on the fabrication, characterization and …...naguib’s lab to do some of the polymer...

On the Fabrication, Characterization and Mechanical Properties

of Melt-Stretched Stochastic Honeycombs

by

Megan Hostetter

A thesis submitted in conformity with the requirements for the degree of

Doctor of Philosophy

Department of Materials Science and Engineering

University of Toronto

© Copyright by Megan Hostetter (2015)

ii

On the Fabrication, Characterization and Mechanical Properties of

Melt-Stretched Stochastic Honeycombs

Megan Hostetter

Doctor of Philosophy 2015

Department of Materials Science and Engineering

University of Toronto

Abstract

This thesis presents a new type of polypropylene (PP) cellular material fabricated through a simple melt-stretching process.

Stochastic honeycombs have an open cell, random honeycomb structure, with webs oriented perpendicular to built-in skins.

This process has the advantage that, for example, PP pellets can be turned into a sandwich panel in one step. It was

demonstrated that despite the randomness in the web structure, the out-of-plane compressive strength of stochastic

honeycombs was repeatable, and exceeded that of commercial PP foams and was comparable to commercial PP

honeycombs.

The key material properties required to produce an this architecture were shown to be a high melt strength and a high

viscosity, branched polymer. The viscosity was shown to affect the total length of the webs in cross-section and the relative

partitioning of material through the skin, transition region and webs. Web thickness was affected by the areal density of the

polymer during fabrication.

Mechanical testing methods were adapted from ASTM standards for honeycombs, and the fabrication method was

advanced from a manual to a machine controlled process. Stochastic honeycombs were shown to buckle elastically,

plastically, and fracture after the peak strength. Elastic and plastic buckling were dominant at lower densities, and plastic

buckling and fracture at higher densities.

A thin-plate buckling model for the strength of stochastic honeycombs was developed and verified experimentally.

The crystallinity of the polymer affected the tensile strength and stiffness, having a linear effect on the buckling strength.

The architecture was composed of webs bound on both sides and webs bound on one side and free on the other. A greater

fraction of bound webs increased the strength of the structure in the buckling model. A fabrication study showed that melt-

stretching the polymer at higher strain rates increased the connectivity and fraction of bound webs. Additionally, higher

density led to a less connected structure. Energy absorption could be improved if the drop in stress from the peak to valley

strength was minimized. Higher melt-stretching rates during fabrication increased the energy absorbed, and higher viscosity

polymers minimized the drop from peak to valley, also increasing the absorbed energy.

iii

Acknowledgements

I would like to extend my sincerest thanks to Prof. Glenn Hibbard, for his support and guidance throughout this

project. Your insights have been invaluable, and you have consistently pushed me to look deeper and think more

closely about the project, and to constantly re-think my assumptions.

I would also like to thank my committee members, Prof. Mansoor Barati, Prof. Keryn Lian and Prof. Hani

Naguib, for their valuable comments and discussions over the years. Additionally, I was granted access to Prof.

Naguib’s lab to do some of the polymer characterization which would not have otherwise been possible. My

thanks also go out to Dr. Dan Grozea, for his patience and assistance when I used his labs for fabrication and

testing, and to Sal Boccia, for his assistance with the scanning electron microscope and polarized light

microscopy.

My thanks also go out to past and current members of the Cellular Hybrid Materials Research Group, for

their valuable discussions and company over the years: Dr. Brandon Bouwhuis, Dr. Evelyn Ng, Dr. Eral Bele,

Bosco Yu, Dr. Balaji Devatha Venkatesh, Karen Chien, Adam Bird, Adam Yaremko, Ante Laušić and Khaled

Abu Samk. Additionally, I would like to thank the summer students who contributed over the years, especially

Vinson Truong who came back for a second year.

Finally, Vajira, thank you for your never-ending faith in me, as well as your understanding and patience

through this process. Thank you to my brothers, who have always and continue to inspire me to do more and go

farther, just to keep up with both of you. And last, but definitely not least, I thank my parents for teaching me

that with enough hard work, nothing is impossible, and to always keep reaching for more.

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Table of Contents

Abstract ......................................................................................................................................................... ii

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

Table of Contents .............................................................................................................................................. iv

List of Figures.................................................................................................................................................. vii

List of Tables .................................................................................................................................................... xv

List of Symbols and Abbreviations ............................................................................................................... xvii

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

1.1 Cellular Materials ............................................................................................................................... 1

1.2 Melt-Stretched Stochastic Honeycombs ............................................................................................ 4

1.3 Overview ............................................................................................................................................ 6

References .............................................................................................................................................. 10

Chapter 2 Background ...................................................................................................................................... 12

2.1 Honeycomb Sandwich Panels .......................................................................................................... 12

2.2 Melt Rheology and Structure of Polypropylene ............................................................................... 15

2.3 Buckling of Thin Plates as it Relates to Honeycombs ..................................................................... 18

2.4 Summary .......................................................................................................................................... 23

References .............................................................................................................................................. 24

Chapter 3 Melt-Stretched Honeycombs ........................................................................................................... 31

Iteration 1: Manual Press ........................................................................................................................ 31

Iteration 2: Fully-Instrumented Process ................................................................................................. 33

References .............................................................................................................................................. 36

Chapter 4 Architecture – Process Relationships in Stochastic Honeycombs ................................................... 37

4.1 Introduction ...................................................................................................................................... 37

4.2 Experimental Methods ..................................................................................................................... 37

4.3 Results and Discussion ..................................................................................................................... 38

4.3.1 Architecture ......................................................................................................................... 38

4.3.2 Fabrication .......................................................................................................................... 42

4.4 Conclusions ...................................................................................................................................... 48

References .............................................................................................................................................. 49

v

Chapter 5 Architectural Characteristics of Stochastic Honeycombs Fabricated from Varying Melt

Strength Polypropylenes ................................................................................................................. 51

5.1 Introduction ...................................................................................................................................... 51

5.2 Experimental Procedures .................................................................................................................. 52

5.3 Results and Discussion ..................................................................................................................... 53

5.3.1 Polymer Characterization .................................................................................................... 53

5.3.2 Architectural Characterization ............................................................................................ 56

5.4 Conclusions ...................................................................................................................................... 62

References .............................................................................................................................................. 63

Chapter 6 Stochastic Honeycomb Sandwich Cores.......................................................................................... 65

6.1 Introduction ...................................................................................................................................... 65

6.2 Experimental .................................................................................................................................... 66

6.3 Cellular Architecture ........................................................................................................................ 67

6.4 Mechanical Properties ...................................................................................................................... 69

6.4.1 Uniaxial Compression ......................................................................................................... 69

6.4.2 Three-Point Bend Testing ................................................................................................... 71

6.5 Conclusions ...................................................................................................................................... 75

References .............................................................................................................................................. 77

Chapter 7 Modeling the Buckling Strength of Polypropylene Stochastic Honeycombs .................................. 78

7.1 Introduction ...................................................................................................................................... 78

7.2 Experimental Details ........................................................................................................................ 78

7.3 Results .............................................................................................................................................. 79

7.3.1 Polymer Characterization .................................................................................................... 79

7.3.2 Honeycomb Architecture .................................................................................................... 81

7.3.3 Mechanical Properties of Stochastic Honeycombs ............................................................. 83

7.4 Discussion ........................................................................................................................................ 84

7.5 Conclusions ...................................................................................................................................... 88

References .............................................................................................................................................. 89

Chapter 8 Post-Peak Collapse and Energy Absorption in Stochastic Honeycombs ......................................... 92

8.1 Introduction ...................................................................................................................................... 92

8.2 Experimental Details ........................................................................................................................ 93

vi

8.3 Results .............................................................................................................................................. 94

8.4 Discussion ...................................................................................................................................... 102

8.5 Conclusions .................................................................................................................................... 107

References ............................................................................................................................................ 108

Chapter 9 Summary and Future Work ............................................................................................................ 110

9.1 Fabrication ...................................................................................................................................... 111

9.2 Internal Architecture ...................................................................................................................... 111

9.3 Mechanical Properties .................................................................................................................... 113

9.4 Future Work ................................................................................................................................... 114

Appendix A Polymer Properties, PP-1 to PP-5 .............................................................................................. 116

References ............................................................................................................................................ 117

Appendix B Melt-Stretched Honeycombs with Various Polymers ................................................................ 118

Appendix C Characterization of Stochastic Honeycomb Sandwich Failure .................................................. 121

C.1 Introduction ................................................................................................................................... 121

C.2 Experimental Details ..................................................................................................................... 122

C.3 Results and Discussion .................................................................................................................. 123

C.4 Conclusions ................................................................................................................................... 133

References ............................................................................................................................................ 134

vii

List of Figures

Figure 1.1 Strength–density property map [4]. .....................................................................................2

Figure 1.2 Honeycombs with varying shape: hexagonal [5] (a), hexagonal and triangular [5] (b), triangular

[5] (c), square [5] (d), circular [6] (e) and Voronoi [7]. .......................................................3

Figure 1.3 A high melt strength PP stochastic honeycomb seen from the side in an unreconstructed X-ray

image. The image shows the as-fabricated skins and the core. ............................................4

Figure 1.4 3D reconstructions of a melt-stretched stochastic honeycomb fabricated with high melt strength

PP (a) and linear PP (b), with the as-fabricated skins edited out. The scale bar in (b) is 5 mm and

applies to both images. .........................................................................................................5

Figure 1.5 Two space-filling model plots of the high melt strength PP sample shown in Figure 1.4, from the

top (a) and a sectioned piece from the centre (b). An example of the buttressing structure is

circled in (a), and a hole in the web indicating an archway is shown with the arrow in (b). The

scale bar in (b) is 5 mm and applies to both images. ...........................................................6

Figure 1.6 Flow chart detailing the four key topics in this thesis and the contributions made in each. The

chapter that these contributions are discussed is indicated through the numbers on the left.

............................................................................................................................................... 9

Figure 2.1 Cellular material sandwich panels, with foam, truss and honeycomb core materials, top to

bottom [8]. .........................................................................................................................13

Figure 2.2 Sandwich panel failure mechanisms [23] ..........................................................................13

Figure 2.3 Closed [34] (a) and open [35] (b) cell foams. ....................................................................14

Figure 2.4 In-plane (x-y) and out-of-plane (z) directions on a honeycomb sandwich panel [10]. ......14

Figure 2.5 2D Voronoi honeycomb with a random but even spacing of points [36] (a) and a completely

random spacing of points [37] (b). .....................................................................................15

Figure 2.6 Schematic of the effect of branching and cross-linking on the tensile strength of PP [100].

...........................................................................................................................................17

Figure 2.7 Schematic of a cross-link [72] (a) and long chain branching [100] (b) in a polymer chain, where

Mbp is the average molecular weight between branch points and MLCB is the average molecular

weight of the branches. ......................................................................................................18

Figure 2.8 Strain hardening (a), strain softening (b) and perfectly plastic (c) stress-strain curves [113].

...........................................................................................................................................19

Figure 2.9 Buckling modes of a square honeycomb in out-of-plane compression [114]. ...................19

Figure 2.10 A schematic of a thin plate, with height a and width b, and thickness t in the x-direction.

...........................................................................................................................................20

Figure 2.11 A thin plate under compression with m = 5 half-waves [121]. ..........................................21

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Figure 3.1 The inside of the oven used for the manual press, showing the upper and lower platens, the

polymer on the lower platen, and the thermocouple leads (highlighted to increase visibility) (a),

and the manual press locked with the bars on either side in the stretched position as the sample

cools, with the upper platen visible (b). .............................................................................32

Figure 3.2 A stochastic honeycomb fabricated with a high melt strength PP (approximately 20 cm x 22 cm).

The circles indicate areas with flaws in the skin, and the arrow points to a corner where there

was not enough polymer and an interconnected structure was not formed. ......................33

Figure 3.3 A Shimadzu AG-I Universal Testing Machine, and the furnace, a Shimadzu TCE-N300

Thermostatic Chamber (a), with the aluminium platens and thermocouple leads within the

furnace (b). Once the thermocouple leads read 180 oC, the upper platen was lowered and the

furnace moved away (c). ....................................................................................................34

Figure 3.4 Stress (σ) as a function of time (t) during fabrication from the Shimadzu AG-I UTM. ....34

Figure 4.1 Examples of the widely different architectures formed over the fabrication space considered,

with melt-stretch rate = 10 mm min-1

and areal density ρA = 0.09 g cm-2

(a) and ρA = 0.23 g

cm-2

(b), and ρA = 0.15 g cm-2

and = 1 mm min-1

(c), and 100 mm min-1

(d). Scale bar indicates

a length of 20 mm. .............................................................................................................39

Figure 4.2 Process Map illustrating the range of fabrication space (melt-stretch rate, , and areal density,

ρA) considered. The points labelled a-d correspond to the 2D projections seen in Figure 4.1.

...........................................................................................................................................39

Figure 4.3 The 2D projections of the webs for five samples fabricated at = 10 mm min-1

and with ρA =

0.10 g cm-2

(a), 0.13 g cm-2

(b), 0.16 g cm-2

(c), 0.19 g cm-2

(d), and 0.23 g cm-2

(e). Scale bar

indicates a length of 20 mm. ..............................................................................................40

Figure 4.4 The 2D projections of the webs for five samples fabricated with ρA = 0.13 g cm-2

and = 1 mm

min-1

(a), 10 mm min-1

(b), 30 mm min-1

(c), 60 mm min-1

(d), and 100 mm min-1

(e). Scale bar

indicates a length of 20 mm. ..............................................................................................40

Figure 4.5 The number of nodes, N (a), and the total length of webs, L (b), plotted against ρA for = 10 mm

min-1

and 100 mm min-1

. ....................................................................................................40

Figure 4.6 The number of nodes, N, plotted against the total web length, L for melt-stretch rates of 10 mm

min-1

and 100 mm min-1

. ....................................................................................................41

Figure 4.7 2D projections for samples having the same areal density and N/L values (ρA = 0.21 g cm-2

and

N/L = 1.18 cm-1

) but with χ2D = -5 (a) and -25 (b). These samples were fabricated at = 10 mm

min-1

and = 100 mm min-1

respectively. Scale bar indicates a length of 20 mm. ...........41

Figure 4.8 Process Map of melt-stretch rate and areal density ρA, with all fabricated samples labelled by

the corresponding Euler number (χ2D), and overlaid with contour lines illustrating the trend in

N/L. ....................................................................................................................................42

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Figure 4.9 Full fabrication curve (σ, stress, and S, extension) showing all four stages, with compressive

stress expressed as negative and tensile stress as positive, with permanent plastic extension

beginning at the origin. The locations marked (i – iv) correspond to the images on the right. At

the beginning of fabrication, the melt (with thickness tblank, and diameter dblank) (i) is compressed

between Al platens in the furnace. In air, the melt (with tmelt, dfinal) is stretched uniaxially (ii), and

the stress increases to a maximum (iii). At the maximum, air ingress begins as the melt fractures,

creating the webs of the stochastic honeycomb (iv). The melt in the webs stretches and some of

the webs fracture, creating partial webs and archways until the sample reaches its final height (h,

dfinal) and cools in air. .........................................................................................................43

Figure 4.10 Stages II, III and IV on the fabrication curve for two samples fabricated with the same

elongation rate ( = 10 mm min-1

) but different densities, ρA = 0.09 g cm-2

and 0.23 g cm-2

(a),

and two samples fabricated with the same density (ρA = 0.14 g cm-2

) and two elongation rates,

= 1 mm min-1

and 100 mm min-1

(b). These curves correspond to the images in Figure 4.1.

...........................................................................................................................................44

Figure 4.11 Peak fabrication stress (σpk) is plotted as a function of areal density (ρA) for = 10 mm min-1

and

100 mm min-1

. ....................................................................................................................44

Figure 4.12 Peak fabrication stress, σpk, and apparent melt stiffness, Emelt (a), and extensional stress, σext (b),

pare plotted as a function of strain rate ( ) for samples with ρA = 0.13 g cm-2

. .................45

Figure 4.13 Apparent elongational viscosity, , is plotted against time, t, for five samples with ρA = 0.13 g

cm-2

and increasing elongation rate, (a), and the peak of the elongational viscosity with time,

, is plotted as a function of strain rate, , on a log–log scale for all samples. ...........46

Figure 4.14 The Process Map plotting elongation rate, , against areal density, ρA, showing the samples

fabricated, with variation in N/L shown in the data points. Contour lines illustrating the trends in

σpk (a), Emelt (b) and (c) are superimposed on top................................................47

Figure 5.1 Parallel-plate rheometry results, giving the viscosity (η) as a function of frequency (ω).

...........................................................................................................................................53

Figure 5.2 Storage modulus (G’) and loss modulus (G") plotted as a function of frequency (ω) for PP-2 and

PP-5. ...................................................................................................................................54

Figure 5.3 The two components of complex viscosity (η' and η") from parallel plate rheology tests: the

curves for PP-1, PP-2, PP-3 and PP-4 each exhibit a linear portion followed by an increase in

slope. This implies that long-chain branching is present in the polymers. PP-5 exhibits a semi-

circular curve, indicating a linear polymer [1, 17]. ............................................................55

Figure 5.4 Mid-height cross-sections for PP-1 (a), PP-2 (b), PP-3 (c), PP-4 (d) and PP-5 (e), each having a

nominal density of ~ 20%. The scale bar in part (e) represents a length of 5 mm and applies to

each figure. ........................................................................................................................56

Figure 5.5 Cross-sectional area fraction (Ac) as a function of position (P) in the sample (from bottom to top)

corresponding to the samples in Figure 5.4. ......................................................................57

x

Figure 5.6 Cross-sectional web length per unit area, lc, (at a nominal relative density of ~ 20%) plotted as a

function of zero-shear viscosity, ηo (a) and crossover frequency, ωc (b). ..........................58

Figure 5.7 Mid-height cross-sections of PP-3 at = 10% (a), 13% (b), 16% (c), and 18% (d). The scale bar

in (d) is 5 mm and applies to all images. ...........................................................................58

Figure 5.8 Mid-height cross-sectional area fraction (Ac) of PP-2 and PP-3 as a function of areal density (ρA).

...........................................................................................................................................59

Figure 5.9 Cross-sectional area fraction (Ac) plotted as a function of position, P, for PP-3 at = 13%. The

central third of the sample was used to determine Ac and lc. .............................................59

Figure 5.10 Cross-sectional area slices from positions a to e in Figure 5.9. The scale bar in (e) is 5 mm and

applies to all figures. ..........................................................................................................59

Figure 5.11 Archway defect in PP-3 ( = 13%) shown in a 3D compilation image (a), with horizontal lines

marking the position of the cross-sections. The cross-section indicated by the top line is shown

in (b), the centre line is (c) and the lowest line is (d). The scale bar is 2 mm and applies to (b), (c)

and (d). ...............................................................................................................................61

Figure 5.12 Buttress defect in PP-1 ( = 11%) shown in 3D compilation image (a), with horizontal lines

marking the position of the cross-sections. The cross-section indicated by the top line is shown

in (b), the centre line is (c) and the lowest line is (d). The scale bar is 2 mm and applies to (b), (c)

and (d). ...............................................................................................................................61

Figure 5.13 Void defects in PP-3 ( = 13%) shown in a 3D compilation image (a), and in three separate

cross-sectional slices (b), (c) and (d). The scale bar is 2 mm and applies to (b), (c) and (d).

...........................................................................................................................................61

Figure 6.1 Top view of an as-fabricated stochastic honeycomb core showing the cellular web structure (a).

Also shown is a three point bend coupon 45 mm x 200 mm, 20 mm core thickness, reinforced

with conventional polypropylene face sheets and an as-fabricated stochastic honeycomb core

compression 30 mm x 30 mm test coupon, 20 mm core thickness (b). .............................65

Figure 6.2 Variation in web thickness, tweb (a) and web length, b (b) over all densities. ....................68

Figure 6.3 SEM images showing the internal structure of the stochastic honeycomb architecture. An

incomplete web with a gap extending to the skin is shown (a), along with a mid-height gap

indicated by the arrow (b), and the complex buttressing of the webs (c). .........................68

Figure 6.4 Typical uniaxial compression curves for stochastic honeycomb cores. ............................70

Figure 6.5 SEM images of the web structure in a ρcore = 11.8% sample (a) and after preloading to strains of

ε ≈ 0.05 (b), ε ≈ 0.25 (c) and ε ≈ 0.35 (d). .........................................................................71

Figure 6.6 Stochastic honeycomb compressive strength, σp (a) and stiffness, E (b) over a range of core

densities. ............................................................................................................................71

Figure 6.7 Typical load-deflection curves for as-prepared stochastic honeycomb in three point bending.

...........................................................................................................................................72

xi

Figure 6.8 Flexural strength, σfl (a) and flexural stiffness, D (b) of stochastic honeycomb cores in three-

point bending. ....................................................................................................................72

Figure 6.9 Failure map plotting regions of dominant sandwich panel failure mechanism. The dominant

failure mechanism at low sheet thickness was face sheet wrinkling, core shearing dominated at

intermediate thickness, and face sheet yielding and fracture dominated at the greatest thickness.

...........................................................................................................................................73

Figure 6.10 Typical three-point bending load–deflection curves illustrating face sheet yield and fracture (tf =

1.75 mm, a), core shearing (tf = 0.94 mm, b) and face sheet wrinkling, (tf = 0.73 mm, c).

...........................................................................................................................................74

Figure 7.1 Heat flow (ΔH/m) plotted as a function of temperature (T) for the cool–heat cycle for PP-1, PP-

2, PP-3 and PP-4 (a), and the two heating curves (for the as-fabricated webs and the base resin)

for PP-1 and PP-4 (b). ........................................................................................................80

Figure 7.2 Stress (σ) – strain (ε) curves for tensile tests on PP-1, PP-2, PP-3 and PP-4, pulled at a rate of 1

mm min-1

. ...........................................................................................................................81

Figure 7.3 Skin thickness as a percentage of height ( ) plotted as a function of total relative density ( ).

...........................................................................................................................................82

Figure 7.4 3D reconstructions of a PP-2 stochastic honeycomb at 9.5% core density. Archways are webs

that do not extend through the entire height of the sample (shown with an arrow), and buttresses

form at the top and bottom the sample where they would join the skin (circled). Each scale bar

represents 5 mm. The images show a top view of the core (a), a side view (b), and views from

the mid-height of the complex structure (c,d). ...................................................................82

Figure 7.5 Stress (σ) – strain (ε) curves for the out-of-plane compression of stochastic honeycombs

fabricated with PP-1, PP-2, PP-3 and PP-4 with core densities (ρcore) of 11% (a), and the peak

strength (σp) of the stochastic honeycombs as a function of ρcore (b). ................................83

Figure 7.6 Surface area over volume (AS/V) as a function of core density (ρcore) for PP-1, PP-2, PP-3 and

PP-4. ...................................................................................................................................85

Figure 7.7 Predicted strength (σcr) from the composite buckling model plotted as a function of web

thickness (tweb) for PP-1 and PP-4, overlaid with the experimental data (σp) plotted as a function

of effective web thickness (teff). .........................................................................................87

Figure 7.8 Normalized web thickness (t/a) plotted as a function of the fraction of bound webs (f) for various

specific strengths (σ/ρ). The core density (ρcore) fraction for a given web thickness is also

displayed. ...........................................................................................................................87

Figure 8.1 3D reconstructions of samples A, B, C and D (a–d, respectively) with core relative densities as

shown in Table 8.1. ............................................................................................................94

Figure 8.2 Successive compressive stress–strain curves for samples A–D with core relative densities as

shown in Table 8.1. The samples were characterized through X-ray tomography in the as-

xii

fabricated condition (εo = 0), after loading to the peak (εp), the beginning of the valley (εv) and

just before the onset of densification (εd). The samples were unloaded and scanned, and then re-

loaded. The stress–strain curves were then stitched together. ...........................................95

Figure 8.3 The mid-height cross-section for sample C (a) and schematics of bound–bound, B (b) and

bound–free, F (c) webs with the length, b and the height, a as shown ..............................96

Figure 8.4 Web CB-1, in the as-fabricated condition (εo), after loading to the peak of the stress–strain curve

(εp), at the beginning of the valley (εv) and the end of the valley before densification (εd). The

first two images at each strain show the web from either side, and the third is a through-thickness

cross-section taken in the centre of the web. .....................................................................97








cross-section taken in thirds along the lower image of the web. .......................................99

Figure 8.7 Web CF-1, in the as-fabricated condition (εo), after loading to the peak of the stress–strain curve







cross-section taken in the centre of the web. ...................................................................100




cross-section taken in the centre of the web. ...................................................................100

Figure 8.10 Webs AB-1 (a), BB-1 (b), AF-1 (c), DF-1 (d), and DB-2 (e), in the as-fabricated state (εo) and at the

valley (εv). These webs demonstrate the three collapse mechanisms: elastic buckling, plastic

buckling near the skin, plastic buckling in the central portion of the web, plastic buckling with

horizontal tears, and plastic buckling with horizontal and vertical tears, from a – e respectively.

.........................................................................................................................................101

Figure 8.11 Energy absorbed, J, calculated up to εf = 0.4 (a), εf = (b), εf = εd (c), and εf = εη (d), plotted as

a function of ρcore..............................................................................................................104

Figure 8.12 Percentage drop (D%) from σp to σv a and energy absorbed, J, up until εf = 0.4 b as a function of

ρcore, for PP-1, PP-2, PP-3 and PP-4. ...............................................................................106

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Figure 8.13 Specific strength (σp/ρ) plotted against energy absorbed up until εf = 0.4 for PP-1, PP-2, PP-3 and

PP-4. .................................................................................................................................107

Figure B.1 Zero-shear viscosity (ηo) plotted as a function of melt flow index (MFI) for a number of PP, PE,

and copolymers of PP and PE (CP), along with side images of representative samples.

.........................................................................................................................................118

Figure B.2 Stochastic honeycombs fabricated with HMS PP (a), linear iPP (b) and LDPE (c). The scale bar

on (c) is 3 cm and applies to all three images. .................................................................119

Figure B.3 SEM images of stochastic honeycombs fabricated with HMS PP (a,b), L-iPP (c,d) and LDPE

(e,f), perpendicular to the webs (top row), and at a tilt (bottom row). The HMS PP has buttresses

that support the webs at the transition between the webs and the skin, while the L-iPP and the

LDPE have smooth transition regions with no buttressing. .............................................120

Figure C.1 3D image reconstructed from a micro-CT scan, with a sample size of 40 mm 40 mm 15 mm

(a) and the web structure as seen from the side (b), with the as-fabricated face sheets removed

digitally in order to more clearly reveal the internal web structure. The scale bars are 5 mm each.

.........................................................................................................................................122

Figure C.2 Micro-CT scans of the mid-height cross-sectional slice (sample size 40 mm 40 mm 15 mm)

for the LD sample (a) and the HD sample (b). The scale bar is 5 mm and applies to both figures.

The corresponding area fraction, Ac, as a function of normalized position in the sample, P, from

bottom to top is given in (c). ............................................................................................123

Figure C.3 Partial web in the LD as-fabricated structure. The scale bar is 2 mm. ............................124

Figure C.4 Schematic of a web bound on one side and free on the other. The length, b, height, a, and

thickness, tweb, are indicated. ............................................................................................125

Figure C.5 Uniaxial compression stress–strain curve for a ρcore = 13% stochastic honeycomb. .......125

Figure C.6 Uniaxial compression stress-strain curve for LD (ρcore = 11%) and HD (ρcore = 18%) stochastic

honeycombs to the valley (ε = 0.20). ...............................................................................125

Figure C.7 As-fabricated LD web bounded on each side (a), and the vertical tear created by the horizontal

tensile strain from the compression of the bounding webs (b). The scale bar is 2 mm and applies

to both figures. .................................................................................................................126

Figure C.8 LD-1 as-fabricated from the side (a), straight on (b), in vertical cross-section (c), and partially

compressed from the side (d), straight on (e), and in vertical cross-section (f). The scale bar is

2.5 mm and applies to all the figures. ..............................................................................127

Figure C.9 Thickness profile of LD-1 and the vertical cross-section for comparison. The dashed line marks

the point of initiation of instability. .................................................................................127

xiv

Figure C.10 LD-2 as-fabricated from the side (a), straight on (b), in vertical cross-section (c), and partially



Figure C.11 Thickness profile of LD-2 and the vertical cross-section for comparison. The dashed lines mark

the locations of initiation of instability. ...........................................................................129

Figure C.12 HD-1 as-fabricated from the side (a), straight on (b), in vertical cross-section (c), and partially



Figure C.13 Thickness profile of HD-1 and the vertical cross-section for comparison. The dashed line marks

the location of initiation of instability. .............................................................................130

Figure C.14 HD-2 as-fabricated from the side (a), from the opposite side (b), straight on (c), in vertical cross-

section (d), and partially compressed from the side (e), from the opposite side (f), straight on (g),

and in vertical cross-section (h). The scale bar is 2.5 mm and applies to all the figures.

.........................................................................................................................................131

Figure C.15 HD-2 buttress as-fabricated (a), in vertical cross-section (b), and partially compressed (c), in

vertical cross-section (d). The scale bar is 1 mm and applies to all the figures. ..............131

Figure C.16 Thickness profile of HD-2 and the buttress (inset) with vertical cross-sections for comparison.

The dashed lines mark the locations of initiation of instability. ......................................132

xv

List of Tables

Table 3.1 The program for the automated process for fabrication of stochastic honeycombs, where Fapp is the

chosen applied force, is the chosen melt stretch rate, and h is the chosen final height of the

sample. ....................................................................................................................................34

Table 4.1 Temperature of the platens during melt elongation at the beginning and end of each stage of

fabrication. ..............................................................................................................................44

Table 5.1 Polymer properties for four high melt strength PP (PP-1 to PP-4) and one conventional (linear) PP

(PP-5). .....................................................................................................................................52

Table 5.2 Cross-sectional area fraction (Ac) and web length per unit area (lc) over the central third of the

stochastic honeycomb samples ( ~ 20%). .............................................................................57

Table 6.1 Average web thickness (tweb) and length (b) for four ranges of core relative densities (ρcore).67

Table 7.1 Zero-shear viscosity (ηo) as measured using parallel-plate rheology tests at 1.1Tm [1]. .........79

Table 7.2 Melting and crystallization temperatures (Tm, Tc) and the crystallinity (xc) of the base polymer and

the webs for all four PPs from differential scanning calorimetry. ..........................................80

Table 7.3 Tensile strength (σTS) and strain at break (εu) for all four PPs. ................................................81

Table 7.4 Buckling model parameters determined from the X-ray images of the webs, and the tangent

modulus, Et, as determined from the tensile testing. ...............................................................87

Table 8.1 Relative density (ρcore), cross-sectional area fraction (Ac), peak strength (σp) and valley strength (σv)

for samples A–D. ....................................................................................................................94

Table 8.2 Dimensions of various webs at mid-height from sample C. ...................................................96

Table 8.3 The percentage of webs from each sample type that exhibited each mechanism of deformation at the

valley. Reported values are based on the characterization of at least 35 webs in each sample.102

Table 8.4 Final strain values as a function of density, for each definition of strain. The references indicate the

study or studies where this integration range was selected. ..................................................103

Table 8.5 Zero-shear viscosity (ηo), melting temperature (Tm), crystallinity (xc) and tensile strength (σTS) of

PP-1, PP-2, PP-3 and PP-4 [8, 9], and peak and valley strength (σp, σv), and the energy absorbed (J)

up until εf = 0.4 at ρcore = 10%. ..............................................................................................106

Table C.1 Compressive properties for the low density and high density stochastic honeycombs.........125

xvi

Table C.2 LD-1 dimensions at failure initiation point. ..........................................................................127

Table C.3 LD-2 dimensions at failure initiation points. ........................................................................129

Table C.4 HD-1 dimensions at failure initiation point. .........................................................................130

Table C.5 HD-2 dimensions at failure initiation points. ........................................................................132

Table C.6 Thickness and aspect ratio correlated with failure mechanism .............................................132

xvii

List of Symbols and Abbreviations

A, B, C Sample designations

Ac Average normalized cross-sectional area over the central third of the

honeycomb height

Acells mm2 Area of apparently closed cells in a 2D image of a stochastic honeycomb

As mm2 Surface area

Atotal mm2 Total area of a stochastic honeycomb sample

Aw mm2 Area of a single web in a given cross-section

Awebs mm2 Area of the webs in a 2D image of stochastic honeycombs

a mm Height of the core of the stochastic honeycomb

B mm Mean breadth

B Boundary condition: web is bound on either side

b mm Length of web at a given cross-section

bB mm Average length of bound webs

bF mm Average length of bound-free webs

CP Copolymer of polypropylene and polyethylene

D N mm2 Flexural stiffness

D% % Percentage drop from peak to valley in compression

d mm Diameter

dblank mm Diameter of a polypropylene blank

dfinal mm Diameter of the final stochastic honeycomb

DSC Differential Scanning Calorimetry/Calorimeter

E MPa Elastic modulus

Emelt MPa Melt stiffness

Es MPa Elastic modulus of the solid material

Et MPa Tangent modulus

F N Applied load

F Boundary condition: web is bound on one side and free on the other

Fapp N Applied compressive force during fabrication

FEA Finite Element Analysis

f Length fraction of bound webs

G* Pa Complex shear modulus

G' Pa Storage modulus

xviii

G" Pa Loss modulus

Gc Pa Crossover modulus

HD High density

HDPE High density polyethylene

HMS High Melt Strength

ΔH W Enthalpy

Δhm J g-1

Specific enthalpy of fusion

Δ J g

-1 Specific enthalpy of fusion of a crystalline polymer

h mm Height of stochastic honeycomb (total thickness)

J J cm-3

Energy absorbed during compression

K Buckling coefficient

KB Buckling coefficient for bound webs

KF Buckling coefficient for bound-free webs

L cm Total length of webs at a given cross-section

l mm Length

lc cm-1

Average length of webs normalized by the cross-sectional area over the central

third of the honeycomb height

LCB Long-Chain Branch

LD Low density

LDPE Low density polyethylene

Ls mm Span length

m Number of half-waves in a thin plate under compression

Mbp g mol-1

Average molecular weight between branch points

MFI g/10 min Melt Flow Index

MLCB g mol-1

Average molecular weight of branches

mSHC g Mass of the stochastic honeycomb

Mw g mol-1

Weight-average molecular weight

MWD Molecular Weight Distribution

N Number of nodes

P mm mm-1

Normalized vertical position in the web

PDI Polydispersity index

PE Polyethylene

PP Polypropylene

PP-1 Daploy WB135 HMS polypropylene

xix




PP-5 Accucomp HP0306L polypropylene

PS Polystyrene

PU Polyurethane

Pw mm Perimeter of a single web at a given cross-section

Pwebs % Percentage of webs

S mm Elongation during fabrication

mm min-1

Elongation or melt-stretch rate during fabrication

s mm Deflection

SEM

Scanning Electron Microscopy/Microscope

T oC Temperature

Tc oC Crystallization temperature

Tm oC Melting temperature

t s Time

tblank mm Thickness of a polypropylene blank

teff mm Effective web thickness

tf mm Thickness of commercial polypropylene sheets

tmelt mm Thickness of a polypropylene blank after force has been applied

tplate Mm Thickness of a plate

tskin mm Average skin thickness

% Skin thickness as a percentage of height

tweb mm Thickness of webs

UTM Universal Testing Machine

V cm3 Volume

w mm Width of square sample

xc % Percent crystallinity

δ mm Deflection

ε mm mm-1

Strain

ε s-1

Strain rate

ε0 mm mm-1

Zero strain

ε mm mm

-1 Strain where stress is equal to twice the peak stress

xx

εd mm mm-1

Strain at the onset of densification

εF mm mm-1

Strain at failure in uniaxial extension of a polymer melt

εf mm mm-1

Arbitrary final strain

εp mm mm-1

Strain at the peak of the stress-strain curve

εU mm mm-1

Strain at break during tensile test

εv mm mm-1

Strain at the beginning of the valley on the stress-strain curve

εη mm mm-1

Strain at the maximum efficiency

η Pa s Viscosity

η' Pa s Real component of complex viscosity

η" Pa s Imaginary component of complex viscosity

ηo Pa s Zero-shear viscosity

ηE Pa s Elongational viscosity

η Pa s Apparent elongational viscosity

η Pa s Maximum apparent elongational viscosity

η(ε) Efficiency of energy absorption

λ s Characteristic relaxation time

ν Poisson's ratio

ξ Area fraction of apparently closed cells

ρ kg m-3

or g cm-3

Density

ρ % Relative density of the stochastic honeycomb

ρA g cm-2

Areal density of a stochastic honeycomb

ρA,blank g cm-2

Areal density of a polypropylene blank

ρ

% Relative density of the stochastic honeycomb core

ρPP g cm-3

Density of polypropylene

MPa Stress

cr MPa Critical buckling strength from thin-plate model

ext kPa Plateau extension stress during fabrication

fl MPa Flexural strength

p MPa Peak strength during compression

pk MPa Peak tensile stress during fabrication

pl MPa Plateau strength of a foam under compression

TS MPa Tensile strength

v MPa Valley strength during compression

xxi

τrelax kPa s-1

Effective relaxation rate of polypropylene in the melt

τmelt kPa s-1

Relaxation rate of polypropylene melt during melt stretching in fabrication

φ Aspect ratio of wall height to wall width, a/b

χ, χ2D, χ3D Euler number, in 2D and 3D

ω rad s-1

Angular frequency

ωc rad s-1

Crossover angular frequency

1

Chapter 1

Introduction

1.1 Cellular Materials

Cellular materials are hybrid structures consisting of a solid material and open space. The solid material can be

partitioned into a wide variety of architectures, from a random foam to a two-dimensional honeycomb to an

ordered three-dimensional truss. Using various geometrical and topological arrangements of the solid material,

spaces on a strength–density property map can be reached that were inaccessible otherwise (Figure 1.1). The

goal is to create materials with higher strength and lower density, filling in the upper left side of the property

map. This gap has an upper bound that is defined by the maximum theoretical strength attainable. In the diagram

shown in Figure 1.1, this limit is based on the strength required to break covalent bonds, which in turn is related

to the bond stiffness [1, 2].

Foams are considered 3D cellular materials because the structure varies in all 3 dimensions. In contrast,

honeycombs can be defined by their 2D projection, since along the perpendicular axis the structure is constant.

Variations in honeycomb topology are shown in Figure 1.2, where it can be seen that honeycombs can be

composed of regular shapes (be it hexagonal, triangular, square or rectangular, circular, or a combination of

these), or irregular shapes, as in the Voronoi honeycomb.

The relative density of cellular materials composed of one material and air is defined as the density of the

cellular material normalized by the density of the parent material ( ). This produces a relative density that is

generally less than 0.20. The mechanical properties, specifically the stiffness and strength, of cellular materials

are thus often reported as a function of this relative density. Similarly, the strength and stiffness under

compression can be normalized by the yield strength (σy) and elastic modulus (Es) of the parent material,

respectively, to yield the specific strength and stiffness of the cellular material. Although foams are 3-

dimensional cellular materials, they are generally considered isotropic in each dimension, unlike honeycombs,

which have very different properties in-plane and out-of-plane. The specific plastic collapse strength and specific

stiffness of foams has been found to scale with relative density as [3]:

(1.1)

2

(1.2)

In contrast, the out-of-plane mechanical properties of honeycombs scale with relative density as [3]:

(1.3)

(1.4)

For foams, these are based on idealized unit cells, while for regular honeycombs, the relationships can be

calculated exactly.

Figure 1.1 Strength–density property map [4].

3

Figure 1.2 Honeycombs with varying shape: hexagonal [5] (a), hexagonal and triangular [5] (b), triangular [5] (c), square [5] (d),

circular [6] (e) and Voronoi [7].

Polymeric cellular materials have enabled important advances in the aerospace, automotive and packaging

industries [8, 9], and have become increasingly important for high-strength low-weight applications over the last

few decades [3]. The most common uses of foams and honeycombs are as the core in sandwich panels, where

two relatively thin and stiff sheets are separated by a thicker low-density cellular core [10]. Historically, polymer

core sandwich panels have primarily had metal face sheets [3], but over the past 20 years it has become more

and more common for the face sheets to be a polymer matrix composite adhesively bonded to the core [11, 12].

There are many types of adhesives, the most common being epoxies or other thermosets [11, 13].

Thermoplastic foams are typically produced by using blowing agents such as carbon dioxide or nitrogen to

create bubbles in the melt [3, 14], or by introduction of volatile compounds (either low melting point liquids or

additives which react and/or decompose) that will form vapour bubbles upon heating [3], and then injection

moulding the final product [14]. Thermoplastic honeycombs can be fabricated through a number of methods,

including extruding sheets of plastic and cutting the sheets into blocks which are then thermally welded or

bonded in narrow strips and expanded into a honeycomb shape [15], or extruded sheets cut into strips and slotted

together [15], or through extrusion into a complex mould [16]. Each of these processes require multiple steps to

create a sandwich panel.

Various methods have been used to improve the mechanical properties of polymer foams and honeycombs

as the core for sandwich panels. Fibre addition has been used to improve the properties of bulk polymers [17]

and has recently been used to improve the properties of honeycomb sandwich structures as well [18]. Clay

nanocomposites are also being used to improve the properties of polymer foams [19]. While these methods can

4

improve the mechanical properties of the cellular material, they typically do so at the expense of recyclability

[20]. The greatest potential for recycling sandwich materials is if a single material could be used for the core,

face sheets and adhesive layers [21, 22]. This would allow the entire structure to be recycled without the need for

additional (and costly) steps to separate the different materials.

1.2 Melt-Stretched Stochastic Honeycombs

Melt-stretched stochastic honeycombs are a new type of cellular material, and are so named due to the

fabrication method and the resulting structure. They are fabricated through uniaxially stretching a molten

polymer between metal platens. This creates a complete sandwich structure, shown in an X-ray image from the

side in Figure 1.3. The as-fabricated skins, which form on the metal platens, create in-situ face sheets for

sandwich panels. Additionally, as the melt is stretched, the polymer is redistributed and air enters the structure,

creating an interconnected set of webs. These webs are analogous to the walls in a honeycomb, and are parallel

to the direction of melt-stretching. The structure is relatively consistent through the cross-section, and as such is

almost a 2D cellular material, like a honeycomb. However, while the webs are interconnected, there are no

closed cells, making this structure more like a Voronoi honeycomb with some missing walls (Figure 1.4).

Finally, the term "stochastic" refers to an element of randomness in the structure, as the melt-stretching produces

a different web pattern in every sample.

The structure does not consist only of webs perpendicular to the skin. Figure 1.4 presents 3D

reconstructions of two stochastic honeycombs, one fabricated with high melt strength (HMS) polypropylene (PP)

and one with linear PP, with the built-in skin layers digitally removed so that the core structure is visible. As can

be seen in Figure 1.4a, near the skin, buttresses appear on either side of some HMS PP webs, but are not present

in the linear PP structure (Figure 1.4b). Also, a significant fraction of the webs do not extend entirely through the

Figure 1.3 A high melt strength PP stochastic honeycomb seen from the side in an unreconstructed X-ray image. The image shows

the as-fabricated skins and the core.

5

Figure 1.4 3D reconstructions of a melt-stretched stochastic honeycomb fabricated with high melt strength PP (a) and linear PP (b),

with the as-fabricated skins edited out. The scale bar in (b) is 5 mm and applies to both images.

structure, but form archways. These are also not present in the linear PP structure. As will be seen in Chapters 4

and 5, the processing conditions and polymer type have a significant effect on the resultant architecture.

Other architectural features are more clearly seen by plotting the open space instead of the solid material.

Figure 1.5 shows part of the high melt strength PP stochastic honeycomb in an inverted space-filling model (the

open area is shown in white and the polymer in the webs is empty space). The buttressing can be seen clearly in

the lower left corner of Figure 1.5a (circled), where the space-filled pore has two indents. The space-filling

models clearly illustrate the range of cell geometries formed. Additionally, an opening in the webs is indicated

with an arrow in Figure 1.5b, illustrating the connectivity of the pores. During fabrication, air enters the structure

from the outer perimeter, creating a continuous pathway through the structure. This leads to an open-cell

honeycomb architecture.

In this thesis, stochastic honeycombs were fabricated using five PPs with variations in molecular structure

and melt and bulk properties to determine the effect of polymer characteristics on the internal architecture and

the mechanical properties of the cellular material. There was a factor of six difference in the melt flow index

from 2.1 to 12 g/10 min (a detailed summary of the PP properties are given in Appendix A). Four branched

(HMS) PPs and one isotactic linear PP were chosen to provide this variation, with the branched PPs having

branch densities on the order of ~ 0.2/1000 C atoms [23]). Other polymers were also initially investigated, and

the early fabrication results for these polymers are summarized in Appendix B.

6

Figure 1.5 Two space-filling model plots of the high melt strength PP sample shown in Figure 1.4, from the top (a) and a sectioned

piece from the centre (b). An example of the buttressing structure is circled in (a), and a hole in the web indicating an

archway is shown with the arrow in (b). The scale bar in (b) is 5 mm and applies to both images.

As with foams and honeycombs, the density of the structure affects the mechanical properties, and as is

demonstrated in Figure 1.4, the molecular structure (and thus the rheological properties) of the polymer will

affect the architecture and the mechanical properties as well, as do the fabrication conditions. The objectives of

this thesis were to develop, quantify and explore the fabrication, architecture, and mechanical properties of

stochastic honeycombs, as well as the determining the effect of material properties and fabrication parameters on

the internal architecture, and the effect of material properties and architecture on the compressive strength and

energy absorption capability of melt-stretched stochastic honeycombs.

1.3 Overview

This thesis presents the first five articles on melt-stretched stochastic honeycombs [24-28], with minor edits for

flow and clarity. The objectives of this document are to demonstrate a compelling motivation for this project and

to determine polymer property–stochastic honeycomb architecture–fabrication variable–mechanical property

relationships. Because this is a new hybrid material, it was important to determine the separate effects of

polymer rheology and fabrication variables on the architecture of stochastic honeycombs, as well as the effect of

architecture on the mechanical properties. The development of a buckling model for the out-of-plane

compressive properties of stochastic honeycombs was desired, as was a determination of collapse mechanisms of

the stochastic honeycomb under compression, along with the energy absorption capabilities. Chapter 2 provides

an overview of key background concepts, namely honeycomb sandwich panels, melt rheology of polypropylene,

and thin-plate buckling.

7

Chapter 3 details the fabrication methods for melt-stretching stochastic honeycombs. It includes the

manual press that was used to fabricate samples for the papers presented in Chapters 5 – 8, and the fully

instrumented press developed to study the polymer response during fabrication, presented in Chapter 4.

Chapter 4 presents a study on the effect of fabrication variables on the architecture for one PP resin (M.

Hostetter, G.D. Hibbard. Architecture-process relationships in stochastic honeycombs, Journal of Applied

Polymer Science 132 (2015) 42174 [27]). Stochastic honeycombs are fabricated by melting PP between

aluminium platens and then uniaxially stretching the polymer melt to a specified height. The areal density of PP

and the strain rate during uniaxial extension both had significant effects on the architecture. Higher strain rates

activated more entanglements leading to a greater number of node points and a more interconnected geometry.

Following this, another study compared the 5 PP resins (4 HMS PPs and one linear PP) to determine the

effect of rheological properties on the resulting stochastic honeycomb architecture. The rheological properties of

the melt were found to have a significant effect on the architecture. The internal architecture changes with the

properties of the melt, such as the melt flow index and the zero-shear viscosity. The presence of long-chain

branches increases the entanglement density with branches acting as entanglement points with primary bonds as

opposed to secondary bonds; branched PPs formed stochastic honeycombs with many more webs and web

branch nodes than linear PP, implying a greater number of activated entanglement points. This study is presented

in Chapter 5 (M. Hostetter, G.D. Hibbard. Architectural characteristics of stochastic honeycombs fabricated from

varying melt strength polypropylenes, Journal of Applied Polymer Science 131 (2014) 40074 [25]).

Chapter 6 (M. Hostetter, B. Cordner, G.D. Hibbard. Stochastic honeycomb sandwich cores, Composites

Part B: Engineering 43 (2012) 1024 [24]) presents the first paper published on melt-stretched stochastic

honeycombs, and demonstrates the excellent mechanical properties of stochastic honeycombs. Stochastic

honeycombs were fabricated over a range of densities and tested mechanically in out-of-plane compression and

three-point bending. A subset of bending samples were reinforced by thermally welding conventional PP sheets

to the skins to create a rigid sandwich structure composed entirely of PP. It was found that despite the simple

fabrication method, stochastic honeycombs had a mechanical strength comparable to commercial PP

honeycombs and greater than commercial PP foams.

Chapter 7 (M. Hostetter, G.D. Hibbard. Modeling the buckling strength of polypropylene stochastic

honeycombs, Journal of Materials Science 49 (2014) 8365 [26]) takes the four HMS PPs and, with the aid of X-

ray tomography, presents a thin-plate buckling model to predict the strength of stochastic honeycombs. The four

HMS PPs were analyzed mechanically over a range of densities to compare the strength across the varying

architectures. Since stochastic honeycombs combine the stochastic character of conventional foams with the

8

aligned webs of a honeycomb, the internal architecture is quite complex. The core structure of the stochastic

honeycomb is similar to an irregular honeycomb with some percentage of missing walls. A buckling model

based on architectural data from X-ray tomography was created and verified for the differing architectures of the

stochastic honeycombs created with each HMS PP.

A final study investigated the post-peak collapse behaviour of stochastic honeycombs under uniaxial

compression. The dominant failure mechanisms over a range of densities were identified, and it was found that

stochastic honeycombs collapse through elastic buckling, plastic buckling, and plastic buckling with fracture.

The effect of different definitions of densification strain on the absorbed energy was presented, thus completing

the study of the compressive mechanical properties of stochastic honeycombs. Additionally, it was shown that

the secondary defect structure in stochastic honeycombs compensates for the lack of the closed-cell structure

common to traditional honeycombs. This work is presented in Chapter 8 (M. Hostetter, G.D. Hibbard. Post-peak

collapse and energy absorption in stochastic honeycombs (2015) In preparation for submission to the Journal of

Materials Science [28]).

Appendix A presents the material properties for the four HMS PPs and the linear isotactic PP, while

Appendix B includes fabrication details of other polyethylenes and polypropylenes. Appendix C presents a

conference paper on the characterization of stochastic honeycomb failure (M. Hostetter, G.D. Hibbard.

Characterization of stochastic honeycomb sandwich failure (2013) In: Proceedings of the International

Conference on Composite Materials 19, Montreal, Canada, 28 July – 2 August, pp. 312–319 [29]).

Figure 1.6 presents a flow chart detailing the key topics covered in this thesis: fabrication, material

properties, internal architecture and mechanical properties, and the contributions made in each category. The

three major contributions are characterization of the stochastic honeycomb internal architecture, creation of the

buckling model, and determination of fabrication windows. As seen in Figure 1.4 and Figure 1.5, the architecture

could be quantified using many different measures, including the angle between webs, the length and thickness

of webs, and the web end constraints, among many others. Additionally, the architectural variables change not

only from web-to-web within one sample, but also in a single web with vertical position. I determined the key

architectural variables that will be shown to vary with material properties (zero-shear viscosity), density of the

structure, melt-stretch rate during fabrication, and vertical position in the test sample. This characterization

allowed the creation of a buckling model. I modified the thin-plate buckling model to include additional

architectural information, such as variation in end constraints between types of webs and variation in

dimensions. This modified model predicted the critical collapse strength quite well. Finally, I developed a fully

instrumented fabrication process, and was able to determine three key fabrication parameters that were affected

by the density of the test sample and the melt-stretch rate during fabrication. Through the fabrication parameters,

9

I was able to determine how the web structure is formed during fabrication, and link melt-stretch rate and density

to the fabrication parameters and the resulting stochastic honeycomb architecture.

A summary of the completed work is presented in Chapter 9, along with suggestions for future work.

Figure 1.6 Flow chart detailing the four key topics in this thesis and the contributions made in each. The chapter that these

contributions are discussed is indicated through the numbers on the left.

Fabrication

Developed methods;

parameters

Determined key material properties

Effect of melt-stretch

rate on architecture

Effect of density on

architecture

Material Properties

Effect of melt material

properties on architecture

Effect of bulk material

properties on compressive

strength

Effect of bulk material

properties on post-peak

collapse and energy

absorption

Internal Architecture

Developed characterization

methods

Quantified the internal

architecture

Effect of internal

architecture on compressive

strength

Effect of internal

archtiecture on post-peak

collapse and energy

absorption

Mechanical Properties

Demonstrated strength is repeatable

Strength comparable to commercial PP honeycombs

Developed buckling model

Post-peak failure mechanisms

Energy absorption

Chapter 4. Architecture–process relationships in stochastic honeycombs

Chapter 5. Architectural characteristics of stochastic honeycombs fabricated from varying melt strength

polypropylenes

Chapter 6. Stochastic honeycomb sandwich panels

Chapter 7. Modeling the buckling strength of polypropylene stochastic honeycombs

Chapter 8. Post-peak collapse and energy absorption in stochastic honeycombs

8

7

8

8

4 – 8

4,6,

6,8

6

6,7

7

7

7

5,7

4

4

5

4,6

10

References

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Okamoto. Foam processing and cellular structure of polypropylene/clay nanocomposites, Polymer Engineering &

Science 42 (2002) 1907-1918.

[20] R.E. Smallman, R.J. Bishop. Plastics and Composites. Modern Physical Metallurgy and Materials Engineering.

Oxford: Butterworth-Heinemann, 1999.

11

[21] K.F. Karlsson, T.B. Åström. Manufacturing and applications of structural sandwich components, Composites Part

A: Applied Science and Manufacturing 28 (1997) 97-111.

[22] R. Gendron, ed. Thermoplastic foam processing : principles and applications, CRC Press, Boca Raton, Fla., 2005.

[23] H. Naguib. Personal Communication, 18 Mar 2015, to M. Hostetter.

[24] M. Hostetter, B. Cordner, G.D. Hibbard. Stochastic honeycomb sandwich cores, Composites Part B: Engineering

43 (2012) 1024-1029.

[25] M. Hostetter, G.D. Hibbard. Architectural characteristics of stochastic honeycombs fabricated from varying melt

strength polypropylenes, Journal of Applied Polymer Science 131 (2014) 40074.

[26] M. Hostetter, G.D. Hibbard. Modeling the buckling strength of polypropylene stochastic honeycombs, Journal of

Materials Science 49 (2014) 8365-8372.

[27] M. Hostetter, G.D. Hibbard. Architecture-process relationships in stochastic honeycombs, Journal of Applied

Polymer Science 132 (2015) 42174.

[28] M. Hostetter, G.D. Hibbard. Post-Peak Collapse and Energy Absorption of Stochastic Honeycombs, In preparation

for submission to Journal of Materials Science (2015).

[29] M. Hostetter, G.D. Hibbard. Characterization of stochastic honeycomb failure. International Conference on

Composite Materials 19. Montreal, Canada, 2013. p.312-319.

12

Chapter 2

Background

It will be shown that polypropylene stochastic honeycombs have similar mechanical properties to conventional

PP honeycombs and superior mechanical properties to PP foams. Polypropylene was chosen as the material for

stochastic honeycombs because it has been considered as a substitute for other closed cell foams such as

polyurethane, polyethylene, and polystyrene, due to its good mechanical strength and relatively high melt

temperature [1, 2]. Equally important is its recyclability [3], as polyurethane is a thermoset [4, 5], many

polyethylene foams are crosslinked [5], and polystyrene foam is quite difficult to recycle [6, 7].

Stochastic honeycombs were conceptualized and characterized based on foundational work in the field of

cellular solids, while the fabrication is heavily dependent on the rheology, and mechanical testing of stochastic

honeycombs was built on work done on foams and honeycombs. Significant results in cellular materials,

particularly sandwich panels with foam or honeycomb cores, are presented here. Additionally, the melt

properties of polypropylene are presented, along with methods to study and improve the rheological properties

for a given application. Finally, previous work on the mechanics of foams and honeycombs is presented, with the

focus on the thin plate buckling model and its applicability to honeycombs.

2.1 Honeycomb Sandwich Panels

Sandwich panels consist of a low density core positioned between two thin, stiff, strong face sheets (Figure 2.1)

[8-11]. The sandwich panels can be fabricated from any material, such as metal foams [12], balsa wood [12],

three-dimensional truss structures [10], or thermoplastic cores [13-17]. Thermoplastic sandwich panels are of

interest for their ease of fabrication [15], and the potential, especially for polypropylene, to fully recycle the end

product [17-19].

In bending, sandwich panels exhibit one or more of five failure mechanisms [12, 20-23] (Figure 2.2):

i. Delamination: the adhesion between the face sheet and the core fails, usually near the ends of

the sandwich panel

13

ii. Face Sheet Failure: the face sheet fails under tension or compression at the bottom or top of

the panel, in the centre

iii. Face Sheet Wrinkling: the face sheet fails on the top panel, and wrinkles under compression

near the centre of the panel, due to a local minimum in core stiffness or the face sheet

properties

iv. Core Crushing: at the supports or in the centre of the beam, the core locally crushes

v. Core Failure: core fails in shear

Figure 2.1 Cellular material sandwich panels, with foam, truss and honeycomb core materials, top to bottom [8].

Figure 2.2 Sandwich panel failure mechanisms [23]

14

A failure mode map is created by plotting the core relative density as a function of the face sheet thickness

normalized by the panel span, and at each point the dominant failure mechanism is identified. Boundaries

between dominant modes of failure can thus be identified as a function of the sandwich panel geometry [12, 20-

22]. It has been shown that the strength depends very little on core thickness [24], but the properties of the panel

can be tuned by adjusting the thickness of the core and face sheets [17].

The core of the sandwich panels are usually foams (Figure 2.3) or honeycombs (Figure 2.4). Foams are

generally considered isotropic materials, while honeycombs are isotropic in-plane, and have a different response

out-of-plane. Closed cell foams are common in automotive parts (polyurethane, PU [4, 12, 25]) and packaging

(polyethylene, PE [12, 26-28], and polystyrene, PS [12, 29, 30]). Foams have many geometric irregularities,

such as cell wall thickness variations, wavy distortions of cell walls [31, 32] for closed cell foams, and variation

in strut-cross-sectional area [32] for open cell foams. Closed and open cell foams exhibit non-uniform shape of

cells [31, 32]. Due to this geometrical and topological variation, foams have scatter in the mechanical properties

[33].

Figure 2.3 Closed [34] (a) and open [35] (b) cell foams.

Figure 2.4 In-plane (x-y) and out-of-plane (z) directions on a honeycomb sandwich panel [10].

15

Figure 2.5 2D Voronoi honeycomb with a random but even spacing of points [36] (a) and a completely random spacing of points

[37] (b).

Voronoi honeycombs are often used to model foams in 2D [36-41]. This model is created by laying points

in 2D, and introducing a line bisecting at 90o a non-existent line between each point, creating a random

honeycomb with various sizes and shapes of cells and with a dispersion in the number of walls per cell (Figure

2.5). Foams have also been modelled as a 3D Voronoi tessellation [32].

Much work has been done on honeycombs, both in-plane [33, 36, 37, 39-50] and out-of-plane [51-67]

(Figure 2.4), analytically, using finite element analysis (FEA) [33, 37-39, 42-45, 57-61], and experimentally [43,

47, 48, 51, 52, 57, 61-64]. A number of studies have looked at the effect of defects [33, 37-44, 46], but primarily

under in-plane compression. In-plane, it was shown that the separation distance between defects had little effect

on the stiffness or strength of the honeycomb [42], and that the stiffness decreased as the fraction of missing

walls increased [43].

A single unit cell model for honeycombs, in out-of-plane compression, cannot be used to study defects

such as missing or fractured walls [39], and as a result this has not been studied extensively. Similarly to foams,

curved walls in honeycombs decrease the strength and stiffness, under both in-plane and out-of-plane

compression [68].

2.2 Melt Rheology and Structure of Polypropylene

Polypropylene is an attractive polymer for use in cellular materials because of its low cost, ease of processing,

environmental stability, non-toxicity and recyclability [17, 69-71]. Additionally, it has high stiffness, high

resistance to corrosive chemicals, and when incinerated does not produce harmful by-products [70-72]. PP is

being considered to replace polyvinyl chloride (PVC), PU and PE in many products, but needs improved melt

strength [70, 71, 73] and greater strain hardening capabilities [70, 71]. The low melt strength of PP results in

16

local instabilities in thermoforming, blow moulding, extrusion coating and foaming [70, 71]. The production of

closed cell foams with linear PP is difficult becasue the melt does not exhibit strain hardening under extensional

flow, resulting in holes in cell walls and a semi-open cell structure [71, 74, 75]. Many of the processing

difficulties of PP come about because it has a sharp melting point [73]. Introducing long chain branches (LCBs)

increases the processability and melt strength [69, 71], and this is termed high melt strength (HMS) PP. Melt

strength is related to the limit of extensibility of the polymer melt [25], and a higher melt strength allows greater

extension before rupture [3]. Strain-hardening is a phenomenon that is linked to melt strength, but occurs when

polymer chains align during melt stretching, increasing the stress required to further extend the polymer [76].

Thus, during foaming, if the polymer strain-hardens it increases the membrane stability and stabilizes cell walls,

preventing them from expanding too quickly and rupturing. Poor melt strength in PP can also lead to melt

fracture in extrusion, which, along with ruptured cell walls, can lead to an open cell foam structure [25, 71, 74,

77, 78]. Increased melt strength also limits shear thinning, which occurs when a polymer is drawn and it thins

and elongates from a single point, with decreasing resistance to elongation [79].

Melt strength and the ability to strain harden are key variables in the foamability of PP. However, melt

strength is not a quantity that is easy or convenient to measure, so often the melt flow index (MFI) is used

instead, especially industrially. Also important to the processability of PP are the extensional properties, as

determined through the extensional viscosity.

The melt flow index (MFI) is determined by how much polymer can be pushed through a capillary of set

diameter and length at a given temperature and pressure, in a set amount of time. It is measured in g/10 min as

per ASTM D1238 [80]. It is a crude, empirical way of determining the relative molecular weight [81, 82], and is

related to the shear viscosity [83]. Polymers with low MFI values have longer molecular chains and higher

average molecular weights which form more entanglements in the melt [81]. The MFI is inversely proportional

to zero-shear viscosity of a given polymer melt [84, 85], and once the MFI is determined, it is possible to

calculate the zero-shear viscosity for most linear polymers [84]. It has been shown that the zero-shear viscosity

obtained from MFI data is quite close to zero-shear viscosity measured directly [84]. However, the zero-shear

viscosity is very sensitive to molecular weight distribution, while the MFI is not [85].

Extensional viscosity (ηE), on the other hand, is a well-defined rheological property that is measured by

drawing a polymer filament in a controlled-temperature medium (usually an oil bath) and measuring the force as

a function of time [83, 86, 87]. This is limited in practice because it must be determined under isothermal,

uniform, low-strain conditions [83] that are difficult to achieve because the melt will often rupture before it

reaches a steady state [86, 88-90]. As a result, the extensional properties of a melt are often measured

isothermally but not at steady extensional flow, using an apparent elongational viscosity ( ) [90, 91]. Under

17

, failure is defined as the maximum on the force–extension curve [86], although melt rupture occurs some

time after this point. Melt rupture is preceded by either ductile failure (necking) or cohesive fracture [87, 89].

High density polyethylene (HDPE) and PP were found to exhibit ductile failure in melt extension [92].

Melt strength, when quantified, is a rough measure of the extensional viscosity of the polymer [76], and is

used exclusively in polymer engineering, specifically extrusion and formation of thin films and fibres [82]. In

this context, it is the resistance of a melt to extension or sag [83]. In contrast to the steady state conditions of

extensional viscosity and the isothermal apparent elongational viscosity, melt strength is measured non-

isothermally and under high strain rates, closer to actual processing conditions [83], and is used to compare the

drawability of polymers [81]. Many methods are used to measure melt strength, the most popular being to use a

capillary rheometer equipped with a haul off device and a force transducer to measure the strength of the

polymer melt [112]. The Gottfert Rheotens test for melt tension (measured in cN) and velocity at break

(measured in mm s-1

) [79] is a form of this test.

On the molecular scale, polymers with more entanglements tend to have higher resistance to extensional

deformation yielding, and so a higher melt strength [81]. Melt strength is improved by the presence of a high

molecular weight tail (where a small fraction of the chains have very high molecular weight, creating a tail on

the MWD), or long chain branches [112]. Figure 2.6 demonstrates the difference between melt extension of a

Figure 2.6 Schematic of the effect of branching and cross-linking on the tensile strength of PP [100].

18

linear PP and a cross-linked or highly entangled PP. In order to extend the cross-linked or entangled melt,

covalent bonds have to be broken, increasing the melt strength.

Long chain branches and cross-links (depicted schematically in Figure 2.7) have a very large effect on the

melt strength and rheological properties of polymer melts [71, 93-100]. LCBs increase the number of

entanglements [71], and even low amounts of LCBs can increase the zero-shear viscosity and reduce shear

thinning [69, 101]. Long chain branching is induced in PP through electron beam irradiation [70, 71, 95-97, 102-

105], the introduction of multi-functional monomers [70, 71, 93, 94, 98, 99], and melt grafting in the presence of

peroxides [69-71, 94, 98, 99, 102, 106, 107]. Many of these methods lead to some measure of cross-linking [72].

Additionally, the melt strength of PP can be increased without long chain branching by the addition very high

molecular weight PE chains [103], organo-clay nanocomposites [108, 109], carbon nanotubes [110, 111], and

the addition of inorganic nano-fillers [69, 107-111].

Figure 2.7 Schematic of a cross-link [72] (a) and long chain branching [100] (b) in a polymer chain, where Mbp is the average

molecular weight between branch points and MLCB is the average molecular weight of the branches.

2.3 Buckling of Thin Plates as it Relates to Honeycombs

There are three types of mechanical responses of a cellular material under uniaxial compression (see Figure 2.8):

[12, 47, 113]

i. Strain hardening: the stress increases steeply in the elastic region, and during plastic

deformation the rate of increase slows but does not stop. At densification, the slope increases

again. This is found in some foams and in-plane compression of honeycombs.

19

ii. Strain softening: the stress increases steeply in the elastic region to a peak, after which the

stress decreases to a valley, and then increases again once densification begins. This is found

in out-of-plane compression of honeycombs.

iii. Perfectly plastic: the stress increases steeply in the elastic region, and then levels off, with a

slope of zero, at the plateau stress. This continues until densification begins. This is found in

ideal foams.

For out-of-plane compression of honeycombs, it is assumed that all of the cell walls are equally loaded, and the

peak compressive strength is determined by competition between elastic buckling and plastic microbuckling

[64]. Out-of-plane buckling of a square honeycomb is demonstrated in Figure 2.9.

Euler determined the buckling strength of a column under compression, and in 1890 Bryan [115] extended

his theory to buckling of a thin plate restrained on one or more unloaded edges. The relationship was put in its

current form by Timoshenko and Gere in 1961 [116]. The critical buckling strength of a thin plate is [117-119]:

Figure 2.8 Strain hardening (a), strain softening (b) and perfectly plastic (c) stress-strain curves [113].

Figure 2.9 Buckling modes of a square honeycomb in out-of-plane compression [114].

20

(2.1)

where: σcr: the critical buckling stress

K: constant for a given system that incorporates the edge constraints

Et : the tangent modulus from the tensile σ–ε curves

ν: Poisson’s Ratio

tplate: thickness of the plate

b: width of the plate

A schematic of a thin plate with the dimensions and axes is shown in Figure 2.10.

The elastic thin plate buckling model assumes that: [117]

i. Deflections are very small (less than the thickness of the plate).

ii. The middle plane of the plate remains a neutral surface.

iii. Plane sections rotate during bending to remain normal to the neutral surface.

iv. Shearing forces are neglected; loads are resisted by the bending and twisting of plate

elements.

v. The thickness of the plate is small compared to the other dimensions.

These assumptions are fairly good in the elastic region, before the peak in the stress-strain curve.

There are three common edge constraint conditions, termed free, simply supported, and built-in (also

termed clamped). Free means that the edge is not restrained in any way; however, if both unloaded edges are

free, the thin plate is considered a column [120]. A simply supported edge condition requires that there is no

curvature or deflection at the edges [117]:

Figure 2.10 A schematic of a thin plate, with height a and width b, and thickness t in the x-direction.

21

Figure 2.11 A thin plate under compression with m = 5 half-waves [121].

(2.2)

where y and z are defined in Figure 2.10, and δ is the deflection in x. The built-in edge conditions state that there

is no deflection or slope at either boundary [117]:

(2.3)

Most models of out-of-plane compression assume simply supported edge conditions (e.g., [52]), although one

showed that the actual edge constraints were between those of simply supported and built-in, but closer to

simply-supported [120].

A simply supported rectangular plate loaded in compression will develop sinusoidal buckles [117, 118]

(see Figure 2.11), and the number of half-waves, m, increases as the aspect ratio a/b increases [117, 118]. It has

also been shown that when a/b >2, K varies very little with aspect ratio [119].

For the simply supported condition, a web bound on both unloaded edges, K, is determined by [117, 118]:

(2.4)

where = a/b. If the web is simply supported on one side and free on the other, K is found by solving the

following relation [117, 118]:

(2.5)

where r, s, p and q are placeholder variables to simplify the expression, and are defined as:

22

(2.6a)

(2.6b)

It has been shown that there is no interaction between neighbouring vertical edges in out-of-plane

compression [21], implying that the application of a thin-plate model to an extended structure like a honeycomb

is valid.

In a sandwich panel, the loaded edges are also restrained by a face sheet, and it was shown that a single

face sheet increases the critical stress supported by the honeycomb over that of the core alone, and that two face

sheets further increases the critical stress [51].

Elastic buckling is followed by inelastic collapse, leading to localized failure [57]. Fracture begins at the

drop from peak to valley [65], and it has been shown that even for low-velocity and low energy impact of a

sandwich panel with carbon fibre face sheets, the residual compressive strength (i.e. strength after buckling has

begun) can be 50% lower than the peak strength [39, 64].

Deformation of the cell walls was shown to be uniform for ε < 0.003 [64], which is in the elastic region.

Wrinkling of the cell walls was observed at larger strains [64], and the overall collapse mechanism varies with

density [39, 64]. At low density, failure is primarily elastic buckling [39, 64], and plastic collapse at high density

[39]. These two mechanism interact at intermediate densities [39].

The energy absorbed during compression is the integral of the stress-strain curve up to a final strain,

usually the densification strain:

(2.7)

where J is the energy absorbed per unit volume and εd is the densification strain. Densification occurs when the

cellular material has collapsed to enough of a degree that the cell walls begin impacting each other, hindering

further compression [12]. For foams, the densification strain is defined as the strain at which the slope of the

stress–strain curve approaches the stiffness of the solid material, Es [122]. Densification strain has also been

defined as the strain at which the stress is equal to twice the peak stress (for out-of-plane compression of

honeycombs) [123], and the strain at the peak efficiency for energy absorption [113, 124, 125]:

23

(2.8)

where η(ε) is the efficiency of the energy absorption as a function of strain. Which densification value is used

will change the quantification of the energy absorbed during compression or impact testing.

2.4 Summary

This work on melt-stretched stochastic honeycombs has built on the diverse bodies of knowledge in cellular

materials, polymer rheological properties and the buckling and collapse of thin plates. Specifically, previous

work on foams, honeycombs, and sandwich panels that pertain to the work done on stochastic honeycombs have

been summarized. Additionally, the melt flow index and melt strength properties of polypropylene, as well as

specific rheological properties and elongational viscosity of molten polymers, as it pertains to the fabricated

stochastic honeycomb and during fabrication are detailed. Finally, previous work on buckling of thin plates and

post-peak buckling of honeycombs and foams was presented.

24

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31

Chapter 3

Melt-Stretched Honeycombs

Melt-stretched honeycombs are fabricated by uniaxially stretching molten polymer between two platens to the

desired thickness of the panel, and letting the polymer cool. The resulting structure consists of two skins, parallel

and initially in contact with the platens, separated by a network of interconnected webs oriented parallel to the

direction of melt-stretching. Hot tack adhesion between the polymer and the platens is strong enough to

overcome the cohesiveness of the melt and create the webbed structure, and then as the sample cools and

contracts, it separates spontaneously from the platens. Initially, the honeycombs were fabricated using a manual

press, and then a fully instrumented process was developed. These are detailed below.

Iteration 1: Manual Press

The manual press consisted of two aluminium platens, one with handles and one without, and a wooden

framework for the melt-stretching process. The Al platens were pre-heated in the oven, set at 180 oC (Figure

3.1a), and the temperature was monitored with thermocouples and an IR thermometer. The polymer was placed

on the lower platen for approximately 20 minutes, until it reached the molten state. The lower platen was then

placed on a wooden base, and the framework for the press was placed on top (Figure 3.1b). The upper platen was

compressed on top of the lower platen, and then raised to the desired height using wooden levers through the

handles, and locked in place as the sample cooled. Consistently fabricating successful samples was difficult, as a

number of challenges presented themselves: temperature, adhesion, and controlling the force applied and the

melt-stretch rate.

The temperature in the oven was monitored, but it was known to have (relatively) hot and cold spots, and

since the platens had to be removed from the oven to place in the press, it was not possible to physically attach

the thermocouple leads to the platens. Instead, they rested on the platens or hung in the air in the oven. Over

large areas, portions of the polymer would reach the molten state sooner than other portions.

32

Figure 3.1 The inside of the oven used for the manual press, showing the upper and lower platens, the polymer on the lower platen,

and the thermocouple leads (highlighted to increase visibility) (a), and the manual press locked with the bars on either side

in the stretched position as the sample cools, with the upper platen visible (b).

In the manual press, the force applied to the polymer melt through compression of the upper platen was

not controllable or even. The upper platen was manually pushed down onto the lower platen, resulting in uneven

applied force and variation in the applied force from sample-to-sample. Similarly, the melt was stretched by

applying a downward force on the levers shown in Figure 3.1b and then locking the upper platen in place with

the levers. This led to little control and a large amount of variability in the melt stretch rate sample-to-sample.

On average, the force applied and melt stretch rate were estimated to be approximately 500 N and 100 mm min-1

,

respectively.

Hot tack adhesion between the molten polymer and the platens is necessary in order to stretch the polymer

uniaxially. Generally, PP was found to have good hot tack adhesion to the aluminium, but occasionally polymer

residue was left on the platens, which could be cleaned with ethanol. PP was also found to adhere well to

galvanized steel in initial fabrication trials, but since steel has lower thermal conductivity than aluminium (< 80

W m-1

oC

-1 and ~ 237 W m

-1 oC

-1 [1], respectively), the steel platens took longer to cool and so aluminium was a

better choice. To increase hot tack adhesion, water was poured on heated Al platens to introduce a hydroxyl

layer. This did increase the hot tack adhesion, but it also increased the adhesion after the fabricated sample had

cooled, making it difficult to remove. As a result, this method was not used.

Figure 3.2 shows a HMS PP sample fabricated in the manual press. Circled are some holes that appeared

in the as-fabricated skin. It was unclear what caused a localized loss of adhesion, but when the platens were

properly cleaned with ethanol and there was no polymer residue on the plates, the skin was much more complete.

Additionally, if the polymer itself was unevenly spread across the platen, and the applied force was not enough

33

to even it out, in some areas the polymer would not form continuous webs. Indeed, as indicated with the arrow,

individual polymer beads would form disconnected columns of polymer during melt-stretching.

Note that despite the above-mentioned inherent uncertainty in fabrication consistency from the manual

press, once conditions were developed, samples could be fabricated with controlled relative densities and

compressive strength that were on the order of ± 10% [2].

Figure 3.2 A stochastic honeycomb fabricated with a high melt strength PP (approximately 20 cm x 22 cm). The circles indicate

areas with flaws in the skin, and the arrow points to a corner where there was not enough polymer and an interconnected

structure was not formed.

Iteration 2: Fully-Instrumented Process

While it was possible to produce reliable samples with the manual press, it was desirable to have more control

over the process, especially the temperature, applied force, and melt stretch rate. A set of platens were fabricated

with attachments such that they could be inserted into a Shimadzu AG-I Universal Testing Machine (UTM). A

Shimadzu TCE-N300 Thermostatic Chamber was wheeled forward to encompass the platens, and then wheeled

back once the polymer was molten. Figure 3.3a shows the thermostatic chamber encompassing the platens, and

Figure 3.3b shows the platens inside the chamber. The thermostatic chamber could control the temperature

within ± 1 oC, and a thermocouple was used to monitor the temperature of the platens in real time. The

aluminium platens were attached with four screws at the corners to steel plates with small spacers leaving an air

34

Figure 3.3 A Shimadzu AG-I Universal Testing Machine, and the furnace, a Shimadzu TCE-N300 Thermostatic Chamber (a), with

the aluminium platens and thermocouple leads within the furnace (b). Once the thermocouple leads read 180 oC, the upper

platen was lowered and the furnace moved away (c).

Table 3.1 The program for the automated process for fabrication of stochastic honeycombs, where Fapp is the chosen applied force,

is the chosen melt stretch rate, and h is the chosen final height of the sample.

Wait for start

Direction Down

Stroke rate 10 mm min-1

Stop when F ≥ Fapp N

Direction Up

Stroke rate mm min-1

Hold stroke when S = h mm

Figure 3.4 Stress (σ) as a function of time (t) during fabrication from the Shimadzu AG-I UTM.

35

gap. The steel plates were attached to rods that linked to the UTM. The thermocouple leads were inserted into

the gap between the steel and Al plates, with two on the top and two on the bottom. The leads were placed such

that they were in contact with the Al platens and not the steel. Once the temperature of the platens reached 180

oC, the upper platen was lowered into contact with the molten polymer (F ≈ 150 N), and the thermostatic

chamber was opened and wheeled back, as in Figure 3.3c. The UTM was programmed to follow a set of steps to

fabricate the melt-stretched honeycomb from this point. Table 3.1 lists the steps in this process. The upper platen

was compressed at a rate of 10 mm min-1

until the force reached a pre-set force, Fapp. The upper platen was then

raised, uniaxially, at a given melt stretch rate, , until it reached a chosen stroke limit, for the height of the final

sample (h). These variables, along with temperature, could then be varied systematically. The output from the

UTM could be plotted as shown in Figure 3.4, where the stress has been plotted as a function of time during

fabrication (where tensile stresses are positive).

At the tensile peak of the fabrication curve in Figure 3.4, the molten polymer undergoes cohesive failure,

and the web structure is created as air rushes in to the centre of the sample. The hot tack adhesion of the polymer

to the platens must exceed the cohesive strength of the polymer to allow this to happen, and the web structure

that is formed is governed in large part by the surface tension of the melt.

36

References

[1] Y.A. Çengel. Heat Transfer: A Practical Approach. 2nd Ed. ed., McGraw-Hill Higher Education, New York, NY,

2003.


43 (2012) 1024-1029.

37

Chapter 4

Architecture – Process Relationships in Stochastic Honeycombs

M. Hostetter, G.D. Hibbard. Architecture-process relationships in stochastic honeycombs, Journal of Applied

Polymer Science 132 (2015) 42174

4.1 Introduction

Chapter 3 detailed the fabrication process of melt-stretched stochastic honeycombs. Using the fully instrumented

process, this study exams the mechanical response of the polymer melt during melt stretching. The response is

linked to the fabrication variables, Fapp and , as is the resulting architecture, and used as a means to characterize

differences in web interconnectivity.

4.2 Experimental Methods

PP sheets were fabricated by melting PP-1 beads (MFI = 2.4 g/10 min, ASTM D1238 [1], see Appendix A) at

1.1Tm (180 oC) between Al platens attached to a Shimadzu AG-I UTM. The temperature was controlled using a

Shimadzu TCE-N300 Thermostatic Chamber. A force of 500 N was applied to the PP melt, and the stroke was

held constant as the PP sheet cooled in air. Disc-shaped blanks of PP were cut from the sheets (thickness, tblank =

1.5 – 4.5 mm and diameter, dblank = 8.2 ± 0.3 cm) with areal densities (ρA,blank) ranging from 0.13 – 0.37 g cm-2

.

Stochastic honeycombs were fabricated from the blanks (again at 1.1Tm), by applying a preliminary compressive

force (Fapp = 200 – 5000 N) at 10 mm min-1

, before melt-stretching at varying elongation rates (melt-stretch

rates) of = 1, 10, 30, 60 and 100 mm min-1

(corresponding to strain rates of 0.005, 0.040, 0.125, 0.235 and

0.370 s-1

) in ambient air to a height of h = 10 mm. The temperature of the platens was monitored using a 4-lead

thermocouple with leads attached at opposing corners. Five samples were fabricated in each set of conditions,

and were then illuminated from underneath and photographed. The photographs were edited in ImageJ to

produce binary images of the 2D projection of the webs that could be used as inputs for analysis in ImageJ and

MatLab.

38

4.3 Results and Discussion

4.3.1 Architecture

While the process to fabricate stochastic honeycombs is simple, their internal architecture is not. This study

focuses on the formation of the initial web pattern; the simple back-lit images given in Figure 4.1 are able to

illustrate this formation, and are used as the basis of architectural characterization. Due to the presence of

internal archways and partial webs, the projection of the webs on either side of the samples varied slightly as

only the webs directly in contact with the skin were imaged, but the variation within one sample was inside the

standard deviation of the samples fabricated under identical conditions.

Figure 4.2 presents a process map of the melt-stretch rates, , and areal densities, ρA, used during

fabrication. The areal densities varied by a factor of three and the melt-stretch rate by two orders of magnitude.

Figure 4.3 shows the webs of samples fabricated with = 10 mm min-1

and ρA over the range of 0.10 g cm-2

to

0.23 g cm-2

. Over this density range, the structure changes dramatically. At ρA = 0.10 g cm-2

the structure has a

large number of branch points, or nodes, and a large total length of webs. As ρA increases, the number of nodes,

N, decreases and the total length of webs, L, decreases as well. Figure 4.4 shows the webs of five samples at

constant areal density (ρA = 0.13 g cm-2

) and with the melt-stretch rate changing from = 1 mm min-1

to 100

mm min-1

. Qualitatively, the distance between nodes becomes smaller and the number of nodes increases as

increases. N and L are plotted as a function of ρA for = 10 mm min-1

and 100 mm min-1

in Figure 4.5. N and L

both decrease as ρA increases, and increase as increases. Both variables depend more strongly on ρA than , and

there appears to be a larger difference between = 10 mm min-1

and = 100 mm min-1

at low densities for N. A

greater number of nodes implies a more branched structure and a greater number of connected webs. Figure 4.6

plots N against L for = 10 mm min-1

and 100 mm min-1

; for each , the samples fit along a linear line, in which

ρA decreases as N and L increase. As decreases from 100 mm min-1

to 10 mm min-1

, the slope of the N–L plot

decreases from 4.3 nodes cm-1

to 2.4 nodes cm-1

. N/L can be calculated for each sample, and this value is in

effect the number of nodes per unit length of web.

39

Figure 4.1 Examples of the widely different architectures formed over the fabrication space considered, with melt-stretch rate = 10

mm min-1 and areal density ρA = 0.09 g cm-2 (a) and ρA = 0.23 g cm-2 (b), and ρA = 0.15 g cm-2 and = 1 mm min-1 (c), and

100 mm min-1 (d). Scale bar indicates a length of 20 mm.

Figure 4.2 Process Map illustrating the range of fabrication space (melt-stretch rate, , and areal density, ρA) considered. The points

labelled a-d correspond to the 2D projections seen in Figure 4.1.

40

Figure 4.3 The 2D projections of the webs for five samples fabricated at = 10 mm min-1 and with ρA = 0.10 g cm-2 (a), 0.13 g cm-2

(b), 0.16 g cm-2 (c), 0.19 g cm-2 (d), and 0.23 g cm-2 (e). Scale bar indicates a length of 20 mm.

Figure 4.4 The 2D projections of the webs for five samples fabricated with ρA = 0.13 g cm-2 and = 1 mm min-1 (a), 10 mm min-1

(b), 30 mm min-1 (c), 60 mm min-1 (d), and 100 mm min-1 (e). Scale bar indicates a length of 20 mm.

Figure 4.5 The number of nodes, N (a), and the total length of webs, L (b), plotted against ρA for = 10 mm min-1 and 100 mm min-1.

Two samples with N/L = 1.18 cm-1

(ρA = 0.21 g cm-2

, = 10 and 100 mm min-1

) are shown in Figure 4.7.

In the 2D projection, the sample at lower (Figure 4.7a) has a more open structure, while the sample at higher

(Figure 4.7b) has more (projected) closed cells. This can be seen through the Euler number (χ) which is a

measure used in 2D cellular solids and 3D foams to give a quantitative value for the connectedness of a structure

[2, 3]. Practically, χ2D is determined through image analysis by subtracting the number of "holes" in an area

(shaded in Figure 4.7) from the number of connected areas [4, 5]. Note that over the entire 3-dimensional

structure of the stochastic honeycombs, χ3D = 1, since each pore must be connected back to the sample perimeter.

41

Figure 4.6 The number of nodes, N, plotted against the total web length, L for melt-stretch rates of 10 mm min-1 and 100 mm min-1.

Figure 4.7 2D projections for samples having the same areal density and N/L values (ρA = 0.21 g cm-2 and N/L = 1.18 cm-1) but with

χ2D = -5 (a) and -25 (b). These samples were fabricated at = 10 mm min-1 and = 100 mm min-1 respectively. Scale bar

indicates a length of 20 mm.

In a 2D projection, however, partial webs give the appearance of closed cells, which relates to the tortuosity of

the path from sample perimeter to centre. For the samples in Figure 4.7, the values of χ2D are -5 and -25,

respectively; Figure 4.7b has a more connected structure with a greater number of 'closed' cells (i.e., partial

webs) than Figure 4.7a. Over the entire range of samples, χ2D ranged from 0 (ρA = 0.23 g cm-2

, = 10 mm min-1

)

to -94 (ρA = 0.14 g cm-2

, = 100 mm min-1

).

The connectedness of the structure can also be quantified by examining the area fraction of the

(apparently) closed cells. At the outer perimeter there is a fractal-like structure that extends some (varying)

distance towards the centre of the sample before the webs become interconnected. This length scale is expressed

as an area fraction of the sample in which the webs are not interconnected, and is calculated through 1 minus the

area fraction of closed cells, disregarding the area of the webs:

42

(4.1)

where Acells is the area of the closed cells, Atotal is the area of the stochastic honeycomb sample, Awebs is the area of

the webs, and ξ is the fractal area fraction. ξ represents the outer area fraction of air ingress through which the

fractal nature of the stochastic honeycomb extends. For the image in Figure 4.7a, ξ = 0.75, while for Figure 4.7b,

ξ = 0.48. In general χ2D and ξ are negatively correlated.

Overall, N/L, χ2D, and ξ all change with the areal density and with the melt-stretch rate. To illustrate how

the architecture varies over the range of samples fabricated, contour lines representing the ratio N/L were

superimposed on the Process Map, where the samples are grouped by their values of χ2D (Figure 4.8). The values

of N/L increase as ρA decreases and increases, and the same is generally true for χ2D.

Figure 4.8 Process Map of melt-stretch rate and areal density ρA, with all fabricated samples labelled by the corresponding Euler

number (χ2D), and overlaid with contour lines illustrating the trend in N/L.

4.3.2 Fabrication

In order to understand how these internal architectures form, one would like to have an in-situ process parameter

that can be used to track the state of the system during fabrication. The force required to maintain the fixed melt

stretching rate as a function of displacement is a useful indicator because it provides a value for the overall

response of the molecular chains. Figure 4.9 presents the force response (normalized by the area of the melt-

compressed blank to give an effective stress) over one complete fabrication cycle ( = 10 mm min-1

and ρA =

0.21 g cm-2

), where tension is defined to be positive. This process can be considered as having four stages. In

Stage I, the molten blank is compressed to a pre-selected force at a compression rate of 10 mm min-1

. The melt

flows outward on the platens as tblank decreases to tmelt and dblank increases to dfinal. In Stage II, the compressive

43

Figure 4.9 Full fabrication curve (σ, stress, and S, extension) showing all four stages, with compressive stress expressed as negative

and tensile stress as positive, with permanent plastic extension beginning at the origin. The locations marked (i – iv)

correspond to the images on the right. At the beginning of fabrication, the melt (with thickness tblank, and diameter dblank)

(i) is compressed between Al platens in the furnace. In air, the melt (with tmelt, dfinal) is stretched uniaxially (ii), and the

stress increases to a maximum (iii). At the maximum, air ingress begins as the melt fractures, creating the webs of the

stochastic honeycomb (iv). The melt in the webs stretches and some of the webs fracture, creating partial webs and

archways until the sample reaches its final height (h, dfinal) and cools in air.

load is removed and melt is stretched uniaxially in air at a selected elongation rate between 1 and 100 mm min-1

.

Initially, the polymer experiences elastic recovery, but once σ > 0 it is under tension at a constant melt stiffness

(Emelt) and the stress increases to a peak tensile stress. Stage III begins at the peak stress (σpk) with the creation of

new internal surface area due to a combination of flow of the polymer chains and air infiltration. Air ingress then

inflates the core, creating the webs of the stochastic honeycomb, and the stress decreases quickly. In Stage IV,

the stress reaches a nearly constant value as the final web pattern has been formed and the webs stretch at the

plateau extensional value (σext). Some of the webs undergo ductile failure as the sample is stretched to its final

height, h, and partial webs and archways are formed [6]. The temperature at each stage as a function of the

elongation rate is detailed in Table 4.1.

Figure 4.10 shows the tensile portion of the fabrication curves for four samples. Figure 4.10a compares

samples fabricated with = 10 mm min-1

at ρA = 0.09 and 0.23 g cm-2

. The lower density sample has a higher σpk

and a larger Emelt, while σext does not vary with density. Figure 4.10b compares two samples with ρA = 0.14 g cm-2

and = 1 mm min-1

and 100 mm min-1

. The sample with the higher elongation rate has a larger σpk, a higher

Emelt, and a larger σext. Figure 4.11 plots σpk as a function of ρA for = 10 and 100 mm min-1

. σpk decreases as a

function of density for = 10 mm min-1

, but as increases, σpk becomes less dependent on ρA, and is

independent of ρA by = 100 mm min-1

. σpk, Emelt, and σext are plotted as a function of for ρA = 0.13 g cm-2

in

Figure 4.12. σpk, Emelt and σext all increase as increases, independent of density.

44

Table 4.1 Temperature of the platens during melt elongation at the beginning and end of each stage of fabrication.

Elongation Rate,

(mm/min)

Temperature, T (oC)

Beginning of

Stage I

Beginning of

Stage II

Beginning of

Stage III

Beginning of

Stage IV End of Stage IV

1 180 179.3 ± 0.7 165.7 ± 1.8 164.2 ± 0.2 141.0 ± 1.6

10 180 179.6 ± 0.6 178.1 ± 0.4 177.9 ± 0.2 175.9 ± 0.5

30 180 179.3 ± 1.5 178.8 ± 0.3 178.6 ± 0.1 178.0 ± 0.4

60 180 179.4 ± 0.2 178.5 ± 0.2 178.4 ± 0.1 178.1 ± 0.1

100 180 179.0 ± 0.7 178.6 ± 0.2 178.4 ± 0.3 178.1 ± 0.2

Figure 4.10 Stages II, III and IV on the fabrication curve for two samples fabricated with the same elongation rate ( = 10 mm min-1)

but different densities, ρA = 0.09 g cm-2 and 0.23 g cm-2 (a), and two samples fabricated with the same density (ρA = 0.14 g

cm-2) and two elongation rates, = 1 mm min-1 and 100 mm min-1 (b). These curves correspond to the images in Figure

4.1.

Figure 4.11 Peak fabrication stress (σpk) is plotted as a function of areal density (ρA) for = 10 mm min-1 and 100 mm min-1.

45

Figure 4.12 Peak fabrication stress, σpk, and apparent melt stiffness, Emelt (a), and extensional stress, σext (b), pare plotted as a function

of strain rate ( ) for samples with ρA = 0.13 g cm-2.

It is well known that the tensile strength and stiffness of polymers below their crystallization temperature

are strain rate dependent [7, 8]; this is also true for polymer melts [9-11], although the strength and stiffness are

not often measured. At the beginning of the melt extension (Stage II), the polymer is relaxing while the net

compressive stress is being reduced. An additional set of tests were conducted where the samples were held at

the compressed thickness for 15 s before extension began (i.e. loading to a preset force and then holding at that

displacement). From these tests, an effective relaxation rate of τrelax = 2.1 ± 0.1 kPa s-1

was determined from the

stress–time curve. This value was compared to the slope of the initial portion of Stage II for the different

elongation rates. For = 1 mm min-1

, τmelt = 1.7 ± 0.2 kPa s-1

, and for = 100 mm min-1

, τmelt = 147 ± 12 kPa s-1

(calculated from the σ–time curve at the onset of melt stretching). Therefore, at the lowest elongation rate, and

when σ < 0 in Stage II, the polymer is experiencing relaxation without any added tensile extension, while at =

100 mm min-1

, the polymer is being extended much faster than the relaxation rate.

At the transition from Stage II to Stage III, the melt morphology changes rapidly as air enters from the

perimeter. In rheological studies of the elongation of polymer melts, PP failed through necking [12-15].

However, those studies were done on narrow filaments where the diameter was much smaller than the length, so

that d/l << 1 (i.e., not flat blanks such as in the present study where d/l ≈ 6.0 – 10.0). A large and quick drop in

stress from the peak indicates either rapid ductile necking or cohesive failure [15, 16]. A molten filament

undergoing cohesive failure would break into two pieces [16], and ductile failure would result in an elongating

neck leading to eventual fracture [14]. In the present case, the melt is neither fracturing across the entire cross

section, nor thinning to a point, but undergoes a combination of the two mechanisms that results in the complex

web architecture of stochastic honeycombs. The surface tension of the molten polymer greatly influences this

46

process, as the surface area of the polymer increases rapidly. The webs also undergo necking during web

extension, likely in both the drop in Stage III and the plateau in Stage IV.

Generally, the apparent elongational viscosity, , and the strain at failure, εF, are reported when referring

to the elongational properties of polymer melts [12, 15-22]. is calculated [20] through:

(4.2)

where t is time, σ is stress, and is the strain rate. Since is a function of time as well as strain rate, it is

continuously changing during fabrication; is plotted as a function of time, so the curve exhibits a peak as seen

in the σ–extension curves (Figure 4.9 and Figure 4.10). –t is shown in Figure 4.13a for five samples at the

same density and varying . At any given time, is larger for samples pulled at a faster , as is also seen in

rheological melt extension studies [15, 16, 21, 22]. The peaks of the –t curves occur at higher

as

decreases, and at the peak (

) is plotted as a function of in Figure 4.13b for all samples. However, the

literature shows that should increase as increases [17], which is opposite to the trend observed here. This

can be explained by taking the change in temperature into account. From the beginning to the end of Stage II, the

temperature of the platens dropped by less than half a degree (0.4 ± 0.2 oC) for a melt stretching rate of = 100

mm/min, while for = 1 mm/min, the temperature dropped by more than ten degrees (13.6 ± 1.8 oC) as shown in

Table 4.1. The decrease in temperature causes the larger apparent elongation viscosity at low , as has been

shown in the literature [16].

Figure 4.13 Apparent elongational viscosity,

, is plotted against time, t, for five samples with ρA = 0.13 g cm-2 and increasing

elongation rate, (a), and the peak of the elongational viscosity with time, , is plotted as a function of strain rate, ,

on a log–log scale for all samples.

47

Figure 4.14 The Process Map plotting elongation rate, , against areal density, ρA, showing the samples fabricated, with variation in

N/L shown in the data points. Contour lines illustrating the trends in σpk (a), Emelt (b) and (c) are superimposed on

top.

48

Contour lines for σpk, Emelt, and are overlaid on the Process Maps in Figure 4.14, with the data points

representing the value of N/L for each sample. The peak PP melt stress and apparent melt stiffness both increase

as ρA decreases and as increases. Similarly, the apparent melt viscosity at the peak increases as increases, and

is also slightly affected by ρA. These response parameters are all affected by the molecular structure of the

polymer. At the onset of air ingress, structural changes are occurring in the polymer at two distinct length scales.

On the molecular scale, polymer chains are shifting and sliding past each other, and stretching between

entanglements, while on the macroscale, webs are forming and branching, and extending between the skins.

Immediately preceding the peak of the fabrication curve, the web must begin to fracture at the outer perimeter,

small cracks (cavities) form in the melt and the fine edge structure around the perimeter of the blank is formed

(e.g. Figure 4.3 and Figure 4.4). In Figure 4.3a, with = 10 mm min-1

and ρA = 0.10 g cm-2

, the fractal area

fraction ξ = 0.5, while in Figure 4.3e ( = 10 mm min-1

, ρA = 0.23 g cm-2

), ξ = 1. The low density sample appears

to have activated a higher percentage of entanglements on a molecular scale, creating a larger number of nodes

than the high density sample. Figure 4.4a ( = 1 mm min-1

, ρA = 0.13 g cm-2

) has ξ = 0.7, while Figure 4.4e ( =

100 mm min-1

, ρA = 0.13 g cm-2

) has ξ = 0.3. Here, as increases, more entanglements are activated and the

average number of nodes in the architecture increases from 245 (Figure 4.4a) to 440 (Figure 4.4e). Figure 4.3a

and Figure 4.4e have σpk > 70 kPa, while Figure 4.3e and Figure 4.4a have σpk ≈ 40 kPa. This higher σpk could

indicate the higher stress required to deform the polymer to allow a path for air into the centre of the sample.

4.4 Conclusions

The melt-stretching of PP stochastic honeycombs was studied in detail and sub-divided into four stages based on

the mechanical response of the polymer melt. At the peak stress, cavitation created air pockets that then

propagated as cracks towards the centre of the structure. The low melt-stretching rates were found to be on the

same order of magnitude as the apparent relaxation rate of the PP, while at higher melt stretching rates the PP

was being extended much more quickly than the polymer chains could realign. Lower melt-stretch rates and

higher density both led to lower peak stress (σpk) and stiffness (Emelt) during melt stretching, as well as higher

apparent elongation viscosity ( ). The melt-stretch rate played an important role in determining the

architecture of the webs. A slower rate gave a less negative Euler number, χ2D, while a faster rate exceeded the

ability of the polymer chains to reorient, creating a structure with more apparent (projected) closed cells (smaller

ξ, and more negative χ2D). At lower melt stretch-rates, density also played a significant role in the architecture,

with a smaller number of nodes per length of webs, N/L, at higher density. These results illustrate that it will be

possible to tune the final architecture by varying the fabrication conditions.

49

References

[1] ASTM D1238 - 13 Standard Test Method for Melt Flow Rates of Thermoplastics by Extrusion Plastometer. vol.

D1328. West Conshohocken, PA: ASTM International, 2014.

[2] G.E. Schröder-Turk, W. Mickel, S.C. Kapfer, M.A. Klatt, F.M. Schaller, M.J.F. Hoffmann, N. Kleppmann, P.

Armstrong, A. Inayat, D. Hug, M. Reichelsdorfer, W. Peukert, W. Schwieger, K. Mechke. Minkowski tensor

shape analysis of cellular, granular and porous structures, Advanced Materials 23 (2011) 2535-2553.



[4] . hser, . M cklich. Statistical Analysis of Microstructures in Materials Science, Wiley, Chichester, West

Sussex, England, 2000.

[5] K. Michielsen, H. De Raedt. Integral-geometry morphological image analysis, Physics Reports 347 (2001) 461–

538.




[8] Y. Zhou, P.K. Mallick. Effects of temperature and strain rate on the tensile behavior of unfilled and talc-filled

polypropylene. Part I: Experiments, Polymer Engineering & Science 42 (2002) 2449-2460.

[9] H. Münstedt. Dependence of the elongational behavior of polystyrene melts on molecular weight and molecular

weight distribution, Journal of Rheology 24 (1980) 847-867.

[10] H.M. Laun. Prediction of elastic strains of polymer melts in shear and elongation, Journal of Rheology 30 (1986)

459-501.

[11] H.M. Laun, H. Münstedt. Elongational behaviour of a low density polyethylene melt I. Strain rate and stress

dependence of viscosity and recoverable strain in the steady-state. Comparison with shear data. Influence of

interfacial tension, Rheologica Acta 17 (1978) 415-425.

[12] A.Y. Malkin, C.J.S. Petrie. Some conditions for rupture of polymer liquids in extension, Journal of Rheology 41

(1997) 1-25.

[13] J.L. White, Y. Ide. Instabilities and failure in elongational flow and melt spinning of fibers, Journal of Applied

Polymer Science 22 (1978) 3057-3074.

[14] Y. Ide, J.L. White. Investigation of failure during elongational flow of polymer melts, Journal of Non-Newtonian

Fluid Mechanics 2 (1977) 281–298.

[15] Y. Ide, J.L. White. Experimental study of elongational flow and failure of polymer melts, Journal of Applied

Polymer Science 22 (1978) 1061-1079.

[16] J. Lee, S.E. Solovyov, T.L. Virkler, C.E. Scott. On modes and criteria of ABS melt failure in extension,

Rheologica Acta 41 (2002) 567-576.



[18] A. Considère. Memoire sur l'emploi du fer et de l'acier dans les constructions, Annales des Ponts et Chaussées 9

(1885).

50

[19] G.H. McKinley, O. Hassager. The Considère condition and rapid stretching of linear and branched polymer

melts, Journal of Rheology 43 (1999) 1195-1212.

[20] F.H. Su, J.H. Yan, H.X. Huang. Structure and melt rheology of long‐chain branching polypropylene/clay

nanocomposites, Journal of Applied Polymer Science 119 (2011) 1230-1238.

[21] R.P. Lagendijk, A.H. Hogt, A. Buijtenhuijs, A.D. Gotsis. Peroxydicarbonate modification of polypropylene and

extensional flow properties, Polymer 42 (2001) 10035-10043.

[22] W. Minoshima, J.L. White, J.E. Spruiell. Experimental investigation of the influence of molecular weight

distribution on the rheological properties of polypropylene melts, Polymer Engineering & Science 20 (1980)

1166-1176.

51

Chapter 5

Architectural Characteristics of Stochastic Honeycombs Fabricated

from Varying Melt Strength Polypropylenes

M. Hostetter, G.D. Hibbard. Architectural characteristics of stochastic honeycombs fabricated from varying melt

strength polypropylenes, Journal of Applied Polymer Science 131 (2014) 40074

5.1 Introduction

The internal architecture of polymer-based cellular materials is determined in part by the rheological behaviour

of the polymer in question. For example, when a polymer is expanded from the melt state to a closed cell foam,

the stability of the walls is paramount, and two of the most important properties for wall stability are the melt

strength [1-3] and the strain hardening ability of the melt [4-6].

Long chain branching in polypropylene has been shown to increase the processability of PP for industrial

purposes. For instance, Stange and Münstedt [7, 8] studied the rheological properties of LCB PPs and the effect

of the LCBs on foaming of PP. They concluded that the presence of branches led to increased cell density and

cell diameter in the PP foam, as well as an increased expansion ratio. Similarly, Nam et al. [9] determined that

the most important variable in determining foam density was the degree of long chain branching of the PP, and

that lower densities of foam were achievable when using LCB PPs. Additionally, Gotsis et al. [10] modified a

linear PP to an LCB PP, and improved the processability of the PP in foaming and thermoforming.

The mechanical properties of closed cell foams depend on the internal cellular architecture of the polymer

foams, which in turn is controlled by a convolution of processing parameters and the physical properties of the

bulk polymer [11]. This study looks at the effect LCB in PP can have on the internal architecture of stochastic

honeycombs. The influence of polymer rheology and areal density of the melt on the architecture of stochastic

honeycombs was investigated by using four different high melt strength PPs and one conventional (control) PP

to fabricate stochastic honeycombs by melt-stretching. Parallel-plate rheometry tests were used to characterize

52

the viscoelastic properties of the starting polymer, and X-ray tomography was used to characterize the

architecture of the as-fabricated honeycombs.

5.2 Experimental Procedures

The five polypropylenes used in this study were Daploy WB135 HMS, WB140 HMS, WB180 HMS, WB260

HMS (Borealis, Austria) and Accucomp HP0306L (Aclo, Canada), referred to, respectively, as PP-1 to PP-5

(details in Appendix A). To complement the manufacturer supplied melt flow index (ASTM Standard D1238

[12]), each of the five types of PP were characterized by differential scanning calorimetry (DSC) and parallel-

plate rheometry. DSC scans were used to determine the melting temperature and define the processing

conditions for each polymer. Samples were put through a heat-cool-heat cycle at 10 oC min

-1 between room

temperature and 200 oC on a Q20 DSC (TA Instruments). The melting temperatures of the five PP types ranged

from 147 oC to 167

oC and are summarized in Table 5.1. Parallel-plate viscosity tests were performed on an

Advanced Rheometric Expansion System (ARES, TA Instruments) using a dynamic shear test (also called a

dynamic test or a frequency sweep) with sample dimensions of 1 mm through-thickness and 25 mm diameter.

The test sweeps the angular frequency from ω = 0.01 to 100 rad s-1

, with a fixed strain rate of 1% at 1.1 times the

melting temperature (Tm). This temperature allowed direct comparison between the rheological properties of the

polymers at the fabrication temperature. The stochastic honeycombs were fabricated as detailed in Chapter 3 (in

the manual press), by first heating each polymer on an Al platen (at 1.1Tm) until a viscous melt was formed.

Plate and polymer were subsequently removed from the furnace and placed in a custom-built press, where a

second Al platen was compressed on top of the molten PP with a force of ~ 500 N. Subsequently the upper plate

was raised to the desired height (in this case, 15 mm) at a rate of ~ 5 mm s-1

and locked in place. Upon cooling,

the sample spontaneously separated from the platens, resulting in an integrated sandwich structure with upper

and lower skins separated by a network of interconnected webs. Four samples for each polymer, with relative

densities ( ) ranging from 11% to 18% were then characterized in a Skyscan 1172 micro-CT X-ray scanner

Table 5.1 Polymer properties for four high melt strength PP (PP-1 to PP-4) and one conventional (linear) PP (PP-5).

Polymer Tm

(oC)

ηo

( 104 Pa s)

λ (s) Gc

( 104 Pa)

ωc (rad s-1

) MFI

+

(g/10 min)

PP-1 162 2.9 ± 0.6 0.081 ± 0.002 1.5 ± 0.2 12.4 ± 0.3 2.4

PP-2 163 2.9 ± 0.7 0.214 ± 0.066 0.7 ± 0.3 5.0 ± 1.7 2.1

PP-3 163 1.9 ± 0.5 0.039 ± 0.005 1.8 ± 0.5 26.2 ± 3.5 6.0

PP-4 147 3.7 ± 0.5 0.137 ± 0.005 1.5 ± 0.3 7.6 ± 2.0 2.4

PP-5 167 0.4 ± 0.1 0.015 ± 0.001 3.7 ± 1.0 66.2 ± 2.5 12 + provided by the manufacturer

53

(Micro Photonics Inc.) at 44 kV source voltage, 188 μA source current, with an exposure of 158 ms and a

rotational step size of 0.4 degrees. The radiographs were reconstructed into a series of cross-sections, with a

voxel edge length of 35 μm, and binarized to aid the analysis. The cross-sectional slices were then stacked,

giving a three-dimensional model of the internal architecture.

5.3 Results and Discussion

5.3.1 Polymer Characterization

Figure 5.1 plots the results of the parallel-plate rheometry tests, with viscosity (η) given as a function of angular

frequency (ω). Zero-shear viscosity (ηo) was determined by extrapolating the low-shear plateau to a hypothetical

"zero" shear. This parameter is used as a benchmark to compare the viscosity of polymers tested under different

conditions, but at the same temperature [13]. All five polymers exhibited a range of viscous properties under

three identical tests, with the variation given in Table 5.1. Under the frequency range available, only PP-5

exhibited a plateau at low shear, and thus a true zero-shear viscosity. The ηo values given in Table 5.1 are thus

the viscosity at 0.01 rad s-1

, which is a lower bound estimate for ηo.

For an ideally elastic solid, Hooke's Law applies, so that when the stress is changed, the strain changes

immediately, with the modulus acting as a constant of proportionality. For a purely viscous solid, the strain is

time-dependent and perfectly out of phase with the stress, by π/2 rad. A viscoelastic solid, as most polymers are,

has a phase-lag between 0 and π/2 rad. The complex modulus (Eqn. (5.1)) has an elastic response (the storage

modulus, G') that is in phase with the stress, and a viscous response (the loss modulus, G") that is out of phase

with the stress [14, 15]:

Figure 5.1 Parallel-plate rheometry results, giving the viscosity (η) as a function of frequency (ω).

54

(5.1)

Figure 5.2 plots the storage modulus, G', and the loss modulus, G", as a function of ω for two polymers

(PP-2 and PP-5). At low frequency, G" is larger than G', implying that the viscous response dominates at low

frequency [14]. The terminal behaviour of PP-5 at low ω for G' (as indicated by the steep decline as ω decreases)

indicates that this is a linear polymer [1]. This allows the characteristic relaxation time, λ, to be determined,

which is the time-scale separating the predominantly viscous response (t < λ) from the predominantly elastic

response (t > λ) of the polymer melt. λ is determined by [16]:

(5.2)

from which it can be seen that λ = ω-1

where G' and G" intersect. This gives the λ values in Table 5.1.

Figure 5.3 illustrates the relationship between the two components of the complex viscosity, η' and η",

where [16]:

(5.3a)

(5.3b)

From here, it can be seen that the curves for PP-1 to PP-4 exhibit a change in slope after the second derivative

goes through an inflection point at higher η'. This is typically seen for long chain branching in PP, while the

semi-circular curve for PP-5 is indicative of a linear PP [1, 17].

Figure 5.2 Storage modulus (G’) and loss modulus (G") plotted as a function of frequency (ω) for PP-2 and PP-5.

55

Figure 5.3 The two components of complex viscosity (η' and η") from parallel plate rheology tests: the curves for PP-1, PP-2, PP-3

and PP-4 each exhibit a linear portion followed by an increase in slope. This implies that long-chain branching is present

in the polymers. PP-5 exhibits a semi-circular curve, indicating a linear polymer [1, 17].

The crossover point between the storage and loss moduli (ωc,Gc) in Figure 5.2 gives some information

about the breadth of the molecular weight distribution (MWD), and about the weight-average molecular weight

(Mw) for linear PP [9, 18-21]. However, the relationships become more complicated when the polymers are

branched. The polydispersity index (PDI) is inversely correlated with Gc [9, 19, 21], and in turn, Gc is inversely

correlated with either MWD or the degree of branching. For polymers with the same MWD but different Gc, this

measure gives information about the branching density [9], but without specific information about the MWD this

is not a useful relationship. However, a higher G' at lower ω implies more elastic behaviour, and thus a higher

branch density [22], and this can be applied to the four HMS PP. From this measure, PP-1, PP-2 and PP-4 have

very similar branch densities, while PP-3 has less branching. This is corroborated through the relaxation times,

where a longer relaxation time implies a more branched structure [22].

The rheometric data can be also compared to the manufacturer supplied melt flow index (MFI). MFI is

defined as the mass of polymer passing through a cylinder of set dimensions under a given load and fixed

temperature (2.16 kg and 230 oC for PP) in ten minutes, and so is reported in g/10 min, after ASTM Standard

D1238 [12]. It is expected that higher viscosity would lead to less polymer flowing through the cylinder, and

vice versa, and indeed it has been found that MFI generally has an inverse relationship with ηo [16]. However, ηo

is sensitive to both branching and molecular weight distribution [16], leading to differences between the

viscosities of PP-1 and PP-4, even though these two polymers have the same MFI (Table 5.1).

56

5.3.2 Architectural Characterization

The varying rheological properties of the five polypropylenes under consideration lead to differences in the self-

assembled internal architecture of the as-fabricated stochastic honeycombs. The initial areal density of polymer

before fabrication (i.e. the mass per unit area before stretching) also affects the architecture. Finally, the structure

also varies in the through-thickness direction, with the formation of various types of web defects. Variations in

local structure due to the starting polymer type, areal density, and position are discussed below.

Figure 5.4 presents mid-height cross-sections for the five different polypropylenes at a nominal relative

density of ~ 20%. The four HMS PP samples appear comparable, while PP-5 has a much less integrated

structure. Similar to the case of melt fracture during conventional foaming, for relative densities below ~ 20%,

the webs for PP-5 (the linear PP) tended to fail, with the molten polymer flowing back to the skins. Although the

branched morphology of the webs of each stochastic honeycomb type was comparable, there were fewer, thicker

webs for the linear PP.

The normalized cross-sectional areas for each sample were plotted as a function of position, from bottom

to top, and are shown in Figure 5.5. PP-1 to PP-4 all had similar curves, with an initially steep transition from

unit area fraction at the skins to the central third of the sample, at which point a relatively constant plateau

developed. In the plateau, an average normalized cross-sectional area (Ac = Awebs/Atotal) could be determined,

where the average was taken over the central third of the sample height and the standard deviation was taken as

the variation over the same. The web length per unit area within the cross-section (lc = L/Atotal) could also be

determined over the plateau, and like the area, the length of the webs was nearly constant over the middle third

of the specimens (Table 5.2). It is important to note that while PP-5 had a similar density and cross-sectional

area in the plateau to the four HMS PPs, the total length of webs was much less (0.07 mm mm-2

compared to

0.17 ± 0.02 mm mm-2

).

Figure 5.4 Mid-height cross-sections for PP-1 (a), PP-2 (b), PP-3 (c), PP-4 (d) and PP-5 (e), each having a nominal density of ~ 20%.

The scale bar in part (e) represents a length of 5 mm and applies to each figure.

57

Figure 5.5 Cross-sectional area fraction (Ac) as a function of position (P) in the sample (from bottom to top) corresponding to the

samples in Figure 5..

Table 5.2 Cross-sectional area fraction (Ac) and web length per unit area (lc) over the central third of the stochastic honeycomb

samples ( ~ 20%).

Polymer Ac (mm2 mm

-2) lc (mm mm

-2)

PP-1 0.113 ± 0.005 0.195 ± 0.007

PP-2 0.118 ± 0.003 0.200 ± 0.003

PP-3 0.074 ± 0.007 0.147 ± 0.004

PP-4 0.086 ± 0.006 0.172 ± 0.013

PP-5 0.095 ± 0.005 0.071 ± 0.001

The cross-sectional area fraction over the central third of the honeycomb is an important parameter when

considering the mechanical performance because bucking tends to occur where the webs are thinnest and

weakest. However, no trend was observed for Ac as a function of the rheological properties: PP-3, PP-4 and PP-5

have similar cross-sectional areas (Table 5.2) but very different viscosities and rheological properties (Table

5.1). In contrast, the total web length per unit area does seem to be determined by the rheological properties of

the polymer. Figure 5.6a shows an increase in the web length with increasing zero-shear viscosity, before a

plateau is reached at higher values of ηo, while the web length was seen to decrease with increasing crossover

frequency (Figure 5.6b). Crossover frequency is governed by the segmental relaxation time, and as such a lower

ωc implies a more branched structure [22], and more branching indicates increasing entanglement density. It may

therefore be the case that a certain local density of entanglement points needs to be activated during melt-

stretching in order to create each new web. If there are more entanglements grouped closer together, the resulting

melt-stretched architecture will have a greater number of relatively short webs, leading to a greater total web

length. In contrast, if there are fewer entanglements, the webs will be longer and fewer in number, as seen with

PP-5.

In addition to the rheological properties, the starting areal density before melt stretching also affects the

58

Figure 5.6 Cross-sectional web length per unit area, lc, (at a nominal relative density of ~ 20%) plotted as a function of zero-shear

viscosity, ηo (a) and crossover frequency, ωc (b).

Figure 5.7 Mid-height cross-sections of PP-3 at = 10% (a), 13% (b), 16% (c), and 18% (d). The scale bar in (d) is 5 mm and

applies to all images.

resultant architecture. Figure 5.7 shows mid-height cross-sections for PP-3 at overall relative densities of =

10%, 13%, 16% and 18% (corresponding to initial areal densities of 0.14, 0.18, 0.21 and 0.25 g cm-2

,

respectively, before melt-stretching to a height of 15 mm). The mid-height web area fraction increases linearly

with the overall relative density, from 0.031 ± 0.005 at 10% to 0.074 ± 0.007 at 18%. This leads to the question

of whether the mid-height area fraction increases with relative density due to an increase in the web length or the

web thickness. Some insight into this question can be obtained by plotting the cross-sectional area (Ac) for PP-2

and PP-3 as a function of areal density in Figure 5.8. For areal density increases of 50% and 80%, the mid-height

web area fractions increased by 50% and 150% for PP-2 and PP-3, respectively. On the other hand, the

normalized lengths were within 10% and 15% of the mean value for PP-2 and PP-3. Thus, the cross-sectional

area fraction increases linearly with density, while the web length is nearly constant. This implies that as the

density increases, the webs generally become thicker, as opposed to becoming more numerous. It was seen

previously that polymer entanglement, as indicated by the viscosity and relative molecular weight, affects the

59

Figure 5.8 Mid-height cross-sectional area fraction (Ac) of PP-2 and PP-3 as a function of areal density (ρA).

Figure 5.9 Cross-sectional area fraction (Ac) plotted as a function of position, P, for PP-3 at = 13%. The central third of the sample

was used to determine Ac and lc.

Figure 5.10 Cross-sectional area slices from positions a to e in Figure 5. The scale bar in (e) is 5 mm and applies to all figures.

60

total length of webs; here it is seen that the mid-specimen cross-sectional area has a nearly linear dependence on

the amount of material present for a given stretching height.

More subtle characteristics of the cellular architecture can be seen by examining a particular sample in

more depth. Figure 5.9 plots the normalized cross-sectional area as a function of position for PP-3 at a relative

density of 13%. The transition from the relatively constant middle plateau (a) to the fully dense outer skins

begins gradually (b) before steeply increasing through a linear area fraction increase (c – e). Sections through the

core at each of these positions are shown in Figure 5.10. By comparing sequential sections through the sample

height it is possible to identify several distinct types of web defects: archways between incomplete webs,

buttresses supporting some webs near the skin, and air pockets, or voids, in the thicker portions of the webs.

Archways are partial webs that do not stretch entirely from one skin to the other. They can be bounded on one

side or on two by adjoining webs, and provide mechanical support to the adjoining webs even though they do not

provide a direct through-thickness path for load transmission. Figure 5.11 shows an example of an archway (in

PP-3, = 13%), with a 3D representation and three horizontal cross-sections through the archway. Buttresses are

small, partial webs that support the complete webs. They are located adjacent, and attached, to the skin. When a

compressive load is added, they act to resist web buckling and rotation. Figure 5.12 shows an example of two

buttresses (from PP-1, = 11%), with a 3D representation and three horizontal cross-sections through the webs.

Figure 5.13 shows an example of three voids in a web from PP-3 ( = 13%), with a 3D representation and

three horizontal cross-sections through the voids. Voids generally occur near the top or bottom of the sample

where the webs are thicker. For PP-3, the total volume of air pockets in the samples increased from 0.25 cm3 to

0.60 cm3 as the density increased from 10% to 18%. PP-1, PP-2 and PP-4 have a higher total volume of voids,

with volumes ranging from 0.60 cm3 to 0.90 cm

3 for densities increasing from = 11% to 18%.

All four HMS PP types, and to a certain extent PP-5, exhibited archways. Arches occurred in gaps

between webs, and were larger near the mid-height of the structure and smaller near the skin. Buttresses and

voids, however, were not distributed uniformly between the stochastic honeycombs fabricated with different

polymer types. Buttresses were almost non-existent in PP-5, and rare in PP-3, the least viscous of the four HMS

PP. They were much more common in PP-1, PP-2 and PP-4. Voids were present for each polymer type, but were

more common in the least viscous polymers (PP-3 and PP-5). Collectively, these web defects represent a

secondary type of structure, superimposed upon the network of continuous webs spanning the distance between

opposing skin surfaces.

61

Figure 5.11 Archway defect in PP-3 ( = 13%) shown in a 3D compilation image (a), with horizontal lines marking the position of the

cross-sections. The cross-section indicated by the top line is shown in (b), the centre line is (c) and the lowest line is (d).

The scale bar is 2 mm and applies to (b), (c) and (d).

Figure 5.12 Buttress defect in PP-1 ( = 11%) shown in 3D compilation image (a), with horizontal lines marking the position of the

cross-sections. The cross-section indicated by the top line is shown in (b), the centre line is (c) and the lowest line is (d).

The scale bar is 2 mm and applies to (b), (c) and (d).

Figure 5.13 Void defects in PP-3 ( = 13%) shown in a 3D compilation image (a), and in three separate cross-sectional slices (b), (c)

and (d). The scale bar is 2 mm and applies to (b), (c) and (d).

62

5.4 Conclusions

Micro-CT characterization of stochastic honeycombs produced over a range of densities and from

polypropylenes having a range of rheological parameters revealed the complex internal architecture of stochastic

honeycombs. Within a given specimen, the structure changed continuously from position to position, but over

the central third of the honeycomb height, the cross-sectional area and the total length of webs were nearly

constant. For a given polypropylene, the mid-height cross-sectional area was seen to increase linearly with the

areal density of the molten polymer before melt-stretching. However, the total length of webs was relatively

constant over the density range examined. This led to the conclusion that as more polymer was available during

melt-stretching, the webs formed tended to be thicker, rather than more numerous. In addition, it was determined

that the total length of webs was largely governed by the rheological properties of the polymer. Finally, all of the

samples, across all densities and polymer types, showed secondary defects such as archways, buttresses and

voids, adding to the complexity of structural characterization.

63

References

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extrusion with various peroxides, Polymer Engineering and Science 50 (2010) 342-351.

[2] S.K. Goyal. The influence of polymer structure on melt strength behavior of PE resins, 51 (1995) 25-28.

[3] J.M. Dealy, K.F. Wissbrun. Melt Rheology and Its Role in Plastics Processing: Theory and Applications, Van

Nostrand Reinhold, Dordrecht, The Netherlands, 1999.

[4] M. Sugimoto, T. Tanaka, Y. Masubuchi, J.I. Takimoto, K. Koyama. Effect of chain structure on the melt

rheology of modified polypropylene, Journal of Applied Polymer Science 73 (1999) 1493-1500.

[5] H.C. Lau, S.N. Bhattacharya, G.J. Field. Melt strength of polypropylene: Its relevance to thermoforming,


[6] S. Li, M. Xiao, D. Wei, H. Xiao, F. Hu, A. Zheng. The melt grafting preparation and rheological characterization

of long chain branching polypropylene, Polymer 50 (2009) 6121-6128.

[7] J. Stange, H. Münstedt. Effect of long-chain branching on the foaming of polypropylene with azodicarbonamide,

Journal of Cellular Plastics 42 (2006) 445-467.

[8] J. Stange, H. Munstedt. Rheological properties and foaming behavior of polypropylenes with different molecular

structures, Journal of Rheology 50 (2006) 907-923.

[9] G.J. Nam, J.H. Yoo, J.W. Lee. Effect of long-chain branches of polypropylene on rheological properties and

foam-extrusion performances, Journal of Applied Polymer Science 96 (2005) 1793-1800.

[10] A.D. Gotsis, B.L.F. Zeevenhoven, A.H. Hogt. The effect of long chain branching on the processability of

polypropylene in thermoforming, Polymer Engineering and Science 44 (2004) 973-982.





[14] A.P. Deshpande, J.M. Krishnan, S. Kumar, eds. Rheology of Complex Fluids, Springer, New York, 2010.

[15] J. Margolis, ed. Engineering plastics handbook: thermoplastics, properties and applications, McGraw-Hill, New

York, 2006.

[16] A. Shenoy, D.R. Saini. Thermoplastic Melt Rheology and Processing, Marcel Dekker, Inc., New York, New

York, 1996.

[17] Z. Zhang, D. Wan, H. Xing, Z. Zhang, H. Tan, L. Wang, J. Zheng, Y. An, T. Tang. A new grafting monomer for

synthesizing long chain branched polypropylene through melt radical reaction, Polymer 53 (2012) 121-129.

[18] J.C. Chadwick, F.P.T.J. van der Burgt, S. Rastogi, V. Busico, R. Cipullo, G. Talarico, J.J.R. Heere. Influence of

Ziegler−Natta catalyst regioselectivity on polypropylene molecular weight distribution and rheological and

crystallization behavior, Macromolecules 37 (2004) 9722-9727.

[19] C. Tzoganakis. A rheological evaluation of linear and branched controlled‐rheology polypropylenes, The

Canadian Journal of Chemical Engineering 72 (1994) 749-754.

[20] . Bernreitner, W. Neiβl, M. Gahleitner. Correlation between molecular structure and rheological behaviour of

polypropylene, Polymer Testing 11 (1992) 89-100.

64



[22] L. Wang, X. Jing, H. Cheng, X. Hu, L. Yang, Y. Huang. Rheology and crystallization of long-chain branched

poly(l-lactide)s with controlled branch length, Industrial and Engineering Chemistry Research 51 (2012) 10731-

10741.

65

Chapter 6

Stochastic Honeycomb Sandwich Cores

M. Hostetter, B. Cordner, G.D. Hibbard. Stochastic honeycomb sandwich cores, Composites Part B: Engineering

43 (2012) 1024

6.1 Introduction

In the present study, a high melt strength (HMS) PP (PP-1) was used to fabricate stochastic honeycombs through

a simple melt-stretching process. The fabrication process will be described, and it will be seen that this process

creates a random structure that varies from sample to sample. The cellular architecture, seen in Figure 6.1a,

resembles an open cell honeycomb. Compression test coupons and sandwich panels (Figure 6.1b) were

fabricated over a range of densities for out-of-plane compression and three-point bending tests.

Figure 6.1 Top view of an as-fabricated stochastic honeycomb core showing the cellular web structure (a). Also shown is a three

point bend coupon 45 mm x 200 mm, 20 mm core thickness, reinforced with conventional polypropylene face sheets and

an as-fabricated stochastic honeycomb core compression 30 mm x 30 mm test coupon, 20 mm core thickness (b).

66

6.2 Experimental

The fabrication method presented here is a new, simple, low-cost process that requires no blowing agents,

additives or adhesives. The operating temperature in the furnace was 180 oC, and the press consisted of a simple

frame with adjustable height settings to guide the position of twin aluminium platens. The internal temperature

of the furnace and the temperature of the Al platens were measured by thermocouple and an infrared laser

thermometer was used to monitor the temperature of the polymer in the furnace and in the press.

The platens were preheated in the furnace until they reached a temperature between 60 and 100 oC. HMS

PP-1 were then placed on the lower platen and left in the furnace until they formed a viscous melt

(approximately 10 to 12 minutes). At this point, the platens along with the PP were removed from the furnace

and placed in the press. The platens were compressed for approximately 30 seconds to ensure hot tack adhesion

between the upper platen and the PP. The upper platen was then raised vertically in the press, expanding the

polymer uniaxially and spontaneously creating the webbed structure. It was locked in place at the desired sample

height and the press was left to cool. Once the press reached a temperature of approximately 35 oC (after 6 – 7

minutes) the PP stochastic honeycomb separated itself from the Al platens.

The fabrication window spans expansion heights of 10 to 35 mm, with relative densities ranging from 5%

to 25%. For this study, a height of 20 mm was selected with relative densities ranging from 7 – 12%. Given the

relatively large variability in local cell structure, at least five samples were tested for each density range. The

relative density of each core was estimated according to Eqn. (6.1), where mSHC is the sample mass, tskin is the

total average skin thickness (over approximately 50 measurements), w is the width of the square sample, V is the

volume, is the relative density of the core, and ρPP (0.905 g cm-3

) is the density of the bulk PP:

(6.1)

30 mm x 30 mm coupons were cut from the as-fabricated samples and used for compression testing at a

cross-head speed of 1 mm min-1

(after [1]) in a Shimadzu AG-I Universal Testing Machine (UTM). Web

collapse mechanisms during compression testing were investigated by pre-loading samples to characteristic

uniaxial strain values and studying the deformed web structure in a scanning electron microscope (SEM). Three

point bend samples were also made in the same manner, and cut into samples 200 mm long and 45 mm wide

(after [2, 3]). The three-point bend tests were conducted at a cross-head speed of 1 mm min-1

and span length (Ls)

of 100 mm.

67

PP face sheets were purchased commercially, having thicknesses of tf = 0.73 mm, 0.94 mm, 1.19 mm and

1.75 mm. The sheets were cut to size and joined to a subset of the three point bend samples using thermal

welding. The reinforcing sheets were heated on the upper and lower Al platens in the furnace until they formed a

partial melt (i.e., the PP was soft but not flowing). The lower platen was then fixed in the press as before, with

the three-point bend sample placed on top, and the upper Al platen fixed in the press at a height of 20 mm. This

provided sufficient pressure to ensure adhesion between the PP face sheets and the as-fabricated sample while

not affecting the web structure.

6.3 Cellular Architecture

Since the cellular architecture spontaneously forms as the opposing Al platens are separated, there is large

variation in the as-fabricated web thickness, web length and angle between adjacent webs. This variation was

characterized by imaging the cross-section of the samples at mid-height and averaging the thickness over the

web length in order to give a single value per web. Given the relatively large dispersions of structures at a

particular density and the relatively narrow range of densities considered (~ 5%), it was difficult to establish a

clear trend with density. This is illustrated in Table 6.1, where the mean and standard deviation in web thickness

(tweb) and length (b) for four density ranges is presented. There was a large coefficient of variance (standard

deviation normalized by the mean) for both the thickness and length parameters; the coefficient of variance was

between 25% and 30% for the thickness values, and for the length values, the coefficient of variance was found

to be between 55% and 75%, over all densities. Figure 6.2 presents histograms of web thickness and web length

as global histograms covering all densities. Measurements from approximately 500 webs are included here. The

thickness histogram can be seen to be almost Gaussian, while the length histogram is positively skewed. The

inset on Figure 6.2b illustrates this variation in web length.

There is a complex network of partial webs that connect the slender webs of the stochastic core to the

built-in skin. This can be seen in Figure 6.3, which presents a series of SEM images of stochastic honeycombs

cut at mid-height. While much of the mass of the structure is contained in either the skin or the complete webs

Table 6.1 Average web thickness (tweb) and length (b) for four ranges of core relative densities (ρcore).

(%) tweb (mm) b (mm)

7.9 – 8.0 0.547 ± 0.138 4.20 ± 2.29

8.0 – 8.5 0.483 ± 0.120 4.27 ± 2.58

9.4 – 9.7 0.477 ± 0.130 4.21 ± 3.14

10.5 – 11.0 0.544 ± 0.156 4.47 ± 2.46

68

Figure 6.2 Variation in web thickness, tweb (a) and web length, b (b) over all densities.

Figure 6.3 SEM images showing the internal structure of the stochastic honeycomb architecture. An incomplete web with a gap

extending to the skin is shown (a), along with a mid-height gap indicated by the arrow (b), and the complex buttressing of

the webs (c).

(i.e. those webs extending continuously from lower to upper skin), there are also numerous incomplete webs.

These webs may have gaps which extend to the built-in skin (Figure 6.3a) or be fully contained within the web

(indicated by an arrow in Figure 6.3b).

There are three key properties of the PP-1 used in this study that allow the architecture to be formed: hot

tack adhesion, a low melt flow index, and the low surface energy of PP. The first allows the PP to form an

immediate bond with the Al when the platen is hot [4]. The PP adheres strongly to the Al in the melt state,

allowing it to be expanded uniaxially in the press. Secondly, the PP has a low melt flow index (MFI) of 2.4 g/10

min (as measured by ASTM D1238 [5]) which limits the flow of PP in the melt state [6]; the high melt strength

of the PP used in this study (0.31 N following [7]), was higher than that of conventional PP, which is typically

between 0.01 N to 0.20 N [8]. Lastly, while PP has good hot tack adhesion to Al, it has very low surface energy,

making it a poor adhesive. Special surface treatments are required, or polar compounds need to be grafted on to

69

the polymer backbone, to increase the surface energy enough for it to function as an adhesive [4]. The low

surface energy is an asset in this fabrication process as it causes the sample to spontaneously separate from the

Al platens when it has cooled.

Finally, because the sandwich core is composed of a single material, the recyclability of stochastic

honeycombs is high; it can be ground up and pelletized without need for separating the face sheet from the core.

In certain cases, it may also be possible to refabricate the samples without the need for repelletization. For

example, preliminary studies have indicated that it is possible to fabricate a stochastic honeycomb core, crush it

in uniaxial compression, and then re-expand the crushed core without any significant loss in the refabricated

mechanical properties. This characteristic of stochastic honeycombs may become important in energy absorption

applications.

6.4 Mechanical Properties

6.4.1 Uniaxial Compression

Figure 6.4 shows typical uniaxial compression stress–strain curves for three different densities of stochastic

honeycombs. The curves consist of an initial peak stress, followed by a broad valley in which the stress is

relatively constant until final densification. The overall form of the curves resembles that seen for out-of-plane

compression in conventional honeycombs (e.g. [9]). To determine how the webs collapsed during compression,

deformation in the same mid-height location of a ρcore = 11.5% sample was tracked from the as-prepared state

through to final densification; Figure 6.5 presents SEM images of the as-fabricated structure (Figure 6.5a) and

after pre-loading to strains of ε ≈ 0.05 (Figure 6.5b), ε ≈ 0.25 (Figure 6.5c) and ε ≈ 0.35 (Figure 6.5d). The

slenderness of the web structure can be seen in the as-prepared state (Figure 6.5a). Local plastic buckling

(indicated by arrows) can be seen in samples pre-loaded to the peak stress (Figure 6.5b). Large scale global

buckling had occurred by the time the valley stress had been reached (Figure 6.5c) and continued through

densification (Figure 6.5d).

Figure 6.6 shows the peak strength (σp) and the stiffness (E) plotted against the relative density. While the

strength increased from 1.0 MPa at 7.2% relative density to 2.4 MPa at 11.5% relative density, the specific

strength values were approximately constant at 21.8 ± 2.0 kPa m3 kg

-1. Similarly the stiffness varied from 60

MPa to 130 MPa over the same density range, and the specific stiffness was nearly constant at 1.18 ± 0.09 MPa

m3 kg

-1. For out-of-plane compression of honeycombs, the following relationships have been found [9]:

70

(6.2)

(6.3)

These relationships are determined geometrically, based on regular hexagons [9]. When the graphs in Figure 6.6

are fit, for stochastic honeycombs, the following relationships are apparent:

(6.4)

(6.5)

These results match the theoretical honeycomb relationships very closely.

Comparing these specific strength and stiffness values to those seen for commercial foams and

honeycombs over a similar density range yields encouraging results. The specific strength for stochastic

honeycombs (21.8 kPa m3 kg

-1) is approximately 3 to 5 times higher than that of conventional, commercial PP

foams (e.g. 3 – 4 kPa m3 kg

-1 [10] and 3 – 8 kPa m

3 kg

-1 [11]); likewise the specific stiffness (1.18 MPa m

3 kg

-1)

was approximately 4 to 10 times higher (e.g. 0.10 – 0.30 MPa m3 kg

-1 [12]) than that of commercial PP foam.

When compared to PP honeycombs of comparable densities, the specific strength was on par with those found

for commercial honeycombs (16 – 28 kPa m3 kg

-1 [13] and 10 – 22 kPa m

3 kg

-1 [14]). Similarly, the specific

stiffness of the stochastic honeycombs was comparable to that of one commercial honeycomb (0.78 – 1.21 MPa

m3 kg

-1 [13]) and slightly higher than that of another commercial PP honeycomb (0.19 – 0.46 MPa m

3 kg

-1 [14]).

Figure 6.4 Typical uniaxial compression curves for stochastic honeycomb cores.

71

Figure 6.5 SEM images of the web structure in a ρcore = 11.8% sample (a) and after preloading to strains of ε ≈ 0.05 (b), ε ≈ 0.25 (c)

and ε ≈ 0.35 (d).

Figure 6.6 Stochastic honeycomb compressive strength, σp (a) and stiffness, E (b) over a range of core densities.

6.4.2 Three-Point Bend Testing

The stochastic honeycombs were first tested in three-point bending in the as-fabricated condition. Figure 6.7

shows typical load–deflection (F–δ) curves for samples having relative densities of 8.8%, 9.8%, and 10.8%. As

the density increased, the samples failed at a lower deflection and a higher load. The built-in face sheets meant

that there was a continuous transition from core to outer skin, making it difficult to distinguish between the

nearly simultaneous face sheet and core fracture mechanisms. Figure 6.8 plots the flexural strength (σfl) and

flexural stiffness (D) of the stochastic honeycomb cores as a function of relative density. The flexural strength

72

Figure 6.7 Typical load-deflection curves for as-prepared stochastic honeycomb in three-point bending.

Figure 6.8 Flexural strength, σfl (a) and flexural stiffness, D (b) of stochastic honeycomb cores in three-point bending.

and flexural stiffness increased from 0.53 MPa to 0.95 MPa and from 125.5 kN mm2 to 328.2 kN mm

2

respectively as the relative density increased from 8.4% to 10.6%. The significantly greater flexural strength and

stiffness of the higher relative density stochastic honeycomb cores is partly explained by differences in the built-

in skin thickness. For example, the average skin thickness on a ρcore = 11.0% sample was ~ 0.24 mm while the

average skin thickness on a ρcore = 8.7% sample was ~ 0.14 mm.

73

The stochastic honeycomb cores were also reinforced with polypropylene sheets having thicknesses

varying from 0.73 mm to 1.75 mm. Sandwich panels under three-point bend testing typically fail by one of four

mechanisms: delamination of the face sheet, wrinkling of the face sheet, core shear failure, and face sheet yield

and fracture [9]. All four of these failure mechanisms were observed in the reinforced sandwich structures of the

present study. However, it was found that through careful sample preparation, delamination of the face sheet

could essentially be eliminated. Figure 6.9 summarizes the failure mechanisms in the form of a failure map for

these samples, where the dashed lines show the transition between dominant failure modes. It is apparent that at

the lower face sheet thicknesses and lower densities, wrinkling was the dominant failure mechanism. As the face

sheet thickness increased, along with the density, core shearing and then face sheet yield and fracture

consecutively became the dominant failure mechanisms. Typical load–displacement curves for each of these

failure mechanisms are shown in Figure 6.10.

The samples that failed primarily through face sheet yield and fracture failed at the highest applied loads

(Figure 6.10a). These samples generally deformed to between 4 and 6 mm deflection and then snapped, failing

abruptly. For tf = 1.75 mm, the flexural strength and stiffness varied from 5.46 MPa and 1140 kN mm2 at 7.6%

relative density to 6.12 MPa and 1300 kN mm2 at a relative density of 8.8%, respectively.

Core shearing (Figure 6.10b) occurred mostly in the samples with face sheet thicknesses of 0.94 mm and

1.19 mm. In both cases, cracking noises could be heard as the core progressively sheared, corresponding to

periodic load drops in the load-displacement curves. For both face sheet thicknesses, the flexural strength and

stiffness increased with core density. When tf = 0.94 mm, the flexural strength and stiffness increased from 4.32

Figure 6.9 Failure map plotting regions of dominant sandwich panel failure mechanism. The dominant failure mechanism at low

sheet thickness was face sheet wrinkling, core shearing dominated at intermediate thickness, and face sheet yielding and

fracture dominated at the greatest thickness.

74

Figure 6.10 Typical three-point bending load–deflection curves illustrating face sheet yield and fracture (tf = 1.75 mm, a), core

shearing (tf = 0.94 mm, b) and face sheet wrinkling, (tf = 0.73 mm, c).

MPa and 808 kN mm2 at 7.4% relative density to 5.24 MPa and 982 kN mm

2 at 9.1% relative density,

respectively. Similarly, when tf = 1.19 mm, the flexural strength and stiffness increased from 4.48 MPa and 982

kN mm2 at 7.5% relative density to 4.65 MPa and 1198 kN mm

2 at 9.4% relative density, respectively.

75

Wrinkling of the face sheet (Figure 6.10c) caused the samples to fold in on themselves, and most did not

fracture but rather slid off of the supports after reaching displacements of more than 65 mm. Samples that failed

by face sheet wrinkling had the thinnest PP sheet reinforcement and failed at the lowest peak loads. When tf =

0.73 mm, the flexural strength and stiffness increased from 1.88 MPa and 558 kN mm2 at 7.0% relative density

to 2.73 MPa and 596 kN mm2 at 8.9% relative density, respectively.

In the reinforced sandwich panels, failure was either largely controlled by the face sheet (in face sheet

yield and fracture) or by the core (in core shearing and wrinkling). When sample failure was controlled by the

face sheet (for tf = 1.75 mm), the specific flexural strength varied by less than 2% and the stiffness varied by

less than 10% over all the samples, at 78.3 ± 0.9 kPa m3 kg

-1 and 16.0 ± 1.7 N m

5 kg

-1 respectively.

Alternatively, when sample failure occurred primarily in the core, much larger variation in the specific flexural

strength and stiffness was seen. For tf = 1.19 mm (failure by core shearing), the specific flexural strength was

60.8 ± 5.5 kPa m3 kg

-1, a variation of 9%, and the specific flexural stiffness was 13.0 ± 2.3 N m

5 kg

-1, a variation

of 17%. For tf = 0.73 mm (failure by wrinkling), the specific flexural strength and stiffness were 30.9 ± 3.1 kPa

m3 kg

-1 and 7.4 ± 2.1 N m

5 kg

-1, variations of 10% and 28% respectively.

At constant core density, the peak load increased with increasing face sheet thickness. Likewise, at

constant face sheet thickness, the peak load increased with increasing core density. It should be noted that

relatively larger sample-to-sample variability was seen in the flexural properties of the as-prepared samples than

the reinforced samples. For example, at a core relative density of 8.8%, the flexural strength values for the as-

prepared samples were within 15%, while at the same density, the reinforced samples varied by 5% on average.

In flexure, fracture occurs at the outer surface and is more sensitive to changes in the local built-in or reinforced

face sheet thicknesses. The reinforced sheets were not only much thicker, but also more uniform than the built-in

skins, accounting for the reduction in variability in the reinforced samples. Further development of stochastic

honeycomb sandwich panels could include using alternative materials for the face sheets. Glass-fibre reinforced

PP sheets, for example, would have higher tensile strength and stiffness, increasing the flexural properties of the

sandwich panels, but could still be thermally welded in place.

6.5 Conclusions

This study has shown that with minimal equipment, and using a simple, one step process, it is possible to

fabricate a PP stochastic honeycomb core that has comparable compressive performance to conventional PP

honeycomb cores and can exceed the performance of conventional PP foams. Mechanical testing methods were

76

developed, adapted from the ASTM Standards for honeycombs (see ASTM C365, C393 and D790 [1, 2, 15]) for

sample size and strain rate. It was demonstrated that the strength of stochastic honeycombs at a given density

was repeatable despite the random structure and sample-to-sample variation. The built-in skin allows external

face sheets to be joined to the core without using an adhesive. Employing a single thermoplastic material allows

the stochastic honeycombs to be recycled without any special preparation, and allows for the possibility of

refabrication (melting down the samples and re-expanding them uniaxially in the press) without repelletization.

When the as-fabricated stochastic honeycomb cores were tested in three-point bending, the panels failed

by face sheet yield and fracture, and the flexural strength and stiffness varied with the core relative density.

When the cores were reinforced with external PP sheets, all four of the typically observed sandwich panel failure

mechanisms were observed: face sheet yield and fracture, core shearing, face sheet wrinkling and delamination.

With careful sandwich preparation it was possible to eliminate delamination in the reinforced samples. For all of

the sandwich panels, the flexural strength and stiffness increased with core relative density at a constant face

sheet thickness, and also increased with increasing face sheet thickness at constant core relative density.

77

References

[1] ASTM C365 Standard Test Method for Flatwise Compressive Properties of Sandwich Cores. ASTM

International, 2005.

[2] ASTM C393 Standard Test Method for Core Shear Properties of Sandwich Constructions by Beam Flexure,

ASTM International (2006).

[3] ASTM D7250 Standard Practice for Determining Sandwich Beam Flexural and Shear Stiffness, ASTM


[4] I. Skeist. Handbook of Adhesives. 2d ed., Van Nostrand Reinhold Co, New York, 1977.



[6] Ides.com -. Typical Properties of Polypropylene (PP). <http://www.ides.com/generics/PP/PP_typical_properties.

htm> Accessed 4 Nov 2011.

[7] J. Zwynenburg. Predicting Polyolefin Foamability Using Melt Rheology. Associated Polymer Labs, 2008.

[8] H.C. Lau, S.N. Bhattacharya, G.J. Field. Melt strength of polypropylene: Its relevance to thermoforming,




[10] JSP. <http://techdocs.jsp.com> JSP Docs. Accessed 4 Nov 2011.

[11] F.P.C.F.P. Corporation. Arplank Technical information - FPC Foam Products Corporation - Expanded

polystyrene, Bead foam. <http://www.fpcfoam.com/eperan-polypro-tech.html> Accessed 4 Nov 2011.

[12] I. Beverte. Deformation of polypropylene foam Neopolen® P in compression, Journal of Cellular Plastics 40

(2004) 191-204.

[13] Hexacor Ltd. Hexacor - Honeycomb Core Materials. <http://www.hexacor.com> Accessed 4 Nov 2011.

[14] Nida-Core Corporation. Polypropylene Honeycomb Core, Composite Reinforcements - Nida-Core Technical

Data. <http://nida-core.com/english/> Accessed 4 Nov 2011.

[15] ASTM D790 Standard Test Methods for Flexural Properties of Unreinforced and Reinforced Plastics and

Electrical Insulating Materials.

78

Chapter 7

Modeling the Buckling Strength of Polypropylene Stochastic

Honeycombs

M. Hostetter, G.D. Hibbard. Modeling the buckling strength of polypropylene stochastic honeycombs, Journal of

Materials Science 49 (2014) 8365 with kind permission from Springer Science and Business Media

7.1 Introduction

Chapter 5 detailed the effect of rheological properties on the architecture of stochastic honeycombs, and Chapter

6 surveyed the out-of-plane compressive and flexural properties. However, the mechanical properties will

depend on both the polymer material properties (i.e. crystallinity, tensile strength) and the architecture (i.e. cross-

sectional area, arrangement of the webs). In order to build a predictive model for their strength, these effects

need to be separated. To accomplish this, the present study used four HMS PPs with different viscosities and

branching structure to fabricate stochastic honeycombs over a range of densities. The samples were

characterized by X-ray tomography and tested in out-of-plane compression in order to develop a composite

buckling model as a first step towards predicting the strength of stochastic honeycombs.

7.2 Experimental Details

The stochastic honeycombs characterized in this study were fabricated from PP-1, PP-2, PP-3 and PP-4. The

rheological properties of these PPs were characterized in Chapter 5 [1] and Appendix A, and their zero-shear

viscosities (ηo) are presented again in Table 7.1. Each of these four PPs contain long chain branches, which

increases their viscosity [1]. To see if melt-stretching during fabrication affected the crystallinity, differential

scanning calorimetry (DSC) was performed using a Q20 DSC (TA Instruments) on the as-received resin as well

as on portions of webs cut from the as-fabricated stochastic honeycombs. The samples in the DSC were put

through heat–cool–heat cycles between 40 oC and 200

oC at 10

oC min

-1. Tensile coupons were moulded with a

79

Table 7.1 Zero-shear viscosity (ηo) as measured using parallel-plate rheology tests at 1.1Tm [1].

Polymer ηo (x104 Pa s)

PP-1 3.24 ± 0.24

PP-2 3.31 ± 0.39

PP-3 2.15 ± 0.21

PP-4 3.94 ± 0.18

gauge length of 18 mm, width of 8 mm and thickness of 2 mm. The tensile tests were performed on a Shimadzu

EZ-L UTM at a stroke rate of 1 mm min-1

for four samples per polymer. The stochastic honeycombs were

fabricated as before by melting the polymer resins at 1.1Tm (as determined by DSC), compressing the melt

between Al platens and then melt-stretching at ~ 5 mm min-1

to a height of 14 mm. The samples were cut into 40

mm 40 mm 14 mm blocks after cooling. The as-fabricated skin thickness (tskin) was measured by taking ~

200 measurements with a point micrometer around the outside edge of each sample. A minimum of four samples

from each polymer, of increasing density, were scanned by X-ray tomography (Skyscan 1172 micro-CT scanner)

with an 181 μA and 44 kV source. The result, after reconstruction using NRecon software, is a digital 3D

representation of the structure in which each voxel has an edge length of 35 μm, allowing the internal

architecture to be analyzed mathematically. A minimum of three samples at six to seven different densities for

each PP were compressed uniaxially in out-of-plane compression at 1 mm min-1

in a Shimadzu AG-I UTM.

7.3 Results

7.3.1 Polymer Characterization

Figure 7.1a gives the DSC cool–heat thermal cycle for all four starting PP resins, representing a calorimetric

baseline for the polymers with no thermal history. The melt temperature (Tm), crystallization temperature (Tc),

and percent crystallinity (xc) were determined from the baseline. Crystallinity is defined as:

(7.1)

where Δhm is the specific enthalpy of fusion, determined from the area under the melting curve in the DSC, and

Δ is the specific enthalpy of fusion of a crystalline polymer [2] (previously reported values for PP have been

on the order of 210 J g-1

[3-6]). PP-1, PP-2 and PP-3 exhibit similar calorimetric profiles while PP-4 has a lower

Tm, Tc, and crystallinity (Table 7.2). Lower Tm and lower crystallinity are linked, as the crystallinity influences

the melting temperature [7]. The crystallinity of all four PPs is also lower than that of isotactic PP (xc ~ 50 –

60%) [8] as expected, since the HMS PPs are substantially branched [1]. Figure 7.1b shows the melting curves

of PP-1 and PP-4 in more detail. It can be seen that PP-1 has a smooth curve, while the melting curve for PP-4

80

Figure 7.1 Heat flow (ΔH/m) plotted as a function of temperature (T) for the cool–heat cycle for PP-1, PP-2, PP-3 and PP-4 (a), and

the two heating curves (for the as-fabricated webs and the base resin) for PP-1 and PP-4 (b).

Table 7.2 Melting and crystallization temperatures (Tm, Tc) and the crystallinity (xc) of the base polymer and the webs for all four

PPs from differential scanning calorimetry.

Polymer Tm (oC) base polymer Tm (

oC) as-fabricated Tc (

oC) xc (%) base xc (%) webs

PP-1 162.1 ± 1.0 165.3 ± 0.7 125.4 ± 2.2 42.9 ± 1.8 44.3 ± 2.7

PP-2 161.2 ± 0.9 163.9 ± 1.3 125.3 ± 1.1 41.5 ± 1.7 42.5 ± 2.5

PP-3 163.0 ± 0.4 165.3 ± 0.4 126.5 ± 0.2 43.2 ± 1.0 43.8 ± 1.5

PP-4 147.7 ± 1.0 150.8 ± 0.6 110.3 ± 1.1 28.5 ± 2.1 28.2 ± 2.5

has a shoulder at lower temperature. The shoulder indicates that this PP likely has at least some syndiotactic

character, leading to two crystalline phases [9].

Additional samples cut from the webs of as-fabricated stochastic honeycombs were also characterized by

DSC. A direct comparison between the as-fabricated webs and the base resin was obtained from the first and

second heating cycles, respectively, on the as-fabricated webs, with the initial heat cycle giving the thermal

history from the melt-stretching fabrication process. The initial and second heat curves for PP-1 and PP-4 are

shown in Figure 7.1b to demonstrate the effect of this thermal history. Note that PP-2 and PP-3 have very similar

curves to PP-1 and so are not shown here. Since fabrication involves melt stretching, it is reasonable to assume

that there may be some degree of strain-induced crystallinity in the webs. In fact, a small change can be seen

through the melting temperature of the as-fabricated webs, which were ~ 2.8 oC higher than that of the base

properties (Table 7.2). A greater percent crystallinity and larger crystal size are both expected to increase Tm, as

determined from the peak of the melting curve. This peak shifts to higher temperatures in the DSC because more

energy is required to melt the crystals [7, 10-14]. However, the larger percent error associated with enthalpy

based crystallinity measurements made it difficult to unambiguously determine an increase in xc.

81

Figure 7.2 Stress (σ) – strain (ε) curves for tensile tests on PP-1, PP-2, PP-3 and PP-4, pulled at a rate of 1 mm min-1.

Table 7.3 Tensile strength (σTS) and strain at break (εu) for all four PPs.

Polymer σTS (MPa) εu (mm mm-1

)

PP-1 32.1 ± 0.8 0.14 ± 0.02

PP-2 30.9 ± 0.9 0.11 ± 0.02

PP-3 29.4 ± 0.5 0.10 ± 0.01

PP-4 18.2 ± 1.6 > 1.0

The tensile stress–strain curves for PP-1 to PP-4 are shown in Figure 7.2, and the maximum strength as

well as the strain at break are given in Table 7.3. It is evident that PP-1, PP-2 and PP-3 have very similar

strengths, stiffnesses and strains at fracture. PP-4, however, has a much lower strength and stiffness, and a much

larger strain at fracture. A typical fracture strain for PP is 4.0, with a strain of ~ 0.12 at yield [15, 16]. PP-4 was

the only PP that approached this amount of deformation, while PP-1, PP-2 and PP-3, with similar strengths, all

fractured just beyond the yielding point, at a strain of ~ 0.1. PP-4 is quite ductile in tension, while PP-1, PP-2

and PP-3 are comparatively brittle; the significantly lower crystallinity of PP-4 likely plays a role in the

increased ductility [17, 18]. However, this difference in ductility is not expected to affect the overall

compressive strength of the stochastic honeycombs, as out-of-plane compressive failure first occurs through

elastic instability followed by plastic buckling [19]. As a result, the tensile properties outside of the elastic region

should have comparatively little role in determining the peak compressive strength. Note that this difference in

ductility may affect the post-peak collapse mechanisms, but this was not the subject of the present study.

7.3.2 Honeycomb Architecture

Conventional honeycomb sandwich panels have a separately bonded face sheet on either side of a low density

core. In contrast, stochastic honeycombs have a facing skin that is formed during the melt-stretching fabrication

step. As a result, the skin itself is a part of the overall cellular architecture. The skin thickness, tskin, was

normalized by the height of the samples to produce a dimensionless face sheet parameter ( ) and plotted as a

82

Figure 7.3 Skin thickness as a percentage of height ( ) plotted as a function of total relative density ( ).

Figure 7.4 3D reconstructions of a PP-2 stochastic honeycomb at 9.5% core density. Archways are webs that do not extend through

the entire height of the sample (shown with an arrow), and buttresses form at the top and bottom the sample where they

would join the skin (circled). Each scale bar represents 5 mm. The images show a top view of the core (a), a side view (b),

and views from the mid-height of the complex structure (c,d).

function of the relative density, , in Figure 7.3. increases with at the same rate for all four resins, but the

line for PP-3 is shifted upwards towards thicker skins from the other three PPs. This PP had the lowest viscosity

(Table 7.1), suggesting that the relative partitioning of melt between skin and core may be largely controlled by

the resistance to flow in the melt state. The skin thickness can also be used to define a density parameter for the

core, as shown in Chapter 6:

83

(7.2)

where ρPP is the density of polypropylene (0.905 g cm-3

), m is the mass of the sample, w2 is the area of the

sample, and h is the total height.

The distribution of material within the honeycomb core was studied by X-ray tomography. Figure 7.4

shows 3D reconstructions of the X-ray scans for a PP-2 sample with = 9.5%. Figure 7.4a shows the web

structure for the central 90% of the core. It can be seen that many of the webs have a connectivity of 3 (defined

as the number of webs that meet at a node), but also that many webs are unbound on one vertical edge. Figure

7.4b shows a side view of a vertical slice of the upper 50% of the structure, where the upper as-fabricated skin

can be seen, along with the edges of the webs. Figure 7.4c and d show different areas in the structure,

highlighting web buttresses, arches and missing walls.

7.3.3 Mechanical Properties of Stochastic Honeycombs

Mechanical properties for the four grades of polypropylene, with core densities ranging from 7 – 14%, were

measured in uniaxial out-of-plane compression (Figure 7.5). Across all samples, the structure first underwent

elastic compression to a peak load (σp). After the peak, plastic post-buckling occurs and the strength decreases to

a valley stress after which densification begins as the webs collapse into each other. Figure 7.5a shows

representative stress–strain curves for each of the four different PPs at a core density of 11%. Even though all

four of these samples have the same core density, PP-1 and PP-2 have peak strengths nearly double that of PP-3

and PP-4. Figure 7.5b plots the peak strength as a function of core relative density for each of the four PPs. Over

Figure 7.5 Stress (σ) – strain (ε) curves for the out-of-plane compression of stochastic honeycombs fabricated with PP-1, PP-2, PP-3

and PP-4 with core densities (ρcore) of 11% (a), and the peak strength (σp) of the stochastic honeycombs as a function of

ρcore (b).

84

the range of densities, PP-1and PP-2 are very similar, as are PP-3 and PP-4. PP-1 and PP-2 show a three-fold

increase in strength over a core density range of 8 – 14%, while PP-3 shows a two-fold increase in strength over

a core density range of 7 – 11%, and PP-4 shows a two-fold increase in strength over a core density range of 9 –

14%. The following section develops a mechanical model to can explain these ranges of properties.

7.4 Discussion

Significant work has been done on the out-of-plane properties of regular honeycombs through experimentation

[20-24], buckling models [21-25] and finite element analysis (FEA) [25-27]. Irregular honeycombs have been

studied in in-plane compression through FEA [28-33], as have regular and irregular honeycombs with defects

(missing walls) [33-36]. Given the complex internal architecture of stochastic honeycombs, the first step is to

provide quantitative relationships for the web geometry and connectivity. The relative significance of material

and architectural effects can then be seen by developing a buckling based model for the compressive strength of

stochastic honeycombs.

The key architectural factors would be the geometry of the webs (length, thickness) and their connectivity.

In order to determine these properties, the structure was analyzed using Minkowski functionals. These

relationships come from integral geometry, and can be used to quantify the connectivity and geometry of

complex shapes [37, 38]. They are often used in the analysis of open cell foams and other porous materials [37,

39], and can be used to extract quantifiable parameters from 3D reconstructions or 2D images [40].

In 3D, the Minkowski functionals are volume, V, surface area, AS, mean breadth, B, and the Euler number,

χ3D [41, 42]. Mean breadth applies only to convex surfaces [42], so it is neglected in the present analysis. V, AS

and χ3D were computed by running binary stacks of images from the X-ray tomography data through a Matlab

program developed by Legland et al. [43]. χ3D is a measure of the connectivity of the structure, and is defined as

the number of disconnected volumes minus the number of fully enclosed pockets [41, 42]. For the as-fabricated

stochastic honeycomb, χ3D = 1 for all PPs, because the structure is fully connected and there are only minor

enclosed air pockets that can be neglected. The volume and surface area were computed over the central third of

the height of the structures, which was shown to have a constant cross-section in Chapter 5 [1]. The V and AS

functionals can be expressed as the ratio AS/V, which increases as the number of webs goes up and decreases as

the webs become thicker. Surface area-to-volume ratios are often used to characterize foams, and specifically as

a measure of porosity [44-47]. This variable can also be seen as the inverse of an effective web thickness:

85

(7.4)

Figure 7.6 plots AS/V as a function of core density; AS/V decreases as core density increases, and at

approximately the same rate for each PP. This implies that as the core becomes denser, the webs generally

become thicker rather than increasing significantly in total length, matching the conclusions drawn in Chapter 5.

Note that PP-3 generally has a higher AS/V for a given than the other polymers (e.g., at 9.5% core density,

PP-3 has an AS/V value of 8 mm-1

, while PP-1 has an AS/V value of 5 mm-1

) which suggests that in addition to

affecting the partitioning between core and skins, the shear viscosity may also affect the total length of webs in

the cross-section.

Buckling of a thin plate is an extension of Euler’s buckling of a column, developed by Bryan in 1890 [48]

and extended by Timoshenko and Gere in 1961 [49]:

(7.5)

where Et (tangent modulus) and ν (Poisson’s ratio) are material property values, K is related to the end

constraints of the thin plate, and tweb (thickness) and b (length) are geometric parameters. The thickness of the

webs, tweb, was chosen as the independent variable, as this would allow relatively simple comparison to the

experimental strength values through use of the Minkowski functionals (in the form of (AS/V)-1

= teff). Since

failure in stochastic honeycombs is initially elastic, the tangent modulus is taken as the slope of the initial linear

elastic portion of the tensile stress–strain curves for each PP. Note that a mean value of ν = 0.4 was taken for the

Poisson’s ratio in the present study [16, 50-53].

Figure 7.6 Surface area over volume (AS/V) as a function of core density (ρcore) for PP-1, PP-2, PP-3 and PP-4.

86

End constraints divide the webs in the stochastic honeycomb into two broad groups: bound on each side

(denoted with a subscript B) and bound on one side, free on the other (denoted with a subscript F). The value of

K will thus be different for each type of web, as will the length of the webs. Mid-height X-ray images were used

to classify each web as either B- or F-type. For each stochastic honeycomb, the length fraction of bound (f) and

bound–free (1 – f) webs were determined, as were the average length of the webs (bB and bF). To determine K,

the type of constraint must first be chosen. In this first study, a conservative simply supported boundary

condition was selected. This choice minimizes the K values, and implies that the webs have no curvature or

deflection at the edges parallel to the loading axis [54]. Given the buckling and folding observed in the study

presented in Chapter 6 [19], this is reasonable since the model applies to the initial peak stress, on the cusp of

buckling failure. In addition, K is a function of the axial aspect ratio of the webs, or a/b, where a is the height of

the sample without the skin. Values for KB and KF are thus obtained using the standard methods [54, 55], as

detailed in Chapter 2.

The thin plate buckling equation can be split into a material properties portion and an architectural portion,

as seen in Eqn. (7.6):

(7.6)

Since tweb is the independent variable and there is a fraction of B and F webs, this can be re-written as Eqn. (7.7):

(7.7)

σcr changes linearly with Et, while f has a smaller effect. Buckling model parameters (KB, KF, bB and bF) varied

by less than 10% and are summarized in Table 7.4 along with the tangent modulus, Et. Overall, there was good

agreement between the composite buckling model and the experimentally measured peak strengths (model and

experimental data for the weakest and strongest honeycombs are shown in Figure 7.7).

Some insight into the role of the geometric parameters in determining the mechanical performance of the

stochastic honeycombs can be seen by plotting contour lines of density and specific strength over the

architectural variables of web thickness (here normalized by the sandwich core height, a) and the fraction of

bound webs, f in Figure 7.8 (Et = 600 MPa and bB, bF, KB and KF taken as the average of the values given in

Table 7.4). These relationships help to explain the experimental observations of the present study.

87

Table 7.4 Buckling model parameters determined from the X-ray images of the webs, and the tangent modulus, Et, as determined

from the tensile testing.

Variable PP-1 PP-2 PP-3 PP-4

KB 4.04 4.05 4.01 4.00

bB (mm) 6.08 7.26 6.38 6.57

f 0.550 0.636 0.489 0.508

KF 0.465 0.506 0.446 0.458

bF (mm) 4.35 5.00 3.96 4.23

Et (MPa) 784 704 691 427

Figure 7.7 Predicted strength (σcr) from the composite buckling model plotted as a function of web thickness (tweb) for PP-1 and PP-

4, overlaid with the experimental data (σp) plotted as a function of effective web thickness (teff).

Figure 7.8 Normalized web thickness (t/a) plotted as a function of the fraction of bound webs (f) for various specific strengths (σ/ρ).

The core density (ρcore) fraction for a given web thickness is also displayed.

88

PP-1 and PP-2 have the highest strengths of the four polymers, with a similar Et and greater fraction of

bound webs, f, than the other two PPs. PP-3 has a similar Et to PP-1 and PP-2, but a lower f. Perhaps more

significantly, however, is the range of web thicknesses in this architecture. PP-1, 2 and 4 have teff ranging from

0.18 mm to 0.25 mm over a core density range of 9 – 11%. Over the same density range, teff for PP-3 ranged

from 0.13 mm to 0.18 mm. Thus, when plotted as a function of core density, PP-3 appears to be much weaker

than PP-1 and PP-2, but when strength is plotted as a function of web thickness, much of the difference

disappears. Finally, PP-4 has a significantly lower Et than the other PPs, likely a factor of the significantly lower

crystallinity observed in PP-4, and accounting for its lower strength.

7.5 Conclusions

Differential scanning calorimetry and tensile tests were performed on four PPs with varying rheological

properties. Stochastic honeycombs with relative densities ranging from 10 – 17% were then fabricated for each

PP, and the as-fabricated skin thicknesses and core density were determined for each sample. A subset of

samples were characterized by X-ray tomography to determine the internal architecture, and the samples were

subsequently crushed in out-of-plane compression. Minkowski functionals were used in conjunction with the

architectural parameters from the X-ray scan data to develop a composite thin plate buckling model as a first step

towards modeling the strength of stochastic honeycombs.

From the Minkowski functionals, a teff (= (AS/V)-1

) value was calculated and it was shown that this

parameter can be more valuable than ρcore when determining the relative strengths of stochastic honeycombs

fabricated from different PPs. PP-3 has a lower teff for a similar ρcore to PP-1 and PP-2, leading to a lower

strength. A comparison of PP-1 and PP-2 with PP-4 demonstrates the importance of the stiffness of the base

polymer. Through the thin plate buckling model, it was shown that the PP-4 stochastic honeycomb is weaker

than the PP-1 and PP-2 stochastic honeycombs due to the stiffness of the polymer and not to a difference in the

architecture.

The composite buckling model separates the effects of material properties from architecture, giving

insight into the mechanisms that strengthen stochastic honeycombs. This also demonstrates that as a first step, a

simple buckling model can be used to study the effect of missing or broken walls in a regular or irregular

honeycomb in out-of-plane compression.

89

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Chapter 8

Post-Peak Collapse and Energy Absorption in Stochastic Honeycombs

M. Hostetter, G.D. Hibbard. Post-peak collapse and energy absorption in stochastic honeycombs (2015) In

preparation for submission to the Journal of Materials Science

8.1 Introduction

Light-weight cellular materials are used in many applications where in addition to high strength– and stiffness–

to–weight ratios, the energy absorption upon collapse is important. In these materials the internal architecture

generally collapses under a combination of deformation mechanisms that result in the overall stress–strain

response after failure has been initiated. Foams, considered ideal energy absorbers [1], have a nearly constant

(plateau) collapse strength up to densification. In contrast, honeycombs in out-of-plane compression have a

much higher initial peak strength, followed by a generally steeper descent to a valley strength, where the stress

eventually rises again during densification [2]. For a given material and density, honeycombs tend to absorb

more total energy in out-of-plane compression than foams, but the stress decrease can be a drawback in certain

applications [1].

Thermoplastic foams are typically produced by using blowing agents such as carbon dioxide or nitrogen to

create bubbles in the melt [2, 3], or by introduction of volatile compounds (either low melting point liquids or

additives which react and/or decompose) that will form vapour bubbles upon heating [2], and then injection

moulding the final product [3]. Thermoplastic honeycombs can be fabricated through a number of methods,

including extruding sheets of plastic and cutting the sheets into blocks which are then thermally welded or

bonded in narrow strips, and expanded into a honeycomb shape [4], or extruded sheets cuts into strips and slotted

together [4], or through extrusion into a complex mold [5]. Each of these processes require multiple steps to

create a sandwich panel.

In contrast, melt-stretched stochastic honeycomb sandwiches can be fabricated in a single step from the

polymer melt, and have comparable mechanical properties to commercially available honeycombs [6]. Not only

does this simple melt-stretching process create a material that can be easily recycled, but it also allows for the

93

possibility of refabrication. For example, in an initial study, samples were fabricated by melt-stretching, tested

mechanically to collapse, re-melted and stretched to form a new honeycomb, and tested mechanically once more

[6]; the strength of the refabricated samples fit within the standard deviation of as-fabricated samples at the same

densities. This means that one need not go through the normal steps of recycling (e.g. separating, regrinding and

pelletizing [7]); one can simply re-melt the collapsed honeycomb, and melt-stretch it once again into a sandwich

panel. This processing characteristic may be particularly attractive for energy absorption applications.

Stochastic honeycombs have a cross-sectional area fraction of unity at the skins which decreases to a

cross-sectional area fraction (Ac) that is related to the core density and is nearly constant over the central third of

the honeycomb height [8]. Between the skin and the plateau, as the cross-sectional area is changing rapidly,

buttresses and archways between neighbouring webs provide added support to resist rotation at the top and

bottom of the stochastic honeycomb. These architectural features are not present in conventional honeycombs

(where the web cross-section is constant over the sample height) and would therefore be expected to affect the

post-peak collapse mechanisms and energy absorption capacity.

Given this architectural complexity and the potential usefulness of melt-stretched honeycombs as energy

absorbers, the following study investigates the post-peak collapse mechanisms by X-ray tomography, and the

results are compared to the compression curves from previously analyzed coupons (Chapter 7, [9]).

8.2 Experimental Details

The stochastic honeycombs were fabricated by melting high melt strength polypropylene (PP-1) at 1.1 times the

melting temperature, compressing the melt, and stretching it uniaxially to the desired height of the sample, h =

14 mm, in the manual press (details in Chapter 3). Square samples with an edge length of w = 40 mm were cut

from fabricated sheets. Samples were fabricated over the core relative density (ρcore) range of 7 – 18%. The skin

thickness for all the samples were measured, and varied from 2% to 5% of the height of the sample. The

stochastic honeycombs were tested in out-of-plane compression at a stroke rate of 1 mm min-1

. The internal

structure of four samples (ρcore = 10.6, 13.0, 15.8 and 17.7%), called A, B, C and D respectively, were

characterized through X-ray tomography using a Skyscan 1172 micro-CT scanner. To investigate the post-peak

collapse mechanisms, samples were pre-loaded to characteristic strain values and examined once again by X-ray

tomography. Samples A and D were compressed to the beginning of the valley (εv = 0.17 and 0.20 respectively),

and unloaded, releasing the stored elastic energy. Samples B and C underwent the same process, but were

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characterized after pre-loading to the peak (εp= 0.05 and 0.06 respectively), the beginning of the valley (εv = 0.13

and 0.18), and just before the onset of densification (εd = 0.33 and 0.36).

8.3 Results

Figure 8.1 presents reconstructed 3-dimensional X-ray tomography views of the 4 sample types. Previous work

has shown that the dimensions of the webs vary within any given sample, in terms of the length and thickness of

the webs, as well as two classes of webs with different end constraints. Additionally, sample-to-sample, the

number and arrangement of the webs differs, and as density increases, the webs become thicker, even as the total

length of webs remains the same [8, 9]. The stress–strain curves for each sample, indicating where the samples

were unloaded and scanned, are presented in Figure 8.2, with Table 8.1 summarizing ρcore, Ac, the peak

compressive stress, σp, and the valley stress, σv, for all 4 sample types. Note that the stress upon re-loading was

lower than the stress upon unloading. In effect, the load–unload–reload testing procedure of the present study is

a type of interrupted cyclic loading. During the first few cycles of a cyclic fatigue test, polypropylene exhibits a

strength decrease on the order of 10 – 20% [11-13]; during stress relaxation, voids are created, beginning the

crack formation process [14]. The lower stress upon re-loading of the compression coupon after X-ray

tomography at εp and εv (Figure 8.2) is likely due to the fact that a fast decay process begins immediately upon

unloading in the portions of the stochastic honeycomb that have deformed the most.

Figure 8.1 3D reconstructions of samples A, B, C and D (a–d, respectively) with core relative densities as shown in Table 8..

Table 8.1 Relative density (ρcore), cross-sectional area fraction (Ac), peak strength (σp) and valley strength (σv) for samples A–D.

Sample ρcore (%) Ac (mm2 mm

-2) σp (MPa) σv (MPa)

A 10.6 0.085 ± 0.004 2.23 0.88 ± 0.01

B 13.0 0.079 ± 0.003 2.90 0.99 ± 0.06

C 15.8 0.099 ± 0.004 3.52 1.53 ± 0.04

D 17.7 0.113 ± 0.005 4.61 2.25 ± 0.11

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Figure 8.2 Successive compressive stress–strain curves for samples A–D with core relative densities as shown in Table 8.. The

samples were characterized through X-ray tomography in the as-fabricated condition (εo = 0), after loading to the peak

(εp), the beginning of the valley (εv) and just before the onset of densification (εd). The samples were unloaded and

scanned, and then re-loaded. The stress–strain curves were then stitched together.

The post-peak regions of the stress–strain curves in Figure 8.2 represent a complex sequence of failure

mechanisms. In what follows, we will first present the failure of sample type C in order to illustrate the

progression of collapse for a given architecture. Then we will move to compare failure in samples of varying

core density.

Figure 8.3a shows the mid-height cross-section (P = 0.5, where P is the fractional vertical position in the

sample, from 0 at the bottom to 1 at the top) of sample C, with ρcore = 15.8%. Six specific webs, covering two

broad categories, are highlighted. The classification scheme of the webs is based on those that are bound at both

ends (denoted with a subscript B) and those that are free on one side and bound on the other (denoted with a

subscript F) (Figure 8.3b,c). Additionally, anywhere two or more webs meet is called a node.

The 45 interior webs in sample C were characterized after excluding exterior webs that were cut when the

as-fabricated sheets were sectioned into samples. Of the 45 webs, 26 (58%) were classified as bound (CB), and

19 (42%) were bound–free (CF). The six webs marked on Figure 8.3a will be discussed in detail. Table 8.2 gives

the mid-height length (b), area (Aw), perimeter (Pw) and average thickness (tweb) for these six webs, as well as the

height/length ratio, a/b. tweb was calculated by averaging 5 measurements of the web thickness along the length

of the web at mid-height.

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Figure 8.3 The mid-height cross-section for sample C (a) and schematics of bound–bound, B (b) and bound–free, F (c) webs with

the length, b and the height, a as shown

Table 8.2 Dimensions of various webs at mid-height from sample C.

Web b (mm) Aw (mm2) Pw (mm) tweb (mm) a/b

CB-1 8.7 5.20 23.5 0.63 ± 0.35 1.55

CB-2 5.6 2.12 13.3 0.37 ± 0.07 2.40

CB-3 8.3 3.48 8.9 0.41 ± 0.16 1.64

CF-1 1.8 0.83 5.1 0.39 ± 0.18 7.34

CF-2 1.4 0.84 4.9 0.56 ± 0.41 9.57

CF-3 3.1 2.11 7.8 0.61 ± 0.16 4.43

Figure 8.4 and Figure 8.5 show 3D reconstructions from two different perspectives on CB-1 and CB-2,

taken at strains corresponding to the points labelled εo, εp, εv and εd in Figure 8.2, and a through-thickness cross-

section from the centre of the webs (bmid) extending from one skin to the other. These two webs demonstrate

typical collapse mechanisms for bound webs in stochastic honeycombs. The web CB-1 has a larger length, b,

than CB-2, and has a much lower aspect ratio (1.55 compared to 2.40). Both webs show a slight initial curvature,

which is more pronounced at the peak (εp). This increase in curvature is due to plastic deformation, but

pronounced buckling has not yet begun. By the beginning of the valley, at εv, CB-2 has a significant fold with

some small cracks that do not extend through the thickness of the web, while CB-1 shows two horizontal

fractures perpendicular to the loading axis at the fold. The ends of the fold and horizontal fractures terminate in

vertical fractures near the nodes. At the end of the valley, in CB-1 the edges of the fold have completely

fractured, while the curvature of CB-2 has increased without fracturing.

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Figure 8.4 Web CB-1, in the as-fabricated condition (εo), after loading to the peak of the stress–strain curve (εp), at the beginning of

the valley (εv) and the end of the valley before densification (εd). The first two images at each strain show the web from

either side, and the third is a through-thickness cross-section taken in the centre of the web.




98

In addition to dimensional variance and curvature, other defects, such as the hole in the centre of CB-3

(Figure 8.6), were present. The through-thickness cross-sections for CB-3 are from the left, centre and right of the

images directly above the cross-sections. CB-3 has a similar length, b, to CB-1, as seen in Table 8.2, and collapses

in much the same way. At the peak, the curvature of the webs has increased, and at the valley, horizontal cracks

are beginning to form, but have not yet fractured through the web thickness. By the end of the valley, at εd, a

vertical tear where the web meets the node has formed, and horizontal fractures have appeared around the hole.

Overall, the mechanism of collapse in this web is the same as in CB-1. The hole may affect the strength of the

web, but it does not affect the collapse mechanism.

Webs that are bound on one side and free on the other had more variation in dimension and in collapse

mechanism. Three webs, termed CF-1, CF-2 and CF-3, are presented in Figure 8.7 – Figure 8.9, along with their

through-thickness profiles. Figure 8.7 presents one web that has a noticeable change in thickness through the

height of the web. At a vertical position, P (the fractional position in the core, from 0 at the bottom to 1 at the

top) of 0.1, the thickness is 0.53 ± 0.10 mm, decreasing to a minimum of 0.33 ± 0.17 mm at P = 0.7, and

increasing to 0.68 ± 0.23 mm at P = 0.9. At the peak strain, the curvature has increased, and by the valley strain,

a large fold and small horizontal fractures have appeared. The upper fold occurs near the minimum thickness at

P = 0.7, and fractures in the thicker portion closer to the bottom. The fractures grow and the curvature at the fold

increases as densification approaches (Figure 8.7, εd). CF-2, in Figure 8.8, has a much more dramatic change in

thickness, as seen in the thickness profiles. The thickness decreases from 1.72 ± 1.10 mm at P = 0.9 to 0.29 ±

0.21 mm at P = 0.3, and increases slightly to 0.47 ± 0.41 mm at P = 0.1. The thickness on any horizontal plane

through this web decreases sharply from the axial node to the free edge, as seen in the large standard deviations,

as well as varying from top to bottom. All of the deformation occurs in the thinner portions (P = 0.1 to 0.4), with

a large kink at the valley (εv) and a horizontal fracture above. CF-3 (Figure 8.9), in contrast, is thicker overall

(maximum tweb = 1.01 ± 0.78 mm at P = 0.9) and has a more uniform thickness profile. This web, however,

contains a small archway, and this causes the web to collapse by localized buckling near the bottom (P = 0.2, tweb

= 0.48 ± 0.14 mm), as opposed to fracturing or buckling near the centre of the web. The bend in the web at

densification occurs near P = 0.4, with tweb = 0.56 ± 0.17 mm.

99



either side, and the third is a through-thickness cross-section taken in thirds along the lower image of the web.

Figure 8.7 Web CF-1, in the as-fabricated condition (εo), after loading to the peak of the stress–strain curve (εp), at the beginning of



100







101

The detailed analysis that was performed on sample C was extended to samples A, B, and D. In these

samples the interior webs were analyzed as before (the number of webs characterized was as follows: AB – 30,

AF – 14; BB – 18, BF – 17; DB – 28, DF – 10). Select webs demonstrating the range of collapse mechanisms are

shown in Figure 8.10, with these images taken as-fabricated (εo) and at the valley (εv). Some webs showed no

plastic deformation at the valley, indicating that these webs had buckled elastically and rebounded when the load

was removed (Figure 8.10a). Webs were also observed to buckle plastically, where some buckled near the skin

(Figure 8.10b), and some in the central portions of the web (Figure 8.10c). In addition to plastic buckling, many

webs fractured. Some only fractured perpendicular to the loading direction, termed horizontally (Figure 8.10d),

and some also fractured parallel to the loading direction, termed vertically (Figure 8.10e).

The post-peak collapse mechanisms can be divided into three main categories: elastic buckling, plastic

buckling, and plastic buckling with fracture. These mechanisms vary from web to web within one sample, from

sample to sample, and with density, but the mechanism evident at εv is the one that will be discussed. Table 8.3

presents the overall fraction of webs in each sample that failed in each manner, averaged between the bound and

bound–free webs. Generally, the lower density sample, A, had thinner webs, and it was only here where a

significant percentage (~ 10%) of webs showed elastic buckling. It is likely that these webs absorbed mainly

elastic energy and released that deformation upon unloading, as seen in Figure 8.10a. 6% of B and 4% of C webs

failed in the same manner, and none of the D webs exhibited only elastic buckling. Plastic buckling can be

broken down into webs that buckled near the skin and those that buckled in the central portion of the core.

Buckling near the skin occurred where there was a significant change in the profile of the web near the upper or

Figure 8.10 Webs AB-1 (a), BB-1 (b), AF-1 (c), DF-1 (d), and DB-2 (e), in the as-fabricated state (εo) and at the valley (εv). These webs

demonstrate the three collapse mechanisms: elastic buckling, plastic buckling near the skin, plastic buckling in the central

portion of the web, plastic buckling with horizontal tears, and plastic buckling with horizontal and vertical tears, from a –

e respectively.

102

Table 8.3 The percentage of webs from each sample type that exhibited each mechanism of deformation at the valley. Reported

values are based on the characterization of at least 35 webs in each sample.

Collapse Mechanism Sample A

ρcore = 10.6%

Sample B

ρcore = 13.0%

Sample C

ρcore = 15.8%

Sample D

ρcore = 17.7%

Elastic Buckling 10% 6% 4% 0%

Plastic Buckling 43% 38% 32% 17%

Plastic Buckling and

Fracture 47% 56% 64% 83%

lower skin. Across B and F webs, as the cross-sectional area fraction increased, the likelihood of fracture

occurring in the web increased as well. This leads to the conclusion that the distribution of material through the

core is important for predicting the core collapse mechanisms.

8.4 Discussion

A hollow cylinder under out-of-plane compression plastically buckles in a series of folds [15, 16] and this

analysis has been extended analytically and through finite element analysis to square honeycomb unit cells [17,

18] and other honeycomb geometries [19-22]. Stochastic honeycombs do not have closed cells as most

honeycombs have, but the collapse mechanisms of two or more webs joined at a node could be expected to be

linked. It has been shown that thin walls joining at a corner, and under compression, have a linked collapse

mechanism. These corners were termed "super-folding elements" [23, 24] that undergo coordinated plastic

buckling across more than one web. However, due to the irregular dimensions of the webs in stochastic

honeycombs, this does not appear to occur. The specific collapse mechanism, i.e., the number of folds as well as

the vertical position of each fold, of every web in sample C were compared across all webs that joined at a given

axial node. No significant link between the vertical location or number of folds was found. The nodes tend to be

thicker than the webs, and in many cases where a thinner web had buckled and met a thicker node, a vertical

fracture developed (see Figure 8.4a, εv).

In a previous study of melt-stretched stochastic honeycombs, we developed a thin-plate buckling model to

examine the peak strength in out-of-plane compression (Chapter 7 [9]). That study looked at four PPs over a

range of core densities (28 sample types in total) but did not examine the post-peak portions of the stress–strain

curves or the energy absorption. The energy absorption capacity of these sample types has now been analyzed

and those results are reported here for the first time.

The energy absorbed per unit volume can be found by integrating under the σ–ε curve to an arbitrary final

strain, εf:

103

(8.1)

Various choices of strain for εf appear in the literature, and the choice of εf can greatly affect J. Four particular

choices for εf will be compared. They are defined as:

i. εf is a fixed strain across all samples and densities, εf = 0.4 [25, 26]

ii. εf is the strain at which σ = 2σp, [27]

iii. εf is the onset of densification, εd, where εd is the strain at the intersection of the tangent to the

σ–ε curve in the valley and the tangent at σ = 2σp [28, 29]

iv. εf is the strain at maximum efficiency (εη), as defined by: [30, 31]

(8.2a)

(8.2b)

The εf values for PP-1 (corresponding to the X-ray tomography samples) are given in Table 8.4, as a function of

density. The total energy absorbed for all PP-1 samples calculated using each of these definitions for εf is plotted

in Figure 8.11. All four definitions show an initially linear increase in J with density, and the change in slope at

the highest density samples is most pronounced when using the εf = εd and εf = εη indices (Figure 8.11c,d).

Around 11 – 12% core density, the percentage drop from peak strength to valley strength exhibits an

increase in sample-to-sample variation, and the average value appears to level off or begin to decrease. This

transition could be a change in dominant collapse mechanism, from primarily elastic and plastic buckling to

Table 8.4 Final strain values as a function of density, for each definition of strain. The references indicate the study or

studies where this integration range was selected.

ρcore (%) εf = 0.40 [25, 26] εf = [27] εf = εd [28, 29] εf = εη [30, 31]

7.9 0.40 0.51 ± 0.07 0.27 ± 0.04 0.42 ± 0.08

8.6 0.40 0.66 ± 0.09 0.39 ± 0.09 0.41 ± 0.09

9.6 0.40 0.78 ± 0.00 0.65 ± 0.00 0.48 ± 0.07

10.6 0.40 0.78 ± 0.00 0.61 ± 0.01 0.52 ± 0.03

11.5 0.40 0.73 ± 0.05 0.55 ± 0.10 0.39 ± 0.10

12.3 0.40 0.73 ± 0.02 0.54 ± 0.02 0.33 ± 0.04

13.0 0.40 0.72 ± 0.00 0.51 ± 0.02 0.46 ± 0.18

104

Figure 8.11 Energy absorbed, J, calculated up to εf = 0.4 (a), εf =

(b), εf = εd (c), and εf = εη (d), plotted as a function of ρcore.

primarily plastic buckling and fracture. The energy absorption plots, when measured up to εd or εη, demonstrate

this transition, while J measured up to ε = 0.4 or does not capture the change. By the time the stochastic

honeycomb reaches densification, both high and low density samples have fractured. Correspondingly, the

energy absorbed for all samples when densification is included is linear, as seen for εf = 0.40 and εf = . When

measuring J with εf = εd or εη, only the higher density samples have fractured, while the lower density samples

have elastically or plastically buckled.

105

Fracturing of the webs requires greater initial energy input than buckling alone, but once a web has

fractured that loading path is no longer able to support further load. Sample A has the least percentage of

fracturing overall, although sample B had a greater percentage of BB webs that did not fracture. Up to εv, these

samples absorbed the same amount of energy, = 0.19 J cm

-3. Samples C and D absorbed

= 0.34 and 0.57 J

cm-3

. From sample A to sample D, density increased by 60% and the energy absorption capacity almost tripled.

However, at εv, buckling and fracture of the webs has just begun, so the effect of fracture on the energy

absorption may not have been felt. Reducing the frequency of fracturing at the valley would improve the energy

absorption per unit volume of material used in the form of stochastic honeycombs. In absolute terms, σp and σv

increase with density [6, 9], but as density increases the percentage drop from the peak strength to the valley

strength stays relatively constant (± 15%).

Foams are considered ideal energy absorbers because there is a nearly constant plateau strength as

opposed to a peak strength decreasing to a valley. As a result, comparing the plateau strength (σpl) of a foam to

the peak strength of a honeycomb is misleading, but we can compare σv to σpl for two commercial PP foams

measured at ε = 0.25 [32, 33]. At ρcore = 8%, the valley strength of the stochastic honeycomb (0.52 ± 0.21 MPa)

is comparable to the plateau strength of the foams (0.53 – 0.55 MPa). Additionally, as ρcore increases, σv

increases more quickly than σpl for the same PP foams (i.e., at = 11.4%, σv = 1.20 ± 0.10 MPa and σpl =

0.75 MPa [32], and at ρcore = 13.0%, σv = 1.39 ± 0.04 MPa and σpl = 0.93 MPa [33]). Thus the stochastic

honeycombs are absorbing more energy than the PP foams, even when the energy absorbed up to the peak stress

is neglected. This demonstrates that stochastic honeycombs have good potential for energy absorbing

applications.

Figure 8.12 plots the percentage drop (D%) from σp to σv and the energy absorbed up to εf = 0.4 as a

function of core relative density for the four PPs from the previous buckling model study [9], where PP-1 is the

polymer presented in the Results section of this article. Table 8.5 presents the material properties of these

polymers detailed in previous studies (zero-shear viscosity, ηo, melting temperature, Tm, crystallinity, xc, and

tensile strength, σTS [8, 9]), as well as the peak and valley strengths, and J for ρcore = 10%. As seen in Figure 8.12

and Table 8.5, there is no statistically significant difference between PP-1 and PP-2. PP-3 is a lower viscosity

PP, and as a result, the internal architecture has fewer buttresses and archways to provide a secondary support

structure [8]. Indeed, while PP-1 and PP-2 each show a 55% decrease from σp to σv (with a coefficient of

variation of 15%), PP-3 shows a 78% decrease, with a coefficient of variation of 7%. Corresponding to this large

decrease, PP-3 absorbed less energy than PP-1 and PP-2. PP-4, on the other hand, has the highest viscosity, and

D% actually decreases from 50% at ρcore = 9.1% to 30% at ρcore = 13.5%. At ρcore = 11%, the cross-sectional area

fraction over the central third of the honeycomb height (Ac) is the same for PP-1, PP-2 and PP-4. However, as

106

ρcore increases, Ac increases by 33% and 48% for PP-1 and PP-2 receptively, while it remains constant for PP-4

with a coefficient of variation of 6%. Since PP-4 was shown to have a similar skin thickness to PP-1 and PP-2

[9], the extra material must be in buttressing and archways, providing a greater secondary support structure, and

leading to a decrease in D% as ρcore increases. However, since PP-4 has a lower crystallinity than PP-1 and PP-2

(Table 8.5), it has lower σp and σv, and thus less energy absorbed overall.

Comparing this data for D% to the literature, it was found that over a variety of materials (Aluminium,

steel, thermosets and fibre-reinforced epoxy) fabricated into regular honeycombs [21, 26, 34-37], D% varied

from ~ 40% – 85%. Within a given material, D% did not vary significantly with density, and the mean, median

and mode for the data set are, respectively, 60%, 50% and 61.3%. Thus, it appears that the secondary defect

structure consisting of buttresses and archways compensates for the lack of closed cells in stochastic

honeycombs, and can make them superior to the traditional closed-cell honeycombs.

Figure 8.12 Percentage drop (D%) from σp to σv a and energy absorbed, J, up until εf = 0.4 b as a function of ρcore, for PP-1, PP-2, PP-3

and PP-4.

Table 8.5 Zero-shear viscosity (ηo), melting temperature (Tm), crystallinity (xc) and tensile strength (σTS) of PP-1, PP-2, PP-3 and PP-

4 [8, 9], and peak and valley strength (σp, σv), and the energy absorbed (J) up until εf = 0.4 at ρcore = 10%.

Polymer ηo

(x104 Pa s)

Tm (oC) xc (%) σTS (MPa)

σp (MPa) at

ρcore = 10%

σv (MPa) at

ρcore = 10%

J (J cm-3

) at

ρcore = 10%,

εf = 0.4

PP-1 3.24 ± 0.24 162.1 ± 1.0 44.3 ± 2.7 32.1 ± 0.8 2.50 ± 0.08 1.09 ± 0.01 0.47 ± 0.01

PP-2 3.21 ± 0.39 161.2 ± 0.9 42.5 ± 2.5 30.9 ± 0.9 2.21 ± 0.40 1.02 ± 0.12 0.50 ± 0.04

PP-3 2.15 ± 0.21 163.0 ± 0.4 43.8 ± 1.5 29.4 ± 0.5 1.29 ± 0.04 0.27 ± 0.09 0.19 ± 0.05

PP-4 3.94 ± 0.18 147.7 ± 1.0 28.2 ± 2.5 18.2 ± 1.6 1.01 ± 0.15 0.55 ± 0.12 0.27 ± 0.05

107

Figure 8.13 Specific strength (σp/ρ) plotted against energy absorbed up until εf = 0.4 for PP-1, PP-2, PP-3 and PP-4.

Finally, Figure 8.13 plots the specific strength (σp/ρ) as a function of energy absorbed (up to εf = 0.4) for

PP-1 to PP-4. For a given specific strength, PP-3 absorbs the least energy and PP-4 absorbs the most. However,

PP-1 and PP-2 have a greater total energy absorption capacity due to their significantly higher strengths

compared to PP-4.

8.5 Conclusions

The post-peak collapse mechanisms were shown to depend on the distribution of material through the stochastic

honeycomb as well as the geometry of the webs themselves. Unlike honeycombs, each stochastic honeycomb

sample will have some webs that exhibit elastic buckling, some plastic buckling and some plastic buckling with

fracture. However, the frequency of each mechanism varies with density. At lower densities, elastic buckling is

more common, and at higher densities, plastic buckling and plastic buckling with fracture dominate. The change

in dominant failure mechanisms occurs between 10% and 12% core density. This is demonstrated by a change in

slope in the energy absorption curves when energy is calculated up to the densification strain, εd, or the strain at

maximum efficiency of energy absorption, εη. Additionally, it was shown that the secondary defect structure,

consisting of buttresses and archways, increases as the viscosity of the polymer increases. That leads to a

decrease in the percentage drop from the peak to the valley during out-of-plane compression (D%), increasing the

energy absorption capacity of the stochastic honeycomb structure. Finally, it was shown that stochastic

honeycombs have a greater energy absorption capacity than commercial PP foams through comparing the valley

strength, σv, to the plateau strength of PP foams.

108

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Materials Science 49 (2014) 8365-8372.



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81-99.

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110

Chapter 9

Summary and Future Work

This thesis presented the first studies on melt-stretched stochastic honeycombs. The mechanical properties

provided a motivation for the further study of stochastic honeycombs, as well as a baseline to improve upon. The

fabrication variables affect the architecture of the melt-stretched stochastic honeycomb, as do the rheological

properties of the polymer melt. The internal architecture and the bulk material properties of the polymer then

determine the mechanical properties of this hybrid material.

Four high melt strength PPs and one linear PP were used to fabricate stochastic honeycombs. The key

material properties to ensure an interconnected web structure were high viscosity and the ability to strain-harden.

Also important to creating a mechanically strong structure was the presence of secondary defects, such as

buttressing and archways. These were more prevalent in PPs with ηo > 2.5 x 104 Pa s and MFI < 3 g/10 min.

Additionally, polymers with good hot-tack adhesion to metals and low surface energy increase the ease of

fabrication. If a polymer is foam-able, it should be able to be successfully melt-stretched.

From a technological standpoint, the fabrication method was advanced from a manual press to a fully

instrumented process. With the original set-up, there was no fine control over the melt-stretch rate, applied force,

or temperature. The controlled process has improved temperature control (within ± 1 oC), and machine control

over the applied force and melt-stretching rate. An understanding of the fabrication curves was developed such

that the mechanical response of the polymer melt was tied to physical changes in the internal architecture after

fabrication. The force and the melt-stretch rate were identified as key process control variables. A method to

thermally weld PP face sheets to the melt-stretched stochastic honeycomb was also developed, involving

applying enough force to form the thermal weld without compromising the integrity of the core. This led to the

creation of all-PP sandwich panels. Characterization methods were developed to analyze the 3D internal

architecture (determined from X-ray tomography) and the 2D projections of the webs. Topological parameters

(in the form of Minkowski functionals) were adapted to be applicable to stochastic honeycombs, as was the

Euler number for connectivity of a cellular structure. Other characterization variables (such as N, N/L, ξ) had to

be developed to distinguish between architectures that were visually different but had similar architectural

parameters. Mechanical testing methods were adapted from the ASTM Standards for honeycombs and sandwich

panels pertaining to sample size, number of repetitions and strain rate. Through the mechanical testing it was

111

shown that despite the random, open-cell structure of stochastic honeycombs, the mechanical properties were

repeatable. An analytical buckling model for stochastic honeycombs was also developed based on the buckling

of a thin plate. Specific results will be discussed further below.

9.1 Fabrication

The fabrication process involves melting PP at 1.1Tm and applying some preliminary force to ensure adhesion

between the upper Al platen and the polymer. When the force is applied, the polymer melt expands radially. The

melt is then stretched uniaxially, and the stress applied increases to a peak, at which point the melt undergoes a

combination of cohesive fracture and ductile failure as air rushes in to the centre. The tunable fabrication

variables are areal density, ρA, and melt-stretch rate, . The fabrication parameters, corresponding to the response

of the polymer melt, were defined to be peak stress, σpk, melt stiffness, Emelt, and the apparent elongational

viscosity at the peak, . These parameters were chosen for the insight they gave into the deformation of the

polymer during melt-stretching.

The peak stress and melt stiffness both increased as density decreased. This is believed to be due to the

higher fraction of polymer chains that must be adhered to the Al platens where there is less mass present. This

would provide anchoring points for the chains, increasing the stress required to move them and increase σpk and

Emelt. Additionally, as density increased the apparent elongational viscosity at the peak increased as well, albeit

to a much lesser degree. Elongational viscosity (ηE) should be independent of density, but in the non-steady state

environment of this fabrication process, some small effect was seen on , likely due to differences in strain

rate.

As the melt-stretch rate increases, the peak stress becomes almost independent of ρA. Increasing

increases σpk and Emelt. At any specific time, is larger for a higher , but because the melt-stretching process is

not temperature controlled, decreases as increases. Additionally, at low , the melt-stretching rate is

lower than the relaxation rate, while at higher , the melt is stretched faster than it can relax.

9.2 Internal Architecture

The internal architecture of stochastic honeycombs is complex, with buttresses, voids and partial webs

superimposed on an open-cell honeycomb structure with bound and bound–free webs. The internal web structure

112

is then sandwiched between two parallel skins. Methods to quantify the architecture from the X-ray scans had to

be developed, to extract meaningful parameters from the countless possible variables in the stochastic

architecture. The cross-sectional area fraction, Ac, decreases from 1 at the skins to a minimum plateau value over

the central third of the honeycomb height. Across that central third, the total length of the webs per area, lc, is

also constant. The webs themselves extend from one skin to the other, and can be classified as bound on both

sides (bound), or bound on one side and free on the other (bound–free). Partial webs, such as archways and

buttresses, serve to support the complete webs and do not extend from skin to skin.

The melt-stretch rate during fabrication has a large effect on the architecture of stochastic honeycombs.

Higher stretch rates increase the number of nodes per length of webs (N/L), implying shorter webs (i.e., lower b)

overall. The total number of nodes increases as increases, and to a lesser extent the total length of webs also

increases as increases. The Euler number in 2D, χ2D, is defined as the number of connected areas minus the

number of apparently closed cells, where the webs consist of 1 connected area. A higher stretch rate leads to a

more negative Euler number, and thus a more tortuous pathway for the air to flow through. ξ was defined as the

outer area fraction of air ingress before the webs become interconnected and form apparently closed cells. ξ thus

increases as χ2D approaches 1, since a larger fraction of air ingress means a less interconnected structure. A

higher stretch rate leads to smaller ξ.

The melt rheology has a large effect on the architecture of stochastic honeycombs. Increased viscosity

leads to a greater total length of webs, and more uniform partitioning of material between the skin and the core.

The PPs with lower viscosity had more material in the skins and the transition to the plateau, and less in the webs

themselves. If the viscosity was too low, like the linear PP-5, many of the webs failed during melt-stretching and

no interconnected structure was formed. On the opposite end, higher viscosity and melt strength led to an

increase in partial webs, such as buttresses, in the structure. Total web length increased as the viscosity

increased, but reached a plateau at the highest viscosity. This implies that an increase in entanglement density

leads to an increase in web length and the conclusion that activation of a certain local density of entanglement

points is required to form a new web.

The applied force during fabrication affects the areal density during melt stretching, and thus the final

density, so Fapp and ρA can be collapsed to one variable. Higher density structures have a less negative Euler

number and as such a larger outer area fraction of air ingress (greater ξ). At lower density, the architecture has

more apparent closed cells, and lower ξ. Lower density structures also have a higher numbers of nodes per length

of webs, N/L. Higher density leads to thicker skin and increased web thickness. The increase in web thickness

can be seen through the increase in Ac as density increased, while lc remained the same. Additionally, over the

113

central third of the honeycomb height, as density increased the surface area-to-volume ratio decreased, and thus

the effective web thickness, teff, increased.

9.3 Mechanical Properties

Under out-of-plane compression, stochastic honeycombs, like regular honeycombs, experience elastic

compression to a peak load, a decrease in stress to a valley, and then an increase in stress at densification. The

structure changes sample-to-sample, but for a given density the strength is consistent. Stochastic honeycombs

collapse through a combination of elastic buckling, plastic buckling, and plastic buckling with fracture, with

elastic buckling more common at lower density and plastic buckling with fracture more prevalent at higher

density. The total energy absorbed depends on the final strain chosen, but the peak strength is comparable to out-

of-plane compressive strength of commercial PP honeycombs and the valley strength is comparable to the

plateau strength of commercial PP foams. The sample-to-sample variation under three-point bending was

reduced when face sheets were thermally welded to the skin creating sandwich panels, and the sandwich panel

failure mechanisms depend on face sheet thickness, core density, and core thickness. The exponents relating

specific strength and stiffness to relative density are 1.59 and 1.01, respectively, and are very close to the

theoretical hexagonal honeycomb exponents of 1.67 (or 5/3) and 1.

The material properties of the PP used to fabricate stochastic honeycombs affect the mechanical properties

of the honeycomb. The crystallinity of the polymer affects the tensile strength and stiffness of the webs, and thus

the buckling strength. The tangent modulus is a key input to the material property portion of the buckling model,

and during buckling the portion of the web at the fold experiences a tensile stress. Increased crystallinity allows

the webs to resist buckling under greater applied stresses.

Thicker webs increases the strength of the stochastic honeycomb in out-of-plane compression, as the

buckling strength is related to the web thickness, squared. Shorter webs (in the in-plane directions, i.e., lower b)

should also increase the strength of the stochastic honeycomb. Additionally, a greater fraction of webs bound on

both sides as opposed to bound on one side and free on the other will increase the buckling strength, as these

webs can support more load. This can be seen through the buckling coefficient, K, that is related to the end

constraints of the webs. KB is approximately an order of magnitude larger than KF, so the bound webs have a

much greater effect on the strength of the stochastic honeycomb than the bound–free webs. An interconnected

structure is also believed to absorb more energy than a less connected structure. Finally, some architectures have

114

more buttresses and archways, and it was shown that these features contribute to an increase in the valley

strength compared to the peak strength (i.e., decreased D%), which increases the energy absorption as well.

9.4 Future Work

The original motivation for this project came from the idea that a sandwich structure could be created with less

environmental impact than what is currently available commercially. Using the simple and low-cost, low-impact

fabrication method, the carbon footprint of the stochastic honeycombs would be expected to be lower than that

of commercial foams and honeycombs. A detailed life-cycle analysis of the impact of foams and honeycombs, as

well as stochastic honeycomb cores, would give the best overview of the relative and absolute environmental

impact. In order to quantify this, the energy requirements to make foams, honeycombs and stochastic

honeycombs must be determined.

Taking this idea a step further, we would like to create stochastic honeycombs using biodegradable

polymers. This would eliminate the need for recycling at end-of-life, and reduce the environmental impact of the

product. From Chapter 5, the effect of melt properties of the polymer on the architecture, it is possible to define a

range of parameters, such as melt flow index and melt strength, from which it can be predicted whether or not

the polymer will be able to form an interconnected stochastic honeycomb structure. Creep tests on the polymers

would allow a better measurement of the zero-shear viscosity than is obtainable from the parallel-plate

rheometry tests, and determining the surface tension of the molten polymers at the fabrication temperature

should give added insight into the mechanism of web formation.

For PP stochastic honeycombs, some questions about the relationships between fabrication, architecture

and mechanical properties remain. It has been shown that a lower viscosity polymer stochastic honeycomb has

more material in the skins and transition region than in the central third of the height of the honeycomb. Also,

higher density stochastic honeycombs have thicker skin as well. The partitioning of material through the

honeycomb is thus affected by the density and the rheology of the polymer; it could also be affected by the melt-

stretching rate during fabrication. If the relative partitioning of material is affected, then the prevalence of

secondary defects in the transition region could be affected as well. Additionally, the varying structures created

at a single density with varying melt-stretching rates may have different strengths, so it should be possible to

determine a structure that is the most efficient, i.e. the highest strength per mass of material. This could be

related to the number of nodes per length of webs, or another characterization variable, and linking those

variables to the strength of the honeycomb would increase the tunability of the stochastic honeycomb properties

115

for a given application. Compression testing structures at the same density fabricated at varying melt-stretching

rates would begin to solve this question.

Mechanically, it is important to expand our existing knowledge of the mechanical properties to include the

impact resistance, and shear and fatigue properties. Depending on the application, any of these properties could

be very important. In order to get reliable data for the shear strength and shear modulus, a series of three- and

four-point bend tests need to be performed.

It has also been shown that during fabrication, air ingress can be directed to come from the top or bottom

(or both) of the structure as opposed to solely from the outer perimeter of the melt through the introduction of

platens with holes. A square or other grid of holes can be used to create an almost regular honeycomb in which

the air enters each cell from the top or bottom, and thus the cells themselves are closed, and the honeycomb

structure has a series of holes in the as-fabricated skin. This closed cell stochastic honeycomb then more

accurately reflects the mathematical construct of a Voronoi honeycomb. With a structure of almost entirely

bound webs, the out-of-plane compressive strength should increase. Additionally, there should be an optimal

hole size and hole spacing to achieve the most efficient architecture.

Finally, all of this work has been towards fabricating flat sheets of a PP stochastic honeycomb. It is

possible to create the open-cell honeycomb structure using two non-parallel platens, or with a more complex

profile, such as corrugated Al platens. The melt would still be stretched uniaxially, so the properties in any

direction other than axially would be related to the shear properties as opposed to the out-of-plane compressive

properties. However, being able to form complex shapes is an advantage that foams have traditionally had over

honeycombs; melt-stretched honeycombs may be able to fill this gap.

116

Appendix A

Polymer Properties, PP-1 to PP-5

The known material properties for PP-1 to PP-5 are presented here. The melting and crystallization temperatures

(Tm, Tc) and crystallinity (xc) were determined from differential scanning calorimetry, and the tensile strength

(σTS), tangent modulus (Et) and ultimate strain (εu) from tensile tests (detailed in Chapter 7). Elongational

viscosity tests were performed on PP-1 and PP-5 (ηE and the temperature of the test, ). Additionally, parallel

plate rheometry tests (ηo, Gc, ωc and λ) were carried out on all five polymers (detailed in Chapter 5).

Polymer PP-1 PP-2 PP-3 PP-4 PP-5

Technical Name WB135HMS+ WB140HMS

+ WB180HMS

+ WB260HMS

+ HP0306L

*

ρ (g cm-3

)‡ 0.905 0.905 0.905 0.905 0.905

ρmelt (g cm-3

) ‡ 0.714 0.714 0.714 0.714

LCB (n/1000 C) ‡ 0.21 0.21

MFI (g/10 min) ‡

2.4 2.1 6 2.4 12

Tensile Modulus

(MPa) ‡

2000 2000 2000 900 1600

Melt Strength (cN) ‡ 32 36 10 27

Melt Extensibility

(mm s-1

) ‡

250 255 245 250

D (MPa) ‡ 1600

fl (MPa) ‡ 46

Tm (oC) 162.1 ± 1.0 161.2 ± 0.9 163.0 ± 0.4 147.7 ± 1.0 166.7

Tm,fab (oC) 165.3 ± 0.7 163.9 ± 1.3 165.3 ± 0.4 150.8 ± 0.6

Tc (oC) 125.4 ± 2.2 125.3 ± 1.1 126.5 ± 0.2 110.3 ± 1.1 124.1

xc (%) 44.3 ± 2.7 42.1 ± 2.5 43.8 ± 1.5 28.2 ± 2.5 43.7

TS (MPa) 32.1 ± 0.8 30.9 ± 0.9 29.4 ± 0.5 18.2 ± 1.6

Et (MPa) 784.4 ± 16.1 703.5 ± 27.7 690.8 ± 25.8 426.6 ± 23.2

εu (mm mm-1

) 0.14 ± 0.02 0.11 ± 0.02 0.10 ± 0.01 > 1

ηE (kPa s) 334.7 120.9

(

oC) 155 165

ηo (x104 Pa s) 3.24 ± 0.24 3.31 ± 0.39 2.15 ± 0.21 3.94 ± 0.18 0.48 ± 0.01

Gc (kPa) 16.2 ± 0.5 7.0 ± 2.7 17.9 ± 5.1 14.7 ± 3.3 37.5 ± 10.0

ωc (rad s-1

) 12.5 ± 0.4 5.0 ± 1.7 26.2 ± 3.5 7.6 ± 2.0 66.2 ± 2.5

λ (s) 0.080 ± 0.002 0.214 ± 0.066 0.039 ± 0.005 0.137 ± 0.031 0.015 ± 0.001 + Borealis AG, Daploy

TM HMS PP [1]

* Aclo Compounders, Accucomp PP [2]

‡ Provided by the manufacturer

117

References

[1] Boralis AG. Daploy(TM) HMS Polypropylene for Foam Extrusion <http://www.borealisgroup.com>

Accessed 26 Nov 2013

[2] Aclo Compounders <http://www.aclocompounders.com/accucomp.htm> Accessed 26 Nov 2013

118

Appendix B

Melt-Stretched Honeycombs with Various Polymers

During fabrication, parallel skins are formed on the Al platens. The skins are separated by webs parallel to the

direction of applied tensile stress. The skin is approximately 40% thicker on the bottom of the sample (in contact

with the lower platen) than on the top of the sample. This is due to a combination of gravitational effects on the

melt and the surface tension of the polymer as the molten polymer adheres to the upper platen and is pulled

upward, redistributing the polymer vertically.

The webs form an interconnected network that is nearly constant in one dimension, and varies in the

others, making in analogous to a 2D cellular material. Figure B.1 plots the zero-shear viscosity (ηo) of a number

of polymers (PP, PE, and copolymers of PE and PP, CP) as a function of the melt flow index (MFI) of these

polymers. Viscosity and MFI have an inverse relationship, as demonstrated here. The fabricated structures, from

the side with the skin visible, of select samples are also shown in Figure B.1. The highest viscosity PE, with MFI

= 0.5 g/10 min, did not expand at all (Image 1). For this polymer, the

Figure B.1 Zero-shear viscosity (ηo) plotted as a function of melt flow index (MFI) for a number of PP, PE, and copolymers of PP

and PE (CP), along with side images of representative samples.

119

cohesive forces in the polymer exceeded the adhesive force to the Al platens. Image 2 is indicative of all four

HMS PP (MFI = 2.1 – 6 g/10 min, ηo = 2 – 4 x 104 Pa s). Images 3 – 6 show the various structures composed of

individual columns or webs, obtained from PE, CP and PP for M I ≥ 5 g/10 min. In Image 5, the upper skin

cooled and separated from the platen and then had no webs or columns to act as supports. For many of these

lower viscosity polymers, it was apparent that the polymer stretched during fabrication and created webs, but

then flowed back to the skin before the structure cooled.

Figure B.2 shows images from the top of melt-stretched honeycombs fabricated with HMS PP, linear

isotactic PP (L-iPP), and low density polyethylene (LDPE). In order to form these L-iPP and LDPE structures, a

greater amount of polymer was used and the platens were cooled quickly. Figure B.3 shows SEM images from

the top and from an angle for the same three polymers. The webs of the HMS PP have buttresses and archways,

which create a secondary defect structure that helps support the webs. In contrast, the L-iPP and LDPE webs

transition smoothly to the skin, with no buttressing at all.

Figure B.2 Stochastic honeycombs fabricated with HMS PP (a), linear iPP (b) and LDPE (c). The scale bar on (c) is 3 cm and applies

to all three images.

120

Figure B.3 SEM images of stochastic honeycombs fabricated with HMS PP (a,b), L-iPP (c,d) and LDPE (e,f), perpendicular to the

webs (top row), and at a tilt (bottom row). The HMS PP has buttresses that support the webs at the transition between the

webs and the skin, while the L-iPP and the LDPE have smooth transition regions with no buttressing.

121

Appendix C

Characterization of Stochastic Honeycomb Sandwich Failure

M. Hostetter, G.D. Hibbard. Characterization of stochastic honeycomb sandwich failure (2013) In: Proceedings

of the International Conference on Composite Materials 19, Montreal, Canada, 28 July – 2 August, pp. 312–319

C.1 Introduction

Foams and honeycombs are composite materials in that they combine the properties of the base material with

empty space through an internal cellular architecture. Stochastic honeycomb structures are a new type of cellular

composite and a variation on the traditional plastic foam or honeycomb sandwich core material. Honeycombs are

essentially two dimensional, with regular, repeating unit cells, while foams have a range of cell size, web

thicknesses and lengths. Stochastic honeycombs are essentially two dimensional, like honeycombs, but have web

lengths and thicknesses that are variable (Figure C.1). Even in the through-thickness direction, the stochastic

honeycomb webs can change in length, thickness, and orientation.

Produced in a simple, low-cost process, the out-of-plane compressive properties of polypropylene (PP)

stochastic honeycombs rival that of PP honeycombs and exceed that of commercial PP foams [1]. For example,

over a relative density range of 7 – 12% the compressive strength of stochastic honeycombs ranged from 1.0 –

2.5 MPa [1], while commercial honeycomb strengths ranged from 1.1 – 2.4 MPa [2-5].

Depending on the architecture and loading configuration, honeycombs have been shown to fail by

buckling, bending, or fracture [6, 7]. Analytical solutions for failure of honeycombs by elastic buckling [6],

plastic collapse (bending) [7], and fracture [6] have been determined and tested experimentally. The failure

mechanism has been found to depend on the density of the honeycomb [6], with elastic buckling typically

occurring at low densities and fracture at higher densities when the local stress exceeds the fracture stress of the

material. Since honeycombs have a regular structure, the geometry of one unit cell is representative of the whole,

and the same failure mechanism is generally seen throughout the structure. This is not the case for stochastic

honeycombs given their complex internal architecture. Multiple failure mechanisms are expected depending on

122

Figure C.1 3D image reconstructed from a micro-CT scan, with a sample size of 40 mm 40 mm 15 mm (a) and the web structure

as seen from the side (b), with the as-fabricated face sheets removed digitally in order to more clearly reveal the internal

web structure. The scale bars are 5 mm each.

the local geometry of the webs. In addition, studies have shown that flaws or defects in a thin-walled structure

(such as the webs in a stochastic honeycomb, or the walls in a conventional honeycomb) can concentrate failure

at that location [8-11]. As such, we would expect that the initiation of these failure mechanisms would be

triggered by local defect structures in the webs. This study is a first look at the detailed failure mechanisms in

stochastic honeycombs.

C.2 Experimental Details

High melt strength polypropylene was heated on a metal platen in an oven at 180 oC until it became a viscous

melt. It was then removed from the oven and placed in a press, where another platen was pressed on top.

Pressure was applied, and the upper platen was raised to the height desired for the sample (in this case, 15 mm)

and locked in place. The sample spontaneously separated from the plates upon cooling, resulting in a sandwich

structure that had integrated upper and lower face sheets separating a network of interconnected webs (Figure

123

C.1). Two samples, with relative densities of ρLD = 11% (low density, LD) and ρHD = 18% (high density, HD),

were scanned in a Skyscan 1172 high resolution micro-CT scanner. The micro-CT scans were reconstructed into

a series of cross-sections in the through-thickness direction, with voxel size of 35 μm 35 μm 35 μm. The

samples were then loaded in out-of-plane compression at a cross-head displacement rate of 1 mm min-1

to a

compressive strain of 0.20, and characterized by micro-CT.

C.3 Results and Discussion

The detailed collapse mechanisms of the LD and HD sandwich cores were investigated by comparing micro-CT

scans of individual webs before and after uniaxial compression. The starting structure of the LD and HD

samples is shown in Figure C.2. Figure C.2a and 2b show mid-height cross-sectional slices. The number of webs

decreased and the thickness of the webs increased as the density increased from 11% to 18%. Figure C.2c plots

the cross-sectional area fraction of the stochastic honeycomb as a function of normalized through-thickness posi-

Figure C.2 Micro-CT scans of the mid-height cross-sectional slice (sample size 40 mm 40 mm 15 mm) for the LD sample (a)

and the HD sample (b). The scale bar is 5 mm and applies to both figures. The corresponding area fraction, Ac, as a

function of normalized position in the sample, P, from bottom to top is given in (c).

124

Figure C.3 Partial web in the LD as-fabricated structure. The scale bar is 2 mm.

tion in the sample (0 indicates the bottom of the sample, 1 indicates the top). Near both the top and bottom, the

cross-sectional area fraction approaches 1 as the webs transform into the built-in skin. From Figure C.2c, it is

also clear that the higher density of the HD sample is due to the greater cross-sectional area in the central 50% of

the overall sandwich thickness. Over this middle half of the sample height, the area fraction of the LD sample

was 0.085 0.006, while the area fraction of the HD sample was 0.114 0.008. Note that even though the

cross-sectional area over the middle half of the sample is effectively constant, the web structure is constantly

changing in the through-thickness direction.

A fraction of the webs in the as-fabricated structure only partially span the distance between opposing face

sheets, as seen in Figure C.3. While not providing a continuous load path of their own, these partial webs do still

provide support to the adjacent complete webs, which are either bounded on both sides, or bounded on one side

and free on the other. The three generally considered restraint conditions for thin plates are termed simply

supported, built-in (or clamped), and free [12, 13]. Simply supported, for a thin plate, is analogous to pin-jointed

for a column or beam, and means that the plate is free to rotate around the hinge, but there is no translational

displacement. In comparison, built-in edge conditions are taken to mean that there is no rotation or translation at

the joint, and free is just that, the edge is unrestrained [12, 13]. When modelling, the edge constraint is often

chosen to be either simply supported or built-in [14]. However, in practice, this boundary is considered to be in

between the classic simply-supported and built-in boundary conditions [6, 15]. A schematic of a web supported

on one side with appropriate dimensions is given in Figure C.4.

The uniaxial compression stress-strain curve of stochastic honeycombs is comparable to that of

conventional honeycombs with an initial peak stress due to plastic buckling, followed by a decrease towards a

valley stress, followed by a final stress increase due to densification (e.g. Figure C.5 [1]). Figure C.6 illustrates

125

Figure C.4 Schematic of a web bound on one side and free on the other. The length, b, height, a, and thickness, tweb, are indicated.

Figure C.5 Uniaxial compression stress–strain curve for a ρcore = 13% stochastic honeycomb.

Figure C.6 Uniaxial compression stress-strain curve for LD (ρcore = 11%) and HD (ρcore = 18%) stochastic honeycombs to the valley

(ε = 0.20).

Table C.1 Compressive properties for the low density and high density stochastic honeycombs.

Sample σp (MPa) σv (MPa) E (MPa)

LD 2.23 0.92 59

HD 4.61 2.23 139

126

stress-strain curves for the LD and HD samples of the present study, which were compressed to the valley stress

beyond the initial peak and then unloaded for subsequent micro-CT characterization. Note that the density

increase from 11% to 18% corresponded to peak strength, valley strength, and elastic modulus increases of

105%, 140%, and 135%, respectively (values summarized in Table C.1). Classification of the web failure

mechanisms was based on the web end constraint conditions. The complete webs that were bounded on both

sides tended to tear vertically, as seen in Figure C.7, from the LD sample. This study focuses on the webs that

are bounded on one side and free on the other. Six webs from each sample were analyzed for their dimensions

and failure mechanisms, termed LD-1 to LD-6, and HD-1 to HD-6. Four specific examples are presented below

(LD-1, LD-2, HD-1, and HD-2), with general results of the overall analysis discussed afterwards.

The web (LD-1) shown in Figure C.8 is from the low density sample. Two views are shown of the as-

fabricated (Figure C.8a and 8b) and partially compressed (Figure C.8d and 8e) webs, while Figure C.8c and 8f

show cross-sections along the height (through-thickness) of the sandwich. Figure C.9 plots the thickness of the

web (x-axis) against the normalized position through the height (y-axis), giving a thickness profile of the web,

also shown in cross-section. The dashed line indicates the point of buckling failure. The web thickness was ~ 0.2

mm and ~ 0.4 mm at the top and bottom (respectively) of the sample and 0.15 mm in the centre. This web did

not buckle at the thinnest point, but rather just above the mid-point along the height dimension. From Figure

C.8c, the cross-section of the as-fabricated web shows that there is already some slight curvature in the area

where the web buckled, so while LD-1 did not buckle at the thinnest point, it did buckle at the area just above the

thinnest point where some curvature was already present. The thickness/length aspect ratio of this web was tweb/b

= 0.10 (Table C.2).

Figure C.7 As-fabricated LD web bounded on each side (a), and the vertical tear created by the horizontal tensile strain from the

compression of the bounding webs (b). The scale bar is 2 mm and applies to both figures.

127

Figure C.8 LD-1 as-fabricated from the side (a), straight on (b), in vertical cross-section (c), and partially compressed from the side

(d), straight on (e), and in vertical cross-section (f). The scale bar is 2.5 mm and applies to all the figures.

Figure C.9 Thickness profile of LD-1 and the vertical cross-section for comparison. The dashed line marks the point of initiation of

instability.

Table C.2 LD-1 dimensions at failure initiation point.

P (mm mm-1

) tweb (mm) tweb/b

0.52 0.15 0.104

128

Another web (LD-2) from the same sample is shown in Figure C.10. This web undergoes a somewhat

more complex failure mechanism, with both web buckling and web fracture being seen in the post-loaded scan.

Figure C.10b and 10c show that the web has some curvature before compression, and afterwards, in addition to

the tear, LD-2 buckled at the bottom (P = 0.11) and the top (P = 0.89). Unlike LD-1, this web was thinner near

the top and bottom (where buckling occurred). Figure C.10a and 10b show that this web was already somewhat

curved before loading, leading to increased bending strains at the thinnest regions, causing the upper and lower

buckles, and then the tear just below the upper buckle. From the thickness profile (Figure C.11), the web tears at

tweb = 0.32 mm, where the aspect ratio is 0.189. The two buckles occur where the web is 0.29 mm and 0.31 mm

thick, and the aspect ratios are 0.103 and 0.152, for the bottom and top, respectively (Table C.3).

The web shown in Figure C.12, in contrast, is from the high-density sample (HD-1). Similarly to LD-1, it

is bound on one side and free on the other. As Figure C.12b and 12c show, this web had some curvature to begin

with, and buckled in the centre in the direction of the initial curvature. At the buckle, this web is at its thinnest

point (tweb = 0.21 mm, Table C.4) as seen in the thickness profile in Figure C.13. The thickness at the top and

bottom of this web is ~ 0.4 mm, so its thinnest point is only 50% of its maximum thickness. This web had a thick

supporting web, as seen straight-on in Figure C.12a. The supporting web buckled at the top and tore, as seen in

Figure C.12d. HD-1 is thicker than LD-1 (0.21 mm compared to 0.15 mm), but has a lower aspect ratio because

it spans a longer distance. This is typical of the HD sample; it had thicker webs, as can be seen from Figure C.2,

but the webs are also longer, leading to lower aspect ratios overall.

HD-2, shown in Figure C.14, is another web from the high-density sample. In addition to being supported

on one side, it is supported by a buttress at the top, which is visible in Figure C.14b and on the opposite side of

the view shown in Figure C.14a. While this web has little initial curvature, it is tilted to one side, as can be seen

in Figure C.14c and 14d. After compression, this web has buckled near the top (tweb = 0.46 mm, Table C.5) and

torn a short distance below the buckle. The buttress supplies extra support for this web, and tears itself, as can be

seen in Figure C.15. The thickness profile for HD-2 (Figure C.16) shows that the buckle occurs at the thinnest

point of the web, and the tear occurs at the upper end of the buttress (Figure C.15c). The thickness profile for the

buttress, shown in the inset for Figure C.16, demonstrates that the buttress also tore near its thinnest point,

stressed by the buckle of the web it was supporting. This web tore near a buckle, as the LD-2 web did. HD-2 had

tweb = 0.50 mm at the tear, while LD-2 had tweb = 0.32 mm. Even though HD-2 was 50% thicker than LD-2, it has

a lower aspect ratio (0.125 for HD-2 and 0.189 for LD-2). Again, this is due to the longer webs in the HD sample

compared to the LD sample.

Table C.6 summarizes the thickness and aspect ratio data at failure for all of the webs considered in this

first study. As some webs had multiple failures, these are each separately recorded in the table. The table is

129

Figure C.10 LD-2 as-fabricated from the side (a), straight on (b), in vertical cross-section (c), and partially compressed from the side


Figure C.11 Thickness profile of LD-2 and the vertical cross-section for comparison. The dashed lines mark the locations of initiation

of instability.

Table C.3 LD-2 dimensions at failure initiation points.

P (mm mm-1

) tweb (mm) tweb/b

0.11 0.29 0.103

0.70 0.32 0.189

0.89 0.31 0.152

130

Figure C.12 HD-1 as-fabricated from the side (a), straight on (b), in vertical cross-section (c), and partially compressed from the side


Figure C.13 Thickness profile of HD-1 and the vertical cross-section for comparison. The dashed line marks the location of initiation

of instability.

Table C.4 HD-1 dimensions at failure initiation point.

P (mm mm-1

) tweb (mm) tweb/b

0.42 0.21 0.042

131

Figure C.14 HD-2 as-fabricated from the side (a), from the opposite side (b), straight on (c), in vertical cross-section (d), and partially

compressed from the side (e), from the opposite side (f), straight on (g), and in vertical cross-section (h). The scale bar is

2.5 mm and applies to all the figures.

Figure C.15 HD-2 buttress as-fabricated (a), in vertical cross-section (b), and partially compressed (c), in vertical cross-section (d).

The scale bar is 1 mm and applies to all the figures.

132

Figure C.16 Thickness profile of HD-2 and the buttress (inset) with vertical cross-sections for comparison. The dashed lines mark the

locations of initiation of instability.

Table C.5 HD-2 dimensions at failure initiation points.

P (mm mm-1

) tweb (mm) tweb/b

0.72 0.50 0.125

0.82 0.31 0.345

0.90 0.46 0.123

Table C.6 Thickness and aspect ratio correlated with failure mechanism

Web tweb (mm) tweb/b Failure Mechanism

HD-1 0.21 0.042 buckle

LD-3 0.18 0.054 buckle

LD-3 0.19 0.058 buckle

HD-4 0.39 0.076 buckle

HD-4 0.32 0.095 buckle

LD-1 0.15 0.103 buckle

LD-2 0.29 0.103 buckle

HD-4 0.30 0.103 buckle

LD-5 0.35 0.115 tear

HD-3 0.57 0.119 buckle

HD-2 0.46 0.123 buckle

HD-2 0.50 0.125 buckle

LD-4 0.60 0.128 tear

LD-4 0.61 0.128 buckle/tear

HD-5 0.93 0.145 tear

LD-2 0.31 0.152 buckle

LD-5 0.38 0.161 buckle

LD-2 0.32 0.189 tear

LD-5 0.45 0.217 tear

LD-4 0.51 0.250 tear

HD-2 0.31 0.345 tear

133

sorted by aspect ratio (tweb/b), and the failure mechanism at each failure is recorded as well. Critical buckling

stress for a thin wall, first developed by Bryan in 1890 [16] is a function of the aspect ratio, tweb/b, squared. A

slender web, with a low aspect ratio, will have a lower critical buckling stress. A web with a higher aspect ratio

will hold more stress, until reaching either its critical buckling stress or the local fracture strength of the material.

From the table, it is clear that if the aspect ratio is greater than 0.17, the web will fail by tearing. Conversely, if

the aspect ratio is less than 0.11, the web will fail by buckling only. Between these values, the web will either

tear or buckle depending on the local environment and geometry. For example, LD-2 with an aspect ratio of

0.152 buckled above a tear, so the web’s constraints had radically changed before that buckle occurred. Critical

buckling stress is also a function of the end constraints, which depend in part on the vertical aspect ratio of the

webs, a/b.

C.4 Conclusions

The strength and stiffness of stochastic honeycombs are determined by the complex internal architecture of the

webs. Micro-CT scans were used to track the detailed local failure mechanisms within the core of the sandwich

structure for ‘low’ (11%) and ‘high’ (18%) relative density variants. Post failure micro-CT characterization

showed that web buckling and web fracture were seen in both sample types. Web buckling tended to occur at the

thinnest portions of the web, whether this was at mid-height or not. The subset of partially constrained webs (i.e.

those supported on one side and free on the other) could be classified into one of two failure modes: web buckle

only and web buckle plus web fracture. Webs with a thickness-to-length aspect ratio of tweb/b < 0.11 tended to

buckle only, while higher aspect ratio webs would both buckle and fracture (tweb/b > 0.18).

134

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