Modeling of fracture and damage in rubber under

dynamic and quasi-static conditions

ELSIDDIG ELMUKASHFI

Doctoral thesis no. 91, 2015

KTH School of Engineering Sciences

Department of Solid Mechanics

Royal Institute of Technology

SE-100 44 Stockholm, Sweden

TRITA HFL-0581ISSN 1104-6813ISRN KTH/HFL/R-15/17-SEISBN 978-91-7595-749-4

KTHSE-100 44 Stockholm

SWEDEN

Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framläggestill offentlig granskning för avläggande av teknologie doktorsexamen i hållfasthets-lära fredagen den 18 december 2015 klockan 10.00 i Sal B2, Brinellvägen 23 (02 tr),Kungl Tekniska högskolan, 100 44, Stockholm. Fakultetsopponent är Professor K.Ravi-Chandar, The University of Texas at Austin, Texas, USA.

© Elsiddig Elmukashfi, December 2015

Tryck: Universitetsservice US AB

Try not to become a man of success, but rather try to become a man of value.

Albert Einstein

iii

Abstract

Elastomers are important engineering materials that have contributed tothe different technical developments and applications since the 19th century.The study of crack growth mechanics for elastomers is of great importanceto produce reliable products and therefore costly failures can be prevented.On the other hand, it is fundamental in some applications such as adhesiontechnology, elastomers wear, etc. In this thesis work, crack propagation inrubber under quasi-static and dynamic conditions is investigated.

In Paper A, theoretical and computational frameworks for dynamic crackpropagation in rubber have been developed. The fracture separation processis presumed to be described by a cohesive zone model and the bulk behavioris assumed to be determined by viscoelasticity theory. The numerical modelis able to predict the dynamic crack growth. Further, the viscous dissipationin the continuum is found to be negligible and the strength and the surfaceenergy vary with the crack speed. Hence, the viscous contribution in the inner-most of the crack tip has been investigated in Paper B. This contributionis incorporated using a rate-dependent cohesive model. The results suggestthat the viscosity varies with the crack speed. Moreover, the estimation ofthe total work of fracture shows that the fracture-related processes contributeto the total work of fracture in a contradictory manner.

A multiscale continuum model of strain-induced cavitation damage andcrystallization in rubber-like materials is proposed in Paper C. The modeladopts the network decomposition concept and assumes the interaction be-tween the filler particles and long-chain molecules results in two networksbetween cross-links and between the filler aggregates. The network betweenthe crosslinks is assumed to be semi-crystalline, and the network between thefiller aggregates is assumed to be amorphous with the possibility of debond-ing. Moreover, the material is assumed to be initially non-cavitated and thecavitation may take place as a result from the debonding process. The cavi-ties are assumed to exhibit growth phase that may lead to complete damage.The comparison with the experimental data from the literature shows thatthe model is capable to predict accurately the experimental data.

Papers D and E are dedicated to experimental studies of the crack prop-agation in rubber. A new method for determining the critical tearing energyin rubber-like materials is proposed in Paper D. The method attempts toprovide an accurate prediction of the tearing energy by accounting for the dis-sipated energy due to different inelastic processes. The experimental resultsshow that classical method overestimates the critical tearing energy by ap-proximately 15%. In Paper E, the fracture behavior of carbon-black naturalrubber material is experimentally studied over a range of loading rates vary-ing from quasi-static to dynamic, different temperatures, and fracture modes.The tearing behavior shows a stick-slip pattern in low velocities with a sizedependent on the loading rate, temperature and the fracture mode. Smoothpropagation results at high velocities. The critical tearing depends stronglyon the loading rate as well as the temperature.

v

Sammanfattning

Elastomerer är en viktig grupp av ingenjörsmaterial som har bidragittill den tekniska utvecklingen alltsedan 1800-talet. Studiet av brottmekanikhos elastomerer är viktigt när det gäller att producera tillförlitliga produk-ter, och dyra brott kan undvikas. Brottmekanik hos elastomerer är ocksåviktigt i tillämpningar som adhesionsteknik, nötning av elastomerer, etc. Idenna avhandling undersöks spricktillväxt i gummi under kvasi-statiska ochdynamiska förhållanden.

I artikel A utvecklas ett teoretiskt och beräkningsmässigt ramverk föratt simulera dynamisk spricktillväxt i gummi. Brottprocessen antogs kunnabeskrivas med en kohesiv zon och bulk-beteendet beskrevs med viskoelasticitets-teori. Den numeriska modellen kunde prediktera den dynamiska spricktil-lväxten i experiment. Den viskösa dissipationen i bulk-materialet befannsvara försumbar, och brottspänningen och ytenergin varierade med sprick-hastigheten.Det viskösa bidraget i regionen närmast sprickspetsen undersök-tes sedan i artikel B. Bidraget togs hänsyn till genom en hastighetsberoendekohesiv lag. Resultaten visar att viskositeten varierar med sprickhastigheten.Undersökningen av den totala brottenergin visade att de direkt brott-relateradeprocesserna bidrog till den totala brottenergin på ett sätt tämligen kom-plicerat sätt.

En flerskale-kontinuum-modell för töjningsinducerad kavitationsskada ochkristallisation i gummi-liknande material presenteras i artikel C. Modellenutgår ifrån en nätverksuppdelning och det antas att interaktionen mellanfyllnadspartiklar och polymermolekyler resulterar i två nätverk: ett mellantvärbindningar och ett mellan fyllnadsaggregat. Näätverket mellan tvärbind-ningar antas vara semi-kristallint, och nätverket mellan fyllnadsaggregat an-tas vara amorft och med lösbara bindningar. Materialet antas vara utankaviteter initialt, och kavitationen kan äga rum genom att bindningar släp-per. Kaviteterna antas kunna växa till och orsaka skada. Jämförelsen medexperimentella data från litteraturen visar att modellen kan prediktera ex-perimentella data väl.

Artiklarna D och E innehåller experimentella studier av spricktillväxti gummi. En ny metod för att bestämma den kritiska ytenergin i gummi-liknande material presenteras i artikel D. Metoden medger bestämning avytenergin där den dissipativa energin i olika icke-elastiska processer tas hänsyntill. De experimentella resultaten visar att den klassiska metoden överskattarytenergin med ca 15%. I artikel E studeras det brottmekaniska beteen-det hos kimrökt naturgummi under olika lasthastigheter, olika temperaturer,och olika brottmoder. Brottbeteendet uppvisar en ojämn hastighetsprofil vidlåga lasthastigheter. Vid höga lasthastigheter erhålls en jämnare spricktil-lväxt. Den kritiska ytenergin beror starkt på lasthastigheten såväl som påtemperaturen.

vii

Preface

The work presented in this thesis has been performed between February 2010 andDecember 2015 at the department of Solid Mechanics at KTH Royal Institute ofTechnology (KTH Hållfasthetslära). The research has been funded by the ProjectGrant no. 2007-5233 of the Swedish Research Council (VR). The financial supportis gratefully acknowledged.

I owe my deepest gratitude to the late professor Fred Nilsson who has introduced,motivated and inspired me to the fracture mechanics field and giving me the chanceto work in this project.

I would like to express my sincerest appreciation to my supervisor Associate Profes-sor Martin Kroon for his encouragement, valuable guidance, and genuine supportduring the whole period enabled me to develop an understanding of the subject.

I am very thankful to M.Sc. Rickard Österlöf for providing the rubber material.M.Sc. Martin Öberg and Dr. Irene Arregui are greatfully acknowledged for theirassistance during the experimental realization of the work. I would like also tothank Mr. kurt Lindqvist and Mr. Göran Rådberg for helping in manufacturingthe specimens.

Special thanks go to my colleagues at solid mechanics department for the friendlyenvironment, permanent readiness, and unlimited help and support.

I am very grateful to my friends, old and new, for keeping in touch, for the nicetimes and good memories and all of you have made a lot of things possible.

I owe immense gratitude to my parents and brothers who have been a constantsource of inspiration ’Thanks for being a wonderful family’. I am particularly grate-ful to my elder brother Ahmed for his suggestions, encouragement, and his patientsupport all the time.

Finally, and most importantly, I would like to thank my wife Fatima. You havebeen my best friend and great companion, loved, supported, encouraged, enter-tained, and helped me get through this period in the most positive way.

Stockholm, December 2015Elsiddig Elmukashfi

ix

List of appended papers and other contributions

Paper A: Numerical analysis of dynamic crack propagation in rubberElsiddig Elmukashfi and Martin KroonInternational Journal of Fracture 177(2), 2012, 163–178

Paper B: Numerical analysis of dynamic crack propagation in biaxially strainedrubber sheetsElsiddig Elmukashfi and Martin KroonEngineering Fracture Mechanics, 124, 2014, 1-17

Paper C: A multiscale continuum modeling of strain-induced cavitation damageand crystallization in rubber-like materialsElsiddig Elmukashfi and Martin KroonReport 579, Department of Solid Mechanics, Royal Institute of Technology (KTH),Stockholm, Sweden

Paper D: An experimental method for estimating the tearing energy in rubber-like materials using the true stored energyElsiddig ElmukashfiReport 580, Department of Solid Mechanics, Royal Institute of Technology (KTH),Stockholm, Sweden

Paper E: Experimental investigations of crack propagation in rubber under dif-ferent loading rates, temperatures and fracture modesElsiddig ElmukashfiReport 586, Department of Solid Mechanics, Royal Institute of Technology (KTH),Stockholm, Sweden

In addition to the appended papers, the work has resulted in the following publi-cations and presentations 1:

Analysis of Dynamic Crack Propagation in Rubber

E. Elmukashfi and M. KroonPresented at 10th World congress on Computational Mechanics (WCCM), SaoPãolo, 2012 (EA,PP)

Modeling and Analysis of Dynamic Crack Propagation in Rubber

E. Elmukashfi and M. Kroonpresented at Svenska Mekanikdagar, Lund, Sweden, 2013 (EA,PP)

1EA = Extended abstract, OP = Oral Presentation, PP = Proceeding paper, P = Poster

Presentation.

x

Numerical Modeling and Analysis of Dynamic Crack Propagation in

Rubber

E. Elmukashfi and M. Kroonpresented at 13th International Conference on Fracture (ICF13), Beijing , 2013(PP,OP)

A multiscale continuum modeling of cavitation damage and strain in-

duced crystallization in rubber materials

E. Elmukashfi and M. KroonPresented at 10th World congress on Computational Mechanics (WCCM), Barcelona,Spain, 2014 (EA,PP)

Characterization of crack growth in rubber materials

E. ElmukashfiPresented at 1st Annual User Meeting of the Odqvist Laboratory for ExperimentalMechanics, Stockholm, Sweden, 2014 (P)

Continuum modeling of cavitation damage and strain induced crystal-

lization in rubber materials

E. Elmukashfi and M. KroonPresented at 27th Nordic Seminar on Computational Mechanics (NSCM), Stock-holm, Sweden, 2015 (EA,PP)

xi

Contribution to the papers

Paper A: Numerical analysis of dynamic crack propagation in rubberElsiddig Elmukashfi: Numerical analysis, Manuscript.Martin Kroon: Supervision, Manuscript.

Paper B: Numerical analysis of dynamic crack propagation in biaxially strainedrubber sheetsElsiddig Elmukashfi: Numerical analysis, Manuscript.Martin Kroon: Supervision, Manuscript.

Paper C: A multiscale continuum modeling of strain-induced cavitation damageand crystallization in rubber-like materialsElsiddig Elmukashfi: Theoretical analysis, Numerical analysis, Manuscript.Martin Kroon: Supervision, Manuscript.

Paper D: An experimental method for estimating the tearing energy in rubber-like materials using the true stored energyElsiddig Elmukashfi: Theoretical analysis, Experimental work, Manuscript.

Paper E: Experimental investigations of crack propagation in rubber under dif-ferent loading rates, temperatures and fracture modesElsiddig Elmukashfi: Theoretical analysis, Experimental work, Manuscript.

Contents

Abstract iii

Sammanfattning v

Preface vii

List of appended papers ix

Contribution to the papers xi

Contents xiii

List of Figures xv

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Outlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Crack propagation in rubber 5

2.1 Fracture mechanics of rubber . . . . . . . . . . . . . . . . . . . . . . 52.2 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Microstructural changes in tearing . . . . . . . . . . . . . . . . . . . 82.4 Tearing behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Modeling crack growth in rubber 11

3.1 Theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Computational framework . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Experimental work 17

4.1 The experimental methods . . . . . . . . . . . . . . . . . . . . . . . . 174.2 A new method for determining the tearing energy . . . . . . . . . . . 20

5 Results and Discussion 23

xiii

xiv CONTENTS

5.1 Modeling dynamic crack propagation . . . . . . . . . . . . . . . . . . 235.2 Experimental determination of crack propagation . . . . . . . . . . . 245.3 The critical tearing energy estimation . . . . . . . . . . . . . . . . . 26

6 Conclusions and Future Work 27

Summary of appended papers 29

Bibliography 33

Paper A

Paper B

Paper C

Paper D

Paper E

List of Figures

1.1 The failure modes in rubber products. . . . . . . . . . . . . . . . . . . . 2

2.1 The tearing test specimens . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 The tearing energy-rate of tear relation . . . . . . . . . . . . . . . . . . 62.3 The critical tearing energy behavior . . . . . . . . . . . . . . . . . . . . 10

3.1 The schematic of the typical dynamic modulus spectrum. . . . . . . . . 123.2 The schematic of the typical crystallinity spectrum. . . . . . . . . . . . 133.3 The fracture processes in quasi-static crack propagation. . . . . . . . . . 153.4 The fracture processes in dynamic crack propagation. . . . . . . . . . . 16

4.1 The fracture specimens. . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 The experimental setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3 The schematic of the load-displacement curves for the pure shear test. . 21

5.1 The total work of fracture. . . . . . . . . . . . . . . . . . . . . . . . . . 245.2 The crack propagation profile. . . . . . . . . . . . . . . . . . . . . . . . . 255.3 The critical tearing energy. . . . . . . . . . . . . . . . . . . . . . . . . . 26

xv

Chapter 1

Introduction

1.1 Background

Elastomers (rubber and rubberlike materials) have played a central role in manyengineering developments since the 1850’s. Their molecular network structure en-ables them to possess unique properties, e.g., their ability to deform elastically byseveral hundred per cent and their too small shear modulus. In addition, theirchemical compositions allow them to perform high resistance in aggressive environ-ments. Consequently, several products in different engineering applications rely ona wide variety of elastomeric materials, e.g., tires, springs, dampers, gaskets, bear-ings, oil seals, etc. Furthermore, it has been shown that many soft human tissuesbehave in a similar fashion as elastomers and their mechanical response can oftenbe modeled in similar ways [16, 29] .

The damage and crack growth in elastomeric materials are fundamentally im-portant in many applications such as wear, adhesive joining applications, etc. Thefailure of elastomeric products due to crack growth severely results in big loss ofcapital and even life, e.g., a tyre explosion, a rubber O-ring seal failure, an adhesivejoint failure, see Figs. 1.1(a) and (b). The phenomena of aorta aneurysm dissectionare due a crack growth mechanism; therefore, the severe damage is often describedby dissection of internal organs, see Fig. 1.1(c).

Despite the crack growth in elastomers problem has a long history of research;it has not reached a success in comparison with other materials. Rivlin andThomas [55] started to investigate this problem and extended the Griffith [26]energetic fracture mechanics approach to rubber and rubber-like materials by in-troducing the tearing energy. The experimental studies show big variation in thereported tearing energy values in which it falls within the wide range 1 − 105 J/m2

[7, 11, 18, 19, 24, 25, 40, 55]. This large discrepancy was the topic of the researchsince then.Experimentally, several techniques using different specimens were proposed for de-termining the critical tearing energy. These methods assume a purely elastic mate-

1

2 CHAPTER 1. INTRODUCTION

(a) (b) (c)

Figure 1.1: Severe failure modes due to: (a) crack growth in a tyre during durability test,(b) cavitation damage and crack growth in rubber O-ring caused by a rapid decrease inpressure, (c) aneurysm dissection of thoracic aorta.

rial and ignore inelastic deformation effects (Mullins softening effect [48], hysteresis,permanent set and induced anisotropy, cavitation damage, and strain-induced crys-tallization). Thus, the main challenge which is discriminating the inelastic deforma-tion effects during the evaluation of the critical tearing energy remains unresolved.Few theoretical and numerical frameworks have been proposed for modeling frac-ture in rubber. Chang [5] and Eshelby [14] (energy-momentum tensor) introducedthe large deformation version of the J-integral which increased the ability of eval-uating the critical tearing for the complex situations, i.e. specimens with complexgeometries and fracture modes. Further, most of the other studies are performedfor specific situations, i.e. crack propagation in viscoelastic media [8, 9, 36, 51], in-vestigations of dynamic crack growth in a wide range of crack velocities [42, 43, 62],exploration of different dynamic instabilities of crack growth [27]. These frame-works were investigating the discrepancy on the tearing energy by modeling otherdissipative mechanisms that are associated with the fracture process, i.e. viscousdissipation. Nevertheless, a clear understanding of the different contributions tothe critical fracture energy is not achieved.

Additionally, under deformation, rubber and rubber-like materials experienceremarkable inelastic changes including the stress softening (Mullins softening effect[48]), hysteresis, permanent set and induced anisotropy. The experimental inves-tigations have shown that these changes are associated with the development ofcavitation damage, breakage of filler-polymer bonds and crystallization. Hence, thedevelopment of highly accurate as well as physically motivated constitutive modelsfor general multiaxial loading conditions is of great interest. Moreover, such a modelcan help in analyzing and predicting complex phenomena such as damage and crackgrowth problems. Several attempts to incorporate these different mechanisms intomodels have been for these phenomena but it remains relatively undeveloped.

1.2 Objectives

The main objective of this thesis work is to investigate different aspects of crackgrowth in natural rubber. These objectives are detailed in which follows:

1.3. OUTLINES 3

• To develop theoretical and numerical frameworks for crack propagation inrubber.

• To investigate the different contributions to the total fracture energy.• To develop experimental method for more accurate estimation of the critical

tearing.

1.3 Outlines

The details of the research work presented in thesis is presented in the appendedpapers. Hence, a background and an overview of the research subject as well as acomprehensive summary of the work are provided in this introductory. The outlinesof this thesis are,

Chapter 2: presents a comprehensive literature review in the fracture me-chanics of rubber and rubber-like materials. The classical concept of the energyrelease rate, other fracture mechanics approaches, specimens, materials, andresults of crack growth are shown.

Chapter 3: details the theoretical and computational frameworks have beendeveloped during this thesis work. The theoretical treatment as well as thecomputational frameworks of the quasi-static and dynamic fracture are dis-cussed.

Chapter 4: shows the experimental work has been performed during thethesis. The experimental techniques used to characterize the crack propagationand the proposed method of estimating the critical tearing are presented.

Chapter 5: highlights the main results have been achieved in this thesis. Thenumerical as well as the experimental results are presented.

Chapter 6: summarizes the conclusion and recommendations of this thesiswork.

Chapter 2

Crack propagation in rubber

A brief literature review of crack growth in rubber and rubber-like materials isprovided in this chapter. The modeling of crack propagation, the experimentalmethods, microstructural changes during tearing, and tearing behavior in rubberand rubber-like materials will be discussed.

2.1 Fracture mechanics of rubber

In this section, the classical approach of the energy release rate is firstly discussedand then we emphasize the other fracture mechanics approaches in rubber andrubber-like materials.

2.1.1 The classical energetic approach

The tearing energy as a fracture mechanics concept was proposed by Rivlin andThomas [55] as an analogy to the energy release rate [26] in order to study the frac-ture in rubber and rubber-like materials. The authors assumed: (i) the approachof Griffith is valid for large strain (in fact, no restriction was made in the origi-nal hypothesis of Griffith), (ii) Irreversible changes in energy due to crack growthtake place only in the vicinity of the crack tip, and (iii) The change in energy isindependent of the shape and dimensions of the body. Therefore, the crack growthgoverned by the critical tearing energy criterion that is defined as

Tc = −∂U

∂A

∣

∣

∣

∣

δc

, (2.1)

where Tc is the critical tearing energy, A is the surface area of one face of the crack,U is the potential energy stored in the system and the suffix (•)

δc

denotes that thedifferentiation is carried out at a constant displacement, i.e. the external forces donot produce work.Several researchers investigated the constancy of the critical tearing energy using

5

6 CHAPTER 2. CRACK PROPAGATION IN RUBBER

Angled Split

Pure Shear Trouser

Figure 2.1: The test specimens used in the early investigation of the tearing energy ap-proach

different specimens [25, 40, 55, 59]. The result of this intensive work was thatthe tearing energy is constant at constant rate of tearing, see Fig. 2.2. Hence,the tearing energy-rate of tearing relation appears to be a fundamental materialproperty, independent of the test specimen shape and configuration.

0.0

1.0

2.0

3.0

10−5 10−4 10−3 10−2

vc [cm/s]

Tc

[KJ/m

2]

bcbc

bc

bc

bc

×

××

××

××××

×

++

++ +

+ ++

bc

bc bc bc

bcbcbcbc

bcbc

bc bc

bc

Figure 2.2: The tearing energy-rate of tear relation for SBR using the test specimensshown in Fig. 2.1 [60]. ’×’ trouser, ’+’ pure shear, ’◦’ split and ’•’ angled.

2.1. FRACTURE MECHANICS OF RUBBER 7

The experimental studies show big variation in the reported tearing energyvalues in which it falls within the wide range 1 − 105 J/m2 [7, 11, 18, 19, 24, 25,40, 55]. The so called intrinsic fracture energy (energy needed to rupture of therubber primary bonds) is estimated experimentally and theoretically to be in therange 1 − 10 J/m2 [1, 2, 20, 23, 38–40, 47]. Thus, the large discrepancy must beattributed to dissipative processes in the material surrounding the crack tip. Thesedissipative processes can be related to the viscoelastic dissipation and dissipationdue to the complex damage processes (strain-induced crystallization, cavitation,molecular slippage from the filler particles) around the crack tip vicinity.

2.1.2 Other fracture mechanics approaches

The first studies were limited to specimens with simple geometries and failure modessuch that the crack remained self-similar. Later, Chang [5] and Eshelby [14] (energy-momentum tensor) proposed a generalization to the path-independent integral forthe case of finite elasticity. Therefore, the critical J-value (Jc) was introduced asequivalence to the critical tearing. Schapery [56] studied the viscous effects onthe fracture of rubber and rubber-like materials. He postulated a J-integral toinclude the viscous effects and applied his results to a particular set of viscoelasticmaterials. The application of the numerical methods such as finite element methodputs forward the computations of J-integral especially for the complex situations,i.e. specimens with complex geometries and fracture modes. Medri and Strozzi [45]used the J-integral approach to investigate cracked seals and obtained encouragingresults. Based on experiments, Lee and Donovan [41] compared the results foundfrom the tearing energy T and J-integral. Later, the J-integral approach has beenused intensively in research with acceptable results [11, 28].

Theoretical studies on different aspects of crack growth in rubber have beenproposed. The crack propagation in viscoelastic solids (considering the viscoelas-tic dissipation as the main dissipative process) was intensively investigated underthe assumption of infinitesimal strains. They have concluded that the viscoelasticdissipation plays notable role in the crack growth and the total work of fractureconsists of two different contributions, i.e. the surface energy and the viscoelasticdissipation. Later, theoretical frameworks for the steady crack propagation growthin rubber under large deformation problem have been proposed [8, 9, 51]. Theseframeworks assumed large strain viscoelasticity and introduced a new contributionto the total work of fracture that comes from the viscoelastic dissipation in thepolymer in front of the crack tip. Kroon [36] proposed a computational frameworkthat analyzes high-speed crack propagation in rubber-like solids under conditions ofsteady-state and finite strain. The framework is able to quantify the contribution ofviscoelastic dissipation to the total work of fracture required to propagate a crackin a rubber-like solid.

In the context of dynamic fracture, theoretical studies using lattice models havebeen used [27, 42, 43, 62]. These studies include investigations of dynamic crack

8 CHAPTER 2. CRACK PROPAGATION IN RUBBER

growth in a wide range of crack velocities and exploration of different dynamicinstabilities of crack growth in rubber such as the oscillatory behavior.

Despite the intensive studies on the crack growth in rubber, an accurate de-scription of the total work of fracture was not achieved. Moreover, no effort wasdirected toward modeling the other fracture-related processes, i.e. cavitation andcrystallization. Thus, a more accurate model can be achieved by including thecontribution of these processes to the total work of fracture.

2.2 Experimental methods

Several experimental techniques using different specimens were proposed for deter-mining the critical tearing energy. Rivlin and Thomas [55] introduced the trouser,pure shear, angled and split specimens, see Fig. 2.1, and since then new spec-imens have been continuously proposed in the literature, e.g. the single edgenotch in tension (SENT) [28], the double cantilever beam (DCB) [50, 58], ten-sile strip test [37, 40], the doubly cracked pure shear specimen (DCPS) [44] andthe circumferentially-cracked cylindrical specimen (CCC) [46]. In spite of the factthat no agreement has been taken for one of these specimens, the trouser and pureshear (edge cracked panel) specimens are the most used specimens.

The evaluation of the critical tearing energy is generally accomplished by thedetermination of the potential decrease due to crack growth, i.e. ∂U/∂A. Analyt-ically, in specimens of simple geometries, the tearing energy is obtained from theenergy balance assuming that the decrease in total energy is due to creation of newcrack surfaces [44, 46, 50, 55, 58]. Other methods are based on constructing a rela-tion between the potential energy stored in the system at the crack initiation andthe crack length experimentally using specimens with different initial crack length[28, 32–34]. These methods assume a purely elastic material and ignore inelasticdeformation effects.

2.3 Microstructural changes in tearing

The investigations of the tear surface revealed a great deal of information about thefracture process in rubber. In the case of non-strain-crystallizing rubber, the tearsurface appears to change with the tear velocity. The lower the tearing velocity therougher the tearing surface is resulted. Further, a transition regime is observed inthe moderate tearing rates such that the surface changes from being a relativelyrough to a smooth surface. In the case of strain-crystallizing rubber the so calledstick-slip and knotty tearing are common in the moderate and low crack velocities.The surface contains smooth and rough regions. The rough surface takes place inthe stick and the knot places while the smooth surface resulted in between the stickand knot positions.

The studies concerning the crack tip morphology during propagation show thatthere are characteristic webs of rubber stretched across the tip as a result of cav-

2.4. TEARING BEHAVIOR 9

itation. The cavities take place ahead of the tear tip, forming strands of rubberbetween the cavities or vacuoles. As the main tear advances, these webs relax andform matching cross-hatched patterns on both of the torn surfaces. Moreover, theadvancing crack front is accompanied with the formation of secondary cracks atthe tear tip at points where the local stress is high. Therefore, the non-uniformcrack propagation is the result of the growth of the secondary cracks in the mi-croscale. These secondary cracks will connect together and form new crack surfaceon a macroscopic scale. The formation of cavities in the crack tip has been re-cently confirmed using small angle X-Ray scattering technique [65]. They wereable to quantified the cavitation volume fraction near the crack tip. In general, thestrain crystallizing elastomers show higher resistance to fracture and the studies ofthe crystallinity around cracks have shown that the strain-induced crystallizationmay occur at the crack tip in both quasi-static and dynamic fracture situations[3, 61, 64].

2.4 Tearing behavior

Under quasi-static loadings, an unfilled strain-crystallizing rubber shows stick-sliptearing behavior with a fairly well defined abrupt increase in crack length as theforce on the test specimen increases (this phenomenon is a characteristic behaviorof strain-crystallizing elastomers). The strain-crystallizing rubber show a muchmore complicated rate and temperature dependence, with regions of high strength(knotty tear). The non-strain-crystallizing rubber show a time dependent crackgrowth behavior with the rate of crack propagation depending on the applied load.This results in a well defined crack velocity during the tearing.

In brittle materials, dynamic fracture theories show that the crack speed is lim-ited by the Rayleigh wave speed in the case of mode I and II and by the shear wavespeed in the mode III case [17, 53]. Additionally, unstable crack propagation, asso-ciated with development of complex branching, occurs as the crack speed exceedsabout half of the Rayleigh wave speed. In rubber materials, cracks propagate atspeeds greater than the speed of sound without branching. Furthermore, under in-creasing transverse stretch the crack speed increases, and at a critical stretch level,oscillatory crack propagation results [6, 10, 21, 27, 35, 42, 43, 52, 62].

The tearing energy shows strong dependency on the temperature and the crackgrowth rate. Smith [57] showed that the tensile properties of rubber can betransformed between different rates and temperatures using Williams-Landel-Ferrytransform [63] (similar to the dynamic properties). Mullins [49] continued and per-formed similar tests in the tearing behavior, and he was able to show that thetearing energy at constant tearing rate is proportional to the loss modulus for arange of non-crystallizing elastomers. These results suggest that internal viscosityis a dominant factor in determination of tearing energy.

Several factors, that affect the crack growth mechanics in rubber, can be relatedto the material rubber formulation and composition. One important factor is the

10 CHAPTER 2. CRACK PROPAGATION IN RUBBER

strain crystallization in which some types of rubber, e.g., natural rubber, exhibitformation of crystals under high loading. The crystallization increases the tearingstrength significantly. Other important factors are the rubber molecular structure,cross-link density, and filler size and concentration [51]. The molecular structuremainly affects the rubber viscoelastic properties. The filler particles concentra-tion and cross-linking have big influence on the tear strength of rubber as well asviscoelastic properties.

The schematic in Fig. 2.3 shows the fracture behavior in the entire range ofthe crack propagation velocity for strain-crystallizing and non-strain-crystallizingrubber [51].

log vc

log Tc

instability inertia regime

with strain crystallizationwithout strain crystallization

Figure 2.3: The schematic of the critical tearing energy Tc and the crack-tip velocity vc

relation in logarithmic scale for strain-crystallizing and non-strain-crystallizing rubber.

Chapter 3

Modeling crack growth in rubber

In this chapter, the modeling methods of the crack propagation in rubber in the caseof dynamic and quasi-static loadings are presented. The theoretical frameworks arefirstly introduced and the computational implementations are then discussed.

3.1 Theoretical framework

In order to investigate crack propagation in rubber and rubber-like solids, the defor-mation around a propagating crack tip is studied. Therefore, a steadily propagatingcrack with a velocity vc is considered. The deformation at a distance r (equal ormore than the separation process zone length) from the crack tip can be charac-terized by the stretch level and the rate of deformation. The deformation rate canbe represented by the frequency ω = vc/r. In following subsections, the roles ofdifferent mechanisms (viscous dissipation, crystallization, cavitation, etc.) in thecrack propagation are analyzed.

3.1.1 The role of viscous dissipation

The dynamic modulus spectrum (relaxation spectrum) for rubber and rubber-likematerials is considered. The relation between the dynamic modulus E(ω) andthe frequency ω shows that there are three different regimes, i.e. the glassy, thetransition, and the rubbery regimes, see Figs. 3.1(a) and (b). At high frequencies,a glassy behavior is observed, such that there is no time for thermally activatedrearrangement of the polymer chain segments, i.e. ω >> 1/τv, where τv denotesthe relaxation time (flipping time). Thus, a stiff material behavior is resulted,characterized by the instant modulus E0. A rubbery response takes place at lowfrequencies, where adiabatic and thermally activated rearrangements of the polymerchains occur, i.e. ω << 1/τv. Hence, a soft material behavior results, characterizedby the low frequency or long term modulus E∞. Further, there is a transition regionthat is centered between the glassy and rubbery zones, where viscous dissipation

11

12 CHAPTER 3. MODELING CRACK GROWTH IN RUBBER

occurs and the loss modulus reaches its maximum value.In the case of propagating crack, closest to the crack tip, the frequency is very highand a glassy region may appear. Far from the crack tip, the frequency is low anda rubbery region is expected. The viscous dissipation region is therefore locatedbetween these two regions. In the case of carbon-filled natural rubber, the typicalglassy and rubbery frequencies are approximately 1010 Hz and 104 Hz, respectively[15]. For crack velocities in the range 0−130 m/s [21], glassy zones are not expectedto exist, while viscous dissipation is expected.

log ω

tan

δlo

gE

rubbery transition glassy

E′

E′′

(a)

(b)

Figure 3.1: The schematic of the typical dynamic modulus spectrum E(ω) = E′ + iE′′ forrubber material: (a) the dynamic moduli (the storage modulus E′ and the loss modulusE′′), (b) the loss tangent tan δ = E′/E′′.

3.1.2 The role of strain-induced crystallization

The relation between the crystallinity ζ(ω) and the frequency ω for rubber andrubber-like materials is considered. In the quasi-static loading, the strain-inducedcrystallization takes place at a critical elongation level. This critical elongation

3.1. THEORETICAL FRAMEWORK 13

increases at higher loading rates implying that there is a characteristic time neededfor the crystallization process to take place, see Fig. 3.2. Denoting the crystal-lization characteristic time τc, at high frequencies and large elongation a smallamount crystallization is expected in which the loading time is smaller than thetime needed for the crystals formation, i.e. ω >> 1/τc. At the low frequencies andlarge elongation, the maximum crystallinity can be achieved ω << 1/τc. Thus,a moderate crystallinity is expected at moderate frequencies. It should be noted,that the amount of crystallinity is very dependent on the stretch level.In the case of propagating crack, closest to the crack tip, the deformation and the

ω

ζ

λ

Figure 3.2: The schematic of the typical crystallinity spectrum for a strain crystallizingrubber material under cyclic loading of different stretch amplitude λ (reproduced fromCandau et al. [4]).

frequency are very high and large amount of crystallinity is expected. Far from thecrack tip, the elongation and the frequency are very small and pure amorphous ma-terial is expected. Hence, a semi-crystalline region of varying crystallinity is formedaround the crack-tip. The shape of this region is dependent on the gradients of thedeformation and the frequency. Moreover, the crystallinity may take its maximumvalues at the onset of the propagation and decreases as the crack speed increases(in strain-crystallizing rubber, the crack typically propagates at low velocity andaccelerates reaching steady state propagation).In the case of quasi-static crack (before the propagation onset, i.e. ω ∼ 0), [61]showed that there are formed a zone of maximum crystallinity and a transition zoneof varying crystallinity around the crack tip in natural rubber. The semi-crystallinezone is found to take a semi-elliptic shape where the long and short axes are deter-mined to be in the range 0.15 − 0.85 mm and 0.08 − 0.48 mm, respectively. Zhanget al. [64] found that there are existed crystallites of the size of tens of nanometers

14 CHAPTER 3. MODELING CRACK GROWTH IN RUBBER

in the case of dynamically propagating crack in natural rubber. Further, Brüninget al. [3] investigated the influence of the deformation frequency on the crystallinitylevel around a crack tip in natural rubber under dynamic load using scanning wide-angle X-ray diffraction (WAXD). They have shown that the crystallinity at thecrack tip is considerably dropped in comparison to the quasi-static crack tip (abouthalf of the crystallinity). Moreover, the height of semi-crystalline zone is obtainedto be of size of ∼ 1.0 mm in the case of quasi-static and ∼ 0.5 mm in the dynamiccase.

3.1.3 The role of filler-polymer disordering and cavitation

damage

The stress softening (Mullins effect) and strain-induced cavitation damage phenom-ena are characteristics of filled rubber and rubber-like material. These phenomenaare explained by mechanisms like the molecular slippage and/or debonding fromthe surface of the filler particles or any other defects (e.g. weak region in the rubbernetwork, actual voids, and undesirable material like dust). Under deformation, thefiller-polymer disordering increases giving rise to nucleation of nano-cavities andresulting in the stress softening phenomenon (at least partially). These cavitiesare expected to deform elastically to a limit and thereafter inelastic growth by afracture process may take place. In the vicinity of a propagating crack the localstress is very high, therefore, the filler-polymer will be disordered and formationand growth of cavities are expected. Further, the experimental results show thatthe new crack surfaces are formed by linking of secondary cracks that are formedfrom these cavities. Therefore, a disordered and cavitated region around the cracktip is expected and its size is dependent on the stretch level around the crack tip.

3.1.4 Modeling quasi-static crack propagation

The quasi-static crack propagation takes place in low velocities (∼ 1 m/s) in whicha blunted crack tip is usually formed. The propagation process takes long time suchthat there is enough time for the crystals to form as well as the viscous dissipationto take place, i.e. ω ∼ 1/τv and ω ∼ 1/τc. In the case blunted crack tip, a bigregion is expected to be subjected to large deformation. Hence, a large regionwill be subjected to filler-polymer disordering and cavitation damage including theregion where the new crack surfaces will be formed. The total work of fractureis equal to the energy dissipated in the viscoelastic, filler-polymer disordering andcavitation processes, and the energy stored in the crystallization, around the cracktip. Figure 3.3 shows the different fracture processes around a quasi-staticallypropagating crack in rubber.

3.2. COMPUTATIONAL FRAMEWORK 15

vc

Semi-crystalline

zoneCavitation and disordering

zone

Glassy zoneViscous dissipation

zone

Rubbery zone

Figure 3.3: The fracture processes around a crack propagating quasi-statically in rubberwith a speed vc.

3.1.5 Modeling dynamic crack propagation

In the case of dynamic fracture, the crack propagates in relatively high velocitiesforming a wedge-like shape crack tip. Hence, the deformation around the crack tipis assumed to be described by viscoelasticity theory, i.e. ω ∼ 1/τv. The propagationprocess time is very short, therefore, the role of the strain-induced crystallizationis assumed to be minor, i.e. ω >> 1/τc. In the wedge-like crack shape, thedeformation is much localized (the crack tip is nearly sharp) in the tip regionimplying that the cavitation damage will take place within that region. Thus, thecavitation role is limited to the formation of the new crack surface where a cohesivezone model is assumed. Hence, the microstructural failure mechanisms are directlyrelated to the bulk deformations where the viscoelastic dissipative mechanisms takeplace, i.e. the size of the fracture process zone, where the damage takes place, isdetermined by length scale (lcz) [54].

The total work of fracture is the sum of different contributions: the surfaceenergy required to create new crack surfaces, the energy dissipated in the viscoelas-tic processes around the crack tip, and the inertia effects. Figure 3.4 shows thedifferent fracture processes around a dynamically propagating crack in rubber.

3.2 Computational framework

In the following subsections the implementations of the previously discussed theo-retical framework of dynamic crack propagations is discussed.

3.2.1 Dynamic crack propagation

A computational framework for dynamic crack propagation in rubber is proposedin which a nonlinear finite element analysis using cohesive zone modeling approach

16 CHAPTER 3. MODELING CRACK GROWTH IN RUBBER

vc

Separation

process

Glassy zoneViscous dissipation

zone

Rubbery zone

0δc

xb

δf

xa

0

σc

2γs

δ, x

T

lcz = xb − xa

δf δcx

y

(a)

(b)

Figure 3.4: The fracture processes around a crack propagating dynamically in rubber witha speed vc: (a) the different zones around the propagating crack tip determined by theviscoelastic behavior; and (b) the cohesive process zone and its traction-separation law(T − δ). 2γs is the surface energy, lcz is the length of the cohesive zone, δc is the criticaldisplacement, δf is the failure displacement, and σc is the cohesive strength.

is used. The rubber material is assumed to be characterized by finite-viscoelasticitytheory and coupled with the fracture processes using a cohesive zone model. Bothrate-independent and rate-dependent cohesive models are implemented such thatthe later accounts for the viscous dissipation at the crack tip vicinity. A nonlinearfinite element analysis using a mixed explicit-implicit integration is applied. Thecomputational framework is able to model and predict the different contributions tothe fracture energy, i.e. surface energy, viscoelastic dissipation and inertia effects.The problem of a suddenly initiated crack at the center of a biaxially stretchedsheet is studied under plane stress conditions. A steadily propagating crack isobtained and the corresponding crack tip position and velocity history as well asthe steady crack propagation velocity are evaluated for different load combinations.The numerical results are compared with experimental data [22] such that thecharacterization of the fracture process is investigated intensively, i.e. differentcohesive model parameters.

Chapter 4

Experimental work

In this chapter, the experimental work performed in a carbon-black natural rubbermaterial is presented. First, the experimental methods are discussed and then anew method for determining the critical tearing energy is introduced.

4.1 The experimental methods

The experimental methods used in determining the tearing behavior and energyunder different loading rates, temperatures, and fracture modes are presented inthis section.

4.1.1 Material

A carbon-black-filled natural rubber material is investigated in these experimentalstudies. The material is manufactured by TrelleborgVibracoustic under the desig-nation NR3233 and its chemical properties are listed in Table 4.1.

Table 4.1: The mix formulation in parts per hundred rubber by weight (phr) and ShoreA hardness of carbon-black-filled natural rubber (NR3233).

NR CB Plasticizer Additives Shore A

100 54 13 19 50

4.1.2 Specimens

Two different types of specimens were used to study crack growth, i.e. the crackedpure-shear specimens were used to study crack propagation pure mode I and thesingle edge notch specimens were used to study mixed mode I and II. The crackedpure-shear specimens were of width W0 = 110 mm, height H0 = 30 mm, andthickness B0 = 2.5 mm, see Fig. 4.1(a). The single edge notch specimens were of

17

18 CHAPTER 4. EXPERIMENTAL WORK

width W0, height 2H0, and thickness B0, see Fig. 4.1(b). The initial cracks in bothspecimens of length a = 30 mm were created using razor blades. In the crackedpure-shear specimens, the cracks were horizontal and an inclined crack with angleβ = 45o with the horizonal line were used in the single edge notch specimens.Both specimens were glued to metallic strips of length W0, height C0 = 30 mm,and thickness 5 mm. Thus, the specimens were mounted to the fixtures in thesemetallic strips where the boundary displacement was applied.

W0

H0

C0

C0

a

W0

H0

H0

C0

C0

a

β

(a)

(b)

Figure 4.1: The specimens for the fracture tests: (a) cracked pure-shear specimen; (b)single edge notch specimen.

In order to determine the critical tearing, uncracked pure-shear specimens weretested. The uncracked specimens were of the same width, height, and thickness ofthe cracked pure-shear specimens.

4.1.3 The experimental procedures

A standard servo-hydraulic test machine of load capacity 50 KN was used. Thedifferent specimens were monotonically loaded at a constant cross head speed untilthe complete failure. Further, the temperature was controlled such that heat was

4.1. THE EXPERIMENTAL METHODS 19

applied to the specimens using a heat lamp controllable with power regulator. Thespecimens were radiated and the temperature was controlled by a probe that wasconnected to the specimens. Therefore, an experimental setup is represented by aloading rate at a certain temperature. For every test, the load-displacement graphswere recorded and the crack growth points were marked during the test, and thecritical displacement was determined. Additionally, a high speed camera at upto 7000 frames/s was used to follow the progress of the crack and later a post-processor was used to obtain the crack trajectory and velocity at different stages.The experimental setup in the case of cracked pure-shear specimen is shown inFig. 4.2.

High speed

camera

White

backgroundSpecimen

Load cell

Flash

δ, δ

x1

x2

x3

(a)

(b)

Figure 4.2: The experimental setup: (a) the loading machine, the cracked pure shearspecimen and the high speed camera; (b) the cracked pure shear specimen, the loadingdirection and the experiment frame.

In the proposed method for determining the tearing energy, the cracked pure-shear specimens were monotonically loaded at cross head speed of 0.3 mm/s untilcomplete failure. The uncracked pure-shear specimens were subjected to a cycle

20 CHAPTER 4. EXPERIMENTAL WORK

of loading and unloading. They were monotonically loaded until the critical dis-placement, i.e. obtained from the cracked pure-shear specimen, was reached andthen they were unloaded completely. The loading cross head speed was kept at0.3 mm/s, as in the case of the cracked specimens. During unloading, the crosshead speed was varied to investigate the effect of the unloading rate. Therefore,three unloading cross head speeds were used, i.e. 0.3, 3.0 and 30.0 mm/s. Threespecimens per unloading rate were tested such that nine specimens were used intotal.

In the investigation of the tearing behavior and the tearing energy, the crackedpure-shear and the single edge notch specimens were used for investigating bothpure mode I and mixed mode I and II fracture behavior, respectively. Further,the loading rate and temperature effects were studied using the cracked pure-shearspecimen. Therefore, the specimens were tested at different cross head speeds of0.3, 30.0 and 450.0 mm/s and at the ambient temperature 22-25◦C and elevatedtemperature of 90◦C. Hence, six different experimental setups were used. The singleedge notch specimens were tested using a single cross head speed of 0.3 mm/s at theambient temperature. The proposed method for the determination of the tearingenergy is used. Therefore, the uncracked pure shear specimens were subjectedto a cycle of loading and unloading such that at least three specimens were usedfor each experimental setup. They were monotonically loaded until the criticaldisplacement (i.e. obtained from the cracked pure-shear specimens at the sameexperimental point) was reached and then they were unloaded completely. Thecross head speed was kept the same for the loading and unloading for the practicalpurpose (the effect of the unloading rate on the elastic energy were found to beminimal [12]).

4.2 A new method for determining the tearing energy

A new method for estimating the critical tearing energy in rubber-like materials isproposed in Paper D. In this method, the energy required for crack propagationin a rubber-like material is determined by the change of the recovered elastic en-ergy. Hence, the dissipated energy due to different inelastic processes (cavitationdamage, breakage of filler-polymer bonds, strain induced crystallization, etc.) isdeducted from the total strain energy applied to a system. For the cracked pureshear specimen in Fig. 4.1(a), the critical tearing energy can be evaluated from

Tc = Ψ · H0, (4.1)

where Ψ is the true elastic free energy stored in the specimen per unit referencevolume. Hence, the change in the total internal energy per unit reference volume,E, can be divided into the change in the true elastic free energy per unit referencevolume Ψ, heat and dissipation energy per unit reference volume Q, and free energyin other forms per unit reference volume Ψ′ (e.g. free energy stored as surfaceenergy between the amorphous and crystalline phase during the strain-induced

4.2. A NEW METHOD FOR DETERMINING THE TEARING ENERGY 21

crystallization), i.e. E = Ψ+Q+Ψ′. The true elastically stored energy in pure shearis determined using cyclic loading such that the recovered energy after unloading isthe true elastic energy. Therefore, consider the load-displacement curves of crackedand uncracked pure shear specimens in Figure 4.3, the elastically stored energy perunit reference volume is the area under the unloading curve and the area betweenthe loading and unloading curves is associated with Q and Ψ′.

δcδp

qp

Ψ V0

(

Q + Ψ′

)

V0

a = 0

a > 0

Crack

initiation

δ

F

Figure 4.3: The schematic of the load-displacement curves for the pure shear test: the lightgray area is the elastic stored energy, Ψ, and the dark gray shaded area is the summationof the heat and dissipation energy, Q, and the free energy in other forms, Ψ′, in theuncracked specimen at the critical displacement (δc). V0 = W0H0B0 is the volume of theuncracked specimen in the reference configuration.

Chapter 5

Results and Discussion

In this chapter, the main results of the research work are reported and discussed.The experimental as well as the modeling work results are presented.

5.1 Modeling dynamic crack propagation

The computational framework for dynamic crack propagation was used to studythe problem of a suddenly initiated crack at the center of a biaxially stretched sheetis studied under plane stress conditions (Papers A and B). A steadily propagatingcrack is obtained and the corresponding crack tip position and velocity history aswell as the steady crack propagation velocity are evaluated for different load com-binations.The total work of fracture at different stretch levels was determined by fitting theexperimental data from Gent and Marteny [22], see Fig. 5.1(a). The total work offracture results in Fig. 5.1(b) shows a controversial behavior implying that theremight be a competition between different fracture related processes. The workneeded for creating new crack surfaces might decrease in the high stretch levelsdue to the highly stretched rubber chains. Further, it has been suggested that thecomplex dissipative process in the bulk material surrounding the crack tip might berelated to the crystallization and cavitation processes [13]. Under increasing trans-verse stretch, the numerical results show that the cavitation decreases suggestingthat the amount of work required to open cavities decreases. The recent exper-imental studies on the strain induced crystallization in a dynamic process showthat there are two different behaviors, depending on the elongation level as well asthe strain rate [4]. At high strain rates, a slow crystallization occurs while a fastcrystallization process takes place at low strain rates and a characteristic time of20 ms is obtained for the test material. At the crack tip vicinity, high stretches areexpected and therefore a complex crystallization pattern might take place in whichboth behaviors are expected.The numerical results show that in the case of fast propagation, i.e. vc > 50 m/s,

23

24 CHAPTER 5. RESULTS AND DISCUSSION

0

20

40

60

80

100

120

140

1 2 3 4 5

λ2 [-]

v c[m

/s]

ut ut ut ut

ut

utut

bc bc bc

bcbc

bc

bcbc

bc

bcbc

rs

rs

rs

rs

rs

rs

rs

rs rs

0

10

20

30

40

50

60

1 2 3 4 5

λ2 [-]G

Ic[K

J/m

2]

bc

bc

bc

bc

bc

bc

bc

bc

bc

bc

bcbc

bc

bc

(a) (b)

Figure 5.1: The estimation of the total work of fracture: (a) Comparison between numeri-cal predictions of the steady crack propagation velocity (the red, blue and green lines) andexperimental data (’N’, ’•’, ’�’) [21] in the stretch cases λ1 = 1.0, 1.5 and 2.0, respectively;(b) The total work of fracture at different stretch levels in X2-direction: the red, blue andblack lines represent the stretch levels in X1-direction λ1 = 1.0, 1.5 and 2.0, respectively.

the fracture can be predicted using a time-dependent traction-separation law ofKelvin-Voigt element type. Therefore, the total work of fracture consists of thesurface energy, viscoelastic dissipation in the inner most of the crack tip, and iner-tia effects. In the case of relatively slow crack propagation, the results show thatfracture-related processes, i.e. creation of new surfaces, cavitation and crystalliza-tion; contribute to the total work of fracture in a contradictory manner.

5.2 Experimental determination of crack propagation

Fracture experiments using cracked pure-shear specimens and single edge notchspecimens were used to study the tearing behavior and the tearing energy (Papers

D and E). The crack propagation profile was determined in different loading rates,temperatures, and fracture modes. In the different experiments, the crack prop-agates in low velocity showing stick-slip pattern and as the loading continues asmooth faster propagation takes place. Further, the lower the loading rate, themore blunted the crack tip and the larger stick-slip region. A wedge-like cracktip is resulted in the fast propagation. Fig. 5.2(a) shows the crack tip at differentinstants in which the region of interest is a domain of length of 50 mm and heightof 20 mm in the case of loading rate of cracked pure-shear specimens loaded atloading rate of 450.0 mm/s and temperature 25◦C. The crack propagation results,including the crack tip position and velocity, are shown in Figs. 5.2(b) and (c),respectively. At t = 0, the crack is of initial length of a = 30 mm, see Fig. 5.2(a).

5.2. EXPERIMENTAL DETERMINATION OF CRACK PROPAGATION 25

The crack propagates at t = 12.96 ms at very low velocity showing stick-slip pat-tern with a blunted tip. The slow propagation continues for approximately 19 msand thereafter the crack accelerates for 13 ms before the fast propagation region.Forming a wedge-like tip, the crack propagates smooth and fast until the specimenis totally fractured. Further, in the fast propagation, the crack tip velocity showssmall oscillations.

0 1cm 0 1cm

0 1cm 0 1cm

(i) t = 0.0 ms (ii) t = 20.0 ms

(iii) t = 40.0 ms (iv) t = 50.0 ms

0

10

20

30

40

50

0 10 20 30 40 50 60

t [ms]

∆a

[mm

]

0

1

2

3

4

5

6

0 10 20 30 40 50 60

t [ms]

v tip

[m/s]

(a)

(b) (c)

Figure 5.2: The experimental results at loading rate of 450.0 mm/s: (a) the propagatingcrack at different instants of time; (b) crack extension ∆a vs time t; and (c) crack tipvelocity vtip vs time t.

26 CHAPTER 5. RESULTS AND DISCUSSION

5.3 The critical tearing energy estimation

A method for determining the critical tearing energy in rubber-like materials wasproposed in Paper D. The method is estimating the critical tearing using the recov-ered elastic energy. Hence, the dissipated energy due to different inelastic processes(cavitation damage, breakage of filler-polymer bonds, strain induced crystallization,etc.) is deducted from the total strain energy applied to a system. In this investi-gation, the pure shear tear test is used to determine the critical tearing where theactual elastic stored energy in pure shear is determined experimentally using cyclicloading. Fig. 5.3 shows comparison between the classical and proposed methods inthe case of the pure shear tear test. The results indicate that the classical methodoverestimates the critical tearing energy by approximately 15% including differentunloading.

0.0

2.0

4.0

6.0

8.0

10.0

10−1 100 101 102

log δ [mm/s]

Tc

[kJ/m

2]

bc bc bc

bc bc bc

Figure 5.3: The critical tearing energy-unloading rate relation. The black and red dotsand lines indicate the classical and modified methods respectively.

Chapter 6

Conclusions and Future Work

In this thesis, we attempted to study different aspects of crack growth in naturalrubber material. The dynamic crack propagation in rubber was firstly investi-gated. A theoretical framework was developed and implemented into a computa-tional framework in the Finite Element environment. These frameworks model thefracture process using a cohesive zone model and the bulk rubber material usingfinite-viscoelasticity theory. Then, a theoretical framework for quasi-static crackpropagation in rubber was developed wherein the different mechanisms, i.e. strain-induced crystallization, cavitation damage, stress softening, and viscoelastic dissi-pation, are assumed to play an important role. Therefore, a multiscale continuummodel that incorporates strain-induced crystallization and cavitation damage wasdeveloped. Further, we performed a series of fracture experiments on carbon-blackfilled natural rubber. A new method for determining the critical tearing energy inrubber-like materials was proposed. Moreover, the tearing behavior and the tearingenergy were investigated under different loading rates, temperatures, and fracturemodes.

The main conclusions that can be drawn from the results are summarized inthese points:

• In the case of dynamic crack propagation at high speed (vc ≥ 50 m/s), the vis-cous dissipation in the bulk material around the crack tip is negligible. Hence,the total work of fracture consists of the surface energy, viscous dissipationin the innermost region at the crack tip, and inertia effects.

• In the case of crack propagation at low speed (vc < 50 m/s), the viscousdissipation in the bulk material and the strain-induced crystallization aroundthe crack tip may come to play a role. Thus, the total work of fractureconsists of the surface energy, viscous dissipation in the bulk material and theinnermost region at the crack tip, and the free energy stored in the strain-induced crystallization process. Further, the fracture-related processes, i.e.

27

28 CHAPTER 6. CONCLUSIONS AND FUTURE WORK

creation of new surfaces, cavitation and crystallization; contribute to the totalwork of fracture in a contradictory manner. The inertia effects are negligible.

• A significant overestimation of the critical tearing energy may result fromusing the total energy stored in a rubber-like material. Using the recoveredelastic energy, i.e. the total energy stored in a rubber-like material is di-vided into elastic and inelastic contributions taking into account the differentinelastic processes, shows that the difference is approximately 15%.

• The tearing behavior of carbon-black filled natural rubber shows a transitionin which the tear surface changes from being a relatively rough at low tearingrates to a much smoother at fast propagation. This behavior is observed atdifferent loading rates, temperatures, and fracture modes. Further, a smallerstick-slip region results at higher loading rates. Moreover, the temperatureaffects the tearing behavior in a similar way. In the mixed mode I and II,the transition from the stick-slip to smooth propagation takes place at thechanging to pure mode I propagation. The dissipative mechanisms associatedwith the crack growth appear to influence the transition.

Based on the thesis findings, the suggestions for the future work are:

• A computational framework for quasi-static crack propagation in rubber couldbe implemented in the Finite Element method. In this framework, the mul-tiscale continuum model for rubber-like materials in Paper C can be usedtogether with large strain viscoelasticity model (i.e., Holzapfel and Simo [31]and Holzapfel [30]). Hence, the rubber material is taken to be viscoelasticwhere the deformation is controlled by cavity nucleation and growth damage,and crystallization. Moreover, the crack propagation process is driven by thecavitation damage. In the Finite Element implementation, the non-local dam-age approach can be used in order to overcome the deficiencies introduced bythe local approach (e.g. mesh dependency).

• In the experimental work, the experiments were performed at two tempera-tures and three loading rates. More experimental setups could be used formore accurate characterization of the fracture behavior of the material. Moreexperimental setups may help in understanding the stick-slip to smooth prop-agation transition as well as the different factors.

Summary of appended papers

Paper A: Numerical analysis of dynamic crack propagation in rubber.

In this paper, the computational framework for dynamic crack propagation inrubber is introduced. The scope is to model dynamic crack propagation in rubbermaterials as well as to investigate the different contributions to the total fractureenergy, i.e. surface energy, viscoelastic dissipation, and inertia effects. The problemof a suddenly initiated crack at the center of an axially stretched sheet is studiedunder plane stress conditions. The bulk material behavior is described by finite-viscoelasticity theory and the fracture separation process is characterized using arate-independent cohesive zone model with a bilinear traction-separation law. Theseparation work per unit area and the strength of the cohesive zone have been pa-rameterized, and their influence on the separation process has been investigated.A full nonlinear dynamic analysis is carried out for the crack propagation analysis,while a nonlinear quasi-static analysis is used for the initial loading.The results show that the contribution from viscoelastic dissipation in the bulkmaterial to the total work of fracture is negligible. The effective fracture energy ofthe rubber material at hand is to be found roughly in the range 20-40 kJ/m2, andthe cohesive strength is expected to be approximately 60-90 MPa. The estimatedfracture energy is taken to be the total work for fracture such that it contains notonly the actual surface energy but also significant amounts of dissipation associ-ated with damage processes in the vicinity of the crack tip. Further, the effectivefracture energy reported in the literature shows good agreement with the values at-tained here. Finally, the maximum principal stretch and stress distribution arounda steadily propagation crack tip suggest that crystallization and cavity formationmay take place and therefore they may be related to the complex damage processesaround the crack tip vicinity.

Paper B: Numerical analysis of dynamic crack propagation in biaxially strainedrubber sheets.

In this paper, the computational framework, that was introduced in (Elmukashfiand Kroon in Int J Fract 169:49-60,2012), is extended by generalizing the loadingcase to a biaxial stretch state and introducing a rate-dependent cohesive model formodeling the viscous dissipation in the crack tip vicinity. A Kelvin-Voigt element is

29

30 CHAPTER 6. CONCLUSIONS AND FUTURE WORK

used to describe the rate-dependent cohesive model wherein the spring is describedby a bilinear law and the dashpot by a constant viscosity is adopted. An explicitintegration is used to incorporate the rate-dependent cohesive model in the FiniteElement environment. Convergence study, i.e. a critical evaluation of the cohesiveparameters effect on the crack propagation and the discretization requirements, isconducted. A parametric study over the cohesive viscosity is performed and thesteady crack propagation velocity is evaluated and compared with experimentaldata. Furthermore, the total work of fracture is estimated using rate-independentcohesive law such that the strength of the cohesive zone is assumed to be constantand the separation work per unit area is determined form the comparison with theexperimental data.The analyses indicate that the viscosity is to be found roughly in the range 0.125-0.750 kPa · s/m and varies with the crack speed. The study of the total work offracture shows a controversial behavior implying that there might be a competitionbetween different fracture related processes. The work needed for creating newcrack surfaces might decrease in the high stretch levels due to the highly stretchedrubber chains while the cavitation decreases as the transverse stretch increases.Furthermore, the experimental studies suggest that a complex crystallization pat-tern might take place in the crack tip vicinity [4].

Paper C: A multiscale continuum modeling of strain-induced cavitation damageand crystallization in rubber-like materials.

A multiscale continuum model for strain-induced cavitation damage and crys-tallization for rubber-like materials is proposed in this paper. The constitutivebehavior is determined by homogenization over different length scales, namely, thenano-scale, micro-scale and macro-scale. The microstructure of a filled rubber-likematerial is seen as interaction between clusters of the filler particles and long-chainmolecules that form two networks, between cross-links and between the filler aggre-gates. The network between cross-links in the nano-scale is modeled using the fullnetwork approach of semi-crystalline chains. A phenomenological law is proposedto describe the crystallite nucleation law. The network between the filler parti-cles is described by statistical mechanics in the nano- and/or micro-scale wherethe polymer chains sliding on and/or debonding from filler aggregate surface isincorporated. The debonding process is regarded as the main mechanism of thenucleation of nano-cavities which introduces non-affine deformation to the networkbetween cross-links. Hence, The nanoscopic initially non-cavitated network betweencross-links is homogenized over the micro-scale assuming a spherical representativevolume element using the kinematics proposed by Hang-Sheng and Abeyaratne(Hang-Sheng and Abeyaratne in Journal of the Mechanics and Physics of Solids 40(3), 571-592, 1992). The constitutive model is presented in form of an averagedstrain energy function. The predictive capabilities of the model are then tested viacomparisons with experimental data from literature.

31

Paper D: An experimental method for estimating the tearing energy in rubber-like materials using the true stored energy.

In this paper, a method for determining the critical tearing energy in rubber-likematerials is proposed. In this method, the energy required for crack propagationin a rubber-like material is determined by the change of the recovered elastic en-ergy. Hence, the dissipated energy due to different inelastic processes (cavitationdamage, breakage of filler-polymer bonds, strain induced crystallization, etc.) isdeducted from the total strain energy applied to a system. Therefore, the classicalmethod proposed by Rivlin and Thomas (Rivlin and Thomas in Journal of Poly-mer Science 10(3):291-318,1953) using the pure shear tear test is modified using theactual stored elastic energy. The elastically stored energy in a pure shear is deter-mined experimentally using cyclic loading. The experimental results show that theclassical method overestimates the critical tearing energy by approximately 15%.Moreover, the effect of the unloading rate on the determination of the elasticallystored energy is investigated and found to be minimal suggesting that the crackpropagation velocity has a minor effect in the change of the elastically stored en-ergy.

Paper E: Experimental investigations of crack propagation in rubber under dif-ferent loading rates, temperatures and fracture modes.

The fracture behavior of carbon-black natural rubber material is experimen-tally investigated in this study. The cracked pure shear and the single edge notchspecimens were used for investigating both pure mode I and mixed mode I and IIfracture behavior, respectively. Further, different testing conditions were employedin the case of the cracked pure shear specimens. The specimens were subjected tothree different loading rates and they were tested in two different temperatures.For studying the crack growth, a high speed camera at up to 7000 frames/s wasused to follow the progress of the crack and later a post-processor was used to ob-tain the crack trajectory and velocity at different stages. The method introducedpreviously by the present author (Elmukashfi in report 580, Department of SolidMechanics, Royal Institute of Technology (KTH), 2015) was used to obtain the crit-ical tearing energy using the cracked pure shear specimens. Hence, the uncrackedpure shear specimens were subjected to cyclic loading history in order to obtain thetrue elastic energy in pure shear. The single edge notch specimens were tested inroom temperature under quasi-static loading. The pure mode I results suggest thatthe critical tearing depends strongly on the loading rate as well as the temperature.The tearing behavior shows stick-slip pattern at low tearing rates and smooth prop-agation at high velocities. The size of the stick-slip region is reduced significantlyby increasing the loading rate as well as the temperature. In the mixed mode I andII, the transition from the stick-slip to smooth propagation and the transition frommixed mode I and II take place approximately simultaneously.

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