onset of yield can be specified by defining a strain value, below which unloading occurs along

the loading curve. For the warp and fill directions, this onset value is respectively set equal to 0.015

and 0.01, based on Figure 3.31 and 3.34.


95

Different stress-strain behavior can be applied in compression and tension, which is a good

characteristic because of the completely different behavior of a coated fabric for opposite stresses.

Under tension, the fabric stretches, while under compression buckling occurs, even for very small

compression loads. Because of the latter, no experimental stress-strain data in compression are

available for the coated fabric. It might seem valid to assume that the membrane does not have any

compression resistance. Though, during both a uniaxial and a biaxial test, contraction is observed

in the direction perpendicular to the loading direction. Because of this, combined with the fact that

the response along warp and fill direction are independent from each other, the sample buckles if

no compression resistance is applied. Therefore, it is inevitable to define a certain resistance in

compression. The stiffness in compression is determined iteratively by comparing the experimental

strain with the numerical strain, which are perpendicular to the loading direction, for a uniaxial test

in the following subsection.

5.7.1. Uniaxial test data based FABRIC model

The applied material behavior in tension is extracted from the experimental uniaxial tensile tests.

Several points of the loading curve are used to define the stress-strain behavior. The original stress-

strain curve in both warp and fill direction, together with the superimposed data points for the

loading behavior in tension are shown in Figure 5.9. Note that the last chosen point does not lie on

the first loading cycle, but defines the maximum stress and strain value at the ultimate loading

cycle. In this way, the FEM model ends up with the total experimental strain in spite of the

shortcoming of the numerical model to take into account the small increase of strain between

different load cycles.

Figure 5.9: Experimental stress-strain curve for a uniaxial test and the superimposed points which define the material behavior in warp direction (a) and fill direction (b)

The experimental unloading and reloading curves have respectively a positive and a negative

curvature, and thus form a hysteresis loop. However, the FABRIC model does not account for the

hysteresis effect, consequently the unloading and reloading curve follow exactly the same course.

For this reason, the numerical unloading data is not chosen as the experimental unloading curve,

but rather as the golden mean between the experimental unloading and reloading curves. The

uniaxial tests with load cycle repetition were unloaded to a pre-stress level of 2.4 MPa, while the

uniaxial tests with increasing load were unloaded until 0 MPa. Therefore, the stress-strain curves of

the latter are used to extract the unloading data, as indicated with the orange line in Figure 5.10.


96

Figure 5.10: Experimental stress-strain curve and chosen unloading behavior for a uniaxial test in warp direction (a) and fill direction (b)

The compression stiffness is temporarily assumed to be 0.025 GPa. The test data in shear for the

numerical model are determined in a similar way. An input file of the uniaxial test data based

FABRIC model is found in appendix A5.

A. Uniaxial test with load cycle repetition

The calculated stress-strain response for a uniaxial test with load cycle repetition in both warp and

fill direction, together with the experimental results, is presented in Figure 5.11. Good agreement

in both warp and fill direction between the experimental and numerical stress-strain curves is

observed. Though, some small deviations at stresses smaller than 10 MPa for the unloading curve

in fill direction can be seen. However, it is concluded that the results are acceptable. Note that the

crimp interchange is implemented in the model in a phenomenological way. The material model

does not capture the reallocations of the fibers and therefore cannot take into account the different

response during crimp interchange in case of another load ratio or load history. Nevertheless, the

present results suggest that the test data based fabric material model is very appropriate for

simulating a coated fabric subjected to a uniaxial tensile stress state. An example of a generated

input file is shown in appendix A5.

Figure 5.11: Calculated and experimentally measured stress-strain curves for a uniaxial tensile test with load cycle repetition having a maximum stress level of 24.1 MPa in warp direction (a) and fill direction (b). The calculations have

been performed by means of the uniaxial test data based fabric model.

Both the calculated FEM strains and the measured experimental strains, in the direction

perpendicular to the loading direction, are plotted as a function of time in Figure 5.12.


97

Figure 5.12: Calculated and experimentally measured strain (perpendicular to the loading direction) during the first loading and unloading cycle of a uniaxial tensile test having a maximum stress level of 24.1 MPa as a function of the

normalized time in warp direction (a) and fill direction (b). The FEM calculations have been performed by means of a uniaxial test data based fabric model with a stiffness in compression of 0.025 GPa.

No agreement is observed between the two curves: the measured experimental strains are in

absolute value clearly higher than the calculated FEM strains. This indicates that the chosen

stiffness of the compression behavior in warp direction was too high. The compression stiffness

was iteratively decreased until an acceptable agreement was obtained between the experiment and

the FEM model. A compression stiffness of 0.031 MPa in warp direction and 0.83 MPa in fill

direction was found. The corresponding strain as a function of normalized time are shown in

Figure 5.13. Still no good agreement is obtained between the experiment and the FEM model, but

at least the strains are in the right range. For the remainder of the present paragraph, the above

compression stiffnesses are used.

Figure 5.13: Calculated and experimentally measured strain (perpendicular to the loading direction) during the first loading and unloading cycle of a uniaxial tensile test having a maximum stress level of 24.1 MPa as a function of the

normalized time in warp direction (a) and fill direction (b). The FEM calculations have been performed by means of a uniaxial test data based fabric model and a stiffness in compression of respectively 0.031 MPa and 0.83 MP in warp and

fill direction.

B. Uniaxial test with increasing load

A second analysis to qualify the appropriateness of the used material behavior, imparts the

simulation of a uniaxial test with increasing load in both warp and fill direction. The results are

presented in Figure 5.14. Very good agreement between the experimental and the FEM stress-

strain curves can be observed.


98

Figure 5.14: Calculated and experimentally measured stress-strain curves for a uniaxial tensile test with increasing maximum stress level in warp direction (a) and fill direction (b). The calculations have been performed by means of a

uniaxial test data based fabric model.

The previous comparisons have proven that the test data based fabric model is appropriate to

simulate a uniaxial tensile test with different loading conditions.

C. Biaxial test

A third method to qualify the suitability of the fabric model fitted to the uniaxial test data concerns

the modeling of a biaxial tensile test. The calculated stress-strain curves are compared to the

experimental stress-strain curves in both warp and fill direction (see Figure 5.15).

Figure 5.15: Calculated and experimentally measured stress-strain curves for a biaxial test with successive load ratios are: 1:1, 2:1, 1:2, 1:0 and 0:1 having a maximum stress level of 24.1 MPa in warp direction (a) and fill direction (b). The

calculations have been performed by means of uniaxial test data based fabric model.

Although the numerically computed strains correspond quite well to the ones obtained in the

experimental biaxial tests, one clearly observes the inability of the numerical model to account for

the different slopes of the various load ratios. Of course, this is not surprising since the fabric

model based on test data assumes independency between warp and fill direction. Furthermore it is

seen that the small increment in permanent strain between different load cycles as well as the

hysteresis are not captured. In warp direction, the error is higher for load ratios 1:2 and 0:1, while

in fill direction it is for load ratios 2:1 and 0:1.

D. Conclusion

It can be stated that the uniaxial test data based fabric model leads to an acceptable agreement with

the experiments for all three investigated loading conditions, namely uniaxial test with load cycle

repetition, uniaxial test with increasing load and biaxial test. In the next subsection, the model is


99

fitted to biaxial instead of uniaxial test data. It is investigated if the agreement between numerical

results and experiments improves.

5.7.2. Biaxial test data based FABRIC model

Here the fabric model is fitted to the stress-strain curve of the biaxial test for a load ratio 1:1. The

data points are superimposed to the experimental stress-strain curve (see Figure 5.16) and have

been chosen in a similar way as in the uniaxial test based fabric model (subsection 5.7.1). These

data points define the material behavior of the biaxial test data based FABRIC model.

Figure 5.16: Experimental stress-strain curve of a biaxial test and the superimposed data points which define the material behavior in warp direction (a) and fill direction (b).


The numerically calculated and experimentally recorded stress-strain curves for a uniaxial test with

load cycle repetition in both warp and fill direction are shown in Figure 5.17. It is observed that the

computed strain in both warp and fill direction is smaller compared to the experimentally obtained

strain. However, the strain differences are rather limited while the overall shape of the stress-strain

curves look very similar to the experiment.

Figure 5.17: Calculated and experimentally measured stress-strain curves for a uniaxial tensile test with load cycle repetition having a maximum stress level of 24.1 MPa in warp direction (a) and fill direction (b). The calculations have

been performed by means of test data based fabric model fitted to the biaxial tests.


The ability of the present model is further investigated with regard to a uniaxial test with increasing

load. The numerical results are compared with the experimental results, and are presented in Figure

5.18. The same observations can be made as was done for the uniaxial test with load cycle

repetition.


100

Figure 5.18: Calculated and experimentally measured stress-strain curves for a uniaxial tensile test with increasing maximum stress level in warp direction (a) and fill direction (b). The calculations have been performed by means of a

biaxial test data based fabric model

C. Biaxial test

The last method to qualify the suitability of the biaxial test data based fabric model concerns the

simulation of a biaxial tensile test. The numerical results are compared with the experimental

results in Figure 5.19.

Figure 5.19: Calculated and experimentally measured stress-strain curves for a biaxial test with successive load ratios: 1:1, 2:1, 1:2, 1:0 and 0:1 having a maximum stress level of 24.1 MPa in both warp direction (a) and fill direction (b). The

calculations have been performed by means of a biaxial test data based fabric model.

Similar conclusions can be drawn as for the uniaxial test data based fabric model, i.e. the

numerically computed strains correspond quite well to the experiment, while the different slopes

of the various load ratios as well as the hysteresis are not captured. Again, this is understood by

taking into account the inability of the current model to account for the interaction between warp

and fill direction. More interesting is the fact that, contrary to the uniaxial test data based fabric

model, the present model effectively predicts the small increments in strain between different load

ratios in a fairly accurate manner.

5.7.3. Conclusion

Two different test data based fabric models have been examined and discussed. The first test data

based fabric model employed the experimental stress-strain curves of uniaxial test, while the

second model is based on the biaxial test results. It was found that both models are capable of

predicting the stress-strain curve. Especially good agreement is found when simulating a uniaxial,

respectively biaxial test with a uniaxial, respectively biaxial test data based fabric model. The

limitations of the test data based fabric models have been identified. First of all, the model cannot

take an interaction between the warp and fill direction into account. Furthermore, it is a


101

phenomenological model and consequently it cannot account for a different crimp interchange

response for different loading conditions. A third limitation concerns the inability to model the

occurrence of hysteresis. The test data based fabric material model definitely provides a much

better prediction of the overall behavior of the coated fabric compared to the commonly used

linear elastic material model. In sections 3.3.2.6 and 3.3.2.7, it is was found that the structural

behavior of a coated fabric is partially governed by plasticity phenomena, even when applying a

modest external load (15%UTS). Hence at this moment a natural question arises: Can a plasticity

material model simulate a coated fabric true-to-nature?

5.8. Combined orthotropic elastic-plastic Hill material

model In chapter 3, it was demonstrated that the slope of the stress-strain curves decreases when the

stress value exceeds 14 MPa. After unloading permanent strain was observed which suggests the

presence of plasticity phenomena. Consequently, it is reasonable to model the coated fabric by

means of a combined elastic and plastic material model. The material constants for the elastic

model are adopted from section 5.3. In Abaqus/Standard, Abaqus/Explicit and Abaqus/CAE, a

classical metal plasticity model is available. It is possible to use either perfect plasticity or work

hardening. Perfect plasticity means that the stress does not change with the plastic strain after

yielding. This is obviously not realistic for a coated fabric. Work hardening implies that the surface

changes size in all directions such that the yield stress increases (or decreases) as plastic straining

occurs. Two yield surface definitions are available, namely the Mises yield surface and the original

Hill yield surface with an associated plastic flow [68]. Both surfaces assume that yielding of the

material is independent of the equivalent pressure stress [69]. The Mises yield surface is used to

model isotropic yielding. In Abaqus the evolution of the Mises yield surface is defined by the

uniaxial yield stress as a function of the uniaxial equivalent plastic strain. The Von Mises stress is

given by:

��= 1√2�£�� − � ¤[ + £� − ��¤[ + �� − ��[ + 6�� [ + 6� �[ + 6��[ (5.19)

The Hill yield surface on the other hand allows the modeling of anisotropic yielding [70]. In this

formulation, a reference yield stress σ0 as well as a set of yield ratios Rij must be specified. The yield

ratios can be used to model materials which exhibit different yield behavior in different directions.

For example, if σij is the only nonzero stress component, the corresponding yield stress can be

calculated as:

�56 =�56� ,� (5.20)

In case of multiple nonzero stress components, the equivalent stress is calculated by means of the

Hill’s potential function:

�� = �4�[[ − �\\�[ + x�\\ − ��[ + �� − �[[�[+2��[\[ + 2Ä�\�[ + 2��[[ (5.21)

in which F, G, H, L, M and N are defined as


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4 = � ,�[2 1�ç[[[ + 1�ç\\[ − 1�ç��[� =12 1�[[[ + 1�\\[ − 1��[�

x = � ,�[2 1�ç\\[ + 1�ç��[ − 1�ç[[[� =12 1�\\[ + 1��[ − 1�[[[�� =� ,�[2 1�ç��[ + 1�ç[[[ − 1�ç\\[� =12 1��[ + 1�[[[ − 1�\\[�

� =32� ,��ç[\�[ =32 1�[\[

Ä =32� ,��ç�\�[ =32 1��\[

=32� ,��ç�[�[ =32 1��[[

(5.22)

In these formula, each σ��. is the measured yield stress value when σ�. is the only nonzero stress

component. σ¬,� is a user-defined reference yield stress, while τ¬,� = ��,�√\ is the reference shear

yield stress. The six yield stress ratios are: R�� = ��,�, R[[ = ��,�, R\\ = ��,�, R�[ = ��,�, R�\ = ��,� andR[\ = ��,�. In case all yield stress ratios are set equal to unity, equation (5.21)

reduces to the Von Mises equivalent stress relation (equation (5.19)).

Abaqus uses the following plastic flow rule [71, 72]:

|�� = |Ø �� =|Ø� ä (5.23)

with λ the stretch ratio and b following matrix:

ä = NOOOOP−x�\\ −�� + �� −�[[�4�[[ −�\\� − �� −�[[�−4�[[ −�\\� + x�\\ −��2��[2Ä�\�2��[\ QR

RRRS (5.24)

For a uniaxial test in the 1-direction, hence σ11 is the only nonzero stress component, the

equivalent Hill stress can be calculated as (see appendix B1):

�� = �� (5.25)

The flow rule on the other hand simplifies to (see appendix B2):


103

|�� = |Ø

NOOOOOOOOOOP

1 ��$− 12�� 1 + ß�� [[$ á[ −ß�� \\$ á[�− 12�� 1 + ß�� \\$ á[ −ß�� [[$ á[�

000 QRRRRRRRRRRS (5.26)

5.8.1. Hardening law

In this subsection, it is determined whether one of the hardening laws, developed for metal

plasticity, can be fitted to the plastic behavior of a coated fabric. The hardening laws give the stress

as a function of the equivalent strain. This equivalent plastic strain comes into existence when the

yield point is exceeded. The yield stress in both warp and fill direction (σk�h�¬ and σ��¬

) was

previously identified to be approximately 14 MPa (see subsections 3.3.2.1 and 3.3.2.2). However,

the exact location of transition between elastic and plastic behavior in the stress-strain curves is not

clear. The most commonly used method to determine the yield point is the so called offset method

[73]. However, this method is not relevant in the fill direction for a coated fabric, mainly because

of the presence of crimp interchange at the beginning of a uniaxial test. In order to determine a

reasonable yield point, a straight line with the same slope as the unloading behavior is shifted

horizontally until the transmission between the straight line and the stress-strain curve at the cross

point is sufficiently smooth. This is shown in Figure 5.20 (a), the added line crosses the stress-

strain curve at a stress level of approximately 11 MPa. The corresponding hardening curve is

shown in Figure 5.20 (b).

Figure 5.20: Experimental stress-strain curve for a uniaxial test in fill direction (green) together with a straight line having the same slope as the unloading curve (orange) (a) and hardening behavior giving the stress as a function of true

equivalent plastic strain (b).

For the stress-strain curve in warp direction, two possible yield points can be identified (see Figure

5.21(a)). The blue line is parallel to the unloading curve, hence its slope determines the elastic

Young’s modulus. However, the initial response of the coated fabric in warp direction is not

tangent to the blue line, but rather to the green line. According to these two lines, two possible

yield points can be determined, namely at 12 MPa for the green line and 17 MPa for the blue line.


104

The corresponding hardening curves are shown in Figure 5.51(b). Both cases are further

investigated in this subsection.

Figure 5.21: Experimentally recorded stress-strain curve for a uniaxial test in warp direction, in which straight lines are added to determine the yield point (a) and hardening behavior giving the stress as a function of true equivalent plastic

strain for both yield points (b).

The parameters of two hardening laws are fitted to the hardening behavior of the coated fabric.

One of the hardening laws concerns the Voce model [74-76]:

�� = ¼ �1 − �exp£−ëq��¤� (5.27)

For metals, this model is known for its saturation of the flow stress at high plastic strains. It is a

physically based model. In other words, the model treats the strain hardening as a micro-

mechanical dislocation theory. Of course, this has no physical meaning for a coated fabric material.

Another hardening law is the Swift model which is a phenomenological hardening law. Hence the

parameters do not have a physical meaning. The Swift model is captured in following relation:

�� = ê£q� + q��¤� (5.28)

The parameters of the two hardening laws have been optimized by minimizing the cost function

C(p) for both warp hardening curves as well as the fill hardening curve. The Cost function C(p) is

defined as:

¼�� = 12T�5�� −�5s�~��W[ (5.29)

with χ�� the column vector of the experimentally measured response and χ�� the column

vector of the numerically computed response which is function of the unknown parameters p. A

schematic of the optimization is shown in Figure 5.22, and was implemented in the mathematical

software package Maple.


105

Figure 5.22: Optimization of the Voce and Swift material parameters

The optimized parameters of the Voce, respectively Swift hardening law for the three curves are

given in Table 5.2 and Table 5.3. The corresponding hardening laws are plotted in Figure 5.23,

together with the experimentally obtained data points.

Table 5.2: Optimized parameters of the Voce hardening law for two warp and one fill hardening curves.

Table 5.3: Optimized parameters of the Swift hardening law for two warp and one fill hardening curves.


106

Figure 5.23: Voce hardening law defined by optimized parameters for two warp curves and a fill curve (a) and Swift hardening law defined by optimized parameters for two warp curves and a fill curve (b) together with the experimental

data points.

Both the optimized Voce hardening laws and the optimized Swift hardening laws show very good

agreement with the experimental hardening data. The difference between the warp curves and the

fill curve in Figure 5.23 indicates that the plastic behavior of the coated fabric cannot be

considered isotropic. The optimization further reveals that the hardening curves are highly

dependent on the chosen yield point. This is illustrated by the difference between the blue and

green curve in warp direction, which only differ in the chosen yield point.

5.8.2. Lankford ratios

A method which is generally used for sheet metals in order to determine the grade of anisotropy is

adopted here. Uniaxial tensile tests provide a tool to measure the Lankford ratio rα, which is

defined as the ratio of the width plastic strain rate £ε�¬¬�»�¤�� to the thickness plastic strain rat£ε� �»�¤��e [77-79]:

!ï = £q � ��"¤��q ��"�� (5.30)

In this relation, α corresponds to the angle between the longitudinal axis of the tensile specimen

and the warp direction (reference direction). Since the thickness strain ε� �»� was not measured

during the experiments, the thickness plastic strain £ε� �»�¤�� is computed from both the length


107

plastic strain rate q ��"�� and the width plastic strain rate £ε�¬¬�»�¤�� by assuming volume conservation

[78, 80]:

q ��"�� =£q � ��"¤�� −q ��"�� (5.31)

With this, equation (5.30) can be written as:

!ï = £q � ��"¤��−£q � ��"¤��−£q ��"¤�� (5.32)

The Lankford ratios r0° (warp direction), r90° (fill direction) and r45° are calculated form the uniaxial

test in warp direction, fill direction and at 45° respectively. As discussed in paragraph 3.3.2.1 and

3.3.2.2, the measured strain during the uniaxial tests might be inaccurate, since the reference figure

for DIC was taken at the moment when no pre-load was applied to the sample. Nevertheless,

those strains are used to compute the Lankford ratios because of the lack of better and more

reliable experiments. The computed Lankford ratios for the coated fabric are presented in Figure

5.24 as a function of time. They are negative during pre-loading (0 s – 500 s), at the end of the pre-

loading, the Lankford ratios suddenly change. After pre-loading, the Lankford ratios are variable

which indicates that the anisotropy of the coated fabric evolves during the uniaxial tensile test. In

order to determine a unique value for each Lankford ratio, the curves in Figure 5.24 are averaged

in the time interval starting at 500 seconds until the end of the experiment. The averaged Lankford

ratios are: r0° = 0.0405, r90° = 0.0489 and r45° = -1.427. The small values of r0° and r90° suggest that

the thickness strain rate £ε� �»�¤�� is larger than the width strain rate £ε�¬¬�»�¤��, at least under the

assumption of volume conservation. The negative value of the Lankford ratio r45° indicates that the

width strain rate £ε�¬¬�»�¤�� becomes positive for a uniaxial test in shear loading.

Figure 5.24: Lankford ratios r0°, r90°, r45° as a function of time.


108

The coefficient of normal anisotropy !̅, or the average Lankford ratio is defined by [81]:

!̅ =!�° + 2!]°° +!$�°4 (5.33)

For the investigated coated fabric, a value of -0.69115 is obtained. The magnitude of this

coefficient defines how much the in-plane material behavior differs from the out-of-plane material

behavior. The negative sign represents the large influence of the negative Lankford ratio !]°° for a

uniaxial test in shear.

The degree of planar isotropy ∆r or the variation, measures how the material behavior varies along

different material directions in the plane of the coated fabric [82]. Usually, ∆r is computed as:

∆! =!�° − 2!]°° +!$�°4 (5.34)

This leads to a value of 0.73585 for the coated fabric. The degree of planar isotropy is fairly higher

than zero. This means that an anisotropic plasticity model is more appropriate than an isotropic

plasticity model to capture the material behavior [83]. The orthotropic Hill plasticity model, which

is implemented in Abaqus, allows for defining 6 yield stress ratios to determine the anisotropy of

the coated fabric. The constants F, G, H and N can be computed, given that:

!�° =�x = �1 − �

!$�° =�4

!]°° =2 − 4 − x2x + 4� = 2 − 4 + � − 121 − � + 4�

(5.35)

With this, following values are obtained for F, G, H and N: 4 = 0.795, x = 0.961,� = 0.0389

and = −2.026. Consequently following yield stress ratios are obtained: R�� = 1, R[[ = 1.095, R\\ = 0.755 and R�[ = 0.861. The uniaxial tensile test is modeled in Abaqus with these yield

stress ratios. Because R11 equals unity, the user defined reference yield stress � ,� as a function of

the plastic strain q��,� corresponds to the yield stress �r~«� as a function of plastic strain qr~«��

in

warp direction.

The numerically calculated and experimentally recorded stress-strain curves for a uniaxial test with

load cycle repetition in both warp and fill direction are shown in Figure 5.25. The FEM model

predicts the behavior in warp direction in a proper way. In fill direction however, the computed

strain is much smaller than the experimental strain.


109

Figure 5.25: Calculated and experimentally measured stress-strain curves for a uniaxial tensile test with load cycle

repetition having a maximum stress level of 24.1 MPa in warp direction (a) and fill direction (b). The calculations have been performed by means of a Hill plasticity material model in which the yield stress ratios are determined by means of

the Lankford ratios.

The presence of the crimp interchange at the beginning of the biaxial test prevents the Lankford

ratio based plasticity model to simulate the structural behavior of the coated fabric in a correct

manner. Because of the poor results, no further simulations are shown for the present material

model. In the next subsection, it is examined whether the Hill’s plasticity material model is more

appropriate to predict the behavior of the coated fabric when ignoring the crimp interchange in fill

direction. In that model no Lankford ratios will be determined, rather a fitting procedure to the

experimental curves is applied.

5.8.3. Hill’s plasticity excluding crimp interchange

The crimp interchange in the experimental stress-strain curve in fill direction is excluded. In this

case, both warp and fill stress-strain curves consist of a linear elastic part until the yield point is

reached, which is followed by a hardening phase. The unloading curve is parallel to the initial linear

elastic zone. In other words, a pure elastic-plastic behavior is obtained. In the numerical model, the � ,� −q�� curve is fitted to the experiments in fill direction while R11, R22 and R12 are fixed at

respectively 1.333, 1 and 1.Note that R33, R13 and R23 are of no importance for a two-dimensional

problem

5.8.3.1. Material behavior fitted to the uniaxial tensile tests


The calculated stress-strain curves together with the experimentally recorded curves are presented

in Figure 5.26 for a uniaxial test in both warp and fill direction. Note that the experimental curve in

fill direction is shifted to the left. In this way, the strain resulting from the crimp interchange is

effectively ignored. The agreement between the FEM and the experiment is satisfactory for both

directions. This suggests that the Hill’s plasticity model is appropriate for simulating the behavior

of a coated fabric in uniaxial tensile tests, on the condition that the strain caused by crimp

interchange is excluded.


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been performed by means of the Hill's plasticity material model, which is fitted to the uniaxial experiments ignoring the crimp interchange. The experimental stress-strain curve in fill direction is shifted to the left to remove the effect of crimp

interchange.

It is important to note that, even though the results are very promising, the model does not predict

the correct behavior of the coated fabric. In Figure 5.27, results are presented in which the

experimental curve in fill direction is not shifted to the left. Although the results in warp direction

are still in close agreement with the experiment, the situation clearly deteriorates in fill direction.

Hence, it is clear that the present model cannot account for the crimp interchange. To check

whether the present model could offer improved results, the present analysis is extended to other

loading conditions.

Figure 5.27: Calculated and experimentally measured stress-strain curves for a uniaxial tensile test with load cycle repetition having a maximum stress level of 24.1 MPa: warp direction (a) and fill direction (b). The calculations have been performed by means of a Hill plasticity material model, which is fitted to the uniaxial experiments ignoring the

crimp interchange.


A second analysis with the present material behavior is performed. A uniaxial test with increasing

load in both warp and fill direction is simulated. The results are presented in Figure 5.28. The same

conclusions can be drawn as for the uniaxial test with load cycle repetition.


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Figure 5.28: Calculated and experimentally measured stress-strain curves for a uniaxial tensile test with an increasing maximum stress level: warp direction (a) and fill direction (b). The calculations have been performed by means of the

Hill's plasticity material model, which is fitted to the uniaxial experiments ignoring the crimp interchange. The experimental stress-strain curve in fill direction is shifted to the left to remove the effect of crimp interchange.

C. Biaxial test

Instead of applying the current material model to simulate uniaxial tests, it was employed to

simulate a biaxial tensile test. The numerically computed stress-strain curves are shown in Figure

5.29, together with the corresponding experimental curves. It can be seen that the numerical model

of the biaxial test correctly accounts for the different slopes in the stress-strain curves for various

load ratios. Despite this achievement, the model can be considered very disappointing. Neither the

permanent strain, which comes into existence during the first loading and unloading cycle, nor the

hardening behavior are modeled in a proper way.

Figure 5.29: Calculated and experimentally measured stress-strain curves for a biaxial test with successive load ratios 1:1, 2:1, 1:2, 1:0 and 0:1 having a maximum stress level of 24.1 MPa: warp direction (a) and fill direction (b). The calculations have been performed by means of a Hill plasticity material model, which is fitted to the uniaxial experiments ignoring the crimp interchange. The experimental stress-strain curve in fill direction is shifted to the left to remove the effect of

crimp interchange.

5.8.3.2. Material behavior fitted to the biaxial tensile test

Instead of fitting uniaxial test data, the orthotropic elastic-plastic model has been further extended

by fitting biaxial test data. Because no substantial improvements were obtained for the simulation

of a coated fabric, only the results are briefly shown in following figures. The interpretation is

similar to the one found in previous sections.


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Figure 5.30: Calculated and experimentally measured stress-strain curves for a uniaxial tensile test with load cycle repetition having a maximum stress level of 24.1 MPa: warp direction (a) and fill direction (b). The calculations have been performed by means of the Hill's plasticity material model, which is fitted to a biaxial experiment ignoring the

crimp interchange. The experimental stress-strain curve in fill direction is shifted to the left to remove the effect of crimp interchange.

Figure 5.31: Calculated and experimentally measured stress-strain curves for a uniaxial tensile test with increasing maximum stress level: warp direction (a) and fill direction (b). The calculations have been performed by means of the

Hill's plasticity material model, which is fitted to a biaxial experiment ignoring the crimp interchange. The experimental stress-strain curve in fill direction is shifted to the left to remove the effect of crimp interchange.

Figure 5.32: Calculated and experimentally measured stress-strain curves for a biaxial test with successive load ratios 1:1, 2:1, 1:2, 1:0 and 0:1 having a maximum stress level of 24.1 MPa: warp direction (a) and fill direction (b). The calculations

have been performed by means of the Hill's plasticity material model, which is fitted to a biaxial experiment ignoring the crimp interchange. The experimental stress-strain curve in fill direction is shifted to the left to remove the effect of crimp

interchange.

5.8.3.3. Conclusion

The above discussed orthotropic elastic-plastic material model uses a stress-strain relationship

which was fitted to the experimentally recorded curve in which the crimp interchange was


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excluded. Poor results are obtained when a uniaxial test respectively a biaxial test is simulated by

means of a orthotropic elastic-plastic Hill model which was fitted to biaxial, respectively uniaxial

experimental results. This strongly indicates that the Hill plasticity model is not capable of

describing the hardening behavior of a coated fabric in a realistic way. In literature, several authors

came to the same conclusion, namely that the Hill model leads to inaccurate results for a variety of

materials [84]. Though, one has to be cautious with such a strong statement, especially because the

crimp interchange was artificially removed, which is a unrealistic simplification of a coated fabric.

It is well possible that exactly this assumption is the driving factor behind the poor results. In fact

it was one of the main challenges of the present thesis to model the effect of crimp interchange.

Therefore, it is investigated in the next sections whether the Hill plasticity model, in which the

crimp interchange is included, is suitable for the simulation of the structural behavior of a coated

fabric.

5.8.4. Hill’s plasticity including crimp interchange

In this subsection, we attempt to construct an anisotropic plasticity model which includes the large

initial strain in fill direction, caused by crimp interchange. Of course, a simple plasticity model

cannot capture the reallocations of the fibers during crimp interchange. Moreover, the exact

reallocation is highly dependent on both the load ratio and the load history. However, the most

important effect of crimp interchange, namely the large initial strain in fill direction during

tensioning, can be included in a phenomenological way. Note that it is impossible to make the

calculated stress-strain curves correspond to the experimental ones in both warp and fill direction.

This is easily understood taking into account the loading curve for a uniaxial test in both the warp

and fill direction. The first is concave, while the latter has a convex-concave shape. Since only 1

factor (R11 or R22) is available, one cannot define the correct shape of the plasticity behavior in

both warp (concave) and fill (convex-concave) direction. Therefore, the Hill plasticity model

including crimp interchange will be an approximation.

5.8.4.1. Material behavior fitted to a uniaxial test in fill (concave) direction

The yielding behavior of the anisotropic plasticity model is fitted to the stress-strain curve of a

uniaxial tensile test in fill direction. By means of a yield stress ratio R11≠1, the stress-strain response

for a uniaxial test in warp direction can effectively differ from the one in fill direction. To this end,

an optimization algorithm is written in the mathematical software Maple. A schematic of the

optimization procedure is shown in Figure 5.33. The procedure iteratively updates the yield stress

ratio R11, with the aim to minimize a cost function. At the end of the optimization, the best

possible agreement between the experimental and numerical stress-strain curve for the uniaxial test

in warp direction is achieved (see appendix C). It is taken care of that the numerical total strain

after the first loading curve corresponds to the experimentally recorded total strain. The optimized

yield stress ratio R11 = 2.38 was found (see Figure 5.34 (a)). The corresponding stress-strain curve

in warp direction is compared to the experimentally recorded curve in Figure 5.34 (b).


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Figure 5.33: Optimization scheme for the yield stress ratio R11

Figure 5.34: Determination of the yield stress factor R11 by minimizing the cost function (a) and comparison of the computed Hill plasticity stress-strain curve in warp direction with the experimentally recorded stress-strain curve in warp

direction (b)

The optimized stress-strain curve is approximately linear, which seems to have poor agreement

with the concave warp curve. Though, it is easy to understand that a linear course is the best

compromise between a concave (warp direction) and a convex (fill direction) stress-strain curve.

With these results, both the uniaxial and the biaxial tests have been numerically simulated.


The stress-strain curves for a uniaxial test with load cycle repetition obtained by means of both the

numerical model and the experiment are shown in Figure 5.35. There is a very good agreement for

the uniaxial test in fill direction, as could be expected since the present material model is supported

by the data of a uniaxial test in fill direction. In warp direction, it is clear that the numerically

obtained results differ from the experiment.. However, the total strain at the end of the first

loading curve, as well as the total permanent strain are predicted in a correct way by the numerical

model. An example of a generated input file of the present material model is shown in appendix

A6.


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been performed by means of a Hill plasticity model, which is fitted to the uniaxial tensile test in fill direction.


The same material model is used to simulate a uniaxial test with increasing load. The results are

shown in Figure 5.36.

Figure 5.36: Calculated and experimentally measured stress-strain curves for a uniaxial tests with increasing load: warp direction (a) and fill direction (b). The calculations have been performed by means of a Hill plasticity model, which is

fitted to the uniaxial test in fill direction.

The behavior of the coated fabric during the uniaxial test in fill direction with increasing load is

predicted very well. The FEM model of the uniaxial test in warp direction on the other hand does

not provide correct results: large deviations between the numerically computed and experimentally

recorded strains can be observed for every load cycle, except for the one in which a maximum load

of 24.1 MPa is reached. This is inherent to the chosen yield stress ratio R11, which was optimized

for a uniaxial test with a maximum stress value of 24.1 MPa. Hence it can be stated that the

anisotropic Hill plasticity model, which was fitted to the uniaxial test in fill direction with a

maximum stress of 24.1 MPa, cannot capture the material behavior of the coated fabric when

other loading conditions are applied.

C. Biaxial test

The last method to qualify the suitability of the Hill plasticity model, fitted to the uniaxial test in fill

direction, concerns the simulation of a biaxial tensile test. The corresponding results are compared

with the experimental results in Figure 5.37. The predicted strains in both warp and fill direction

are not exactly correct, but at least it is of the right order of magnitude. Besides, the model

correctly accounts for the different slopes in the stress-strain curves for different load ratios.


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Figure 5.37: Calculated and experimentally measured stress-strain curves for a biaxial test with successive load ratios are: 1:1, 2:1, 1:2, 1:0 and 0:1 having a maximum stress level of 24.1 MPa in warp direction (a) and fill direction (b). The

calculations have been performed by means of a Hill plasticity model fitted to the uniaxial experimental test in fill direction.

D. Conclusion

The anisotropic Hill plasticity model fitted to the uniaxial test in fill direction does not capture the

material behavior of the coated fabric perfectly, but the overall results are not too bad. In

comparison to the commonly used linear elastic material models, the anisotropic Hill plasticity

model is definitely a huge improvement. Note that the same procedure can be repeated in case the

model is optimized to a uniaxial tensile test in warp direction, instead of fill direction. Here, we

suffice with the statement that analogous results were obtained.

5.8.4.2. Material behavior fitted to a biaxial experiment

In the following, the anisotropic Hill plasticity material model is fitted to an experimental biaxial

test. The user-defined reference yield stress � ,� as a function of plastic strain q�� is fitted to the

fill direction and the yield stress ratio was optimized according to Figure 5.33, a value of R11 = 1.5

is obtained.


The numerically and experimentally obtained stress-strain curves for a uniaxial test with load cycle

repetition in both warp and fill direction are presented in Figure 5.38. Poor agreement can be

observed between the numerical and experimental results: the computed strain is almost twice as

high as the experimental strain.

Figure 5.38: Calculated and experimentally measured stress-strain curves for a uniaxial tensile test with load cycle

repetition having a maximum stress level of 24.1 MPa: warp direction (a) and fill direction (b). The calculations have been performed by means of a Hill plasticity model, which is fitted to the experimental biaxial stress-strain curves.


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The results, when simulating a uniaxial tensile test with increasing load, are shown in Figure 5.39.

As could be expected from the results of the uniaxial test with load cycle repetition, the calculated

strain is twice as high as the experimentally recorded strain.

Figure 5.39: Calculated and experimentally measured stress-strain curves for a uniaxial tests with increasing load: warp direction (a) and fill direction (b). The calculations have been performed by means of a Hill plasticity model, which is

fitted to the experimental biaxial stress-strain curves.

C. Biaxial test

The third examination concerns the simulation of a biaxial test. In Figure 5.40, the numerically

obtained stress-strain curves are shown, together with the corresponding experimental curves. The

permanent strain, obtained after the first loading and unloading cycle, is predicted in a proper way.

It is furthermore seen that the numerically computed curves have a slope which closely resembles

the experiment for all load ratios. However, the small strain increments, when passing to another

load cycle, are not captured well by the numerical model. In addition, the shape of computed first

loading cycle in warp direction does not correspond to the experimental one.

Figure 5.40: Calculated and experimentally measured stress-strain curves for a biaxial test with successive load ratios 1:1, 2:1, 1:2, 1:0 and 0:1 having a maximum stress level of 24.1 MPa: warp direction (a) and fill direction (b). The calculations

have been performed by means of a Hill plasticity model, which is fitted to the experimental biaxial test.

5.8.4.3. Conclusion

The parameters of the Hill plasticity model have been fitted to three different experimental stress-

strain curves: (i) the uniaxial test in fill direction, (ii) the uniaxial test in warp direction (for brevity,

the results are not shown) and (iii) the biaxial test. To this end, an optimization scheme was written

in the mathematical language Maple. Results have been obtained which are partially satisfying.

Several numerically computed stress-strain curves correspond well to the experiment. The varying


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slopes of the stress-strain curves, when considering different load ratios, is well reproduced by the

numerical model. However the model has certain undeniable limitations. First of all, only one yield

stress factor can be defined. This makes it impossible to simultaneously account for the concave,

respectively convex stress-strain curve in warp, respectively fill direction. Hence, the anisotropic

plasticity cannot be modeled in a realistic manner. Secondly, the crimp interchange is implemented

in an artificial way. Consequently, the model cannot account for a different crimp interchange

response for different loading conditions. Nevertheless, it should be clear that the combined

elastic-plastic Hill material model provides a much better prediction of the overall behavior of the

coated fabric compared to the commonly used linear elastic material model.

5.9. Overall strain field Up to now, the computed strains have been compared with the experimentally recorded strains at

the center of the central part of the cruciform sample. In order to get an impression of the overall

picture of the strain distribution in the cruciform sample, both processed DIC images and contour

plots of the FEM results for a biaxial test are shown.

5.9.1. Strain in warp direction

In this subsection, the strain field in warp direction is presented. To facilitate the interpretation, a

rainbow color scale is used to correlate with the strain values: the purple, respectively red color

corresponds to a low, respectively a high strain value (see Figure 5.41).

Figure 5.41: Color code used for the strain field in warp direction.

The following figures show the strain field in warp direction for a biaxial test setup: (a) the

processed DIC image, (b) the contour plot obtained by means of the uniaxial test data based fabric

model and (c) the contour plot obtained by means of the combined orthotropic elastic-plastic Hill

model, which is fitted to the uniaxial test in fill direction. In the cruciform sample, the horizontal

direction corresponds to the warp direction, while the vertical direction corresponds to the fill

direction. For each load ratio, results are shown for 3 different load levels, namely at 200 N (pre-

stress), at 1000 N (half of the maximum load level) and at 2000 N (maximum load level). For

clarity, the points at which the strain field in warp direction is extracted, are superimposed on the

biaxial load cycle (blue dots in Figure 5.42).


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Figure 5.42: Biaxial load cycle, with the superimposed points of extraction (blue dots).

Figure 5.43: Contour plots of the strain field in warp direction for load ratio 1:1 at the specified load level:

experiment (a), fabric model (b) and Hill model.




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121

The results clearly show that good agreement is obtained between the experimentally recorded

strain field in warp direction and both numerically computed strain fields in warp direction.

Though, some discrepancies can be observed between experiment and numerical modeling,

especially in the top arm (and because of symmetry also the bottom arm). Indeed, when the

experimental contour plot has a purple color, the numerical model often shows a blue color. This

indicates that the contraction in warp direction, when applying a load in fill direction, is

underestimated by the orthotropic elastic-plastic Hill model. Indeed, the Hill model goes with a

small Poisson ratio of 0.04, and consequently results in a limited contraction in warp direction. In

the experiments on the other hand, the contraction is higher because of the effect of crimp

interchange.

5.9.2. Strain in fill direction

In analogy with the previous subsection, both the experimentally recorded and numerically

computed contour plots of the strain in fill direction are presented. A similar color scale is used

and is shown in Figure 5.41. In the cruciform sample, the horizontal direction corresponds to the

warp direction, while the vertical direction corresponds to the fill direction.

Figure 5.48: Color code used for the strain field in fill direction.

Figure 5.49: Contour plots of the strain field in fill

direction for load ratio 1:1 at the specified load level: experiment (a), fabric model (b) and Hill model.




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direction for load ratio 0:1 at the specified load level: experiment (a), fabric model (b) and Hill model

Slightly worse results, compared to the strains in warp direction, are obtained for the

experimentally recorded and numerically computed strains in fill direction. Especially, the


123

orthotropic elastic-plastic Hill model leads to some major differences. The strain field in fill

direction in the top arm of the cruciform sample is characterized by a red color for the elastic-

plastic Hill model, while the experimentally recorded strain in fill direction is coded with a green

color. This implies that the elastic-plastic Hill model severely overestimates the strain in fill

direction.

5.10. Conclusion Several built-in material models of Abaqus have been studied and discussed. Their suitability for

modeling the behavior of a coated fabric was judged, based on several objective parameters. The

most promising material models were examined into detail and were applied to a numerical

model, simulating the uniaxial and biaxial experiments. The stress-strain curves obtained from the

experiments served as a reference for the numerically calculated results.

A linear elastic material model is not at all capable to model the structural behavior of a coated fabric.

Permanent strain, nonlinear behavior due to crimp interchange and hysteresis cannot be included

in the model. In practice, this model has become popular mainly because its capability to predict

the behavior in a stabilized state, i.e. when both crimp interchange and plasticity are removed. In

reality however, such a stabilized state rarely occurs.

The multilinear material model on the other hand, can be fitted well to the experimental results.

Therefore it is possible to include nonlinearity, plastic strain as well as hysteresis, but only for a

certain specified loading condition. The main drawback of the multilinear model concerns the

fact that it is not applicable for a real tent structure because of the different computation steps.

Once a certain stress level in a certain material point is reached, the multilinear elastic model has

to switch its material model instantaneously to the following material model (different E-

modulus), in that certain material point. But in advance, it is not known when this moment will

occur, nor at which place it will occur.

The isotropic hyperelastic material model with permanent set does reproduce all possible nonlinear loading

curves as well as permanent strain. However, the inherent isotropic formulation of the material

model makes it unsuitable for the simulation of the material behavior of a coated fabric.

Contrary to the isotropic hyperelastic material model with permanent set, the anisotropic hyperelastic

material model is able to account for different structural behavior in warp and fill direction.

However, the numerically computed stress-strain curves are limited to an exponential shape

because of the implicit definition of the energy potential U. The experimentally recorded stress-

strain curves in both warp and fill direction reveal rather a logarithmic shape and thus cannot be

reproduced by the energy potential U. Moreover, no permanent strain can be captured in the

anisotropic hyperelastic material model.

The test data based fabric material model is one of the most promising built-in material models of

Abaqus. It is based on the input of experimental test data in warp direction, in fill direction and

in shear. Orthotropy, nonlinearity as well as permanent strain can be implemented in this model.

For a certain loading condition, the fabric material model is very suitable to model the behavior

of a coated fabric. Though, the main problem arises when the loading condition changes. The

material model is not capable of predicting the correct strain for a variety of different loading


124

conditions. Furthermore, it does not reproduce the different slopes for varying load ratios, as can

be observed in the experimental biaxial tests. Two main reasons are responsible for these

shortcomings. First of all, the crimp interchange is implemented in a phenomenological way, and

thus cannot account correctly for the reallocation of the fibers which is highly dependent on the

loading behavior. Secondly, the test data based fabric model explicitly assumes no interaction

between warp and fill direction. Hence an ordinary Poisson effect is not taken into account,

which obviously leads to poor results.

The last examined material model is the combined orthotropic elastic-plastic Hill material model. This

model is very interesting since the experiments revealed that part of the structural response of the

coated fabric is dominated by plasticity phenomena, even for modest load levels (~15% UTS). In

contrast with the previously described models, the plasticity phenomena are actually integrated in

the material model instead of trying to capture the hardening behavior by means of a nonlinearity

in the loading curve. Though, a plasticity model is not capable of capturing the crimp interchange

and thus must be either ignored or artificially included. In addition, the so called plasticity

anisotropy in 2D is defined by a single factor, namely the yield stress ratio R11 or R22. Since one

single yield stress ratio cannot transform the concave warp curve into a combined convex-

concave fill curve or vice versa, the application of the combined anisotropic elastic-plastic Hill

material model is being undermined. The determination of the yield stress ratios Rij by means of

the Lankford ratios rα, which is principally done for metal sheets, was not successful because of

the influence of crimp interchange. Application of the anisotropic Hill plasticity model, in which

the crimp interchange is ignored, resulted in reasonable agreement with the experimentally

recorded stress-strain curves. Though, it is clear that in reality the crimp interchange has a major

influence on the structural behavior of a coated fabric, and thus cannot be omitted. When

accounting for the crimp interchange, by an artificial implementation in the orthotropic

hardening behavior, the behavior in one of the directions is captured in a correct way, but never

in both directions. For example, the material model, fitted to the fill direction, does predict the

correct stress-strain behavior in fill direction, but shows deviations in warp direction. It can be

stated that the combined anisotropic elastic-plastic Hill model does not predict the material

behavior of the coated fabric perfectly, but at least the computed strains are in the correct range.

It has certainly major improvements compared to frequently used material models in current

research to simulate a coated fabric. Indeed, a lot of seasoned material models neither include

nonlinear effects, nor permanent strain. Moreover, the combined anisotropic elastic-plastic Hill

material model is straightforward and easily applicable in FEM calculations. Besides, it has a large

computational efficiency: the computational time is more or less 1 minute for the simulated

uniaxial and biaxial tests.

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Chapter 6.Conclusions and recommendations Experimental tensile tests have been performed in both uniaxial and biaxial regime. The results of

the uniaxial tensile tests revealed that the investigated coated fabric (T2107) has a completely

different response to an external load for fill direction and warp direction. Moreover, the

experiments strongly suggest that the coated fabric exhibits both elastic and plastic behavior.

Once the stress level exceeds 14 MPa, which is a very moderate stress level for typical structural

applications, yielding phenomena are observed. In fill direction, another effect of plasticity, which

plays an important role for the structural behavior of a coated fabric, is observed: crimp

interchange, i.e. the reallocation of the fibers under influence of an external load. The same

experiments furthermore revealed that the hysteresis is not only triggered by the effect of the

crimp interchange, i.e. energy losses due to the friction of the fibers during reallocation, but also

by the yielding.

A numerical study of the influence of both the geometry and the size of the cruciform sample on

the biaxial experimental results was carried out. It was found that the presence of slits in the arms

causes stress oscillations at the corresponding locations in the central part. The amplitude of the

oscillations decreases, while a higher mean stress value is obtained in the central part, when

increasing the number of slits. This implies that more slits provide a more uniform and higher

stress distribution, which is advantageous for the correct and unambiguous interpretation of the

(DIC) results. Though, the addition of slits in the cruciform sample is a tedious and time-

consuming process. It is concluded that the improved stress introduction in the central part does

not outweigh the increased preparation time, hence biaxial experiments have been performed on

cruciform samples without slits. In these experiments, several load cycles and load ratios have

been considered. When proceeding to a different load ratio, the coated fabric undergoes a small

increase or decrease in permanent strain. More important, the stress-strain curve shows different

slopes for varying load ratios, which indicates that the fill direction is coupled to the warp

direction and vice versa. It is observed that the structural behavior of a coated fabric is dependent

on the load history, in other words the material 'memorizes' the last load cycle. In literature, it is a

rule of thumb that the load history can be excluded by repeating each load cycle 3 times.

However, a biaxial experiment with several load ratios, in which each load cycle is repeated 3

times, is performed. The first and the last load ratio was taken as 1:1. According to the rule of

thumb, the coated fabric should be in the same state after the first 1:1 cycles, as after the last 1:1

cycles. The experiment clearly showed that the material did not behave as expected. Even more,

this suggests that the load history introduces, besides the reversible crimp interchange, some

irreversible plasticity phenomena.

Several software packages were developed during the last decades, though most of them are

found on largely simplified material models and consequently cannot capture the correct

behavior of coated fabrics. On the other hand, packages were developed which use more

advanced modeling techniques. The main drawback of these packages concerns the extremely

high computation time making the simulation of a complete tent structure unrealistic. Hence a

package is needed which strikes the golden mean, i.e. an acceptable material model and a

Chapter 6. Conclusions and recommendations

126

reasonable computation time. Several built-in material models of Abaqus have been studied and

discussed. Their appropriateness for modeling the behavior of a coated fabric was judged based

on objective parameters. The most promising material models were examined into detail and

were applied to a FEM model which serves as the numerical counterpart of the uniaxial and

biaxial experiments. The stress-strain curves obtained from the experiments served as a reference

for the numerically calculated results.

Although a linear elastic material model is frequently used for the design of tent structures, it is not

capable to realistically model the material behavior of a coated fabric. It does not account for

permanent strain, nonlinear effects or hysteresis. The multilinear material model on the other hand,

is capable of including nonlinearity as well as permanent strain. The main drawback of this model

concerns the fact that it is practically not applicable, because the stress evolution has to be known

in advance in every material point. The third examined model concerns the isotropic hyperelastic

material model with permanent set, which is able to model both nonlinearity and permanent strain.

However, the inherent isotropic formulation of the material model makes it unsuitable for the

simulation of an orthotropic coated fabric. The logic successor of the isotropic hyperelastic

material model with permanent set, concerns the anisotropic hyperelastic material model because it is

able to account for a different structural behavior in warp and fill direction. However, the model

makes use of a strain energy potential. Both the Fung potential and the Holzapfel-Gasser-Ogden

potential have an exponential shape, and consequently cannot reproduce the experimentally

recorded stress-strain curves which are rather logarithmic of nature. Moreover, the anisotropic

hyperelastic material model cannot capture plasticity phenomena. The test data based fabric material

model is one of the most promising built-in material models of Abaqus for simulating a coated

fabric. Orthotropy, nonlinearity as well as permanent strain can be implemented. However,

because of both the independency of the response in warp and fill direction and the

phenomenological implementation of crimp interchange, the numerically computed stress-strain

curves exhibit limited agreement with the experimentally obtained stress-strain curves.

Nevertheless, it is definitely an improvement for simulating coated fabrics, compared to the

widespread linear elastic material model. The last examined material model concerns the combined

anisotropic elastic-plastic Hill material model, in which a realistic physical description of plasticity

phenomena is integrated. Though, a plasticity model is not capable of capturing the crimp

interchange (since it is not a real plasticity phenomenon) and thus must be either ignored or

artificially included. Moreover, the so called plasticity anisotropy in 2D is defined by a single

factor, namely the yield stress ratio, which limits the correct implementation of the anisotropic

stress-strain behavior. When crimp interchange is artificially implemented in the orthotropic

hardening behavior, the simulated material behavior of the coated fabric is in reasonable

agreement with the experiments. Compared to more conventional material models, the combined

orthotropic elastic-plastic Hill model has an interesting formulation which closely leans towards

the physical material behavior of a coated fabric under an external load. The Hill model

distinguishes itself also with regard to the computational efficiency: the computational time for

the shown simulations correspond to more or less one minute.

In this master thesis, several material models have been studied on small-scale samples. It is

recommended to evaluate both the test data based fabric model and the combined orthotropic

elastic-plastic Hill model on a larger scale and for more complex geometries. In the end, a full-

Chapter 6. Conclusions and recommendations

127

scale foldable tent structure should be simulated in order to expose the full capabilities of the

material model for practical applications. Of course, additional experiments have to be carried

out in order to gather a more thorough knowledge of the structural behavior of a coated fabric.

Especially the biaxial tests have to be extended with various sequences of different load ratios as

well as with several extra load ratios (e.g. 5:1 and 1:5). Subjecting the coated fabric to a state of

multi-axial stress will definitely lead to several new insights as well as an improved understanding

of the material. Without any doubt, it can be stated that a coated fabric is an incredibly complex,

but very fascinating material.

128

Appendices

Appendix A

Appendix A1: Typical input file for an orthotropic linear elastic material model

*Heading ** Job name: Biaxial Model name: Biaxial ** Generated by: Abaqus/CAE 6.11-1 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** ** PARTS ** *Part, name=fabric *Node 1, 0., 0.210362613, 0.

… 6822, 0.0476894304, 0.0727364048, 0. *Element, type=M3D8R 1, 544, 545, 946, 942, 2309, 2310, 2311, 2312 … 2207, 2306, 327, 1810, 2307, 6698, 6706, 6636, 4056 *Nset, nset=_PickedSet30, internal, generate 1, 6822, 1 *Elset, elset=_PickedSet30, internal, generate 1, 2207, 1 ** Section: Section-1 *Membrane Section, elset=_PickedSet30, material=linearelastic 0.00083, *End Part ** ** ** ASSEMBLY ** *Assembly, name=Assembly ** *Instance, name=fabric-1, part=fabric *End Instance ** *Node 1, 0.25, 0., 0. *Node 2, 0., 0.25, 0. *Nset, nset=_PickedSet47, internal 1, *Nset, nset=_PickedSet48, internal 2, *Nset, nset=_PickedSet59, internal, instance=fabric-1 18, *Nset, nset=_PickedSet60, internal, instance=fabric-1 1, … 6792 *Elset, elset=_PickedSet60, internal, instance=fabric-1 31, … 2084 *Nset, nset=_PickedSet61, internal, instance=fabric-1 18, … 5791 *Elset, elset=_PickedSet61, internal, instance=fabric-1 15, … 1367 *Nset, nset=_PickedSet62, internal, instance=fabric-1, generate 1, 6822, 1

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*Nset, nset=_PickedSet62, internal 1, 2 *Elset, elset=_PickedSet62, internal, instance=fabric-1, generate 1, 2207, 1 *Nset, nset=_PickedSet63, internal 1, *Nset, nset=_PickedSet64, internal 2, *Nset, nset=_PickedSet69, internal, instance=fabric-1 35, … 5486 *Elset, elset=_PickedSet69, internal, instance=fabric-1 214, … 1160 *Nset, nset=_PickedSet70, internal 1, *Nset, nset=_PickedSet71, internal, instance=fabric-1 17, … 5495 *Elset, elset=_PickedSet71, internal, instance=fabric-1 1, … 1167 *Nset, nset=_PickedSet72, internal 2, *Surface, type=NODE, name=_PickedSet69_CNS_, internal _PickedSet69, 1. *Surface, type=NODE, name=_PickedSet71_CNS_, internal _PickedSet71, 1. ** Constraint: Constraint-1 *Coupling, constraint name=Constraint-1, ref node=_PickedSet70, surface=_PickedSet69_CNS_ *Kinematic 1, 1 ** Constraint: Constraint-2 *Coupling, constraint name=Constraint-2, ref node=_PickedSet72, surface=_PickedSet71_CNS_ *Kinematic 2, 2 *End Assembly *Amplitude, name=load-right 0., 0.1, 0.1, 0.2, 0.2, 0.3, 0.3, 0.4 0.4, 0.5, 0.5, 0.6, 0.6, 0.7, 0.7, 0.8 0.8, 0.9, 0.9, 1., 1., 0.9, 1.1, 0.8 1.2, 0.7, 1.3, 0.6, 1.4, 0.5, 1.5, 0.4 1.6, 0.3, 1.7, 0.2, 1.8, 0.1, 1.9, 0.2 2., 0.3, 2.1, 0.4, 2.2, 0.5, 2.3, 0.6 2.4, 0.7, 2.5, 0.8, 2.6, 0.9, 2.7, 1. 2.8, 0.9, 2.9, 0.8, 3., 0.7, 3.1, 0.6 3.2, 0.5, 3.3, 0.4, 3.4, 0.3, 3.5, 0.2 3.6, 0.1, 3.7, 0.2, 3.8, 0.25449, 3.9, 0.28948 4., 0.32446, 4.1, 0.35945, 4.2, 0.39444, 4.3, 0.42943 4.4, 0.46661, 4.5, 0.5, 4.6, 0.46661, 4.7, 0.42943 4.8, 0.39444, 4.9, 0.35945, 5., 0.32446, 5.1, 0.28948 5.2, 0.25449, 5.3, 0.2, 5.4, 0.1, 5.5, 0.2 5.6, 0.3, 5.7, 0.4, 5.8, 0.5, 5.9, 0.6 6., 0.7, 6.1, 0.8, 6.2, 0.9, 6.3, 1. 6.4, 0.9, 6.5, 0.8, 6.6, 0.7, 6.7, 0.6 6.8, 0.5, 6.9, 0.4, 7., 0.3, 7.1, 0.2 7.2, 0.1, 7.3, 0.1, 7.4, 0.1, 7.5, 0.1 7.6, 0.1, 7.7, 0.1, 7.8, 0.1, 7.9, 0.1 8., 0.1, 8.1, 0.1, 8.2, 0.1, 8.3, 0.1 8.4, 0.1, 8.5, 0.1, 8.6, 0.1, 8.7, 0.1 8.8, 0.1, 8.9, 0.1, 9., 0.1 *Amplitude, name=load-top 0., 0.1, 0.1, 0.2, 0.2, 0.3, 0.3, 0.4 0.4, 0.5, 0.5, 0.6, 0.6, 0.7, 0.7, 0.8 0.8, 0.9, 0.9, 1., 1., 0.9, 1.1, 0.8

Appendices

130

1.2, 0.7, 1.3, 0.6, 1.4, 0.5, 1.5, 0.4 1.6, 0.3, 1.7, 0.2, 1.8, 0.1, 1.9, 0.2 2., 0.25449, 2.1, 0.28948, 2.2, 0.32446, 2.3, 0.35945 2.4, 0.39444, 2.5, 0.42943, 2.6, 0.46661, 2.7, 0.5 2.8, 0.46661, 2.9, 0.42943, 3., 0.39444, 3.1, 0.35945 3.2, 0.32446, 3.3, 0.28948, 3.4, 0.25449, 3.5, 0.2 3.6, 0.1, 3.7, 0.2, 3.8, 0.3, 3.9, 0.4 4., 0.5, 4.1, 0.6, 4.2, 0.7, 4.3, 0.8 4.4, 0.9, 4.5, 1., 4.6, 0.9, 4.7, 0.8 4.8, 0.7, 4.9, 0.6, 5., 0.5, 5.1, 0.4 5.2, 0.3, 5.3, 0.2, 5.4, 0.1, 5.5, 0.1 5.6, 0.1, 5.7, 0.1, 5.8, 0.1, 5.9, 0.1 6., 0.1, 6.1, 0.1, 6.2, 0.1, 6.3, 0.1 6.4, 0.1, 6.5, 0.1, 6.6, 0.1, 6.7, 0.1 6.8, 0.1, 6.9, 0.1, 7., 0.1, 7.1, 0.1 7.2, 0.1, 7.3, 0.2, 7.4, 0.3, 7.5, 0.4 7.6, 0.5, 7.7, 0.6, 7.8, 0.7, 7.9, 0.8 8., 0.9, 8.1, 1., 8.2, 0.9, 8.3, 0.8 8.4, 0.7, 8.5, 0.6, 8.6, 0.5, 8.7, 0.4 8.8, 0.3, 8.9, 0.2, 9., 0.1 ** ** MATERIALS ** *Material, name=linearelastic *Density 963., *Elastic, type=ENGINEERING CONSTANTS 1.13e+09, 8.25e+08, 2e+07, 0.04, 0.25, 0.25, 3e+08, 2e+06 2e+06, ** ** BOUNDARY CONDITIONS ** ** Name: right Type: Displacement/Rotation *Boundary _PickedSet47, 2, 2 … _PickedSet47, 6, 6 ** Name: top Type: Displacement/Rotation *Boundary _PickedSet48, 1, 1 … _PickedSet48, 6, 6 ** ---------------------------------------------------------------- ** ** STEP: Step-1 ** *Step, name=Step-1 *Static 0.1, 9., 9e-20, 0.1 ** ** BOUNDARY CONDITIONS ** ** Name: pt Type: Displacement/Rotation *Boundary _PickedSet59, 1, 1 … _PickedSet59, 6, 6 ** Name: xsymm Type: Symmetry/Antisymmetry/Encastre *Boundary _PickedSet60, XSYMM ** Name: ysymm Type: Symmetry/Antisymmetry/Encastre *Boundary

Appendices

131

_PickedSet61, YSYMM ** Name: z Type: Displacement/Rotation *Boundary _PickedSet62, 3, 3 ** ** LOADS ** ** Name: load-right Type: Concentrated force *Cload, amplitude=load-right _PickedSet63, 1, 2000. ** Name: load-top Type: Concentrated force *Cload, amplitude=load-top _PickedSet64, 2, 2000. ** ** OUTPUT REQUESTS ** *Restart, write, frequency=10 ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT, time interval=0.1 ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT, time interval=0.1 *End Step

Appendix A2: Typical input file for an orthotropic multilinear material

model

Part1:

*Heading ** Job name: FillPart1 Model name: FillPart1 ** Generated by: Abaqus/CAE 6.11-1 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** ** PARTS ** *Part, name=fabric *End Part ** ** ** ASSEMBLY ** *Assembly, name=Assembly ** *Instance, name=fabric-1, part=fabric *Node 1, 0., 0., 0. … 10, 0.0250000004, 0.100000001, 0. *Element, type=M3D4R 1, 1, 2, 4, 3 … 4, 7, 8, 10, 9 *Nset, nset=_PickedSet9, internal, generate 1, 10, 1 *Elset, elset=_PickedSet9, internal, generate 1, 4, 1 ** Section: Section-1

Appendices

132

*Membrane Section, elset=_PickedSet9, material=part1 0.00083, *End Instance ** *Node 1, 0., 0.109999999, 0. *Nset, nset=_PickedSet15, internal 1, *Nset, nset=_PickedSet16, internal 1, *Nset, nset=_PickedSet17, internal, instance=fabric-1, generate 1, 9, 2 *Elset, elset=_PickedSet17, internal, instance=fabric-1, generate 1, 4, 1 *Nset, nset=_PickedSet18, internal, instance=fabric-1 1, 2 *Elset, elset=_PickedSet18, internal, instance=fabric-1 1, *Nset, nset=_PickedSet19, internal, instance=fabric-1, generate 1, 10, 1 *Elset, elset=_PickedSet19, internal, instance=fabric-1, generate 1, 4, 1 *Nset, nset=_PickedSet20, internal 1, *Nset, nset=_PickedSet24, internal, instance=fabric-1 9, 10 *Elset, elset=_PickedSet24, internal, instance=fabric-1 4, *Surface, type=NODE, name=_PickedSet24_CNS_, internal _PickedSet24, 1. ** Constraint: Constraint-1 *Coupling, constraint name=Constraint-1, ref node=_PickedSet15, surface=_PickedSet24_CNS_ *Kinematic 2, 2 *End Assembly *Amplitude, name=Amp-1 0., 0., 1., 0.2 ** ** MATERIALS ** *Material, name=part1 *Elastic, type=ENGINEERING CONSTANTS 1.364e+09, 1.61e+08, 2e+08, 0.04, 0.3, 0.3, 3e+08, 2e+08 2e+08, ** ---------------------------------------------------------------- ** ** STEP: Step-1 ** *Step, name=Step-1, nlgeom=YES *Static 1., 1., 1e-05, 1. ** ** BOUNDARY CONDITIONS ** ** Name: pt Type: Displacement/Rotation *Boundary _PickedSet20, 1, 1 _PickedSet20, 3, 3 _PickedSet20, 4, 4 _PickedSet20, 5, 5 _PickedSet20, 6, 6 ** Name: xsym Type: Symmetry/Antisymmetry/Encastre

Appendices

133

*Boundary _PickedSet17, XSYMM ** Name: ysym Type: Symmetry/Antisymmetry/Encastre *Boundary _PickedSet18, YSYMM ** Name: z Type: Displacement/Rotation *Boundary _PickedSet19, 3, 3 ** ** LOADS ** ** Name: Load-1 Type: Concentrated force *Cload, amplitude=Amp-1 _PickedSet16, 1, 0. _PickedSet16, 2, 500. ** ** OUTPUT REQUESTS ** *Restart, write, frequency=10 ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step

Part 2: *Heading ** Job name: FillPart2 Model name: FillPart2 ** Generated by: Abaqus/CAE 6.11-1 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** ** PARTS ** ** ** ASSEMBLY ** *Assembly, name=Assembly ** *Instance, library=FillPart1, instance=fabric-1 ** ** PREDEFINED FIELD ** ** Name: Predefined Field-1 Type: Initial State *Import, state=yes, update=no *Nset, nset=_PickedSet9, internal, generate 1, 10, 1 *Elset, elset=_PickedSet9, internal, generate 1, 4, 1 *End Instance ** *Node 1, 0., 0.109999999, 0. *Nset, nset=_PickedSet15, internal 1, *Nset, nset=_PickedSet16, internal 1, *Nset, nset=_PickedSet17, internal, instance=fabric-1, generate 1, 9, 2

Appendices

134

*Elset, elset=_PickedSet17, internal, instance=fabric-1, generate 1, 4, 1 *Nset, nset=_PickedSet18, internal, instance=fabric-1 1, 2 *Elset, elset=_PickedSet18, internal, instance=fabric-1 1, *Nset, nset=_PickedSet19, internal, instance=fabric-1, generate 1, 10, 1 *Elset, elset=_PickedSet19, internal, instance=fabric-1, generate 1, 4, 1 *Nset, nset=_PickedSet20, internal 1, *Nset, nset=_PickedSet24, internal, instance=fabric-1 9, 10 *Elset, elset=_PickedSet24, internal, instance=fabric-1 4, *Surface, type=NODE, name=_PickedSet24_CNS_, internal _PickedSet24, 1. ** Constraint: Constraint-1 *Coupling, constraint name=Constraint-1, ref node=_PickedSet15, surface=_PickedSet24_CNS_ *Kinematic 2, 2 *End Assembly *Amplitude, name=Amp-1 0., 0.2, 1., 0.6 ** ** MATERIALS ** *Material, name=part1 *Elastic, type=ENGINEERING CONSTANTS 1.364e+09, 3.45e+08, 2e+08, 0.04, 0.3, 0.3, 3e+08, 2e+08 2e+08, ** ---------------------------------------------------------------- ** ** STEP: Step-1 ** *Step, name=Step-1, nlgeom=YES *Static 1., 1., 1e-05, 1. ** ** BOUNDARY CONDITIONS ** ** Name: pt Type: Displacement/Rotation *Boundary _PickedSet20, 1, 1 _PickedSet20, 3, 3 _PickedSet20, 4, 4 _PickedSet20, 5, 5 _PickedSet20, 6, 6 ** Name: xsym Type: Symmetry/Antisymmetry/Encastre *Boundary _PickedSet17, XSYMM ** Name: ysym Type: Symmetry/Antisymmetry/Encastre *Boundary _PickedSet18, YSYMM ** Name: z Type: Displacement/Rotation *Boundary _PickedSet19, 3, 3 ** ** LOADS ** ** Name: Load-1 Type: Concentrated force

Appendices

135

*Cload, amplitude=Amp-1 _PickedSet16, 1, 0. _PickedSet16, 2, 500. ** ** OUTPUT REQUESTS ** *Restart, write, frequency=10 ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step

Part 3: *Heading ** Job name: FillPart3 Model name: FillPart3 ** Generated by: Abaqus/CAE 6.11-1 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** ** PARTS ** ** ** ASSEMBLY ** *Assembly, name=Assembly ** *Instance, library=FillPart2, instance=fabric-1 ** ** PREDEFINED FIELD ** ** Name: Predefined Field-1 Type: Initial State *Import, state=yes, update=no *Nset, nset=_PickedSet9, internal, generate 1, 10, 1 *Elset, elset=_PickedSet9, internal, generate 1, 4, 1 *End Instance ** *Node 1, 0., 0.109999999, 0. *Nset, nset=_PickedSet15, internal 1, *Nset, nset=_PickedSet16, internal 1, *Nset, nset=_PickedSet17, internal, instance=fabric-1, generate 1, 9, 2 *Elset, elset=_PickedSet17, internal, instance=fabric-1, generate 1, 4, 1 *Nset, nset=_PickedSet18, internal, instance=fabric-1 1, 2 *Elset, elset=_PickedSet18, internal, instance=fabric-1 1, *Nset, nset=_PickedSet19, internal, instance=fabric-1, generate 1, 10, 1 *Elset, elset=_PickedSet19, internal, instance=fabric-1, generate 1, 4, 1 *Nset, nset=_PickedSet20, internal 1,

Appendices

136

*Nset, nset=_PickedSet24, internal, instance=fabric-1 9, 10 *Elset, elset=_PickedSet24, internal, instance=fabric-1 4, *Surface, type=NODE, name=_PickedSet24_CNS_, internal _PickedSet24, 1. ** Constraint: Constraint-1 *Coupling, constraint name=Constraint-1, ref node=_PickedSet15, surface=_PickedSet24_CNS_ *Kinematic 2, 2 *End Assembly *Amplitude, name=Amp-1 0., 0.6, 1., 1. ** ** MATERIALS ** *Material, name=part1 *Elastic, type=ENGINEERING CONSTANTS 3.21e+08, 1.83e+08, 2e+08, 0.04, 0.3, 0.3, 3e+08, 2e+08 2e+08, ** ** PREDEFINED FIELDS ** ** Name: Predefined Field-2 Type: Stress *Initial Conditions, type=STRESS, file=FillPart2, step=2, inc=1 ** ---------------------------------------------------------------- ** ** STEP: Step-1 ** *Step, name=Step-1, nlgeom=YES *Static 1., 1., 1e-05, 1. ** ** BOUNDARY CONDITIONS ** ** Name: pt Type: Displacement/Rotation *Boundary _PickedSet20, 1, 1 … _PickedSet20, 6, 6 ** Name: xsym Type: Symmetry/Antisymmetry/Encastre *Boundary _PickedSet17, XSYMM ** Name: ysym Type: Symmetry/Antisymmetry/Encastre *Boundary _PickedSet18, YSYMM ** Name: z Type: Displacement/Rotation *Boundary _PickedSet19, 3, 3 ** ** LOADS ** ** Name: Load-1 Type: Concentrated force *Cload, amplitude=Amp-1 _PickedSet16, 1, 0. _PickedSet16, 2, 500. ** ** OUTPUT REQUESTS ** *Restart, write, frequency=10 ** ** FIELD OUTPUT: F-Output-1

Appendices

137

** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step

Part 4: *Heading ** Job name: FillPart4 Model name: FillPart4 ** Generated by: Abaqus/CAE 6.11-1 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** ** PARTS ** ** ** ASSEMBLY ** *Assembly, name=Assembly ** *Instance, library=FillPart3, instance=fabric-1 ** ** PREDEFINED FIELD ** ** Name: Predefined Field-1 Type: Initial State *Import, state=yes, update=no *Nset, nset=_PickedSet9, internal, generate 1, 10, 1 *Elset, elset=_PickedSet9, internal, generate 1, 4, 1 *End Instance ** *Node 1, 0., 0.109999999, 0. *Nset, nset=_PickedSet15, internal 1, *Nset, nset=_PickedSet16, internal 1, *Nset, nset=_PickedSet17, internal, instance=fabric-1, generate 1, 9, 2 *Elset, elset=_PickedSet17, internal, instance=fabric-1, generate 1, 4, 1 *Nset, nset=_PickedSet18, internal, instance=fabric-1 1, 2 *Elset, elset=_PickedSet18, internal, instance=fabric-1 1, *Nset, nset=_PickedSet19, internal, instance=fabric-1, generate 1, 10, 1 *Elset, elset=_PickedSet19, internal, instance=fabric-1, generate 1, 4, 1 *Nset, nset=_PickedSet20, internal 1, *Nset, nset=_PickedSet24, internal, instance=fabric-1 9, 10 *Elset, elset=_PickedSet24, internal, instance=fabric-1 4, *Surface, type=NODE, name=_PickedSet24_CNS_, internal _PickedSet24, 1. ** Constraint: Constraint-1 *Coupling, constraint name=Constraint-1, ref node=_PickedSet15, surface=_PickedSet24_CNS_ *Kinematic

Appendices

138

2, 2 *End Assembly *Amplitude, name=Amp-1 0., 1., 1., 0. ** ** MATERIALS ** *Material, name=part1 *Elastic, type=ENGINEERING CONSTANTS 1.13e+09, 8.25e+08, 2e+08, 0.04, 0.3, 0.3, 3e+08, 2e+08 2e+08, ** ** PREDEFINED FIELDS ** ** Name: Predefined Field-2 Type: Stress *Initial Conditions, type=STRESS, file=FillPart3, step=3, inc=1 ** ---------------------------------------------------------------- ** ** STEP: Step-1 ** *Step, name=Step-1, nlgeom=YES *Static 1., 1., 1e-05, 1. ** ** BOUNDARY CONDITIONS ** ** Name: pt Type: Displacement/Rotation *Boundary _PickedSet20, 1, 1 _PickedSet20, 3, 3 _PickedSet20, 4, 4 _PickedSet20, 5, 5 _PickedSet20, 6, 6 ** Name: xsym Type: Symmetry/Antisymmetry/Encastre *Boundary _PickedSet17, XSYMM ** Name: ysym Type: Symmetry/Antisymmetry/Encastre *Boundary _PickedSet18, YSYMM ** Name: z Type: Displacement/Rotation *Boundary _PickedSet19, 3, 3 ** ** LOADS ** ** Name: Load-1 Type: Concentrated force *Cload, amplitude=Amp-1 _PickedSet16, 1, 0. _PickedSet16, 2, 500. ** ** OUTPUT REQUESTS ** *Restart, write, frequency=10 ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step

Appendices

139

Appendices

140

Appendix A3: Typical input file for an isotropic hyperelastic material model

calibrated from test data

*Heading ** Job name: YEOH Model name: YEOH ** Generated by: Abaqus/CAE 6.11-1 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** ** PARTS ** *Part, name=Part-1 *Node 1, 0., 0., 0. … 10, 0.100000001, 0.0250000004, 0. *Element, type=M3D4R 1, 1, 2, 7, 6 … 4, 4, 5, 10, 9 *Nset, nset=_PickedSet10, internal, generate 1, 10, 1 *Elset, elset=_PickedSet10, internal, generate 1, 4, 1 ** Section: Section-1 *Membrane Section, elset=_PickedSet10, material=Material-1 0.00083, *End Part ** ** ** ASSEMBLY ** *Assembly, name=Assembly ** *Instance, name=Part-1-1, part=Part-1 *End Instance ** *Node 1, 0.109999999, 0., 0. *Nset, nset=_PickedSet15, internal, instance=Part-1-1 5, 10 *Elset, elset=_PickedSet15, internal, instance=Part-1-1 4, *Nset, nset=_PickedSet16, internal 1, *Nset, nset=_PickedSet17, internal 1, *Nset, nset=_PickedSet18, internal, instance=Part-1-1 1, 6 *Elset, elset=_PickedSet18, internal, instance=Part-1-1 1, *Nset, nset=_PickedSet19, internal, instance=Part-1-1, generate 1, 5, 1 *Elset, elset=_PickedSet19, internal, instance=Part-1-1, generate 1, 4, 1 *Nset, nset=_PickedSet20, internal, instance=Part-1-1, generate 1, 10, 1 *Nset, nset=_PickedSet20, internal 1, *Elset, elset=_PickedSet20, internal, instance=Part-1-1, generate 1, 4, 1 *Nset, nset=_PickedSet21, internal

Appendices

141

1, *Surface, type=NODE, name=_PickedSet15_CNS_, internal _PickedSet15, 1. ** Constraint: Constraint-1 *Coupling, constraint name=Constraint-1, ref node=_PickedSet16, surface=_PickedSet15_CNS_ *Kinematic 1, 1 *End Assembly *Amplitude, name=Amp-1 0., 0., 1., 0.2, 2., 0., 3., 0.4 4., 0., 5., 0.6, 6., 0., 7., 0.8 8., 0., 9., 1., 10., 0., 11., 1.2 12., 0., 13., 1.4, 14., 0. ** ** MATERIALS ** *Material, name=Material-1 *Density 1000., *Hyperelastic, yeoh, test data input *Uniaxial Test Data 50000., 0. 99395.2, 7.59713e-05 151829., 0.000157879 194554., 0.000167866 246019., 0.000208089 295542., 0.00024295 343123., 0.000267683 385848., 0.00032444 431487., 0.000341397 479065., 0.000370796 527617., 0.000387013 580053., 0.000419081 633460., 0.000443113 673272., 0.000457349 713084., 0.000473627 765518., 0.000489374 809214., 0.000502221 861651., 0.000534373 910202., 0.000555217 954870., 0.000575554 1.00925e+06, 0.000607638 1.06654e+06, 0.000632682 1.11315e+06, 0.0006382 1.15879e+06, 0.000670105 1.20734e+06, 0.000691728 1.24618e+06, 0.000737886 1.2957e+06, 0.000750791 1.34328e+06, 0.000757309 1.4064e+06, 0.000784 1.44135e+06, 0.00081215 1.49088e+06, 0.000834153 1.53457e+06, 0.000841618 1.58409e+06, 0.000851007 1.63556e+06, 0.000877629 1.68411e+06, 0.000881088 1.73169e+06, 0.000909986 1.82394e+06, 0.000941674 1.87346e+06, 0.000965457 1.9191e+06, 0.000977401 1.97153e+06, 0.00100964 2.01912e+06, 0.00104116

Appendices

142

2.06281e+06, 0.00105641 2.11719e+06, 0.00108059 2.15506e+06, 0.00110093 2.21235e+06, 0.00112053 2.26576e+06, 0.001154 2.30848e+06, 0.0011641 2.34927e+06, 0.00120028 2.40947e+06, 0.00120101 2.56095e+06, 0.00125426 2.84352e+06, 0.00136489 3.21154e+06, 0.00145411 3.43196e+06, 0.0015316 3.54848e+06, 0.00165015 3.79804e+06, 0.00171991 4.08255e+06, 0.00178111 4.32919e+06, 0.00187482 4.58554e+06, 0.00200109 4.8351e+06, 0.00209415 5.07882e+06, 0.00216932 5.3109e+06, 0.00227956 5.57308e+06, 0.00238235 5.79836e+06, 0.00248366 6.0314e+06, 0.00258404 6.26542e+06, 0.00264729 6.49847e+06, 0.00275966 6.83056e+06, 0.00291522 7.0238e+06, 0.0030011 7.25199e+06, 0.00306403 7.48989e+06, 0.00315365 7.7074e+06, 0.00324425 7.9589e+06, 0.00337077 8.15699e+06, 0.00351752 8.4716e+06, 0.00357973 8.693e+06, 0.00366497 8.90856e+06, 0.00383598 9.1319e+06, 0.00392641 9.44943e+06, 0.00404908 9.66403e+06, 0.0041061 9.86115e+06, 0.00428221 1.01661e+07, 0.00437521 1.03719e+07, 0.00446028 1.06535e+07, 0.00461956 1.0839e+07, 0.0047508 1.11254e+07, 0.00484054 1.13808e+07, 0.00498513 1.15605e+07, 0.00505985 1.18896e+07, 0.00524976 1.20421e+07, 0.00537509 1.23635e+07, 0.00554207 1.25062e+07, 0.00564796 1.28199e+07, 0.00580176 1.30296e+07, 0.00593159 1.32607e+07, 0.00610731 1.34831e+07, 0.00627643 1.3754e+07, 0.00642036 1.40094e+07, 0.00659317 1.42425e+07, 0.00679148 1.446e+07, 0.00702947 1.47367e+07, 0.00719522 1.49503e+07, 0.00745502 1.52387e+07, 0.00768155 1.54349e+07, 0.00792367

Appendices

143

1.57087e+07, 0.00817918 1.59408e+07, 0.00844796 1.61758e+07, 0.00873173 1.64341e+07, 0.00905825 1.66632e+07, 0.00934739 1.69118e+07, 0.00971589 1.71381e+07, 0.0100598 1.73799e+07, 0.0104316 1.76265e+07, 0.0108939 1.78518e+07, 0.011277 1.81023e+07, 0.0117087 1.83451e+07, 0.0121883 1.85917e+07, 0.0126173 1.88335e+07, 0.0130712 1.90811e+07, 0.0135404 1.93152e+07, 0.0140067 1.95618e+07, 0.0145119 1.98055e+07, 0.0149922 2.00502e+07, 0.0155213 2.02891e+07, 0.0160087 2.05319e+07, 0.0165373 2.07649e+07, 0.0170223 2.10038e+07, 0.0175422 2.12475e+07, 0.0180688 2.14767e+07, 0.018618 2.17224e+07, 0.0191339 2.19739e+07, 0.0196839 2.22166e+07, 0.0202431 2.24604e+07, 0.0207949 2.26925e+07, 0.0213479 2.29401e+07, 0.0219037 2.31789e+07, 0.0224527 2.33965e+07, 0.0229974 2.36616e+07, 0.0235602 2.38927e+07, 0.0241356 2.4e+07, 0.024136 2.45e+07, 0.0241365 2.5e+07, 0.024137 2.55e+07, 0.0241375 2.6e+07, 0.024138 2.7e+07, 0.024139 2.8e+07, 0.02414 2.9e+07, 0.02415 4e+07, 0.038 *Plastic 1.4e+07, 0. 4e+07, 0.03 ** ---------------------------------------------------------------- ** ** STEP: Step-1 ** *Step, name=Step-1, nlgeom=YES, inc=10000 *Static 0.1, 14., 4e-100, 0.1 ** ** BOUNDARY CONDITIONS ** ** Name: pt Type: Displacement/Rotation *Boundary _PickedSet21, 2, 2 … _PickedSet21, 6, 6

Appendices

144

** Name: xsym Type: Symmetry/Antisymmetry/Encastre *Boundary _PickedSet18, XSYMM ** Name: ysym Type: Symmetry/Antisymmetry/Encastre *Boundary _PickedSet19, YSYMM ** Name: z Type: Displacement/Rotation *Boundary _PickedSet20, 3, 3 ** ** LOADS ** ** Name: Load-1 Type: Concentrated force *Cload, amplitude=Amp-1 _PickedSet17, 1, 500. ** ** OUTPUT REQUESTS ** *Restart, write, frequency=0 ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT, time interval=0.05 ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT, time interval=0.05 *End Step

Appendix A4: Typical input file for an orthotropic hyperelastic Fung model

*Heading ** Job name: fungvierkant Model name: Model-1 ** Generated by: Abaqus/CAE 6.11-1 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** ** PARTS ** *Part, name=Part-1 *Node 1, 500., 500., 0. … 2601, -500., -500., 0. *Element, type=M3D4R 1, 1, 2, 53, 52 … 2500, 2549, 2550, 2601, 2600 *Nset, nset=_PickedSet2, internal, generate 1, 2601, 1 *Elset, elset=_PickedSet2, internal, generate 1, 2500, 1 *Nset, nset=_PickedSet8, internal, generate 1, 2601, 1 *Elset, elset=_PickedSet8, internal, generate 1, 2500, 1 ** Section: Section-1 *orientation, name=ori-1,local directions=2 1.0,0.0,0.0,0.0,1.0,0.0 3,0.0 1.0, 0.0, 0.0 0.0, 1.0, 0.0

Appendices

145

*Membrane Section, elset=_PickedSet2, material=Material-1, orientation=Ori-1 0.6, *End Part ** *Part, name=Part-2 *Node 1, -500., 500., 0. … 11, 500., 500., 0. *Element, type=RB3D2 1, 1, 2 … 10, 10, 11 *Node 12, 0., 520., 0. *Nset, nset=Part-2-RefPt_, internal 12, *Elset, elset=Part-2, generate 1, 10, 1 *End Part ** ** ** ASSEMBLY ** *Assembly, name=Assembly ** *Instance, name=Part-1-1, part=Part-1 *End Instance ** *Instance, name=Part-2-1, part=Part-2 *End Instance ** *Nset, nset=f1, instance=Part-1-1, generate 1, 51, 1 *Elset, elset=f1, instance=Part-1-1, generate 1, 50, 1 *Nset, nset=w1, instance=Part-2-1, generate 1, 11, 1 *Elset, elset=w1, instance=Part-2-1, generate 1, 10, 1 *Nset, nset=_PickedSet8, internal, instance=Part-2-1 12, *Nset, nset=_PickedSet9, internal, instance=Part-1-1, generate 2551, 2601, 1 *Elset, elset=_PickedSet9, internal, instance=Part-1-1, generate 2451, 2500, 1 *Nset, nset=_PickedSet10, internal, instance=Part-2-1 12, *Surface, type=NODE, name=w1_CNS_, internal w1, 1. *Surface, type=NODE, name=f1_CNS_, internal f1, 1. *Rigid Body, ref node=Part-2-1.Part-2-RefPt_, elset=Part-2-1.Part-2 ** Constraint: Constraint-1 *Tie, name=Constraint-1, adjust=yes f1_CNS_, w1_CNS_ *End Assembly *Amplitude, name=Amp-1 0.1, 140., 0.2, 280., 0.3, 420., 0.4, 560. 0.5, 700., 0.6, 840., 0.7, 980., 0.8, 1120. 0.9, 1260., 1., 1400. **

Appendices

146

** MATERIALS ** *Material, name=Material-1 *Density 0.0001, *Anisotropic Hyperelastic, fung-orthotropic 1371.17, 288.81, 1.1418e+06, 28.44, 26.1108, 81.071, 23.33, 2. 2., 1.2, 0.0002 ** ---------------------------------------------------------------- ** ** STEP: Step-1 ** *Step, name=Step-1, nlgeom=YES *Static 1., 1., 1e-05, 0.1 ** ** BOUNDARY CONDITIONS ** ** Name: BC-1 Type: Displacement/Rotation *Boundary _PickedSet9, 1, 1 … _PickedSet9, 6, 6 ** Name: BC-2 Type: Displacement/Rotation *Boundary _PickedSet10, 1, 1 … _PickedSet10, 6, 6 ** ** LOADS ** ** Name: Load-1 Type: Concentrated force *Cload, amplitude=Amp-1 _PickedSet8, 2, 1400. ** ** OUTPUT REQUESTS ** *Restart, write, frequency=0 ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step

Appendix A5: Typical input file for a test data based fabric model

*Heading ** Job name: fabric-uniaxial-fill Model name: fabric-uniaxial-fill ** Generated by: Abaqus/CAE 6.11-1 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** ** PARTS ** *Part, name=Part-1 *End Part ** **

Appendices

147

** ASSEMBLY ** *Assembly, name=Assembly ** *Instance, name=Part-1-1, part=Part-1 *Node 1, 0., 0., 0. … 10, 0.0250000004, 0.100000001, 0. *Element, type=M3D4 1, 1, 2, 4, 3 … 4, 7, 8, 10, 9 *Nset, nset=_PickedSet2, internal, generate 1, 10, 1 *Elset, elset=_PickedSet2, internal, generate 1, 4, 1 ** Section: Section-1 *Membrane Section, elset=_PickedSet2, material=Fabric 0.00083, *End Instance ** *Node 1, 0., 0.109999999, 0. *Nset, nset=_PickedSet5, internal, instance=Part-1-1 9, 10 *Elset, elset=_PickedSet5, internal, instance=Part-1-1 4, *Nset, nset=_PickedSet6, internal 1, *Nset, nset=_PickedSet15, internal 1, *Nset, nset=_PickedSet16, internal, instance=Part-1-1 1, *Nset, nset=_PickedSet17, internal 1, *Nset, nset=_PickedSet18, internal, instance=Part-1-1, generate 1, 9, 2 *Elset, elset=_PickedSet18, internal, instance=Part-1-1, generate 1, 4, 1 *Nset, nset=_PickedSet19, internal, instance=Part-1-1 1, 2 *Elset, elset=_PickedSet19, internal, instance=Part-1-1 1, *Nset, nset=_PickedSet20, internal, instance=Part-1-1, generate 1, 10, 1 *Elset, elset=_PickedSet20, internal, instance=Part-1-1, generate 1, 4, 1 *Surface, type=NODE, name=_PickedSet5_CNS_, internal _PickedSet5, 1. ** Constraint: Constraint-1 *Coupling, constraint name=Constraint-1, ref node=_PickedSet6, surface=_PickedSet5_CNS_ *Kinematic 2, 2 *End Assembly *Amplitude, name=Amp-1 0., 0., 0.5, 1, 1., 0 ** ** MATERIALS ** *Material, name=Fabric *Density

Appendices

148

1000., *Damping, Beta=1.e-6 *Fabric, stress free initial slack=yes *Uniaxial, Component=1 *Loading data, type=permanent deformation, direction=tension, yield onset=0.01, extrapolation=linear 0, 0.000000 1009246, 0.000916 2019116, 0.001570 3211537, 0.002193 5078824, 0.003271 7023795, 0.004525 9131901, 0.005920 11125431, 0.007299 13029629, 0.008944 15940795, 0.012738 17138084, 0.015169 19081137, 0.020417 21003802, 0.026451 23892667, 0.038292 *Unloading data, definition=shifted curve 0, 0 23694074.94, 0.02 *Loading data, type=permanent deformation, direction=compression, yield onset=0.4, extrapolation=linear 0., 0. 15000000., 1000 *Unloading data, definition=shifted curve 0., 0. 15000000., 1000 *Uniaxial, Component=2 *Loading data, type=permanent deformation, direction=tension, yield onset=0.01, extrapolation=linear 0, 0.000000 1035995, 0.008549 2002672, 0.015624 3390696, 0.022825 5311935, 0.032131 7265800, 0.039092 9169333, 0.044338 11104561, 0.049050 13022082, 0.054116 15452313, 0.063179 17374506, 0.073349 19295771, 0.084003 21756788, 0.095620 23636116, 0.106071 34000000, 0.14 *Unloading data, definition=shifted curve 0, 0 2500000, 0.01077 5000000, 0.0188475 7500000, 0.024771 10000000, 0.0296175 12500000, 0.034464 15000000, 0.0379104 17500000, 0.0411414 20000000, 0.045234 22500000, 0.048465 25000000, 0.051696 27500000, 0.054927 *Loading data, type=permanent deformation, direction=compression, yield onset=0.4, extrapolation=linear 0., 0. 15000000., 200 *Unloading data, definition=shifted curve

Appendices

149

0., 0. 15000000., 200 *Uniaxial, Component=shear *Loading data, type=permanent deformation, direction=tension, yield onset=0.01, extrapolation=linear 0, 0 1042816.867, 0.023883246 2002971.084, 0.055159132 3034901.205, 0.085882118 4013698.795, 0.116957776 5139320.482, 0.14766422 6155409.639, 0.170358113 7164975.904, 0.189879236 8168019.277, 0.20664293 9184115.663, 0.221259538 10143816.87, 0.233604556 11112843.37, 0.244897436 12073949.4, 0.25552153 13042983.13, 0.264968703 *Unloading data, definition=shifted curve 0, 0 15000000, 0.05 ** ---------------------------------------------------------------- ** ** STEP: Step-1 ** *Step, name=Step-1 *Dynamic, Explicit, element by element , 1. *Bulk Viscosity 0.4, 1.2 ** ** BOUNDARY CONDITIONS ** ** Name: pt Type: Displacement/Rotation *Boundary _PickedSet16, 1, 1 … _PickedSet16, 6, 6 ** Name: rfpt Type: Displacement/Rotation *Boundary _PickedSet17, 1, 1 … _PickedSet17, 6, 6 ** Name: xsymm Type: Symmetry/Antisymmetry/Encastre *Boundary _PickedSet18, XSYMM ** Name: ysymm Type: Symmetry/Antisymmetry/Encastre *Boundary _PickedSet19, YSYMM ** Name: z Type: Displacement/Rotation *Boundary _PickedSet20, 3, 3 ** ** LOADS ** ** Name: load-top Type: Concentrated force *Cload, amplitude=Amp-1 _PickedSet15, 2, 500. ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO

Appendices

150

** ** FIELD OUTPUT: F-Output-1 ** *Output, field, time interval=0.05 *Node Output A, RF, U, V *Element Output, directions=YES EVF, LE, PE, PEEQ, PEEQVAVG, PEVAVG, S, SVAVG ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT, time interval=0.05 *End Step

Appendix A6: Typical input file for a combined orthotropic elastic-plastic material model

*Heading ** Job name: Uniaxial-fill-goodfill Model name: Uniaxial-fill-goodfill ** Generated by: Abaqus/CAE 6.11-1 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** ** PARTS ** *Part, name=Part-1 *End Part ** ** ** ASSEMBLY ** *Assembly, name=Assembly ** *Instance, name=Part-1-1, part=Part-1 *Node 1, 0.0250000004, 0.0250000004, 0. … 196, 0., 0., 0. *Element, type=M3D4R 1, 1, 2, 16, 15 … 169, 181, 182, 196, 195 *Nset, nset=_PickedSet2, internal, generate 1, 196, 1 *Elset, elset=_PickedSet2, internal, generate 1, 169, 1 ** Section: Section-1 *Membrane Section, elset=_PickedSet2, material=plasticbehavior 0.00083, *End Instance ** *Node 1, 0., 0.0299999993, 0. *Nset, nset=_PickedSet48, internal 1, *Nset, nset=_PickedSet62, internal, instance=Part-1-1, generate 1, 14, 1 *Elset, elset=_PickedSet62, internal, instance=Part-1-1, generate 1, 13, 1 *Nset, nset=_PickedSet63, internal 1,

Appendices

151

*Nset, nset=_PickedSet64, internal 1, *Nset, nset=_PickedSet65, internal, instance=Part-1-1 196, *Nset, nset=_PickedSet66, internal, instance=Part-1-1, generate 14, 196, 14 *Elset, elset=_PickedSet66, internal, instance=Part-1-1, generate 13, 169, 13 *Nset, nset=_PickedSet67, internal, instance=Part-1-1, generate 183, 196, 1 *Elset, elset=_PickedSet67, internal, instance=Part-1-1, generate 157, 169, 1 *Surface, type=NODE, name=_PickedSet62_CNS_, internal _PickedSet62, 1. ** Constraint: Constraint-3 *Coupling, constraint name=Constraint-3, ref node=_PickedSet63, surface=_PickedSet62_CNS_ *Kinematic 2, 2 *End Assembly *Amplitude, name=load-top 0., 0., 1., 1., 2., 0. ** ** MATERIALS ** *Material, name=plasticbehavior *Density 963., *Elastic, type=ENGINEERING CONSTANTS 1.13e+09, 8.25e+08, 2e+07, 0.04, 0.25, 0.25, 3e+08, 2e+06 2e+06, *Plastic 200000., 0. 1e+06, 0.008 2.4e+06, 0.015 4.8e+06, 0.024 7.3e+06, 0.03 9.6e+06, 0.034 1.2e+07, 0.037 1.45e+07, 0.041 1.7e+07, 0.049 1.9e+07, 0.058 2.17e+07, 0.068 2.36e+07, 0.075 5e+07, 0.3 *Potential 2.65,1.,1.,1.,1.,1. ** ** BOUNDARY CONDITIONS ** ** Name: top Type: Displacement/Rotation *Boundary _PickedSet48, 2, 2 … _PickedSet48, 6, 6 ** ---------------------------------------------------------------- ** ** STEP: Step-1 ** *Step, name=Step-1, inc=1000 *Static 0.05, 2., 2e-100, 0.05 **

Appendices

152

** BOUNDARY CONDITIONS ** ** Name: pt Type: Displacement/Rotation *Boundary, op=NEW _PickedSet65, 1, 1 … _PickedSet65, 6, 6 ** Name: top Type: Displacement/Rotation *Boundary, op=NEW _PickedSet48, 1, 1 … _PickedSet48, 6, 6 ** Name: xsymm Type: Symmetry/Antisymmetry/Encastre *Boundary, op=NEW _PickedSet66, XSYMM ** Name: ysymm Type: Symmetry/Antisymmetry/Encastre *Boundary, op=NEW _PickedSet67, YSYMM ** ** LOADS ** ** Name: load-top Type: Concentrated force *Cload, amplitude=load-top _PickedSet64, 1, 0. _PickedSet64, 2, 500. ** ** OUTPUT REQUESTS ** *Restart, write, frequency=10 ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT, time interval=0.05 ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT, time interval=0.05 *End Step

Appendices

153

Appendix B

Appendix B1: Calculation of equivalent stress if σ11 is the only nonzero stress component > restart:> F:=1/2*(1/R22^2+1/R33^2-1/R11^2); G:=1/2*(1/R33^2+1/R11^2-1/R22^2); H:=1/2*(1/R11^2+1/R22^2-1/R33^2); L:=3/2/R23^2; M:=3/2/R13^2; N:=3/2/R12^2; f:=sqrt(F*(sigma22-sigma33)^2+G*(sigma33-sigma11)^2+H*(sigma11-sigma22)^2+2*L*sigma23^2+2*M*sigma13^2+2*N*sigma12^2);

> sigma22:=00: sigma33:=0: sigma12:=0: sigma23:=0: sigma13:=0: simplify(f);

:= F + − 1

2 R222

1

2 R332

1

2 R112

:= G + − 1

2 R332

1

2 R112

1

2 R222

:= H + − 1

2 R112

1

2 R222

1

2 R332

:= L3

2 R232

:= M3

2 R132

:= N3

2 R122

f

+ − 1

2 R222

1

2 R332

1

2 R112 ( ) − σ22 σ33 2

:=

+ − 1

2 R332

1

2 R112

1

2 R222 ( ) − σ33 σ11 2 +

+ − 1

2 R112

1

2 R222

1

2 R332 ( ) − σ11 σ22 2 3 σ232

R232

3 σ132

R132

3 σ122

R122 + + + +

( )/1 2

σ112

R112

Appendices

154

Appendix B2: Calculation of the flow rule and calculation of the flow rule if σ11 is the only nonzero stress component > restart: > f:=sqrt(F*(sigma22-sigma33)^2+G*(sigma33-sigma11)^2+H*(sigma11-sigma22)^2+2*L*sigma23^2+2*M*sigma13^2+2*N*sigma12^2);

> de11:=simplify(value(d(lambda)*Diff(f,sigma11))); de22:=simplify(value(d(lambda)*Diff(f,sigma22))); de33:=simplify(value(d(lambda)*Diff(f,sigma33))); de12:=simplify(value(d(lambda)*Diff(f,sigma12))); de13:=simplify(value(d(lambda)*Diff(f,sigma13))); de23:=simplify(value(d(lambda)*Diff(f,sigma23))); simplify(de11+de22+de33);

> F:=1/2*(1/R22^2+1/R33^2-1/R11^2): G:=1/2*(1/R33^2+1/R11^2-1/R22^2): H:=1/2*(1/R11^2+1/R22^2-1/R33^2): L:=3/2/R23^2: M:=3/2/R13^2: N:=3/2/R12^2: sigma22:=0: sigma33:=0: sigma12:=0: sigma23:=0:sigma13:=0: > simplify(de11); simplify(de22); simplify(de33); simplify(de12);

f F σ222 2 F σ22 σ33 F σ332 G σ332 2 G σ33 σ11 G σ112 H σ112 − + + − + + ( :=

2 H σ11 σ22 H σ222 2 L σ232 2 M σ132 2 N σ122 − + + + + )( )/1 2

de11 ( )d λ ( )− + + − G σ33 G σ11 H σ11 H σ22 F σ222 2 F σ22 σ33 F σ332 − + (/ :=

G σ332 2 G σ33 σ11 G σ112 H σ112 2 H σ11 σ22 H σ222 2 L σ232 + − + + − + +

2 M σ132 2 N σ122 + + )( )/1 2

de22 ( )d λ ( ) − − + F σ22 F σ33 H σ11 H σ22 F σ222 2 F σ22 σ33 F σ332 − + (/ :=


2 M σ132 2 N σ122 + + )( )/1 2

de33 ( )d λ ( ) − − + F σ22 F σ33 G σ33 G σ11 F σ222 2 F σ22 σ33 F σ332 − + (/− :=


2 M σ132 2 N σ122 + + )( )/1 2

de12 2 ( )d λ N σ12 F σ222 2 F σ22 σ33 F σ332 G σ332 2 G σ33 σ11 − + + − (/ :=

G σ112 H σ112 2 H σ11 σ22 H σ222 2 L σ232 2 M σ132 2 N σ122 + + − + + + + )^( )/1 2

de13 2 ( )d λ M σ13 F σ222 2 F σ22 σ33 F σ332 G σ332 2 G σ33 σ11 − + + − (/ :=


de23 2 ( )d λ L σ23 F σ222 2 F σ22 σ33 F σ332 G σ332 2 G σ33 σ11 − + + − (/ :=


0

Appendices

155

simplify(de13); simplify(de23);

( )d λ σ11

R112 σ112

R112

12

( )d λ ( )− − + R332 R222 R332 R112 R112 R222 σ11

R332 R112 R222 σ112

R112

− 12

( )d λ ( ) + − R112 R222 R332 R222 R332 R112 σ11

R332 R112 R222 σ112

R112

0

0

0

Appendices

156

Appendix C

Optimization of R11 for anisotropic plasticity with crimp interchange fitted to the uniaxial tensile test in fill direction > restart:with(plots):with(LinearAlgebra):with(Optimization):with(CurveFitting): > Warp:=Matrix(<<0.05|0>, <0.91|0.0008>, <1.82|0.0014>, <3.8|0.0026>, <8.69|0.0055>, <13.48|0.0095>, <15.94|0.0127>, <18.35|0.0184>, <20.29|0.0241>, <22.22|0.0305>, <23.89|0.039>>): Fill:=Matrix(<<0.04|0>, <0.89|0.0075>, <1.86|0.0146>, <3.87|0.0253>, <6.28|0.0359>, <11.1|0.0491>, <13.52|0.0556>, <15.94|0.0655>, <18.33|0.0788>, <20.75|0.0913>, <23.64|0.1043>>): EFill:=860: EWarp:=1000: tabel:=Matrix(10,2): for i from 1 to 10 do tabel[i,1]:=Fill[i,1]; tabel[i,2]:=Fill[i,2]-tabel[i,1]/EFill; end do: > N:=2000: StandAfw:=Matrix(N,3): for i from 1 to N do R11:=0+i/200; WarpHill[i]:=Matrix(11,2): WarpInterpolated[i]:=Matrix(11,2): for j from 1 to 11 do WarpHill[i][j,1]:=Warp[j,1]: WarpHill[i][j,2]:=ArrayInterpolation(tabel[1..-1,1],tabel[1..-1,2],WarpHill[i][j,1]/R11)/R11+WarpHill[i][j,1]/EWarp; end do: for k from 1 to 11 do WarpInterpolated[i][k,1]:=Fill[k,1]; WarpInterpolated[i][k,2]:=ArrayInterpolation(WarpHill[i][1..-1,1],WarpHill[i][1..-1,2],WarpInterpolated[i][k,1]); end do: StandAfw[i,1]:=R11; StandAfw[i,2]:=sqrt((WarpInterpolated[i][1,2]-Warp[1,2])^2+ (WarpInterpolated[i][2,2]-Warp[2,2])^2+ (WarpInterpolated[i][3,2]-Warp[3,2])^2+ (WarpInterpolated[i][4,2]-Warp[4,2])^2+ (WarpInterpolated[i][5,2]-Warp[5,2])^2+ (WarpInterpolated[i][6,2]-Warp[6,2])^2+ (WarpInterpolated[i][7,2]-Warp[7,2])^2+

Appendices

157

(WarpInterpolated[i][8,2]-Warp[8,2])^2+ (WarpInterpolated[i][9,2]-Warp[9,2])^2+ (WarpInterpolated[i][10,2]-Warp[10,2])^2+ (WarpInterpolated[i][11,2]-Warp[11,2])^2*50); StandAfw[i,3]:=max((WarpInterpolated[i][1,2]-Warp[1,2]), (WarpInterpolated[i][2,2]-Warp[2,2]), (WarpInterpolated[i][3,2]-Warp[3,2]), (WarpInterpolated[i][4,2]-Warp[4,2]), (WarpInterpolated[i][5,2]-Warp[5,2]), (WarpInterpolated[i][6,2]-Warp[6,2]), (WarpInterpolated[i][7,2]-Warp[7,2]), (WarpInterpolated[i][8,2]-Warp[8,2]), (WarpInterpolated[i][9,2]-Warp[9,2]), (WarpInterpolated[i][10,2]-Warp[10,2]), (WarpInterpolated[i][11,2]-Warp[11,2])); end do: > pointplot([StandAfw[1..-1,1],StandAfw[1..-1,2]], axes=boxed);

> N:=2000: StandAfw:=Matrix(N,3): for i from 1 to N do R11:=2.+i/2000; WarpHill[i]:=Matrix(11,2): WarpInterpolated[i]:=Matrix(11,2): for j from 1 to 11 do WarpHill[i][j,1]:=Warp[j,1]: WarpHill[i][j,2]:=ArrayInterpolation(tabel[1..-1,1],tabel[1..-1,2],WarpHill[i][j,1]/R11)/R11+WarpHill[i][j,1]/EWarp; end do: for k from 1 to 11 do WarpInterpolated[i][k,1]:=Fill[k,1]; WarpInterpolated[i][k,2]:=ArrayInterpolation(WarpHill[i][1..-1,1],WarpHill[i][1..-1,2],WarpInterpolated[i][k,1]); end do: StandAfw[i,1]:=R11; StandAfw[i,2]:=sqrt((WarpInterpolated[i][1,2]-Warp[1,2])^2+ (WarpInterpolated[i][2,2]-Warp[2,2])^2+ (WarpInterpolated[i][3,2]-Warp[3,2])^2+ (WarpInterpolated[i][4,2]-Warp[4,2])^2+ (WarpInterpolated[i][5,2]-Warp[5,2])^2+ (WarpInterpolated[i][6,2]-Warp[6,2])^2+ (WarpInterpolated[i][7,2]-Warp[7,2])^2+ (WarpInterpolated[i][8,2]-Warp[8,2])^2+ (WarpInterpolated[i][9,2]-Warp[9,2])^2+ (WarpInterpolated[i][10,2]-Warp[10,2])^2+

Appendices

158

(WarpInterpolated[i][11,2]-Warp[11,2])^2*50); end do: > pointplot([StandAfw[1..-1,1],StandAfw[1..-1,2]], axes=boxed,labels=["R11","Cost function"],labeldirections=[horizontal,vertical],title="Optimization of R11");

> R:=2.38; nr:=(R-2)*2000; WarpAndFill:=<WarpHill[1225],Warp>:

> plot1:=pointplot([WarpHill[760][1..-1,2],WarpHill[600][1..-1,1]], color=green, connect): plot2:=pointplot([Warp[1..-1,2],Warp[1..-1,1]], color=blue, connect): plot3:=pointplot([WarpHill[760][1..-1,2],WarpHill[600][1..-1,1]], color=green, legend = "Experimental warp curve"): plot4:=pointplot([Warp[1..-1,2],Warp[1..-1,1]], color=blue, legend = "Hill plasticity warp curve"): display(plot3,plot4,plot1,plot2,labels=["Engineering strain warp (-)","Stress warp (MPa)"],labeldirections=[horizontal,vertical],title="Best fitting warp curve");

:= R 2.38

:= nr 760.00

159

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162

List of figures

Figure 1.1: Indian teepee. ............................................................................................................................ 2

Figure 1.2: Emergency tent used by the Red Cross [1]. .......................................................................... 2

Figure 1.3: Festival tent [2]. ......................................................................................................................... 2

Figure 1.4: Loom to weave a fabric. .......................................................................................................... 3

Figure 1.5: Woven fabric with straight warp fibers and curved fill fibers [4]. ..................................... 3

Figure 1.6: Plain weave (a), Satin weave (b) and twill weave (c). ........................................................... 4

Figure 1.7: The coated fabric T2107 that is used in this master thesis. ................................................ 5

Figure 1.8: Photo of the surface of the coated fabric of the outer surface (a) and the inner surface

(b). ................................................................................................................................................................... 5

Figure 1.9: Microscopic image (x200) of the coated fabric, cut out along the warp direction. ........ 6

Figure 1.10: Microscopic image (x200) of the coated fabric, cut out along the fill direction. .......... 6

Figure 1.11: Photos of the embedded samples in Epofix with a fluorescent yellow color. .............. 7

Figure 1.12: : Microscopic image of the coated fabric, cut out along the warp direction after

embedding and polishing with magnification x55 (a) and x200 (b). ..................................................... 7

Figure 1.13: Microscopic image of the coated fabric, cut out along the fill direction after

embedding and polishing with magnification x55 (a) and x200 (b). ..................................................... 7

Figure 1.14: Microscopic image of the coated fabric, cut out along the warp direction after

embedding, polishing and further polishing by hand with magnification x55 (a) and x200 (b). ...... 8


embedding, polishing and further polishing by hand with magnification x55 (a) and x200 (b). ...... 8

Figure 1.16: Microscopic image of the coated fabric, cut out along the warp direction after

embedding and cutting by means of a cut-off machine with magnification x55 (a) and x200 (b). .. 9


embedding and cutting by means of a cut-off machine with magnification x55 (a) and x200 (b). .. 9

Figure 1.18: Anticlastic surface consisting of arching (A) and hanging (B) elements [5]. ............... 10

Figure 1.19: Saddle structure (a), high point structure [2] (b), ridge and valley structure [5] (c) and

hypar structure [5] (d). ............................................................................................................................... 11

Figure 1.20: Home-made soap film models, representing the minimal surface for the chosen

boundaries. .................................................................................................................................................. 11

Figure 1.21: Soap film model [8] (a) and a realized membrane structure with the same shape [9]

(b). ................................................................................................................................................................. 12

Figure 1.22: Minimal surfaces for two parallel rings: hyperboloid (a) and two parallel rings (b)

[10]. ............................................................................................................................................................... 12

Figure 1.23: Membrane structure simplified as a cable network. ........................................................ 13

Figure 1.24: Tokyo dome ‘Big egg’ football stadium [12]. ................................................................... 14

Figure 1.25: Modern teahouse 2007 (Frankfurt) [13]. ........................................................................... 15

Figure 1.26: Tensairity used as bearing element of a bridge [14]. ....................................................... 15

Figure 1.27: Fuji group pavilion, expo 1970 Osaka [15]. ...................................................................... 16

Figure 1.28: The Eden project [16]. ......................................................................................................... 17

Figure 1.29: ETFE Facade as second skin for the Training center for the Bavarian mountain

rescue (Bad Tolz, Germany) [17]. ............................................................................................................ 17

List of figures

163

Figure 1.30: Eco membrane at the inside of the Deckelhalle as thermal insulation (Munich,

Germany) [18]. ............................................................................................................................................ 17

Figure 1.31: Deformation of a cable to a more efficient structure due to the load [6]. ................... 18

Figure 1.32: Hovering archives: temporary art project (Hamburg, Germany) [19]. ......................... 18

Figure 1.33:Green void: a temporary construction in the Sydney custom house (Sydney, Australia)

[20]. ............................................................................................................................................................... 18

Figure 1.34: Venezuelan pavilion [21]. .................................................................................................... 19

Figure 1.35: Convertible cover of a swimming pool in Seville, Spain [22]. ....................................... 19

Figure 1.36: Retractable roof of the Toyota stadium (Nagoya, Japan) [23]. ...................................... 19

Figure 1.37: Integration of glazed areas in the supporting steelwork to introduce direct

illumination in the enclosure, APPP church (Maassluis, the Netherlands) [24]. .............................. 21

Figure 1.38: The water cube, Bejing [25]. ............................................................................................... 21

Figure 2.1: Load-strain curve for uniaxial test in fill (a) and warp (b) direction [6]. ......................... 23

Figure 2.2: Crimp interchange. ................................................................................................................. 23

Figure 2.3: Stress-strain behavior for a biaxial test with load ratio 1:1 on a PTFE-glass fiber

membrane. Warp and fill yarns show a different behavior [4]. ........................................................... 24

Figure 2.4: Stress - strain curves for a biaxial test with load ratio 1:5 (a) and 5:1 (b) on a PTFE-

glass fiber membrane [4]. .......................................................................................................................... 24

Figure 2.5: Influence of cycle repetition for a 1:1 load ratio for a PVC-polyester membrane with a

maximum stress of 12 kN/m [35]. .......................................................................................................... 25

Figure 2.6: Loading cycles, used to investigate the influence of load history [35]. ........................... 26

Figure 2.7: Influence of the load history in warp direction for a load ratio 1:1 for a PVC coated

polyester fabric [35]. ................................................................................................................................... 26

Figure 2.8: Influence of the loading rate measured with uniaxial tensile tests in warp and fill

direction [35]. .............................................................................................................................................. 27

Figure 2.9: Influence of initial pre-stress level for a 1:1 loading in warp direction (a) and in fill

direction (b) [35]. ........................................................................................................................................ 27

Figure 2.10: Stress-strain behavior under different temperatures for a PTFE coated glass fiber

fabric material in warp direction (a) and fill direction (b) [38]. ............................................................ 28

Figure 2.11: Load errors (a) and strain errors (b) [34]. ......................................................................... 30

Figure 2.12: Optimized regression line and switched regression line [34]. ........................................ 31

Figure 2.13: Experimental and calculated (according to MSAJ standard) stress strain curves for

different load ratios [34]. ........................................................................................................................... 32

Figure 2.14: Experimental and calculated (according to a nonlinear elastic material model, taking

the load ratio into account) stress strain curves for a PTFE coated glass fiber fabric for different

load ratio and a maximum load of 12kN/m [30]. ................................................................................. 34

Figure 2.15: Response surfaces: experimental data point in stress-stress-strain space (a) and fitted

surface (b). ................................................................................................................................................... 35

Figure 2.16: Experimental data points in plotted strain-strain space showing the bounds of the

feasible membrane response for a PVC coated polyester fabric (a) and for a PTFE coated glass

fiber fabric (b) [4]. ...................................................................................................................................... 39

Figure 3.1: Photo of a uniaxial experiment on a membrane [47]. ....................................................... 41

Figure 3.2: Schematic drawing of the uniaxial test in warp direction (a), in fill direction (b) and in

shear (c). ....................................................................................................................................................... 43

Figure 3.3: Uniaxial tests up to failure in warp direction. ..................................................................... 43

List of figures

164

Figure 3.4: Uniaxial tests up to failure in fill direction. ......................................................................... 44

Figure 3.5: Uniaxial tests up to failure in shear. ..................................................................................... 44

Figure 3.6: Load as a function of time for a uniaxial cycle repetition test in warp direction. ......... 45

Figure 3.7: Elongation as a function of time for a uniaxial cycle repetition test in warp direction.

....................................................................................................................................................................... 45

Figure 3.8: Strain (in warp and fill direction) as a function of time for a uniaxial cycle repetition

test in warp direction. ................................................................................................................................ 46

Figure 3.9: Stress (warp) as a function of strain (warp) for a uniaxial cycle repetition test in warp

direction. ...................................................................................................................................................... 46

Figure 3.10: Stress (warp) as a function of strain (warp) of the first loading for a uniaxial tensile

test in warp direction for a glass fiber ETFE foil (a) and for the PVC-polyester fabric (b). .......... 47

Figure 3.11: Poisson ratio νwf as a function of time for a uniaxial cycle repetition test in warp

direction. ...................................................................................................................................................... 48

Figure 3.12: Load as a function of time for a uniaxial cycle repetition test in fill direction. ........... 48

Figure 3.13: Elongation as a function of time for a uniaxial cycle repetition test in fill direction. 48

Figure 3.14: Strain in fill and warp direction as a function of time for a uniaxial cycle repetition

test in fill direction ..................................................................................................................................... 49

Figure 3.15: Stress (fill) as a function of strain (fill) for a uniaxial cycle repetition test in fill

direction. ...................................................................................................................................................... 49

Figure 3.16: Stress (fill) as a function of strain (fill) of the first loading for a uniaxial tensile test in

fill direction for a glass fiber ETFE foil (a) and for the PVC-polyester fabric (b). .......................... 49

Figure 3.17: Poisson ratio νfw as a function of time for a uniaxial cycle repetition test in fill

direction. ...................................................................................................................................................... 50

Figure 3.18: Load as a function of time for a uniaxial cycle repetition test in shear. ....................... 51

Figure 3.19: Elongation as a function of time for a uniaxial cycle repetition test in shear. ............ 51

Figure 3.20: Deformation of the sample during a uniaxial tensile test in shear. ............................... 51

Figure 3.21: Strain as a function of time for a uniaxial cycle repetition test in shear. ...................... 52

Figure 3.22: Stress as a function of strain for a uniaxial cycle repetition test in shear. .................... 52

Figure 3.23: Load (up to 500 N) as a function of time for a uniaxial cycle repetition test in warp

direction. ...................................................................................................................................................... 53

Figure 3.24: Strain (warp and fill) as a function of time for a uniaxial cycle repetition test in warp

direction up to a load of 500 N. ............................................................................................................... 53

Figure 3.25: Stress – strain curve for a uniaxial cycle repetition test in warp direction up to a load

of 500 N. ...................................................................................................................................................... 53

Figure 3.26: Load (up to 500 N) as a function of time for a uniaxial cycle repetition test in fill

direction. ...................................................................................................................................................... 54

Figure 3.27: Strain (warp and fill) as a function of time for a uniaxial cycle repetition test in fill

direction up to a load of 500 N. ............................................................................................................... 54

Figure 3.28: Stress – strain curve for a uniaxial cycle repetition test in fill direction up to a load of

500 N............................................................................................................................................................ 54

Figure 3.29: Load as a function of time for a uniaxial test in warp direction with increasing load.

....................................................................................................................................................................... 55

Figure 3.30: Strain (warp and fill) as a function of time for a uniaxial test in warp direction with

increasing load. ........................................................................................................................................... 55

Figure 3.31: Stress-strain curve for a uniaxial test in warp direction with increasing load. ............. 56

List of figures

165

Figure 3.32: Load as a function of time for a uniaxial test in fill direction with increasing load. ... 56

Figure 3.33: Strain (warp and fill) as a function of time for a uniaxial test in fill direction with

increasing load. ........................................................................................................................................... 57

Figure 3.34: Stress-strain curve for a uniaxial test in fill direction with increasing load. ................. 57

Figure 3.35: Geometry and loading of the uniaxial sample (a) and a quarter of the geometry with

symmetry boundary conditions (b). ......................................................................................................... 58

Figure 3.36: Coupling between the boundary nodes of the sample and the RP in the loading

direction. The load is applied in the reference point. ........................................................................... 58

Figure 4.1: Experimental setup for a biaxial test [35]. .......................................................................... 60

Figure 4.2: Possible load history (MSAJ test protocol) [35]. ................................................................ 61

Figure 4.3: Radial load paths in the warp-fill stress space [4]. ............................................................. 61

Figure 4.4: Bubble inflation test device for performing biaxial tests [48]. ......................................... 62

Figure 4.5: Geometry of the cruciform sample used in experimental biaxial tests (a) and a quarter

of the sample with boundary conditions and loading (b). .................................................................... 63

Figure 4.6: Loading the ends of the arms by coupling the boundary nodes to a reference point in

which the load is applied. .......................................................................................................................... 63

Figure 4.7: Geometry of the finite element simulation, including symmetry boundary conditions

and reference points. .................................................................................................................................. 64

Figure 4.8: Value of the stresses S11 (a) and S22 (b) at the central point of the cruciform sample

for 7 meshes with a different number of mesh elements. .................................................................... 65

Figure 4.9: Horizontal, vertical and diagonal path in central part along which the stresses and

strains are compared. ................................................................................................................................. 65

Figure 4.10: Stress S11 along the diagonal path (a), horizontal path (b) and vertical path (c) for 7

different meshes. ........................................................................................................................................ 66

Figure 4.11: Mesh used in the FEM for a sample with four slits in each arm. ................................. 67

Figure 4.12: Three different shapes of the tips of the slits: rectangular, round, peak. ..................... 67

Figure 4.13: Contour plots of stresses S11 in biaxial test sample under 1:1 load ratio (25 MPa) for

orthotropic linear elastic material behavior: rectangular tips (a), round tips (b), peak tips (c) and

legend (Pascal) (d) ...................................................................................................................................... 68



legend (Pascal) (d) ...................................................................................................................................... 68



legend (Pascal) (d). ..................................................................................................................................... 68

Figure 4.16: Stress curves S11 along diagonal path for rectangular tips, round tips and peak tips 69

Figure 4.17: Stress curves S22 along diagonal path for rectangular tips, round tips and peak tips 69

Figure 4.18: Stress curves S12 along diagonal path for rectangular tips, round tips and peak tips.

....................................................................................................................................................................... 69


(magnification) ............................................................................................................................................ 69


(magnification) ............................................................................................................................................ 70


(magnification). ........................................................................................................................................... 70

List of figures

166

Figure 4.22: Comparison size of geometry of two cruciform samples: small geometry (a) and big

geometry (b). ............................................................................................................................................... 70

Figure 4.23: Stress curves S11 along diagonal path for the big geometry and the small geometry

(a) and magnification (b). .......................................................................................................................... 71





Figure 4.26: Geometry of samples with different number of slits studied in order to determine

the influence of the number of slits on the stress state in the central part. ....................................... 72

Figure 4.27: Stresses S11 along diagonal path for geometries with different numbers of slits (a)

and magnification (b). ................................................................................................................................ 73





Figure 4.30: Correspondence between location of slits in the geometry and oscillations of the

stresses S11 in the central part. ................................................................................................................ 74

Figure 4.31: Mean values of stresses S11 over a range of 80 % of the diagonal path for geometries

with a different number of slits. ............................................................................................................... 75

Figure 4.32: Standard deviation of stresses S11 over a range of 80 % of the diagonal path for

geometries with a different number of slits. ........................................................................................... 75

Figure 4.33: Configurations with different locations of two slits. ....................................................... 76

Figure 4.34: Stresses S11 along diagonal path for configurations with different location of slits (a)






Figure 4.37: Mean values of stresses S11 over a range of 80 % of the diagonal path for geometries

with different location of slits. ................................................................................................................. 78

Figure 4.38: Standard deviations of stresses S11 over a range of 80 % of the diagonal path for

geometries with different locations of slits. ............................................................................................ 79

Figure 4.39: Exact geometry of the biaxial test sample used for the experiments. .......................... 79

Figure 4.40: Load as a function of time for a biaxial test with successive load ratios 1:1, 2:1, 1:2,

1:0 and 0:1. .................................................................................................................................................. 80

Figure 4.41: Strain (warp and fill) as a function of time for a biaxial test with successive load

ratios 1:1, 2:1, 1:2, 1:0 and 0:1. ................................................................................................................. 80

Figure 4.42: ‘Applied stress’-strain curve in warp direction for a biaxial test with successive load

ratios 1:1, 2:1, 1:2, 1:0 and 0:1 and a maximum stress of 24.1 MPa. .................................................. 81

Figure 4.43: ’Applied stress’-strain curve in fill direction for a biaxial test with successive load

ratios 1:1, 2:1, 1:2, 1:0 and 0:1 and a maximum stress of 24.1 MPa. .................................................. 81

Figure 4.44: Stress - strain curve in warp direction for a biaxial test with successive load ratios 1:1,

2:1, 1:2, 1:0, 0:1 and 1:1. ............................................................................................................................ 82

List of figures

167

Figure 4.45: Stress - strain curve in fill direction for a biaxial test with successive load ratios 1:1,

2:1, 1:2, 1:0, 0:1 and 1:1. ............................................................................................................................ 82

Figure 5.1: Calculated and experimentally measured stress-strain curves for a uniaxial tensile test

with load cycle repetition having a maximum stress level of 24.1 MPa: warp direction (a) and fill

direction (b). The calculations have been performed by means of an orthotropic linear elastic

material model. ........................................................................................................................................... 85

Figure 5.2: Calculated and measured strain in the direction opposite to the loading direction for a

uniaxial test in warp direction (a) and in fill direction (b) as a function of the normalized time

during the last loading and unloading cycle. ........................................................................................... 86

Figure 5.3: Calculated and experimentally measured stress-strain curves for a biaxial test with 5

identical cycles having a maximum stress of 24.1 MPa. The material model is an orthotropic

linear elastic material model, the Poisson ratio corresponds to 0.04. ................................................. 86

Figure 5.4: Calculated and experimentally measured stress-strain curves for a uniaxial test in warp

direction (a) and in fill direction (b). The material model is an orthotropic multilinear material

model............................................................................................................................................................ 87

Figure 5.5: Calculated and experimentally measured stress-strain curves for a uniaxial test with

increasing load in warp direction (a) and fill direction (b). The calculations have been performed

by means of a test data based Yeoh hyper elasticity model fitted to the warp direction combined

with a permanent set law. .......................................................................................................................... 90

Figure 5.6: Calculated and experimentally measured stress-strain curves for a uniaxial test with

increasing load in warp direction (a) and fill direction (b). The calculations have been performed

by means of a hyper elastic Fung model with a scaling factor. ........................................................... 91

Figure 5.7: Material orthogonal basis and yarn local directions for the reference configuration (a)

and the deformed configuration (b) [67] ................................................................................................ 93

Figure 5.8: The actual unloading curve (BCD) is obtained by shifting the user-specified unloading

curve horizontally [67] ............................................................................................................................... 94

Figure 5.9: Experimental stress-strain curve for a uniaxial test and the superimposed points which

define the material behavior in warp direction (a) and fill direction (b) ............................................ 95

Figure 5.10: Experimental stress-strain curve and chosen unloading behavior for a uniaxial test in

warp direction (a) and fill direction (b) ................................................................................................... 96


with load cycle repetition having a maximum stress level of 24.1 MPa in warp direction (a) and fill

direction (b). The calculations have been performed by means of the uniaxial test data based

fabric model. ............................................................................................................................................... 96

Figure 5.12: Calculated and experimentally measured strain (perpendicular to the loading

direction) during the first loading and unloading cycle of a uniaxial tensile test having a maximum

stress level of 24.1 MPa as a function of the normalized time in warp direction (a) and fill

direction (b). The FEM calculations have been performed by means of a uniaxial test data based

fabric model with a stiffness in compression of 0.025 GPa. ............................................................... 97

Figure 5.13: Calculated and experimentally measured strain (perpendicular to the loading

direction) during the first loading and unloading cycle of a uniaxial tensile test having a maximum

stress level of 24.1 MPa as a function of the normalized time in warp direction (a) and fill

direction (b). The FEM calculations have been performed by means of a uniaxial test data based

fabric model and a stiffness in compression of respectively 0.031 MPa and 0.83 MP in warp and

fill direction. ................................................................................................................................................ 97

List of figures

168


with increasing maximum stress level in warp direction (a) and fill direction (b). The calculations

have been performed by means of a uniaxial test data based fabric model. ..................................... 98

Figure 5.15: Calculated and experimentally measured stress-strain curves for a biaxial test with

successive load ratios are: 1:1, 2:1, 1:2, 1:0 and 0:1 having a maximum stress level of 24.1 MPa in

warp direction (a) and fill direction (b). The calculations have been performed by means of

uniaxial test data based fabric model. ...................................................................................................... 98

Figure 5.16: Experimental stress-strain curve of a biaxial test and the superimposed data points

which define the material behavior in warp direction (a) and fill direction (b). ................................ 99



direction (b). The calculations have been performed by means of test data based fabric model

fitted to the biaxial tests. ........................................................................................................................... 99


with increasing maximum stress level in warp direction (a) and fill direction (b). The calculations

have been performed by means of a biaxial test data based fabric model ....................................... 100


successive load ratios: 1:1, 2:1, 1:2, 1:0 and 0:1 having a maximum stress level of 24.1 MPa in

both warp direction (a) and fill direction (b). The calculations have been performed by means of a

biaxial test data based fabric model. ...................................................................................................... 100

Figure 5.20: Experimental stress-strain curve for a uniaxial test in fill direction (green) together

with a straight line having the same slope as the unloading curve (orange) (a) and hardening

behavior giving the stress as a function of true equivalent plastic strain (b). .................................. 103

Figure 5.21: Experimentally recorded stress-strain curve for a uniaxial test in warp direction, in

which straight lines are added to determine the yield point (a) and hardening behavior giving the

stress as a function of true equivalent plastic strain for both yield points (b). ................................ 104

Figure 5.22: Optimization of the Voce and Swift material parameters ............................................ 105

Figure 5.23: Voce hardening law defined by optimized parameters for two warp curves and a fill

curve (a) and Swift hardening law defined by optimized parameters for two warp curves and a fill

curve (b) together with the experimental data points. ........................................................................ 106

Figure 5.24: Lankford ratios r0°, r90°, r45° as a function of time........................................................... 107



direction (b). The calculations have been performed by means of a Hill plasticity material model

in which the yield stress ratios are determined by means of the Lankford ratios. .......................... 109



direction (b). The calculations have been performed by means of the Hill's plasticity material

model, which is fitted to the uniaxial experiments ignoring the crimp interchange. The

experimental stress-strain curve in fill direction is shifted to the left to remove the effect of crimp

interchange. ............................................................................................................................................... 110



direction (b). The calculations have been performed by means of a Hill plasticity material model,

which is fitted to the uniaxial experiments ignoring the crimp interchange. ................................... 110

List of figures

169


with an increasing maximum stress level: warp direction (a) and fill direction (b). The calculations

have been performed by means of the Hill's plasticity material model, which is fitted to the

uniaxial experiments ignoring the crimp interchange. The experimental stress-strain curve in fill

direction is shifted to the left to remove the effect of crimp interchange. ...................................... 111


successive load ratios 1:1, 2:1, 1:2, 1:0 and 0:1 having a maximum stress level of 24.1 MPa: warp

direction (a) and fill direction (b). The calculations have been performed by means of a Hill

plasticity material model, which is fitted to the uniaxial experiments ignoring the crimp

interchange. The experimental stress-strain curve in fill direction is shifted to the left to remove

the effect of crimp interchange. ............................................................................................................. 111



direction (b). The calculations have been performed by means of the Hill's plasticity material

model, which is fitted to a biaxial experiment ignoring the crimp interchange. The experimental

stress-strain curve in fill direction is shifted to the left to remove the effect of crimp interchange.

..................................................................................................................................................................... 112


with increasing maximum stress level: warp direction (a) and fill direction (b). The calculations

have been performed by means of the Hill's plasticity material model, which is fitted to a biaxial

experiment ignoring the crimp interchange. The experimental stress-strain curve in fill direction is

shifted to the left to remove the effect of crimp interchange. .......................................................... 112



direction (a) and fill direction (b). The calculations have been performed by means of the Hill's

plasticity material model, which is fitted to a biaxial experiment ignoring the crimp interchange.

The experimental stress-strain curve in fill direction is shifted to the left to remove the effect of

crimp interchange. .................................................................................................................................... 112

Figure 5.33: Optimization scheme for the yield stress ratio R11 ...................................................... 114

Figure 5.34: Determination of the yield stress factor R11 by minimizing the cost function (a) and

comparison of the computed Hill plasticity stress-strain curve in warp direction with the

experimentally recorded stress-strain curve in warp direction (b) .................................................... 114



direction (b). The calculations have been performed by means of a Hill plasticity model, which is

fitted to the uniaxial tensile test in fill direction. .................................................................................. 115

Figure 5.36: Calculated and experimentally measured stress-strain curves for a uniaxial tests with

increasing load: warp direction (a) and fill direction (b). The calculations have been performed by

means of a Hill plasticity model, which is fitted to the uniaxial test in fill direction. ..................... 115


successive load ratios are: 1:1, 2:1, 1:2, 1:0 and 0:1 having a maximum stress level of 24.1 MPa in

warp direction (a) and fill direction (b). The calculations have been performed by means of a Hill

plasticity model fitted to the uniaxial experimental test in fill direction. ......................................... 116



List of figures

170

direction (b). The calculations have been performed by means of a Hill plasticity model, which is

fitted to the experimental biaxial stress-strain curves. ........................................................................ 116

Figure 5.39: Calculated and experimentally measured stress-strain curves for a uniaxial tests with

increasing load: warp direction (a) and fill direction (b). The calculations have been performed by

means of a Hill plasticity model, which is fitted to the experimental biaxial stress-strain curves.

..................................................................................................................................................................... 117



direction (a) and fill direction (b). The calculations have been performed by means of a Hill

plasticity model, which is fitted to the experimental biaxial test. ...................................................... 117

Figure 5.41: Color code used for the strain field in warp direction. ................................................. 118

Figure 5.42: Biaxial load cycle, with the superimposed points of extraction (blue dots). .............. 119

Figure 5.43: Contour plots of the strain field in warp direction for load ratio 1:1 at the specified

load level: experiment (a), fabric model (b) and Hill model. ............................................................. 119









Figure 5.48: Color code used for the strain field in fill direction. ..................................................... 121

Figure 5.49: Contour plots of the strain field in fill direction for load ratio 1:1 at the specified load

level: experiment (a), fabric model (b) and Hill model. ...................................................................... 121








level: experiment (a), fabric model (b) and Hill model ....................................................................... 122

171

List of tables

Table 4.1: Elasticity constants and other material constants for the membrane material. .............. 64

Table 5.1: Young’s moduli in warp and fill direction for all linearized parts of the stress-strain

curve ............................................................................................................................................................. 87

Table 5.2: Optimized parameters of the Voce hardening law for two warp and one fill hardening

curves. ........................................................................................................................................................ 105

Table 5.3: Optimized parameters of the Swift hardening law for two warp and one fill hardening

curves. ........................................................................................................................................................ 105

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