c© 2011 Tapan Sabuwala

A SKELETON-AND-BUBBLE MODEL OF ELASTIC OPEN-CELL FOAMS FOR FINITEELEMENT ANALYSIS AT LARGE DEFORMATIONS

BY

TAPAN SABUWALA

DISSERTATION

Submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Theoretical and Applied Mechanics

in the Graduate College of theUniversity of Illinois at Urbana-Champaign, 2011

Urbana, Illinois

Doctoral Committee:

Associate Professor Gustavo Gioia, Chair, Director of ResearchProfessor James PhillipsProfessor Nancy SottosProfessor Iwona Jasiuk

Abstract

We formulate a new unit-cell model of elastic open-cell foams. In this model, the conventional

skeleton of open-cell foams is supplemented by fitting a thin-walled bubble within each cavity

of the skeleton, as a substitute for the membranes that occlude the openings of the skeleton

in elastic polyether polyurethane foams. The model has 9 parameters, and the value of each

parameter may be readily estimated for any given foam. We implement the model as a

user-defined material subroutine in the finite-element code ABAQUS.

To calibrate the model, we carry out fully nonlinear, three-dimensional finite-element

computational simulations of the experiments of Dai et al., in which a set of five polyether-

polyurethane EOC foams covering the entire range of commercially available relative densi-

ties was tested under five loading conditions: compression along the rise direction, compres-

sion along a transverse direction, tension along the rise direction, simple shear combined with

compression along the rise direction, and hydrostatic pressure combined with compression

along the rise direction. We show that, with a suitable choice of the values of the parameters

of the model, the model is capable of reproducing the most salient trends evinced in the

experimental stress-strain curves. We also show that the model can no longer reproduce all

of these trends if the bubbles be excluded from the model, and conclude that the bubbles

play a crucial role at large deformations, at least under certain loading conditions.

Next, we turn our attention to the stretch fields. Of special interest to us are the two-

phase stretch fields associated with a phase transition. These fields consist of mixtures of

two configurational phases of the foam, a high-deformation phase and a low-deformation

ii

phase. We show that the stretch fields that obtain in our computational simulations are

in good accord with the digital-image-correlation measurements of Dai et al., except for

simple shear combined with compression along the rise direction. For this loading condition,

Dai et al. concluded that the stretch fields remained continuous and there was no evidence

of distinct configurational phases in the foam. And yet, Dai et al. might have concluded

otherwise had they probed the stretch fields close to the lateral faces of the specimen, where

according to our computational simulations they would have found circumscribed nuclei of

the high-deformation phase. To settle the matter, we subject a foam specimen to simple

shear combined with compression along the rise direction, measure the stretch fields via a

digital-image-correlation technique, and find discontinuities in the stretch fields—but only

close to the lateral faces of the specimen, just as expected on the basis of our computational

simulations. All of the major features of the stretch fields turn out to be well reproduced in

the computational simulations.

In the last part of this thesis, we assess the capacity of the new model to yield reliable

predictions of the mechanical response of foams under punching, a common type of loading

in applications of elastic polyether polyurethane foams. In punching problems the geometry

is complex, the stress fields are highly spatially heterogeneous, the deformations are very

large, and the use of finite-element simulations is indispensable.

We have recourse to data from numerous punching experiments in which tall and short

specimens of foams of three low values of relative density were penetrated by a wedge-shaped

punch and a conical punch. For each experiment we run a fully nonlinear, three-dimensional

finite-element computational simulation and find the simulated force–penetration curve to

be in good accord with the corresponding experimental force–penetration curve. Wedge-

shaped punches and conical punches lack an intrinsic characteristic length, a fact that has

been exploited to carry out a simple but powerful theoretical analysis of the mechanical

response of foams under punching. Some of the assumptions and simplifications of this

iii

theoretical analysis have remained inaccessible to direct experimental verification, and we

use our computational simulations to show that these assumptions and simplifications are

justified.

Our results on punching serve to underscore the importance of phase transitions in the

mechanics of EOC foams. The force–penetration curves display a number of striking features,

and we can relate each one of these features to the presence of two configurational phases

in the foam. Feature by feature the relation is so specific and intricate that we are lead

to conclude that the force–penetration curves could be taken by themselves as proof of the

occurrence of phase transitions, even in the absence of any direct experimental evidence of

the stretch fields. A number of models of EOC foams might not allow for the occurrence of

phase transitions and still be able to account for numerous stress–stretch curves measured

under compression along the rise direction, for example. But the force–penetration curves

are inextricably linked to the prevalence of two-phase stretch fields, and it is likely that no

model can account for these curves unless it allows for the occurrence of phase transitions.

iv

Acknowledgments

The author would like to acknowledge Prof. Gustavo Gioia who helped shape the ideas that

form the core of this thesis. Thanks to Prof. Nancy Sottos, Prof. Iwona Jasiuk and Prof. Jim

Phillips for taking the time to serve on the examination committee and providing valuable

suggestions along the way. Thanks also to Xiangyu Dai who carried out the experiments

that are integral to this thesis. Special thanks to the friends and acquaintances who helped

me through the rough patches and made my experience as a graduate student more than

just an academic exercise.

The financial support from the National Science Foundation under grant CMS-0092849

(Ken P. Chong, program director) is acknowledged with gratitude.

Finally thanks to my wife Payal whose love and support through the years has been

unwavering and unconditional and to my parents and brother who planted the seeds that

made this Ph.D possible.

v

Table of Contents

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Elastic open-cell foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Microstructural models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Mechanical behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.1 Polymeric EOC foams under compression along the rise direction . . 51.4.2 Multiaxial loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Mechanical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Chapter 2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1 Microstructural model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Unit cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Mechanics of the skeleton . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.1 Compatibility and continuum . . . . . . . . . . . . . . . . . . . . . . 182.3.2 Hinges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.3 Strain energy of the skeleton . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Mechanics of the bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5 Model parameters and scale invariance . . . . . . . . . . . . . . . . . . . . . 252.6 Finite element implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 252.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Chapter 3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3.1 Test 1: Compression along the rise direction . . . . . . . . . . . . . . 293.3.2 Test 2: Compression along a transverse direction . . . . . . . . . . . 393.3.3 Test 3: Tension along the rise direction . . . . . . . . . . . . . . . . . 413.3.4 Test 4: Simple shear combined with compression along the rise direc-

tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

vi

3.3.5 Test 5: Hydrostatic pressure combined with compression along the risedirection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4 Example of application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Chapter 4 Punching of EOC foams . . . . . . . . . . . . . . . . . . . . . . . 584.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3.1 The self-similar regime . . . . . . . . . . . . . . . . . . . . . . . . . . 624.3.2 Beyond the self-similar regime. . . . . . . . . . . . . . . . . . . . . . 64

4.4 Computational simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.4.1 Wedge-shaped punch . . . . . . . . . . . . . . . . . . . . . . . . . . 654.4.2 Conical punch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Chapter 5 Summary, discussion and concluding remarks . . . . . . . . . . . 78

Appendix A UMAT implementation . . . . . . . . . . . . . . . . . . . . . . . 87A.1 Stress and elasticity tensor for the skeleton. . . . . . . . . . . . . . . . . . . 87A.2 Stress and elasticity tensor for the bubble . . . . . . . . . . . . . . . . . . . . 91A.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Appendix B Digital Image Correlation . . . . . . . . . . . . . . . . . . . . . 93

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

vii

List of Figures

1.1 Examples of solid foams. (a) Cancellous bone (an open-cell foam) [1] (b)Honeycomb (a two-dimensional foam) [2] (c) Metallic foam (a closed-cell foam)[3] (d) Polyether polyurethane foam (an open-cell foam) [4]. . . . . . . . . . 2

1.2 Microphotographs of two sections of a typical polyether polyurethane EOCfoam of measured apparent density ρa = 51.6 kg/m3 (General Plastics EF-4003) [9]. The micrographs were shot using a transmission electron micro-scope. (a) Section parallel to the rise direction. (b) Section normal to the risedirection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Five tetrakaidecahedra [16]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Typical strain field on the surface of a low-density polyether polyurethane

EOC foam tested under compression along the rise direction, from digital-image-correlation measurements [9]. This type of strain field occurs in con-junction with a stress plateau in the stress–strain curve. The rise directionis the direction of the X–axis. Each contour line corresponds to a constantvalue of u, where u is the displacement along the X–axis. The value of uon any given contour line differs by a fixed increment from the value of u onthe two adjacent contour lines. Where successive contour lines are in closeproximity to one another, the local strain, εX ≡ ∂u/∂X, is relatively large;where successive contour lines are far apart from one another, the local strainεX is relatively small. The contour lines in the figure suggest the existence oftwo preferred values of strain; in this view, the strain field may be describedas set of subparallel bands in which bands of one of the preferred values ofstrain alternate with bands of other preferred value of strain. The preferredvalues of strain have been interpreted as the characteristic values of strain oftwo configurational phases of the foam [9, 41]. . . . . . . . . . . . . . . . . . 6

1.5 An assembly of tetrakaiecahedral cells compressed along the rise direction [54].(a) Undeformed (b) Deformed . . . . . . . . . . . . . . . . . . . . . . . . . . 9

viii

1.6 Non-monotonic theoretical stress–stretch (P–λ) curve for a foam of relativedensity ρ = 0.03. The stress is normalized by the elastic modulus of thefoamed material Es = 65 MPa. The foam was subjected to compression alongthe rise direction (inset), and the P–λ curve was computed using the unit-cell model of Dai et al. [26]. The Maxwell stress PM and the characteristicstretches λL and λH were determined by applying the Erdmann equilibriumcondition of equal shaded areas, S = S ′. . . . . . . . . . . . . . . . . . . . . 11

2.1 Micrographs of a polyether polyurethane EOC foam shot using (a) a reflected-light microscope and (b) a scanning-electron microscope. The relative densityof the foam is 0.03. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Idealized microstructure of an EOC foam. . . . . . . . . . . . . . . . . . . . 162.3 Model of the microstructure. (a) View parallel to the rise direction. (b) Cross-

sectional view of the transverse plane. The dashed line marks the boundaryof the unit cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 The unit cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5 (a) Schematic of the skeleton after the tips have been displaced in accord

with (2.6) and (2.7), so that uk = −uk′, where k and k′ are paired tips. (We

exclude the rigid-body displacement U , as it does not lead to deformation.)Note that when we displace the tips we do not allow them to rotate. (b)Schematic of the skeleton after the tips have been displaced and rotated. (c)Reactions. (d) Action–reaction pairs. . . . . . . . . . . . . . . . . . . . . . . 20

2.6 (a) The lower half of the skeleton. The skeleton is hinged at the tips and atO. (b) Schematic of springs (in gray) used to simulate bending at an interiornode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.7 Deformation of a bubble. λ1, λ2 and λ3 are the principal stretches. (a)Undeformed bubble (b) Deformed bubble . . . . . . . . . . . . . . . . . . . . 23

2.8 Smooth approximation of heaviside function . . . . . . . . . . . . . . . . . . 24

3.1 Experimental (dashed lines) and simulated (solid lines) stress–stretch curvesfor foams of five relative densities (ρ) tested under compression along the risedirection, Test 1. Inset: schematic of the experiments and of the finite-elementmesh used in the simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Experimental results for the foam of relative density ρ = 0.038 tested undercompression along the rise direction, Test 1, but with a larger specimen thanin Fig. 3.1. (a) Experimental stress–stretch curve. Inset: schematic of theexperiment. (b) Plots of the local stretch λ along the dashed-line transectof the inset of (a). Plots A, B, C, etc. correspond respectively to the pointsmarked “A,” “B,” “C,” etc. on the stress–stretch curve of (a). The arrowsindicate the broadening of the band of high deformation. . . . . . . . . . . . 31

ix

3.3 Simulated stress–stretch curve for the foam of relative density ρ = 0.038 testedunder compression along the rise direction, Test 1, but with a larger specimenthan in Fig. 3.1. Inset: schematic of the finite-element mesh used in thesimulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4 Contour plots of the local stretch λ on the faces of the specimen in the simu-lation of Fig. 3.3. The contour plots of (a), (b), etc. correspond respectivelyto the points marked “A,” “B,” etc. on the stress–stretch curve of Fig. 3.3.The contour plots are mapped on the undeformed configuration. . . . . . . 33

3.5 Plots of the local stretch λ along the dashed-line transect of Fig. 3.4a. Theplots of (a), (b), etc. correspond respectively to the contour plots of Fig. 3.4a,Fig. 3.4b, etc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.6 Simulated stress–stretch curves for the foam of relative density ρ=0.03 testedunder compression along the rise direction. The finite-element mesh consistsof (a) 64 elements (b) 240 elements and (c) 1024 elements. . . . . . . . . . . 35

3.7 Evolution of the deformed shape of the skeleton of an individual cell. Theposition of this cell is marked “+” on Fig. 3.4a; note that the cell is locatedwithin the first band of phase H that nucleates in the specimen (Fig. 3.4c).The deformed shapes of (a), (b) and (c) correspond respectively to Fig. 3.4a,Fig. 3.4b and Fig. 3.4c (and to the points marked A, B, and C on the stress–stretch curve of Fig. 3.3). The deformation proceeds continuously from (a) to(b) and discontinuously from (b) to (c). . . . . . . . . . . . . . . . . . . . . 36

3.8 Simulated stress–stretch curves for the foam of relative density ρ = 0.038tested under compression along the rise direction. The simulated stress–stretch curve shown as a solid line is the same as the simulated stress–stretchcurve of Fig. 3.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.9 Plots of W vs. λ and plots of P vs. λ, for a single cell, where W is thestrain energy density, λ is the local stretch in the rise direction, and P isthe local stress in the rise direction, P = ∂W/∂λ. (a) The cell is compresseduniaxially along the rise direction, and W does not include the contribution ofthe bubbles to the strain energy density. (b) The cell is compressed uniaxiallyalong the rise direction, and W includes the contribution of the bubbles tothe strain energy density. (c) The cell is stretched uniaxially along the risedirection, and W includes the contribution of the bubbles to the strain energydensity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.10 Experimental results for the foam of relative density ρ = 0.065 tested undercompression along the rise direction, Test 1, but with a larger specimen thanin Fig. 3.1. (a) Experimental stress–stretch curve. Inset: schematic of theexperiment. (b) Plots of the local stretch λ along the dashed-line transect ofthe inset of (a). Plots A, B, C, and D correspond respectively to the pointsmarked “A,” “B,” “C,” and “D” on the stress–stretch curve of (a). . . . . . 39

x

3.11 Computational results for the foam of relative density ρ = 0.065 tested undercompression along the rise direction, Test 1, but with a larger specimen than inFig. 3.1. Cf. Fig. 3.10. (a) Simulated stress–stretch curve. Inset: schematic ofthe simulation. (b) Plots of the local stretch λ along the dashed-line transectof the inset of (a). Plots A, B, C, and D correspond respectively to the pointsmarked “A,” “B,” “C,” and “D” on the stress–stretch curve of (a). . . . . . 40

3.12 Experimental (dashed lines) and simulated (solid lines) stress–stretch curvesfor foams of five relative densities (ρ) tested under compression along a trans-verse direction, Test 2. Inset: schematic of the experiments and of the finite-element mesh used in the simulations. . . . . . . . . . . . . . . . . . . . . . . 41

3.13 Simulated stress–stretch curves for foams of five relative densities (ρ) testedunder compression along two mutually perpendicular transverse directions (asindicated in the inset). The simulated stress–stretch curves, shown here asthick lines, are the same as the simulated stress–stretch curves of Fig. 3.12. 42

3.14 Experimental (dashed lines) and simulated (solid lines) stress–stretch curvesfor foams of five relative densities (ρ) tested under tension along the risedirection, Test 3. Inset: schematic of the experiments and of the finite-elementmesh used in the simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.15 Experimental (dashed lines) and simulated (solid lines) force–displacementcurves for foams of five relative densities (ρ) tested under simple shear com-bined with compression along the rise direction, Test 4. Inset: schematic ofthe experiments and of the finite-element mesh used in the simulations. . . . 44

3.16 Simulated force–displacement curve for the foam of relative density ρ = 0.038tested under simple shear combined with compression along the rise direction,Test 4, but with a larger specimen than in Fig. 3.15. Inset: schematic of thefinite-element mesh used in the simulation. . . . . . . . . . . . . . . . . . . . 45

3.17 Contour plots of the local stretch λ on one of the lateral faces of the specimenin the simulation of Fig. 3.16. The contour plots of (a), (b), etc. correspondrespectively to the points marked “A,” “B,” etc. on the stress–stretch curveof Fig. 3.16. The contour plots are mapped on the undeformed configuration. 46

3.18 Plots of the local stretch λ along the dashed-line transects of Fig. 3.17a. Theplots of (a) correspond to transect A of Fig. 3.17a, the plots of (b) correspondto transect B of Fig. 3.17a, and so on. . . . . . . . . . . . . . . . . . . . . . 46

3.19 Simulated force–displacement curve for the foam of relative density ρ = 0.03tested under simple shear combined with compression along the rise direction,Test 4, but with a larger specimen than in Fig. 3.15. Inset: schematic of thefinite-element mesh used in the simulation. . . . . . . . . . . . . . . . . . . . 47

3.20 Contour plots of the local stretch λ on one of the lateral faces of the specimenin the simulation of Fig. 3.19. The contour plots of (a), (b), etc. correspondrespectively to the points marked “A,” “B,” etc. on the stress–stretch curveof Fig. 3.19. The contour plots are mapped on the undeformed configuration. 48

xi

3.21 Plots of the local stretch λ along the dashed-line transects of Fig. 3.20a. Theplots of (a) correspond to transect A of Fig. 3.20a, the plots of (b) correspondto transect B of Fig. 3.20a, and so on. . . . . . . . . . . . . . . . . . . . . . 48

3.22 Contour plots of the local stretch λ on one of the lateral faces of the specimenin the experiment from DIC measurements. The contour plots are mappedon the undeformed configuration. Cf. the computational results of Fig. 3.20 . 49

3.23 Plots of the local stretch λ along the dashed-line transects of Fig. 3.22a. Theplots of (a) correspond to transect A of Fig. 3.22a, the plots of (b) corre-spond to transect B of Fig. 3.22a, and so on. Cf. the computational results ofFig. 3.21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.24 Schematic of the loading of the specimens in Test 5. (a) Step 1. (b) Step 2. . 513.25 Experimental (dashed lines) and simulated (solid lines) stress–stretch curves

for foams of five relative densities (ρ) tested under hydrostatic pressure com-bined with compression along the rise direction, Test 5. po is the averagehydrostatic pressure. (a) ρ = 0.03. (b) ρ = 0.038. (c) ρ = 0.046. (d)ρ = 0.065. (e) ρ = 0.086. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.26 The local stretch λ on the surface the specimen of relative density 0.03 inthe simulation with p = 1.59 kPa. The contour plots are mapped on theundeformed configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.27 Experimental (dashed grey lines) and simulated (solid black lines) force–penetration curves for foams of three relative densities (ρ). Inset: schematicof the experiments and of the finite-element mesh used in the simulations; thewhite punch is cylindrical, the black punch spherical. . . . . . . . . . . . . . 55

3.28 Distribution of the local strain in the rise direction. (a) Before and (b) afterpunching. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1 (a) Crash test [68]. (b) Idealized model of crash test. The body of the caroccupant is modeled as a collection of rigid punches that penetrate the car seat. 59

4.2 Experimental setup for punching experiments [75] with (a) a wedge-shapedpunch and (b) a conical punch. . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3 Punching force–penetration curves from experiments with a wedge-shapedpunch. (a) Tall (cubic) EOC foam specimens. (b) Short (brick-shaped) EOCfoam specimens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4 Punching force–penetration curves from experiments with a conical punch.(a) Tall (cubic) EOC foam specimens. (b) Short (brick-shaped) EOC foamspecimens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.5 Punching force–penetration curves from experiments with a conical punch,plotted in Log-Log scale. (a) Tall (cubic) EOC foam specimens. (b) Short(brick-shaped) EOC foam specimens. . . . . . . . . . . . . . . . . . . . . . . 61

4.6 Schematic of a punching experiment with a wedge-shaped punch. (a) Thesharp interface between the high-deformation configurational phase and thelow-deformation configurational phase. (b) The self-similar regime. . . . . . 62

xii

4.7 Scaled force–penetration curves for (a) tall (cubic) and (b) short (brick-shaped)EOC foam specimens penetrated by a wedge-shaped punch. . . . . . . . . . 63

4.8 Schematic of punching experiments with a conical punch. The high-deformationphase is separated from the low-deformation phase by a semi-spherical sharpinterface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.9 Scaled force–penetration curves for (a) tall (cubic) and (b) short (brick-shaped)EOC foam specimens penetrated by a conical punch. . . . . . . . . . . . . . 65

4.10 Finite element meshes used in the computational simulations with a wedge-shaped punch. (a) Tall (cubic) specimens and (b) short (brick-shaped) speci-mens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.11 Force–penetration curves from simulations with a wedge-shaped punch andtall EOC foam specimens of relative density ρ. As we have seen in chapter 3,the oscillations of the force–penetration curves must be ascribed to the dis-creteness of the finite element mesh. In fact, the amplitude of these oscillationswill lessen if the finite element mesh is made more dense. . . . . . . . . . . . 67

4.12 (a) Contour plot of the local stretch λ for a tall EOC foam specimen of relativedensity ρ = 0.03 under a wedge-shaped punch. The penetration d = 29.1 mm.The plot is in the undeformed geometry. (b) Plot of the local stretch λ alonga line of nodes below the tip of the punch. The discontinuity in λ marks thesharp interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.13 (a) Plots of λ along a line of nodes below the tip of the punch for d = 15.0 mm,17.4 mm, 21.1 mm, 24.9 mm, 27.1 mm, 31.1 mm ,33.0 mm and 37.2 mm. (b)Plot of R vs. d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.14 Simulated force–penetration curve and attendant contour plots of the stretchλ for a tall EOC foam specimen of relative density 0.03 under a wedge-shapedpunch. Points A through F correspond to d = 10.0 mm, 16.3 mm, 24.1 mm,32.2 mm, 38.0 mm, 41.0 mm and 51.0 mm, respectively. . . . . . . . . . . . . 69

4.15 Force–penetration curves for short EOC foam specimens of relative density ρunder a wedge-shaped punch. . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.16 Simulated force–penetration curve and attendant contour plots of the stretchλ for a short EOC foam specimen of relative density 0.03 under a wedge-shaped punch. Points A through F correspond to d = 10.3 mm, 13.5 mm,16.3 mm, 20.0 mm, 24.3 mm, and 28.1 mm respectively. . . . . . . . . . . . . 71

4.17 Scaled force–penetration curves for (a) tall and (b) short specimen under awedge-shaped punch. The force is normalized by the plateau force FM . . . . 71

4.18 Finite element meshes used in the computational simulations with a conicalpunch. (a) Tall (cubic) specimens and (b) short (brick-shaped) specimens. . 72

4.19 Force–penetration curves from simulations with a conical punch and tall EOCfoam specimens of relative density ρ, in (a) linear-linear scale and (b) log-logscale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

xiii

4.20 (a) Plots of λ along a line of nodes below the tip of the punch for d = 15.0 mm,20.0 mm, 25.1 mm, 30.2 mm, 40.1 mm, 45.0 mm and 50.0 mm. (b) Plot of Rvs. d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.21 Force–penetration curve and the attendant contour plots of the stretch λ fora cubic EOC foam specimen of relative density 0.03 under a conical punch.Points A through F correspond to d = 15.14 mm, 22.0 mm, 29.1 mm, 36.1 mm,43.5 mm, 48.6 mm and 57.4 mm, respectively. . . . . . . . . . . . . . . . . . 74

4.22 Force–penetration curves from simulations with a conical punch and shortEOC foam specimens of relative density ρ, in (a) linear-linear scale and (b)log-log scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.23 Simulated force–penetration curve and attendant contour plots of the stretchλ for a short EOC foam specimen of relative density 0.03 under a conicalpunch. Points A through F correspond to d = 12.2 mm, 17.5 mm, 20.7 mm,25.6 mm, 29.3 mm, and 32.4 mm respectively. . . . . . . . . . . . . . . . . . 75

4.24 Scaled force–penetration curves for (a) tall and (b) tall specimens under aconical punch. The force is normalized by the plateau force FM . . . . . . . 76

B.1 Markers used for DIC on a lateral surface of a low-density foam (ρ = 0.03). . 93

xiv

Chapter 1

Introduction

1.1 Foams

A foam is a solid cellular material that may be described as numerous cells joined together

to fill space. Examples of foams are abundant in nature in the form of honeycombs, cork,

cancellous bone, pumice, sponge, coral, etc. Other foams are man made: from bread and

meringue to aerogels, the list brings together the ordinary and the exotic. To capitalize

on the unique blend of properties of a cellular microstructure, synthetic foams have been

manufactured from a wide variety of materials including polymers, metals, carbon, graphite,

and ceramics. Although liquid cellular materials—such as, for instance, shaving cream—

are sometimes called foams, here we shall use the word “foam” to refer exclusively to solid

cellular materials.

The cells of a foam may consist of bars or membranes. A few examples are shown in

Fig. 1.1. Foams with cells consisting of bars are known as open-cell foams. A typical example

of naturally occurring open-cell foam is cancellous bone. Foams with cells consisting of

membranes are known as closed-cell foams. A typical example of naturally occurring closed-

cell foam is cork.

It is apparent from Fig. 1.1 that foams are usually very light and porous. In fact, foams

have been termed “castles in the air” [5], and the relative density of a foam is frequently

lower than 0.01. The relative density, denoted ρ, is defined by the expression ρ ≡ ρa/ρs,

where ρa is the apparent density of the foam (that is, the mass per unit volume of foam)

1

(a)

(c)

(b)

(d)

Figure 1.1: Examples of solid foams. (a) Cancellous bone (an open-cell foam) [1] (b) Hon-eycomb (a two-dimensional foam) [2] (c) Metallic foam (a closed-cell foam) [3] (d) Polyetherpolyurethane foam (an open-cell foam) [4].

and ρs is the density of the solid material. Thus, a foam of relative density 0.01 is 100 times

less dense than the solid material of which the foam is made.

1.2 Elastic open-cell foams

Here, we shall concentrate on elastic open-cell (EOC) foams (also known commercially as

“flexible” open-cell foams), and in particular on polymeric EOC foams. Polymeric EOC

foams are employed in many engineering applications. For example, in the aeronautical

industry, these foams are among the most commonly used materials in the cores of sandwich

panels, where their low density is advantageous. Due to their ability to absorb impact energy

and sustain large deformations at relatively low compressive stresses, polymeric EOC foams

are used as impact-resistant fillers or sheets in the packaging of fragile products; as protective

linings in helmets and knee pads; and as cushioning cores in furniture, mattresses and car

seats.

2

(a) (b)

Rise

Transverse

Transverse

Figure 1.2: Microphotographs of two sections of a typical polyether polyurethane EOCfoam of measured apparent density ρa = 51.6 kg/m3 (General Plastics EF-4003) [9]. Themicrographs were shot using a transmission electron microscope. (a) Section parallel to therise direction. (b) Section normal to the rise direction.

The first polymeric foam was made in 1941 [6], in Germany. Industrial production started

in 1952. Since then, polymeric foams have become a large industry with a global annual

output that is projected to reach 18.0 million tons by 2015 [7].

Polymeric foams are manufactured by using a foaming agent that promotes the growth of

numerous gas bubbles within a liquid or solid layer of polymer. During the foaming process

the layer of polymer expands anisotropically, predominantly along a direction normal to its

mid-plane—the so-called rise direction [8]. Depending on the polymer and the processing

conditions, the outcome is a foam which can be either open cell or closed cell.

In Fig. 1.2 we show microphotographs of two sections of a typical polyether polyurethane

EOC foam. The section of Fig. 1.2a is parallel, whereas the section of Fig. 1.2b is normal,

to the rise direction. We can describe the microstructure of the foam of Fig. 1.2 as a three-

dimensional network of bars of similar length and cross section. In this thesis, we shall

frequently refer to this network of bars as the “skeleton.”

3

Figure 1.3: Five tetrakaidecahedra [16].

1.3 Microstructural models

In modeling a foam, it is common to define an idealized microstructure of the foam. A

typical idealized microstructure in two dimensions is a hexagonal honeycomb [10–14]. A

typical idealized microstructure in three dimensions consists of tetrakaidecahedra, that is,

14-sided polyhedra with 8 hexagonal and 6 quadrilateral faces (Fig. 1.3) [15–17].

If a uniform surface tension is the dominant force during the foaming process, as it is in

liquid foams made of soapy water, the rules of Plateau will be satisfied. According to these

rules, three cell faces meet at equal angles of 120◦ to form cell edges, and four edges join at

each vertex at the tetrahedral angle, cos−1(−1/3) ≈ 109.47◦.

In two dimensions the hexagonal honeycomb satisfies the rules of Plateau. In three

dimensions the rhombic dodecahedron and the tetrakaidecahedron come close to satisfying

the rules of Plateau. For many years a conjecture was entertained to the effect that the

tetrakaidecahedron (also known as the Kelvin cell) provided the most efficient partition of

three-dimensional space [18]. This conjecture was refuted in 1994 by Weaire and Phelan [19],

who used computer simulations to find a compound structure, now known as the Weaire-

Phelan cell, which effects a better partition than the tetrakaidecahedron.

In practice, foam cells do not adhere to the theoretically efficient shapes mentioned

4

above. In his pioneering experimental work on the geometry of soap bubbles, Matzke [20]

did not find a single Kelvin cell. Similar results have been obtained more recently for solid

foams using modern imaging techniques [21]. Cubes [22], stagerred cubes [23], tetradehdra

[9, 24–26], rhombic dodecahedra [27], pentagonal dodecahedra [28–30] and terakaidecahedra

[15, 17, 22, 23, 31–35] have all been used to define idealized microstructures in 3D.

1.4 Mechanical behavior

An extensive review of the mechanical behavior of cellular materials (including wood, cork,

cancellous bone, sponge, coral, honeycombs, metallic foams, and polymeric foams) can be

found in Ref. [36].

For polymeric EOC foams, experimental studies have mostly focused on two subjects,

the elastic properties [23, 24, 37] (which are of scant interest in applications of EOC foams,

where the strains are typically large under service conditions) and the mechanical response

under compression along the rise direction and up to large strains [9, 23, 31, 38].

1.4.1 Polymeric EOC foams under compression along the rise

direction

The stress–strain curves of low-density polymeric EOC foams tested under compression along

the rise direction are distinctive in that they have a stress plateau [9, 23, 31, 38]. The stress

plateau is an extended portion of the stress–strain curve where the stress remains constant

and equal to the so-called plateau stress .

It has long been known experimentally—but for the most part ignored in modeling—

that stress plateaus are accompanied by spatially heterogeneous strain fields [9, 25, 39–41],

even though the stress fields remain spatially homogeneous under compression along the rise

5

X

Figure 1.4: Typical strain field on the surface of a low-density polyether polyurethane EOCfoam tested under compression along the rise direction, from digital-image-correlation mea-surements [9]. This type of strain field occurs in conjunction with a stress plateau in thestress–strain curve. The rise direction is the direction of the X–axis. Each contour linecorresponds to a constant value of u, where u is the displacement along the X–axis. Thevalue of u on any given contour line differs by a fixed increment from the value of u on thetwo adjacent contour lines. Where successive contour lines are in close proximity to oneanother, the local strain, εX ≡ ∂u/∂X, is relatively large; where successive contour lines arefar apart from one another, the local strain εX is relatively small. The contour lines in thefigure suggest the existence of two preferred values of strain; in this view, the strain fieldmay be described as set of subparallel bands in which bands of one of the preferred values ofstrain alternate with bands of other preferred value of strain. The preferred values of strainhave been interpreted as the characteristic values of strain of two configurational phases ofthe foam [9, 41].

direction. In these spatially heterogeneous strain fields, bands of high local strain alternate

with bands of low local strain (Fig. 1.4). As the mean applied strain is increased during an

experiment, the bands of high local strain become broader, and the bands of low local strain

become concomitantly thinner [9, 40, 41].

1.4.2 Multiaxial loading

Experimental research into the mechanical behavior of polymeric foams under multiaxial

loading has been largely aimed at mapping the failure envelope in stress space of rigid (or

“brittle”) polymeric foams (see, for example, [39] for rigid polystyrene foams; [42–44] for rigid

polyurethane foams; [45–47] for rigid PVC foams; and [48] for rigid polypropylene foams).

6

This failure envelope is also known by the name of yield envelope. This name may appear

confusing, but rigid polymeric foams often undergo a progressive form of brittle failure where

tested under compression. In this progressive form of brittle failure, a large deformation

accumulates at a constant value of stress, which may be construed as an apparent yield

stress.

For polymeric EOC foams, extensive multiaxial tests (including uniaxial, biaxial, and

hydrostatic tests) have been carried out by Triantafyllidis et al. [44]. The purpose of these

tests was to trace the yield envelope in stress space. Here, the word “yield” refers to the

beginning of the stress plateau that is characteristic of the stress–strain curves of some

polymeric EOC foams tested under compression. (This stress plateau has been discussed in

section 1.4.1.)

As the stress plateau occurs only in relatively light foams, all of the foams tested by

Triantafyllidis et al. were of very low apparent density, namely 14.3, 21.8, 28.0, 41.5, and

51.6 kg/m3. Unfortunately, only a few stress–strain curves were published, to illustrate the

method whereby the yield stress was extracted from the stress–strain curves.

More recently, Dai et al. [41] carried out extensive experimental work to elucidate the

mechanical response of polyether polyurethane EOC foams over the full range of commer-

cially available apparent densities, up to large strains and for a variety of loading conditions.

To that end, specimens of a commercial set of foams were subjected to five tests, namely

(1) compression along the rise direction, (2) compression along a transverse direction, (3)

tension along the rise direction, (4) simple shear combined with compression along the rise

direction, and (5) hydrostatic pressure combined with compression along the rise direction.

The stress field was spatially homogeneous (at least to a good approximation) in all these

tests except Test 4. In Test 4, simple shear combined with compression along the rise direc-

tion, the stress field was spatially heterogeneous. (Stress fields associated with simple shear

are intrinsically heterogeneous, unless the specimen is infinitely large.) The commercial set

7

of polyether polyurethane EOC foams consisted of foams of five apparent densities ranging

from 50.3 to 221 kg/m3.

For each test and foam density, Dai et al. reported the mechanical response in the form

of a complete stress–strain curve. For a number of tests and foam densities, they employed

a Digital Image Correlation (DIC) technique to measure strain fields on the surface of the

specimens.

1.5 Mechanical models

Meinecke and Clark [49], Hilyard and Cunningham [50], and Gibson and Ashby [36] have

provided extensive reviews of mechanical models of foams. Other references include [51–54].

Here, we shall briefly review unit-cell models of EOC foams. In these models, an idealized

microstructure is defined as a regular, periodic network of bars. A suitable subset of the

idealized microstructure is chosen as a unit cell and used to study the mechanical response of

the foams. The tips of the unit cell, where the unit cell is severed from the periodic network

of bars, are subjected to displacements affine with the applied deformation gradient. The

strain energy of the unit cell may then be computed and used to evaluate the stress attendant

on the applied deformation gradient.

There exist several unit-cell models of EOC foams [15, 23–25, 34]). Unit-cell models have

been used to predict elastic properties such as the Young’s modulus of a foam [15, 17, 23,

24, 32, 33, 37, 55–59]. But unit-cell models have seldom been used to predict the mechanical

response up to large strains [16, 25, 38], and the predictions have seldom been compared

with experimental data.

A notable exception is the work of Kyriakides et al. [31, 60]. Using an elaborate model in

which three-dimensional finite elements made up the bars of a large assembly of tetrakaidec-

ahedral cells (Fig. 1.5a), Kyriakides et al. computed stress–strain curves for compressive

8

(a) (b)Rise

Figure 1.5: An assembly of tetrakaiecahedral cells compressed along the rise direction [54].(a) Undeformed (b) Deformed

.

loading along the rise direction and compressive loading along a transverse direction, up

to large strains, and compared these stress–strain curves with experimental data for five

polyether polyurethane EOC foams of low relative densities in the narrow range, ρ = 0.021

to 0.028. For compression along the rise direction, Kyriakides et al. found that the assembly

of unit cells buckled in a long wavelength mode, much like an elastic column (Fig. 1.5b). The

buckling stress was identified with the plateau stress—that is, the stress level of the stress

plateau that is observed in the experimental stress–strain curves of low-density EOC foams

tested under compression along the rise direction. (The stress plateau has been discussed in

section 1.4.1.)

More recently, Dai et al. [26] formulated a unit-cell model of EOC foams. In this model, a

foam consisted of four-bar tetrahedra arranged in the hexagonal diamond structure known as

Lonsdaleite. The bars of the unit cell were uniform in cross section, and they were governed

by conventional small-strain, large-rotation theory of beams. Dai et al. used the model

to compute theoretical stress–stretch curves, which they compared with the stress–stretch

curves from their experiments [41]. (These experiments were described in section 1.4.1.)

9

Further, in order to appraise the goodness of their model, Dai et al. took into account the

relation between the stress–stretch curve measured in an experiment and the nature of the

attendant stretch fields.

Where the experimental stress–stretch curve had a stress plateau, Dai et al. found that

the corresponding theoretical stress–stretch curve was non-monotonic: the stress increased,

decreased, and increased once more, so that the theoretical stress–stretch curve displayed

the wiggly outline of a rollercoaster, as seen in the example of Fig. 1.6. Dai et al. applied

the Erdmann equilibrium condition [9] to determine the Maxwell stress PM and the two

characteristic values of stretch, λL and λH , of each non-monotonic theoretical stress–stretch

curve (Fig. 1.6). The Maxwell stress was identified with the plateau stress of the experi-

mental stress–stretch curve. The two characteristic values of stretch were associated with

two configurational phases of the foam: λL with a low-deformation configurational phase

and λH with a high-deformation configurational phase. These configurational phases can

only coexist at the stress level PM . On the basis of this analysis, Dai et al. interpreted the

spatially heterogeneous stretch fields measured in the experiments as mixtures of the two

configurational phases of the foam (Fig. 1.4).

From the brief descriptions above, it is apparent that the models of Dai et al. and Kyr-

iakides et al. lead to disparate interpretations of the mechanical response of low-density

polyether polyurethane EOC foams compressed along the rise direction. On the one hand,

both models have been shown to predict a stress plateau, and in this sense both models are

equally in accord with the experimental stress–strain curves. On the other hand, there is a

sharp contrast between the predictions of the models in regard to the nature of the strain

fields that accompany a stress plateau.

In the model of Kyriakides et al., the stress plateau is the manifestation of a bifurcation

of equilibrium (or Euler buckling), a global phenomenon that encompasses all of the mi-

crostructure of the foam at once. Here, the plateau stress corresponds to an eigenvalue and

10

0.80.91.00.0

0.1

0.2

0.3

Plateau stress (PM

)

λL

λH

λ

P/E

s (

x 1

0-3 )

S = S'

∆ Rise

ρ = 0.03

P

S

S'

Figure 1.6: Non-monotonic theoretical stress–stretch (P–λ) curve for a foam of relativedensity ρ = 0.03. The stress is normalized by the elastic modulus of the foamed material Es= 65 MPa. The foam was subjected to compression along the rise direction (inset), and theP–λ curve was computed using the unit-cell model of Dai et al. [26]. The Maxwell stressPM and the characteristic stretches λL and λH were determined by applying the Erdmannequilibrium condition of equal shaded areas, S = S ′.

the attendant strain fields correspond to an eigenfunction. The domain of the eigenfunction

is the entire microstructure of the foam, and its amplitude is arbitrary. Therefore, the strain

field remains invariant except for its amplitude, which increases to accommodate the increase

in mean applied strain. But a strain field that remains invariant except for its amplitude

can hardly be reconciled with the experimental evidence of section 1.4.1, where we have seen

that an increase in the mean applied strain is accommodated by the widening of a number

of bands of high local strain at the expense of the width of the adjacent bands of low local

strain.

In the model of Dai et al., the stress plateau is the manifestation of a phase transition, a

local phenomenon that sweeps progressively through the microstructure of a foam. Here, the

plateau stress corresponds to a Maxwell stress and the attendant strain fields correspond to

two-phase strain fields governed by the rule of mixtures. Therefore, an increase in the mean

11

applied strain is accommodated by an increase in the volume fraction of a high-deformation

phase and a concomitant decrease in the volume fraction of a low-deformation phase. This

theoretical scenario is in good accord with the experimental evidence of section 1.4.1.

1.6 Outline of the thesis

In chapter 2, we formulate a new unit-cell model of EOC foams. In this model, the mi-

crostructure of a foam consists of a three-dimensional, regular skeleton and a set of thin-

walled bubbles. The skeleton is the same network of bars as in the unit-cell model of Dai et

al. To this skeleton we add the set of bubbles. A single bubble is fitted tightly within each

cavity of the skeleton. The bubbles are meant to substitute for the membranes that span

the openings (or “windows”) of the skeleton of polyether polyurethane EOC foams. These

membranes were first documented more than 40 years ago [61, 62]. It has been shown that

the membranes contribute significantly to the elastic modulus of a foam, and as part of the

present work we show that the membranes play a crucial role in the mechanical response of

foams at the large deformations that prevail in typical applications of EOC foams. And yet,

to our knowledge the membranes have so far been ignored in models of EOC foams.

In chapter 3, we calibrate the model by comparing numerous experimental stress-stretch

curves with the corresponding stress–stretch curves computed using the model. The calibra-

tion procedure and the experimental data are those adopted by Dai et al. to calibrate their

unit-cell model. Note, however, that Dai et al. computed stress-stretch curves by subjecting

a single cell to a deformation gradient that was suitably chosen to reflect the experimental

details. Thus, for example, for compression along the rise direction, the experiments had

been carried out with specimens in which the size measured along the rise direction was a

fraction of the size measured perpendicular to the rise direction. Further, it was reasonable

to assume that in the experiments there had been no slip between the loading plates and the

12

specimens. In view of these experimental details, Dai et al. computed stress-stretch curves

by uniaxially stretching a single cell along the rise direction [26].

This method might seem appropriate where the stress field is spatially homogeneous,

much less so where the stress field is spatially heterogeneous (as it is in test 4, simple

shear combined with compression along the rise direction). To be able to to account better

for the actual conditions that prevailed when the experimental stress–stretch curves were

measured, we implement our model in a general-purpose finite-element code and compute

stress-stretch curves by running a three-dimensional, fully nonlinear finite-element simulation

of each experiment.

In many applications of polyether polyurethane EOC foams, the purpose of the foam is

to react to outside forces that impinge on the foam. Thus, it is of interest to determine what

happens when a punch penetrates an elastic foam, and how the foam reacts mechanically.

In chapter 4, we use our model to investigate the punching of polyether polyurethane EOC

foams. We run computational simulations of an extensive set of experiments in which speci-

mens of foams of three different relative densities were penetrated by a wedge-shaped punch

and a conical punch. We also use the model to test a theory of punching in the absence of

a length scale.

We close the thesis with an extensive discussion, in chapter 5

13

Chapter 2

Model

In this chapter, we formulate a new unit-cell model of elastic open-cell (EOC) foams. The

model is rich in features and may be adapted to most types of EOC foams. As we are

especially interested in polyether polyurethane EOC foams (known commercially as flexible

polyether polyurethane foams), the model includes novel features which are peculiar to this

type of foams.

Our objective is to be able to predict, up to large deformations and for arbitrary loading

and boundary conditions, the mechanical response of three-dimensional bodies made of EOC

foams, including complete stress and stretch fields. Thus, at the end of the chapter we briefly

discuss the implementation of the model in a general-purpose finite-element code.

2.1 Microstructural model

We have seen in chapter 1 that the microstructure of EOC foams is usually modeled as

a three-dimensional network of bars. It has long been known, however, that in polyether

polyurethane EOC foams the network of bars is accompanied by membranes. These mem-

branes can be discerned under a transmission-electron microscope (Fig. 1.2) or a reflecting-

light microscope (Fig. 2.1a) but become strikingly apparent under a scanning-electron micro-

scope (Fig. 2.1b). Yasunaga et al. [62] have ascertained that about 75–80% of the “windows”

of the network of bars are spanned by membranes. Although many of the membranes have

holes (Fig. 2.1b), it has been shown experimentally that the membranes contribute signifi-

14

(a) (b)

Figure 2.1: Micrographs of a polyether polyurethane EOC foam shot using (a) a reflected-light microscope and (b) a scanning-electron microscope. The relative density of the foamis 0.03.

cantly to the elastic modulus of a foam [61]. More important to applications of EOC foams,

the membranes are likely to affect the mechanical response under large strains. And yet, to

our knowledge, the membranes have been ignored in models of EOC foams.

To account for the membranes, we propose a microstructural model with two components,

namely a skeleton and a set of bubbles (Fig. 2.3). The skeleton is the network of bars. Each

node of the skeleton may be thought of as the center of a tetrahedron whose vertices are the

midpoints of the four bars that converge on the node. The tetrahedra are arranged in the

hexagonal diamond structure Lonsdaleite [63].

The bubbles are thin membranous ellipsoids. A single bubble is fitted tightly within

each cavity of the skeleton (Fig. 2.3). For simplicity we shall make the bubbles spherical

(and without holes), and calculate the radius of the spherical bubbles so that the volume

contained in a spherical bubble equals the volume contained in an ellipsoidal bubble.

15

Figure 2.2: Idealized microstructure of an EOC foam.

BubblesSkeleton

(a)

(b)

Microstructure

Microstructure

BubblesSkeleton

Rise

Transverse

Transverse

Figure 2.3: Model of the microstructure. (a) View parallel to the rise direction. (b) Cross-sectional view of the transverse plane. The dashed line marks the boundary of the unitcell.

16

Skeleton

60

O11'

2

2'

3

3'

4

4'

5

5'Rb

O

L1

L/2

α

Bubble

(b)

(a)

o

Figure 2.4: The unit cell.

2.2 Unit cell

We chose the unit cell marked with a dashed line in figure 2.3. This unit cell is contained in

a parallelepiped (Fig. 2.4a) of volume

V0 = 3√

3(L sinα)2(L1 − L cosα), (2.1)

where L1 is the length of the bars parallel to the rise direction and L is the length of the

bars that form an angle α with the rise direction (Fig. 2.4b). Ten points of the skeleton lie

on the surface of the parallelepiped. We shall refer to these points as the “tips” of the unit

cell.

The volume of solid material contained in the unit cell is Vs = 2πr2(L1 + 3L) + 8πR2bt,

17

where r is the cross-sectional radius of the bars, and Rb is the radius of the bubbles,

Rb = 0.77 3√

(L sinα)2(L1 − L cosα), (2.2)

and t is the thickness of the bubbles.

The relative density of a foam, ρ, is defined as the ratio between the apparent density of

the foam, or mass per unit volume of foam, and the density of the solid material. With our

model the relative density of a foam can be calculated as

ρ =VsV0

=2πr2(L1 + 3L) + 8πR2

bt

3√

3(L sinα)2(L1 − L cosα). (2.3)

2.3 Mechanics of the skeleton

As noted earlier, the unit cell has ten tips. The tips are those points of the skeleton that the

cell shares with contiguous cells.

2.3.1 Compatibility and continuum

The tips are paired; using the notation of Fig. 2.4a, tip 1 is paired with tip 1′, tip 2 is paired

with tip 2′, and so forth. To assemble the foam by putting together numerous cells, we

attach tip k of a cell to tip k′ of a contiguous cell, where k might take any integer value from

1 to 5. Because of the symmetry with respect to point O (Fig. 2.4a), if X k be the position

vector of tip k, X k′be the position vector of tip k′, and the position vectors radiate from

O, then X k = −X k′.

In order to be able to assemble the deformed foam by putting together numerous deformed

cells, it must be that the displacement of tip k relative to tip n equals the displacement of tip

n′ relative to tip k′. This reasoning leads to the compatibility condition uk+uk′= un+un′

,

18

where uk, uk′, un and un′

are respectively the displacement vectors of tip k, tip k′, tip n,

and tip n′. As the compatibility condition must hold for all k and n, we can write

uk + uk′= 2U for all k, (2.4)

where U does not depend on k.

Suppose that we pin the tips of the unit cell to a continuum medium in which there

prevails a given displacement gradient F. In this case, uk−uk′= F·(X k−X k′

)−(X k−X k′).

As X k = −X k′,

uk − uk′= 2(F ·X k −X k) (2.5)

From (2.4) and (2.5), we conclude that

uk = U + (F ·X k −X k) (2.6)

and

uk′= U − (F ·X k −X k) (2.7)

It is clear that U represents a rigid-body displacement. In Fig. 2.5a we show a schematic of

the skeleton after the tips have been displaced in accord with (2.6) and (2.7).

We have been concerned so far with the displacements of the tips. But the tips may also

rotate, and the rotation of a tip is independent of the displacement of the tip. Thus, there

is a second compatibility condition: In order to be able to assemble the deformed foam by

putting together numerous deformed cells, it must be that

φk = φk′for all k, (2.8)

where φk and φk′are the rotation vectors of tip k and tip k′, respectively. In Fig. 2.5b

19

fk

fk'

= -fk

mk

mk' = mk

(c) (d)

uk

uk'

= -uk

Ok

k'

(a)

uk

uk'

= -uk

Ok

k'

(b)

φk'

= φk

φk

O fk'

mk'

O

fk

mk

O

Figure 2.5: (a) Schematic of the skeleton after the tips have been displaced in accord with(2.6) and (2.7), so that uk = −uk′

, where k and k′ are paired tips. (We exclude the rigid-body displacement U , as it does not lead to deformation.) Note that when we displace thetips we do not allow them to rotate. (b) Schematic of the skeleton after the tips have beendisplaced and rotated. (c) Reactions. (d) Action–reaction pairs.

we show a schematic of the skeleton after the tips have been displaced and rotated. The

skeleton is symmetric with respect to O, the displacement vectors are antisymmetric with

respect to O, and the rotation vectors are symmetric with respect to O. If follows that the

skeleton remains symmetric with respect to O after the tips have been displaced and rotated.

2.3.2 Hinges

Attendant on the application of displacements and rotations at the paired tips k and k′,

there may arise reactions in the form of forces f k and f k′, and moments mk and mk′

. For

consistency with the prevailing symmetries, the forces must be antisymmetric with respect

to O, and the moments symmetric, so that f k = −f k′

and mk = mk′for all k (Fig. 2.5c).

Note, however, that tip k of a cell is attached to tip k′ of a contiguous cell. It follows that

(f k,f k′) and (mk,mk′

) are action-reaction pairs, so that f k = −f k′

and mk = −mk′for all

20

O O O

(a) (b)

Figure 2.6: (a) The lower half of the skeleton. The skeleton is hinged at the tips and at O.(b) Schematic of springs (in gray) used to simulate bending at an interior node.

k (Fig. 2.5d). We conclude that mk = mk′= 0 for all k. The skeleton is hinged at the tips.

As mk = mk′= 0 for all k, we can write the equation of equilibrium of moments for the

skeleton in the form∑

k Xk × f k +

∑k′ X

k′ × f k′

= 0; as X k = −X k′and f k = −f k

′, it

follows that∑

k Xk× f k = 0. Now we write the equation of equilibrium of moments for the

lower half of the skeleton,∑

k Xk× f k = mO (where mO is the bending moment at O), and

conclude that mO = 0. The skeleton is hinged at O.

2.3.3 Strain energy of the skeleton

As the symmetry with respect to O is preserved after the tips have been displaced and

rotated, we need only concern ourselves with the lower half of the skeleton (Fig. 2.6a). The

displacement of the tips can be computed from the displacement gradient F by using (2.6),

where we set U = 0. Point O does not move (Fig. 2.6a).

The strain energy density of the skeleton is the sum of the axial strain energy density

and the bending strain energy density, W s = W sA +W s

B. To compute the axial strain energy

density we use the formula

W sA =

1

2V0

∑i

KiA(di −Di)2, (2.9)

where the sum on i extends over the 7 bars in the lower half of the skeleton. Here di is the

21

distance between the endpoints of bar i after the deformation, Di is the distance between

the endpoints of bar i before the deformation, and KiA is the axial stiffness of bar i, and

Kia = γEπr2/Di, where γ is a dimensionless parameter, r is the cross-sectional radius of the

bars, and Es is the Young’s modulus of the solid material. For the nominal foam in which

every bar has the same circular cross section, γ = 1. But in EOC foams the cross sections

of the bars are hardly circular; further, the cross section may vary from bar to bar and even

along the length of any given bar [17]. To account in a simple way for deviations from the

nominal foam, we shall treat γ as a free dimensionless parameter.

To compute the bending energy density of the skeleton, W sB, we make the simplifying

assumption that the bars themselves remain straight, and simulate the effect of bending by

means of the set of springs shown schematically for one of the interior nodes of the lower

skeleton in Fig. 2.6b. We use the formula

W sB =

1

2V0

∑j

KjB(θj −Θj)2, (2.10)

where the sum on j extends over all the springs. Here θj is the angle between the bars

spanned by spring j after the deformation, Θj is the angle between the bars spanned by

spring j before the deformation, and KjB is the stiffness of spring j. If spring j spans two

of the bars that form an angle α with the rise direction, KjB = β1Esπr

4/4L, where β1 is a

dimensionless parameter; otherwise, KjB = β2Esπr

4/4L, where β2 is another dimensionless

parameter. Here, we shall treat β1 and β2 as dimensionless fitting parameters. We expect

the values of these parameters to be of order 1.

In principle, W s depends on F and v , where v is the 6 × 1 vector of the components

of the displacements of the two interior nodes in the lower part of the skeleton (2.6b), and

we can write W s = W s(F, v). The force vector ∂W s/∂v

∣∣∣∣F

is the work-conjugate of v ;

as the interior nodes must remain force-free, we determine v by imposing the equilibrium

22

Rb

Rb

Rb

13

2

(a)

λ1Rbλ2Rb

λ3Rb

(b)

Figure 2.7: Deformation of a bubble. λ1, λ2 and λ3 are the principal stretches. (a) Unde-formed bubble (b) Deformed bubble

condition ∂W s/∂v

∣∣∣∣F

= 0. The stress contributed by the skeleton can be obtained by taking

a derivative of W s with respect to F at equilibrium (See appendix A).

2.4 Mechanics of the bubbles

Suppose that we pin each material point of a bubble to the continuum medium in which

there prevails the displacement gradient F. Under the action of principal stretches λ1,λ2 and

λ3, an initially spherical bubble deforms into an ellipsoid (2.7). The bubble is thin-walled,

and we can therefore neglect bending and compute the strain energy density of the bubble

by using the formula [64]

W b =Esh

V0(1− ν2s )

∫S

(E21′1′ + E2

2′2′ + 2νsE1′1′E2′2′)dS (2.11)

where S is the surface of the undeformed bubble and E1′1′ , E2′2′ are the in-plane strain

components at any point on S. We approximate the integral in (2.11) by averaging over

values obtained at poles 1 to 3 (Fig. 2.7), which gives,

23

0.0 0.25

0.0

0.5

1.0

-0.25

k = 100k = 50

k = 500

E11

c 1

Figure 2.8: Smooth approximation of heaviside function

W b =8EsπR

2bt

3V0(1− ν2s )

[E211 + E2

22 + E233 + ν(E11E22 + E11E33 + E22E33)] (2.12)

where E11 = (λ21 − 1)/2,E22 = (λ2

2 − 1)/2 and E33 = (λ23 − 1)/2.

Thin membranes tend to wrinkle under in-plane compressive strains and thus offer

minimal resistance under compression. To account for the lack of compressive stiffness,

we introduce coefficients c1 = (1 + tanh(kE11))/2, c2 = (1 + tanh(kE22))/2 and c3 =

(1 + tanh(kE33))/2 to the strain energy density W b,

W b =8EsπR

2bt

3V0(1− ν2s )

[c1E211 + c2E

222 + c3E

233 + ν(c1c2E11E22 + c1c3E11E33 + c2c3E22E33)] (2.13)

Coefficients c1, c2, c3 are smooth approximations of the corresponding heaviside function

H(E11),H(E22) and H(E33) respectively (Fig. 2.8). With k = 500, the coefficients c1, c2 and

c3 act as a switch that eliminate the contribution of compressive strains to W b.

Next, we take appropriate derivatives of the strain-energy density (2.13) with respect to

F to obtain the attendant stress and elasticity tensor (See appendix A).

24

2.5 Model parameters and scale invariance

Suppose that for a given foam we know ρ, L1/L, t/r, and α. By using (2.2) and (2.3), we

can determine Rb/L and r/L, respectively. Then, any length scale of the model will be set

by L, and the strain energy density of the foam, W = W s + W b, will depend, in principle,

on L, Es, νs, and the nine components of F, Fij.

Now, from Buckingham’s theorem [65] and the dimensional equations [W ] = [Es]1[L]0,

[νs] = [Es]0[L]0, and [Fij] = [Es]

0[L]0, we conclude that the functional relation among the

variables W , L, Es, νs, and Fij, can may be expressed in the form of an equivalent functional

relation among the dimensionless variables W/Es, νs, and Fij. Thus, it is possible to write

W = EsΠ(νs, Fij), with the implication that the strain energy density (and therefore the

stress) is invariant to changes in L. The complete list of parameters of the model is ρ,

L1/L, t/r, α, Es, νs, β1, β2, and γ.

2.6 Finite element implementation

We have implemented the model as a user-defined material subroutine (UMAT) in the com-

mercial general-purpose finite-element code ABAQUS. The finite-element code invokes the

UMAT subroutine to solve the constitutive relation at each of the integration points of a

finite element. We provide details of the UMAT implementation in appendix A.

2.7 Discussion

In this chapter, we have formulated and implemented in a general-purpose finite-element

code a unit-cell model of elastic open-cell foams. The model includes novel features which

are peculiar to polyether polyurethane EOC foams.

In the model the microstructure of a foam consists of a three-dimensional, regular skeleton

25

(or network of bars) and a set of thin-walled bubbles. The skeleton and the set of bubbles

are coupled in parallel, and contribute additively to the strain energy density, and therefore

to the stress.

The thin-walled bubbles substitute for certain membranes that can be seen in polyether

polyurethane EOC foams under a scanning-electron microscope. To our knowledge, these

membranes have so far been ignored in models of EOC foams.

The model can be fully described using 9 parameters. Each one of these parameters has

a clear geometrical or mechanical significance. The values of the parameters may be readily

estimated for any given foam.

As in common applications of EOC foams the deformations are large and the mechanical

response of a foam is dominated by nonlinear geometric effects, we have used a linear elastic

constitutive relation for the skeleton and bubbles. Thus, the mechanical parameters include

the Young’s modulus and the Poisson’s ratio of the skeleton and the bubbles.

We have shown theoretically that the mechanical response predicted using the model is

invariant to isomorphic changes in the dimensions of the microstructure of the foam. Thus,

there is no intrinsic characteristic length in our model.

In the next chapter, we turn to the calibration of the model.

26

Chapter 3

Calibration

In chapter 2, we formulated and implemented in a general-purpose finite-element code a

unit-cell model of elastic open-cell foams. In this chapter, we calibrate the model.

3.1 Experiments

To calibrate the model we have recourse to the experiments of Dai et al. [41], which were

carried out on a set of polyether polyurethane EOC foams of relative densities 0.030, 0.038,

0.046, 0.065 and 0.086. These foams are known to the manufacturer General Plastics of

Tacoma, Washington, by the commercial codes EF-4003, EF-4004, EF-4005, TF-5070-10,

and TF-5070-13, respectively. A specimen of each foam was tested under five loading condi-

tions, namely compression along the rise direction (test 1); compression along a transverse

direction (test 2); tension along the rise direction (test 3); simple shear combined with com-

pression along the rise direction (test 4); and hydrostatic pressure combined with compression

along the rise direction (test 5). For each combination of loading condition and relative den-

sity of the foam, we compute a stress–stretch curve (or, for some tests, a force–displacement

curve) using the model and compare it with the stress–stretch curve (or force–displacement

curve) measured by Dai et al. in their experiments.

27

3.2 Simulations

To compute a stress–stretch curve, it would be possible for us to subject a single cell to

an average deformation gradient suitably chosen to reflect the experimental details. This

method might be appropriate where the stress field is spatially homogeneous (as it nearly is

in test 1), but not where the stress field is spatially heterogeneous (as it is in test 4, simple

shear combined with compression along the rise direction). To be able to to account for

the actual conditions that prevailed in the experiments, we compute stress–stretch curves

(or, depending on the test, force–displacement curves) by running a fully three-dimensional,

nonlinear finite-element simulation of each experiment.

We model the loading plates of the experiments using ABAQUS’ “analytical rigid surface”

[66]. The frictional coefficient is 0.3 between the foam and the loading plates. The frictional

coefficient does not have a significant quantitative effect on the computed stress–stretch

curves, but a higher value leads to spurious oscillations in the stress–stretch response for a

few cases.

3.3 Results

The complete list of parameters of the model is ρ, L1/L, t/r, α, Es, νs, β1, β2, and γ

(chapter 2). To determine the values of the parameters, we start by setting L1/L = 1.5 and

t/r = 0.04 (which are rough estimates from direct observation of micrographs of the foams),

α = 109.5◦ (which is the angle of a perfect tetrahedron), Es = 65 MPa (which is the Young’s

modulus of polyether polyurethane as provided by the manufacturer), and νs = 0.5 (which

is an estimate of the Poisson’s ratio of polyether polyurethane). We assume that the values

of L1/L and t/r remain fixed regardless of the relative density of foam. We treat γ, β1, and

β2 as dimensionless fitting parameters, but expect the values of these parameters to be of

28

order 1. Therefore, we start by setting γ = β1 = β2 = 1.

We compute stress–stretch curves, compare them visually with the experimental stress–

stretch curves, and make adjustments to the values of the parameters. If experimental

measurements of the stretch fields be available, we also compare the nature of the stretch

fields measured in the experiments with the nature of the stretch fields predicted using the

model. The results discussed in what follows are for Es = 65 MPa, ν = 0.5, L1/L = 1.2,

t/r = 0.032, α = 108.5◦, γ = 1.15, β1 = 1.6 and β2 = 5.2.

3.3.1 Test 1: Compression along the rise direction

We show a schematic of the experiments of Dai et al. [41] in the inset of Fig. 3.1. The size

of the specimens is 10 cm × 10 cm × 5 cm, and the height (of 5 cm) is aligned with the rise

direction. The specimens are compressed along the rise direction, through rigid plates.

Stress–stretch curves

We compare the simulated and experimental stress–stretch curves in Fig. 3.1. The stress P

is the compressive force per unit area of contact between the plates and a specimen (10 cm2),

and the stretch λ is the height of the deformed specimen normalized by the height of the

undeformed specimen (in other words, λ is the average stretch in the specimen). For the

three foams of lower relative density (ρ = 0.03, 0.038, and 0.046) the simulated stress–stretch

curves have stress plateaus, consistent with the experimental curves. Note, however, that

these stress plateaus are serrated. We shall show that the serrations are but a by-product of

the discreteness of the finite-element mesh.

For the two foams of higher relative density (ρ = 0.065 and 0.086) the simulated stress–

stretch curves do not have stress plateaus, again consistent with the experimental curves.

29

0.60.70.81.00.0

0.4

0.8

1.2

1.6

2.0

Simulated

Experimental

Rise ρ = 0.086

ρ = 0.065

P/E

s (

x 1

0-3

)

ρ = 0.046

ρ = 0.03

ρ = 0.038

0.9

H ∆

P

λ

Figure 3.1: Experimental (dashed lines) and simulated (solid lines) stress–stretch curves forfoams of five relative densities (ρ) tested under compression along the rise direction, Test 1.Inset: schematic of the experiments and of the finite-element mesh used in the simulations.

Stretch fields

Dai et al. used the digital image correlation (DIC) technique to measure the stretch fields in

a representative foam of higher relative density and in a representative foam of lower relative

density. To facilitate the DIC measurements, they repeated Test 1 with larger specimens of

size 10 cm × 10 cm × 10 cm. (The DIC measurements were carried out on one of the lateral

faces of the specimens.)

Low-density foam. For ρ = 0.038, Dai et al. found that in conjunction with the stress

plateau of the experimental stress–stretch curve (Fig. 3.2a), the stretch fields were spatially

heterogeneous and consisted of a band of high deformation sandwiched between two bands

of low deformation. These bands were about perpendicular to the rise direction (Fig. 3.2b).

Within the band of high deformation the local stretch λ = λH ≈ 0.75; within the bands of

30

0.50.60.70.80.91.0

20

40

60

80

Y(m

m)

0.40.60.81.00.0

0.1

0.2

0.3

0.4

0.5

Rise direction

DIC frame

Y

AB GFEDC

H

(a) (b)

P/E

s(x 1

0-3)

P

λ λ

A B GFEDC H

Figure 3.2: Experimental results for the foam of relative density ρ = 0.038 tested undercompression along the rise direction, Test 1, but with a larger specimen than in Fig. 3.1. (a)Experimental stress–stretch curve. Inset: schematic of the experiment. (b) Plots of the localstretch λ along the dashed-line transect of the inset of (a). Plots A, B, C, etc. correspondrespectively to the points marked “A,” “B,” “C,” etc. on the stress–stretch curve of (a). Thearrows indicate the broadening of the band of high deformation.

low deformation the local stretch λ = λL ≈ 0.95. The band of high deformation nucleated

when the stress attained the value of the plateau stress, at the leftmost end of the stress

plateau. At this stage in the experiment, the applied average stretch λ ≈ λL. As λ was

lessened further during the experiment, the band of high deformation broadened, and the

bands of low deformation thinned concomitantly. In time, the band of high deformation came

to encompass the entire lateral face of the specimen, so that the stretch field became spatially

homogeneous once again. This occurred when the rightmost end of the stress plateau was

reached. At this stage in the experiment, λ ≈ λH , and any additional lessening of λ lead to

an increase in the stress (Fig. 3.2a).

In view of these experimental findings, Dai at al. interpreted the plateau stress as a

Maxwell stress and the accompanying stretch fields as two-phase stretch fields—that is,

mixtures of two configurational phases of the foam. In this interpretation, λH was the

31

0.70.80.910.0

0.1

0.2

0.3

0.4

0.5

0.6

= 0.038

Y

Ris

e

ρ

A

B

H

C

FEG

D

P/E

s(x 1

0-3)

P

Contour frame

λ

Figure 3.3: Simulated stress–stretch curve for the foam of relative density ρ = 0.038 testedunder compression along the rise direction, Test 1, but with a larger specimen than inFig. 3.1. Inset: schematic of the finite-element mesh used in the simulation.

characteristic stretch of phase H (the high-deformation phase), and λL was the characteristic

stretch of phase L (the low-deformation phase). As the stress plateau was traced from left

to right during the experiment, the volume fraction of phase H grew at the expense of the

volume fraction of phase L. Thus, Dai et al. reached the conclusion that the low-density

foam had undergone a phase transition. This conclusion is consistent with our simulation of

the experiment, to which we turn next.

In Fig. 3.3 we show the simulated stress–stretch curve, and in Fig. 3.4 we show the

attendant evolution of the stretch field on the faces of the specimen. Each contour plot of

Fig. 3.4 corresponds to one of the points (A, B, etc.) marked on the stress–stretch curve of

Fig. 3.3: the contour plot of Fig. 3.4a to pt. A, the contour plot of Fig. 3.4b to pt. B, and

so on.

In Fig. 3.5 we show plots of the local stretch λ along the dashed-line transect of the inset

of Fig. 3.4a. The plot of Fig. 3.5a corresponds to the contour plot of Fig. 3.4a, the plot of

32

(a) (d)(c)(b)

(e) (h)(g)(f)

0.59

1.00

Figure 3.4: Contour plots of the local stretch λ on the faces of the specimen in the simulationof Fig. 3.3. The contour plots of (a), (b), etc. correspond respectively to the points marked“A,” “B,” etc. on the stress–stretch curve of Fig. 3.3. The contour plots are mapped on theundeformed configuration.

Fig. 3.5b corresponds to the contour plot of Fig. 3.4b, and so on.

As in the experiment of Dai et al., a band of phase H (Fig. 3.4c) nucleates when P attains

the value of the Maxwell stress, at the leftmost end of the stress plateau (pt. C of Fig. 3.3).

From Fig. 3.5c, we estimate λH ≈ 0.7 and λL ≈ 0.9; these values are in reasonable accord

with the ones measured by Dai et al. (λH ≈ 0.75 and λL ≈ 0.95). As λ is lessened further

during the simulation, the band of phase H does not broaden, as it did in the experiment of

Dai et al. (Fig. 3.2b); instead, new bands of phase H nucleate (e.g., Fig. 3.4d). (Two-phase

stretch fields with multiple bands of phase H have been observed in other experiments, for

example [40], [9], and [67].) In any event, the volume fraction of phase H increases at the

expense of the volume fraction of phase L, while P remains invariant, consistent with the

occurrence of a phase transition. And in time, the rightmost end of the stress plateau is

reached (pt. H of Fig. 3.3), and phase H comes to encompass the entire specimen (Fig. 3.4h).

Serrations of the stress–stretch curves. To the process of nucleation of one of the bands of

phase H (for instance, Fig. 3.4b–c and Fig. 3.5b–c) there corresponds one of the serrations of

33

0.60.81.00

40

80

Y(mm)

λ(b)

0.60.81.00

40

80Y(mm)

λ(a) (c)

0.60.81.00

40

80

Y(mm)

λ(d)

0.60.81.00

40

80

Y(mm)

λ(e)

0.60.81.00

40

80

Y(mm)

λ(f)

0.60.81.00

40

80

Y(mm)

λ(g)

0.60.81.00

40

80

Y(mm)

λ(h)

0.60.81.00

40

80

Y(mm)

λ

Figure 3.5: Plots of the local stretch λ along the dashed-line transect of Fig. 3.4a. The plotsof (a), (b), etc. correspond respectively to the contour plots of Fig. 3.4a, Fig. 3.4b, etc.

the stress–stretch curve (for instance, pts. B and C of Fig. 3.3). As the thickness of each new

band of phase H coincides with the height of a single finite element (Fig. 3.6), the number of

serrations should equal the number of finite elements that span the height of the specimen.

This condition can indeed be verified in Fig. 3.6. Further, the amplitude of the serrations

lessens as the number of finite elements is increased (Fig. 3.6), and we conclude that the

serrations are but artifacts of the discreteness of the finite-element mesh. Note, however,

that the serrations may be conveniently interpreted as a signature of the presence of multiple

configurational phases in the foam.

Micromechanical nature of the phases. In Fig. 3.7 we show the evolution of the deformed

shape of the skeleton of an individual cell. This individual cell is located on one of the lateral

faces of the specimen and within the band of phase H of Fig. 3.4c.

From Fig. 3.7 we conclude that the transition from phase L (Fig. 3.7b) to phase H

(Fig. 3.7c) takes place locally where the three slanted bars that converge on a node of the

skeleton switch dynamically from the upward slant to the downward slant (or the other way

34

0.60.70.80.910.0

0.1

0.2

0.3

0.4

PM

0.60.70.80.910.0

0.1

0.2

0.3

0.4

(c)

PM

A+

A-

A+A-

0.60.70.80.910.0

0.1

0.2

0.3

0.4

PM

(b)

A+

A-

A+A-

(a)

A+A-

A+

A-

P/E

s(x

10

-3)

P/E

s(x

10

-3)

P/E

s(x

10

-3)

λ

λ

λ

Figure 3.6: Simulated stress–stretch curves for the foam of relative density ρ=0.03 tested un-der compression along the rise direction. The finite-element mesh consists of (a) 64 elements(b) 240 elements and (c) 1024 elements.

35

(a) (b) (c)

Figure 3.7: Evolution of the deformed shape of the skeleton of an individual cell. Theposition of this cell is marked “+” on Fig. 3.4a; note that the cell is located within the firstband of phase H that nucleates in the specimen (Fig. 3.4c). The deformed shapes of (a),(b) and (c) correspond respectively to Fig. 3.4a, Fig. 3.4b and Fig. 3.4c (and to the pointsmarked A, B, and C on the stress–stretch curve of Fig. 3.3). The deformation proceedscontinuously from (a) to (b) and discontinuously from (b) to (c).

around). A cell of a low-density foam may be likened to a light switch, the cap of a sham-

poo bottle, and other bistable elastic structures that snap between two stable equilibrium

configurations.

Effect of the bubbles. If we exclude the contribution of the bubbles from the strain energy

density and run the simulation again, the simulated stress–stretch curve will not have a

stress plateau (Fig. 3.8). Without bubbles, there is no phase transition.

To elucidate further the effect of the bubbles on the mechanical response of the foam,

we must realize that a cell cannot snap unless there be some lateral confinement, so that

the three slanted bars that converge on a node of the skeleton are prevented from splaying

out. For a cell located halfway through the height of the specimen (the cell of Fig. 3.7, say),

the loading plates and the rest of the foam can hardly provide any lateral confinement from

outside the cell. But the bubbles could provide enough lateral confinement from inside the

cell.

To test this possibility, consider an individual cell that is compressed uniaxially along

the rise direction, so that the external lateral confinement is null. For this single cell we

compute the strain energy density W and the local stress in the rise direction, P = ∂W/∂λ,

36

0.70.80.91.00.0

0.15

0.30

0.45

P/E

s(x

10

-3)

Simulated with bubble

Simulated without bubble

Experiment

∆Rise

ρ = 0.038

P

λ

Figure 3.8: Simulated stress–stretch curves for the foam of relative density ρ = 0.038 testedunder compression along the rise direction. The simulated stress–stretch curve shown as asolid line is the same as the simulated stress–stretch curve of Fig. 3.3.

and plot W vs. λ and P vs. λ. If we set W = W s (to exclude the contribution of the bubbles

to the strain energy density) we will find that the strain energy density is a convex function

of function of λ, P is a monotonic function of λ, and the cell does not snap (Fig. 3.9a).

If we set W = W s +W b (to include the contribution of the bubbles to the strain energy

density) we will find that the strain energy density is a non-convex function of λ, P is a

non-monotonic function of λ, and the cell snaps (Fig. 3.9b). We conclude that the snapping

of a cell is made possible by the bubbles, which provide lateral confinement from inside the

cell.

Now, for a cell close to a loading plate, the loading plate provides some (external) lateral

confinement from outside the cell in addition to the lateral confinement that the bubbles

provide from inside the cell. To ascertain the effect of the external lateral confinement,

consider a single cell that is stretched uniaxially along the rise direction, so that the external

lateral confinement is maximum. For this single cell we compute the strain energy density

W (including the contribution from the bubbles) and the local stress in the rise direction,

37

(c)

0.0

0.1

W/Ε

s (x

10

-3)

0.60.81.0

λ

0.0

0.4

P/Ε

s (x

10

-3)

(b)

0.60.81.0

λ

0.0

0.4

P/Ε

s (x

10

-3)

0.0

0.1

W/Ε

s (x

10

-3)

(a)

0.60.81.0

λ

0.0

0.4

P/Ε

s (x

10

-3)

0.0

0.1W/Ε

s (x

10

-3)

Figure 3.9: Plots of W vs. λ and plots of P vs. λ, for a single cell, where W is the strainenergy density, λ is the local stretch in the rise direction, and P is the local stress in therise direction, P = ∂W/∂λ. (a) The cell is compressed uniaxially along the rise direction,and W does not include the contribution of the bubbles to the strain energy density. (b)The cell is compressed uniaxially along the rise direction, and W includes the contributionof the bubbles to the strain energy density. (c) The cell is stretched uniaxially along the risedirection, and W includes the contribution of the bubbles to the strain energy density.

P = ∂W/∂λ, and plot W vs. λ and P vs. λ (Fig. 3.9c). From a comparison of Fig. 3.9c (which

corresponds to maximum external lateral confinement) and Fig. 3.9b (which corresponds to

null external lateral confinement), we conclude that the external lateral confinement leaves

the Maxwell stress unchanged but broadens the immiscibility gap (that is, the value of

λL − λH). Keeping these conclusions in mind, we turn once again to the simulation of

Fig. 3.5. In that simulation, as well as in the experiment of Dai et al., the switch from the

phase L to phase H progresses from the interior of the specimen towards the loading plates,

so that the first cell to snap does so under scant external lateral confinement, whereas the

last cell to snap does so under sizable external lateral confinement. Thus, the immiscibility

gap increases as the stress plateau is traced from left to right during the simulation (Fig. 3.5).

This effect is also apparent in the experimental measurements of Dai et al. (Fig. 3.2b).

38

0.50.60.70.80.91.0

30

40

50

60

70

Y(m

m)

λλ

0.40.60.81.00.0

0.4

0.8

1.2

1.6

AB C

D

ρ = 0.065

P

Rise direction

DIC frame

Y

(a) (b)

P/E

s (

x 1

0-3 )

A B C D

Figure 3.10: Experimental results for the foam of relative density ρ = 0.065 tested undercompression along the rise direction, Test 1, but with a larger specimen than in Fig. 3.1. (a)Experimental stress–stretch curve. Inset: schematic of the experiment. (b) Plots of the localstretch λ along the dashed-line transect of the inset of (a). Plots A, B, C, and D correspondrespectively to the points marked “A,” “B,” “C,” and “D” on the stress–stretch curve of (a).

High-density foam.

For the representative foam of higher relative density, ρ = 0.065, our model yields stretch

fields (Fig. 3.11) that are consistent with the measurements of of Dai et al., who found no

evidence of the presence of two configurational phases in the foam (Fig. 3.10).

3.3.2 Test 2: Compression along a transverse direction

We show a schematic of the experiments of Dai et al. [41] in the inset of Fig. 3.12. The size of

the specimens is 10 cm × 10 cm × 10 cm. The specimens are compressed along a transverse

direction, through rigid plates.

We compare the simulated and experimental stress–stretch curves in Fig. 3.12. Here

the stress P is the compressive force per unit area of contact between the plates and the

specimens (10 cm2), and the stretch λ is the height of the deformed specimen normalized

by the height of the undeformed specimen—that is, the average stretch in the specimen.

39

0.60.70.80.91.00.0

0.5

1.0

1.5

ρ = 0.065

D

CB

A

Contour

frameY

Ris

eP

0.60.70.80.91.00

20

40

60

80

λ

P/E

s(x

10

-3)

Y (

mm

)

λ

DCBA

(a) (b)

Figure 3.11: Computational results for the foam of relative density ρ = 0.065 tested undercompression along the rise direction, Test 1, but with a larger specimen than in Fig. 3.1. Cf.Fig. 3.10. (a) Simulated stress–stretch curve. Inset: schematic of the simulation. (b) Plotsof the local stretch λ along the dashed-line transect of the inset of (a). Plots A, B, C, and Dcorrespond respectively to the points marked “A,” “B,” “C,” and “D” on the stress–stretchcurve of (a).

The simulated stress–stretch curves are monotonic for all the foams, consistent with the

experimental stress–stretch curves.

Dai et al. found that where a foam was tested under compression along two mutually

perpendicular transverse directions, the experimental stress–stretch curves differed slightly

from each other, in particular at relatively high deformation. The differences were within

the margin of error of the experimental measurements, however, and Dai et al. stated that

the stress–stretch curves were “practically indistinguishable,” and concluded that the foams

were transversely isotropic. In accord with the experimental findings of Dai et al., the

simulated stress–stretch curves differ slightly depending on the transverse direction, and

only at relatively high deformation (Fig. 3.13).

Dai et al. measured the stretch fields in the foam of the lowest relative density, ρ = 0.030,

and found no evidence of the presence of two configurational phases in the foam. Our model

40

0.80.850.90.9510.0

0.2

0.4

0.6

0.8

1.0Simulated

Experimental

∆Rise

ρ = 0.086

ρ = 0.065

ρ = 0.046

ρ = 0.03

ρ = 0.038

P/Ε

s (

x 1

0-3 )

P

λ

Figure 3.12: Experimental (dashed lines) and simulated (solid lines) stress–stretch curvesfor foams of five relative densities (ρ) tested under compression along a transverse direction,Test 2. Inset: schematic of the experiments and of the finite-element mesh used in thesimulations.

yields continuous stretch fields (not shown here).

3.3.3 Test 3: Tension along the rise direction

We show a schematic of the experiments of Dai et al. [41] in the inset of Fig. 3.14. The size

of the specimens is 10 cm × 5 cm × 5 cm, and the height (of 10 cm) is aligned with the rise

direction. The specimens are glued to rigid plates.

We compare the simulated and experimental stress–stretch curves in Fig. 3.14. Here the

stress P is the tensile force per unit area of contact between the plates and the specimens

(25 cm2), and the stretch λ is the height of the deformed specimen normalized by the height

of the undeformed specimen—that is, the average stretch in the specimen. The simulated

stress–stretch curves are monotonic for all the foams, consistent with the experimental stress–

41

0.840.880.920.961.00.0

0.2

0.4

0.6

0.8

ρ = 0.086

ρ = 0.065

ρ = 0.046

ρ = 0.03

ρ = 0.038

Transverse

Tra

nsv

erse

Simulated Px

Simulated Pz

P/Ε

s (

x 1

0-3

)P

x

Pz

λ

Figure 3.13: Simulated stress–stretch curves for foams of five relative densities (ρ) testedunder compression along two mutually perpendicular transverse directions (as indicated inthe inset). The simulated stress–stretch curves, shown here as thick lines, are the same asthe simulated stress–stretch curves of Fig. 3.12.

stretch curves. The simulated curves are systematically stiffer than the experimental ones,

in particular at relatively large deformation. Dai et al. noted that for λ > 1.05 there was

“some indication of microstructural damage” on the surface of the specimens, and the stiffer

response predicted by the model may therefore be ascribed to the fact that the model does

not include any way of accounting for microstructural damage.

3.3.4 Test 4: Simple shear combined with compression along the

rise direction

We show a schematic of the experiments of Dai et al. [41] in the inset of Fig. 3.15. The size

of the specimens is 10 cm × 10 cm × 5 cm, and the height (of 5 cm) is aligned with the rise

42

1 1.02 1.04 1.06 1.08 1.10

6.0

2.0

0.8

0.4

0.0

8.0Simulated

Experimental

∆

Rise

ρ = 0

.086

ρ = 0.065

ρ = 0.046

ρ = 0.03

ρ = 0.038

P/Ε

s (

x 1

0-3

)

λ

Figure 3.14: Experimental (dashed lines) and simulated (solid lines) stress–stretch curvesfor foams of five relative densities (ρ) tested under tension along the rise direction, Test 3.Inset: schematic of the experiments and of the finite-element mesh used in the simulations.

direction. The specimens are loaded through rigid wedges. The angle of the wedges is 30◦

(inset of Fig. 3.15).

Force–displacement curves

We compare the simulated and experimental force–displacement curves in Fig. 3.15. Here

the force F is the compressive force applied through the wedges, and the displacement ∆ is

the displacement of the lower wedge (inset of Fig. 3.15).

The simulated force–displacement curves of Fig. 3.15 display serrations for the three foams

of lower relative density (ρ = 0.03, 0.038, and 0.046). These serrations may be interpreted as

a signature of the presence of two configurational phases in the foam (recall our discussion

of test 1). Thus, we expect to find discontinuous, two-phase stretch fields in the three foams

of lower density.

43

0.0 0.1 0.2 0.3 0.40

200

400

600

800

1000

1200Simulated

Experimental

∆/H

F (

N)

ρ = 0.086

ρ = 0.065

ρ = 0.046

ρ = 0.03

ρ = 0.038

Rise

H∆

Figure 3.15: Experimental (dashed lines) and simulated (solid lines) force–displacementcurves for foams of five relative densities (ρ) tested under simple shear combined with com-pression along the rise direction, Test 4. Inset: schematic of the experiments and of thefinite-element mesh used in the simulations.

Stretch fields

To facilitate the analysis of the stretch fields, we repeat the simulations of Test 4 with larger

specimens of size 10 cm × 10 cm × 10 cm. We show some of the results in Figs. 3.16, 3.17,

and 3.18 (for ρ = 0.038).

From Fig. 3.17 we conclude that two roughly horizontal bands of phase H nucleate at

diagonally opposite corners on the face of the specimen. The nucleation of these bands occurs

concurrently with the marked loss of stiffness that is apparent on the force–displacement

curve at ∆/H ≈ 0.1 (Fig. 3.16). As the simulation proceeds, the bands broaden progressively

(note the way in which the stretch discontinuity propagates downward in Fig. 3.18c), and the

leading edges of the bands spread toward the centerline of the specimen (that is, transect A).

The same sequence of events can be verified in the simulation for ρ = 0.03 (Figs. 3.19, 3.20,

44

0 0.1 0.2 0.30

100

200

300

A

EDBF

∆/H

F(N

)

CG H

Rise

H ∆

Contour frameY

Figure 3.16: Simulated force–displacement curve for the foam of relative density ρ = 0.038tested under simple shear combined with compression along the rise direction, Test 4, butwith a larger specimen than in Fig. 3.15. Inset: schematic of the finite-element mesh usedin the simulation.

and 3.21).

Dai et al. used the DIC technique to measure the stretch fields in the foam of relative

density ρ = 0.038. From an analysis of plots similar to our plots of Fig. 3.18a (where there

is no clear indication of discontinuity in the stretch fields), they concluded that there was

no evidence of two configurational phases in their experiment. Note, however, that if we

limit ourselves to the plots of Fig. 3.18a, and if we disregard the plots of Figs. 3.18b and

3.18c (where there is clear indication of discontinuity in the stretch fields), we will fail to

notice the presence of two configurational phases in our simulation. We submit that Dai

et al. would have seen a discontinuity in the stretch fields of their experiments if they had

inspected the distribution of local stretch along a vertical transect close to one the lateral

faces of the specimen.

To settle the matter, we carry out an experiment of our own. We subject a foam specimen

45

1.00

0.55

(a) (b) (c) (d)

(e) (f) (g) (h)

ABC

Figure 3.17: Contour plots of the local stretch λ on one of the lateral faces of the specimenin the simulation of Fig. 3.16. The contour plots of (a), (b), etc. correspond respectively tothe points marked “A,” “B,” etc. on the stress–stretch curve of Fig. 3.16. The contour plotsare mapped on the undeformed configuration.

0.60.70.80.91.00

20

40

60

80

λ

Y (

mm

)

0.60.70.80.91.00

20

40

60

80

λ

Y (

mm

)

0.60.70.80.91.00

20

40

60

80

λ

Y (

mm

)

(b)(a) (c)

Figure 3.18: Plots of the local stretch λ along the dashed-line transects of Fig. 3.17a. Theplots of (a) correspond to transect A of Fig. 3.17a, the plots of (b) correspond to transect Bof Fig. 3.17a, and so on.

46

0

60

120

180

F(N

)

0.1 0.2 0.3∆/H

0.0

Rise

H ∆

Contour frameY

AEDB FC

G H

Figure 3.19: Simulated force–displacement curve for the foam of relative density ρ = 0.03tested under simple shear combined with compression along the rise direction, Test 4, butwith a larger specimen than in Fig. 3.15. Inset: schematic of the finite-element mesh usedin the simulation.

of size 10 cm × 10 cm × 10 cm to simple shear combined with compression along the rise

direction and use a DIC technique to measure the stretch fields in the foam (See appendix

B for details of the DIC technique). We chose the foam of the lowest relative density (ρ =

0.03), in which any stretch discontinuity is likely to be readily apparent.

The experimental results (Fig. 3.22 and Fig. 3.23) closely resemble the attendant com-

putational results (Fig. 3.20 and Fig. 3.21). There is evidence of two configurational phases,

but only close to the lateral faces of the specimen. Thus, for example, a propagating stretch

discontinuity can be seen in the distribution of local stretch along vertical transects B and

C (Fig. 3.23b and Fig. 3.23c) but not in the distribution of local stretch along vertical

transect A (Fig. 3.23a).

We conclude that the presence of two configurational phases of the foam in our compu-

tational simulations of test 4 is in accord with the experimental evidence.

47

1.00

0.55

(a) (b) (c) (d)

(e) (f) (g) (h)

ABC

Figure 3.20: Contour plots of the local stretch λ on one of the lateral faces of the specimenin the simulation of Fig. 3.19. The contour plots of (a), (b), etc. correspond respectively tothe points marked “A,” “B,” etc. on the stress–stretch curve of Fig. 3.19. The contour plotsare mapped on the undeformed configuration.

0.60.70.80.91.00

20

40

60

80

λ

Y (

mm

)

0.60.70.80.91.00

20

40

60

80

λ

Y (

mm

)

0.60.70.80.91.00

20

40

60

80

λ

Y (

mm

)

(c)(b)(a)

Figure 3.21: Plots of the local stretch λ along the dashed-line transects of Fig. 3.20a. Theplots of (a) correspond to transect A of Fig. 3.20a, the plots of (b) correspond to transect Bof Fig. 3.20a, and so on.

48

1.00

0.50(a) (b) (c) (d)

(e) (f) (g) (h)

ABC

Figure 3.22: Contour plots of the local stretch λ on one of the lateral faces of the specimen inthe experiment from DIC measurements. The contour plots are mapped on the undeformedconfiguration. Cf. the computational results of Fig. 3.20

0.60.70.80.91.0

10

40

70

λ

Y(mm)

0.60.70.80.91.0

10

40

70

λ

Y(mm)

0.60.70.80.91.0

10

40

70

λ

Y(mm)

(c)(b)(a)

Figure 3.23: Plots of the local stretch λ along the dashed-line transects of Fig. 3.22a. Theplots of (a) correspond to transect A of Fig. 3.22a, the plots of (b) correspond to transect Bof Fig. 3.22a, and so on. Cf. the computational results of Fig. 3.21

49

In both test 1 and test 4 the presence of two configurational phases is signalized by

serrations in the computational stress–stretch (or force–displacement) curves. Note, however,

that in test 1 the serrations–and, therefore, the two-phase stretch fields—occur always in

conjunction with a stress plateau (Fig. 3.1), whereas in test 4 neither the simulated nor the

experimental force–displacement curves have clear force plateaus, except perhaps for ρ =

0.03 (Fig. 3.15).1

To explain these contrasting results, we note that in test 1 the stress field is spatially

homogeneous and congruent with the boundary conditions (that is, at any given point the

principal directions of stress are parallel to the free boundaries of the specimen), whereas in

test 4 the stress field is spatially heterogeneous (because it includes a component of simple

shear) and incongruent with the boundary conditions. This disparity in the nature of the

stress fields has a reflection on the attendant stretch fields:

On the one hand, in test 1, phase H nucleates on a band that spans the entire width

of a specimen, from a free boundary through the centerline of the specimen to the other

free boundary (Fig. 3.4). The phase is either H or L at any given point of the specimen

throughout the test. As the test proceeds. the band of phase H broadens, the volume

fraction of phase H increases, and the external loading supplies the energy required to effect

the transition from phase L to phase H. As this energy is proportional to the increase in

the volume fraction of phase H, which in turn is proportional to the average stretch, the

stress-stretch curve has a stress plateau.

On the other hand, in test 4, phase H nucleates on two bands at diagonally opposite

corners on the face of the specimen. These bands are initially short and spread gradually

towards the centerline of the specimen (Fig. 3.17), so that each band has a propagating

leading edge surrounded by phase L. Consider, for example, the leading edge that propagates

1Dai et al. remarked that, for ρ = 0.038 and 0.046, the experimental force–displacement curves remainedmonotonic throughout. The same can be said, for ρ = 0.038 and 0.046, of the simulated force–displacementcurves.

50

p0

p0

p0 - ∆p/2

p0

p0

∆

H

P1 P2

(a) (b)

Figure 3.24: Schematic of the loading of the specimens in Test 5. (a) Step 1. (b) Step 2.

from left to right. On the left side of the edge, the phase is H and the stretch along the edge,

λH . On the right side of the edge, the phase is L and the stretch along the edge, λL. To

preserve compatibility across the edge, the edge must be straddled by a boundary layer in

which the stretch is neither λL nor λH . This boundary layer carries an excess energy which

may be ascribed to the edge in the form of an energy per unit length of edge. As the test

proceeds, the edge lengthens (because the band broadens), and the excess energy increases.

Thus, the external loading must supply an increasing amount of excess energy in addition

to the energy required to effect the transition from phase L to phase H. As a result, the

force–displacement curve is monotonic.

3.3.5 Test 5: Hydrostatic pressure combined with compression

along the rise direction

We show a schematic of the experiments of Dai et al. [41] in Fig. 3.24. The height of

the cylindrical specimens, H = 4.9 cm, is oriented along the rise direction; the diameter

is 5.1 cm. A specimen is loaded in two steps. In Step 1 (Fig. 3.24a), the specimen is

placed in a pressure chamber, and the surface of the specimen is subjected to a confining

pressure p0 + ∆p(1− 2Y/H), where p0 is the average pressure on the surface of the specimen

51

0

0.1

0.2

0.3

0.60.70.80.91.0

Simulated

Experimental

ρ = 0.03

po = 0.0 kPa

1.59 kPa

3.72 kPa

P/Ε

s (

x 1

0-3 )

0

0.1

0.2

0.3

0.4

0.60.70.80.91.0

Simulated

Experimental

ρ = 0.038

po = 0.0 kPa

2.21 kPa

3.10 kPa

0

0.2

0.4

0.6

0.60.70.80.91.0

Simulated

Experimental

ρ = 0.046

po = 0.0 kPa

2.07 kPa

4.55 kPa

0

0.4

0.8

1.2

0.60.70.80.91

Simulated

Experimental

ρ = 0.065

po = 0.0 kPa

10.3 kPa

24.1 kPa

0

0.4

0.8

1.2

1.6

2.0

0.60.70.80.91

Simulated

Experimentalρ = 0.086

po = 0.0 kPa

2.07 kPa

4.55 kPa

λ λ

λ λ

λ

P/Ε

s (

x 1

0-3 )

P/Ε

s (

x 1

0-3 )

P/Ε

s (

x 1

0-3 )

P/Ε

s (

x 1

0-3 )

Figure 3.25: Experimental (dashed lines) and simulated (solid lines) stress–stretch curvesfor foams of five relative densities (ρ) tested under hydrostatic pressure combined with com-pression along the rise direction, Test 5. po is the average hydrostatic pressure. (a) ρ = 0.03.(b) ρ = 0.038. (c) ρ = 0.046. (d) ρ = 0.065. (e) ρ = 0.086.

52

1.00

0.60

Figure 3.26: The local stretch λ on the surface the specimen of relative density 0.03 in thesimulation with p = 1.59 kPa. The contour plots are mapped on the undeformed configura-tion.

and ∆p = 0.46 kPa is the pressure difference between the bottom the specimen (which

corresponds to Y = 0) and the top of the specimen (which corresponds to Y = H), due to

the weight of the oil that fills the pressure chamber. In Step 2 (Fig. 3.24b), the top of the

specimen is forced to move by a displacement ∆ by applying a deviatoric compressive force

F through a mobile rigid plate while the bottom of the specimen rests on a fixed rigid plate.

We compare the simulated and experimental stress–stretch curves in Fig. 3.25. The stress

P is the compressive deviatoric force F per unit cross–sectional area of the undeformed

specimens. (In Fig. 3.25, P is normalized by Es, the Young modulus of the polyether

polyurethane.) The stretch λ is the height of the specimen after Step 2 normalized by the

height of the specimen before Step 1. For each foam, we show results for three different

values of pressure p0, including p0 = 0. The model is able to reproduce the most notable

trends evinced in the experiments, for example, that the stress–stretch curves are depressed

by an increase in p0.

From the serrations in the computational stress–stretch curves of the three foams of lower

relative density (ρ = 0.03, 0.038, and 0.046) we expect the presence of two configurational

phases in those foams. We show an example of the two-phase stretch fields in Fig. 3.26.

53

3.4 Example of application

In typical applications of EOC foams, a foam is loaded with variously shaped punches.

Consider, for example, a car occupant: his legs, back, and buttocks impinge on the foam

of the seat. In this example, the occupant’s legs and back may be thought of as cylindrical

punches, and the buttocks as spherical punches.

With this example in mind, we carry out a set of experiments and simulations. We

subject specimens of the three foams of relative density < 0.05 to the concurrent actions of

a cylindrical punch and a spherical punch (inset of Fig. 3.27). The size of the specimens is

20 cm × 10 cm × 10 cm. In the experiments we measure, and in the simulations we compute,

the total force on the punches, F , vs. the penetration of the punches, d. The penetration d is

parallel to the rise direction of the foams in both the experiments and the simulations. The

punches are modeled using ABAQUS’s “analytical rigid surfaces” with a friction coefficient

of 0.3 between the punches and the foam.

We compare the simulated and experimental force–penetration curves in Fig. 3.27, and

show plots of the undeformed and the deformed finite-element mesh in Fig. 3.28. Once more,

the model reproduces all of the major trends evinced in the experiments. To our knowledge,

this is the first study in which a unit-cell model of EOC foams has been implemented in

a finite-element code and used to carry out a simulation of direct relevance to a common

application of EOC foams.

3.5 Discussion

In this chapter, we have calibrated the unit-cell model of EOC foams of chapter 2.

To calibrate the model, we have carried out fully nonlinear, three-dimensional finite-

element computational simulations of the experiments of Dai et al., in which flexible polyether-

54

0 15 30 450

3

6

9

d (mm)

F (

N)

ρ = 0.03

ρ = 0.038

ρ = 0.046d

Figure 3.27: Experimental (dashed grey lines) and simulated (solid black lines) force–penetration curves for foams of three relative densities (ρ). Inset: schematic of the experi-ments and of the finite-element mesh used in the simulations; the white punch is cylindrical,the black punch spherical.

(a) (b)

Figure 3.28: Distribution of the local strain in the rise direction. (a) Before and (b) afterpunching.

55

polyurethane foams of five relative densities were tested under five loading conditions (tests 1

to 5). We have compared the experimental stress–stretch curves with the simulated stress–

stretch curves and verified that, with a suitable choice of the values of the parameters of

the model, the simulated stress–stretch curves reproduce all of the major trends seen in the

experiments.

Thus, for example, both the simulated and the experimental stress–stretch curves have

stress plateaus for low-density foams (ρ < 0.05) tested under compression along the rise

direction (test 1). We have found, however, that if the bubbles be excluded from the model,

the simulated stress–stretch curves do not have a stress plateau, not even for ρ = 0.03. We

have therefore concluded that the bubbles play a crucial role at large deformations, at least

under certain loading conditions.

We have shown that our model yields two-phase stretch fields only for low-density foams

(ρ < 0.05), and only for test 1 (compression along the rise direction), test 4 (simple shear

combined with compression along the rise direction), and test 5 (hydrostatic pressure com-

bined with compression along the rise direction). These tests can be characterized as tests

in which there is a component of compression along the rise direction.

Dai et al. carried out experimental measurements of the stretch fields for tests 1 and 4.

For test 1, Dai et al. reached the conclusion that the stretch fields were two-phase stretch

fields (for ρ < 0.05). This conclusion is in accord with our computational simulations of

test 1.

For test 4, Dai et al. reached the conclusion that the stretch fields showed no evidence of

the presence of two configurational phases of the foams, regardless of relative density. This

conclusion appears to be at odds with our computational simulations of test 4.

We have noted, however, that in our simulations of test 4 the high-deformation phase

nucleates on the lateral faces of a specimen and spreads towards the centerline of the specimen

only gradually, as the test proceeds. From this observation, we have argued that it should be

56

hard to detect evidence of a phase transition except close to the lateral faces of a specimen.

As Dai et al. plotted the distribution of local stretch only along the centerline of the specimen,

they might have missed the evidence.

To test this possibility, we have carried out an experiment of our own. We have subjected

a foam specimen of relative density 0.03 to simple shear combined with compression along

the rise direction, measured the stretch fields in the foam via a digital image correlation

technique, and found evidence of a phase transition—but only close to the lateral faces of

the specimen, just as predicted on the basis of our computational simulations. We have

therefore concluded that the two-phase stretch fields of our computational simulations of

test 4 are supported by the experimental evidence.

57

Chapter 4

Punching of EOC foams

4.1 Introduction

Polymeric EOC foams are used in sandwich panels, packaging, car seats, footware, protective

gear such as helmets and knee pads, and upholstered furniture such as sofas and mattresses.

In these and other applications, EOC foams are typically subjected to punching. For ex-

ample, the legs and buttocks of a car occupant act respectively as cylindrical and spherical

punches on the foam of the seat of the car (Fig 4.1). In the case of an accident, the foam is

relied upon to absorb energy at a relatively low compressive stress. Thus, it is of interest to

determine what happens when a punch penetrates an EOC foam, and how the foam reacts

mechanically.

Nevertheless, there has been scarce research on the punching of EOC foams, and in most

of this research punching has been only a means of estimating indentation resilience [69]

or Young’s modulus [70], or of validating hyperelastic constitutive models ([71–74]). The

physical mechanisms that underlie the mechanical response of EOC foams under punching

remain largely unknown.

Here, we use our model of EOC foams as calibrated in chapter 3 to carry out finite-

element simulations of a set of experiments in which polyether polyurethane EOC foams

were subjected to punching using a wedge-shaped punch and a conical punch. We compare

simulated and experimental force–penetration curves and use our computations to test a

theoretical analysis of the experiments.

58

(a) (b )

Figure 4.1: (a) Crash test [68]. (b) Idealized model of crash test. The body of the caroccupant is modeled as a collection of rigid punches that penetrate the car seat.

(a) (b)Rise

Figure 4.2: Experimental setup for punching experiments [75] with (a) a wedge-shaped punchand (b) a conical punch.

4.2 Experiments

The experiments were carried out by Dai [75] on polyether polyurethane EOC foams of

relative densities 0.03, 0.038, and 0.046. These are the same foams that we used to calibrate

our model (chapter 3).

Two types of specimens were punched in the experiments: tall (of height 10 cm) and

short (of height 5 cm). The tall specimens were cubic specimens of size 10 cm × 10 cm ×

10 cm. The short specimens were brick-shaped specimens of size 10 cm × 10 cm × 5 cm. The

specimens were punched along the rise direction, which was aligned with the height of the

specimens.

59

0 0.2 0.4 0.60

100

200

300F

(N

)

d/H

(a)

ρ = 0.03

ρ = 0.038ρ = 0.046

0 0.2 0.4 0.60

100

200

300

ρ = 0.03

ρ = 0.038

ρ = 0.046F (

N)

d/H

(b)

F

d

F

dH H

Figure 4.3: Punching force–penetration curves from experiments with a wedge-shaped punch.(a) Tall (cubic) EOC foam specimens. (b) Short (brick-shaped) EOC foam specimens.

0.0 0.2 0.4 0.60

100

200

0 0.2 0.4 0.6 0.80

100

200

F (

N)

F (

N)

d/H d/H

F

d

F

dH H

ρ = 0.03

ρ = 0.038

ρ = 0.046

ρ = 0.03

ρ = 0.038

ρ = 0.046

(a) (b)

Figure 4.4: Punching force–penetration curves from experiments with a conical punch. (a)Tall (cubic) EOC foam specimens. (b) Short (brick-shaped) EOC foam specimens.

60

10-2 10-1 1001

10

100 1

2

F (

N)

d/H

F

dH

ρ = 0.03

ρ = 0.038

ρ = 0.046

(a) (b)

1

2

F

dH

ρ = 0.03ρ = 0.038

ρ = 0.046

1

10

100

F (

N)

10-1 100

d/H

Figure 4.5: Punching force–penetration curves from experiments with a conical punch, plot-ted in Log-Log scale. (a) Tall (cubic) EOC foam specimens. (b) Short (brick-shaped) EOCfoam specimens.

Two punches were used: a wedge-shaped punch and a conical punch. The punches were

much stiffer than the foams and may be considered rigid. The angle of attack of both was

90◦ for both the wedge-shaped punch and the conical punch (Fig. 4.2).

We show the force-penetration curves from the experiments with the wedge-shaped punch

in Fig. 4.3 and the force-penetration curves from the experiments with the conical punch in

Fig. 4.4. The force-penetration curves display some striking features. For the wedge-shaped

punch the force–penetration curves remain linear up to a penetration of about 40% of the

height of the specimen. For the conical punch the force–penetration curves remain quadratic

up to a penetration of about 40% of the height of the specimen (Fig. 4.5). At a penetration of

about 40% of the height of the specimens, there is a sudden change in the force–penetration

curves: for the tall specimens, there is a momentary loss of stiffness; for the short specimens,

there is a sizable increase in stiffness. To explain these curious experimental results, Dai [75]

proposed a simple theory, which we repeat here for completeness.

61

H - High deformation phase

L - Low deformation phase

InterfaceL

Hd

R d1 R1

FF

R

(a) (b)

Figure 4.6: Schematic of a punching experiment with a wedge-shaped punch. (a) Thesharp interface between the high-deformation configurational phase and the low-deformationconfigurational phase. (b) The self-similar regime.

4.3 Theory

4.3.1 The self-similar regime

Wedge-shaped punch. Consider a foam specimen as it is being penetrated by a wedge-

shaped punch (Fig. 4.6). Close to the tip of the punch, the foam is in the high-deformation

phase, and the foam cells have already snapped; far from the tip, the foam is in the low-

deformation phase, and the foam cells are yet to snap. There cannot be a smooth transition

between the configurational phases. Instead, there is a sharp interface. For simplicity, we

assume that the sharp interface is a semi-cylinder of radius R, as shown in Fig. 4.6a.

A wedge-shaped punch lacks a characteristic length, and during a punching experiment

the only prevailing length scale is the penetration d of the punch. Thus, the radius R of the

sharp interface must scale with d. As the penetration increases during the experiment, there

is at first a regime in which the radius of the sharp interface increases in direct proportion

to the penetration, as shown in Fig. 4.6b. Dai called this the self-similar regime, because in

this regime the sharp interface remains always a semi-cylinder.

62

0.2 0.4 0.6

0.4

0.8

1.2

1.6

0.0 0.2 0.4 0.60.0

0.4

0.8

1.2

1.6

0.00.0

d/H d/H

F/Fp

F/Fp

F

d

F

dH

H

(a) (b)

Figure 4.7: Scaled force–penetration curves for (a) tall (cubic) and (b) short (brick-shaped)EOC foam specimens penetrated by a wedge-shaped punch.

Any two foam cells that happen to be immediately on each side of the sharp interface

can only be in equilibrium with one another under the Maxwell stress. Thus the interface

stress is the Maxwell stress PM , which is strictly a property of the foam and does not depend

on the radius of the interface.

Now, the surface area of the interface is proportional to R, and from equilibrium it

must be that F ∝ PMR. But R is proportional to d in the self-similar regime, and PM is

independent of R; it follows that F ∝ PMd in the self-similar regime. Thus, we predict that

for a wedge-shaped punch the force should be proportional to the penetration in the self-

similar regime. As we have seen, this prediction is in accord with the experiments. Further,

we predict that the force–penetration curves can be properly scaled so that the scaled force–

penetration curves of all the experiments will collapse on a single curve in the self-similar

regime. To test this prediction, for each punching experiment we scale the punching force by

the plateau force FM = PM×Ao, where Ao is the cross-sectional area of the undeformed foam

specimen and PM is the plateau stress measured under compression along the rise direction.

63

F

L

H

Interface

R

H - High deformation phase

L - Low deformation phase

Figure 4.8: Schematic of punching experiments with a conical punch. The high-deformationphase is separated from the low-deformation phase by a semi-spherical sharp interface.

The scaled force-penetration curves fall one on top of the other in the self-similar regime

(Fig. 4.7).

Conical punch. Consider a foam specimen as it is being penetrated by a conical punch.

The sharp interface is now a semi-sphere (Fig. 4.8) of radius R and surface area proportional

to R2. A conical punch lacks a characteristic length, and during a punching experiment the

only prevailing length scale is the penetration d of the punch. It follows that F ∝ PMd2

in the self-similar regime. Thus, we predict that for a conical punch the force should be

proportional to the square of the penetration in the self-similar regime. As we have seen,

this prediction is in accord with the experiments. Further, the force–penetration curves can

be properly scaled so that the scaled force–penetration curves of all the experiments will

collapse on a single curve in the self-similar regime (Fig. 4.9).

4.3.2 Beyond the self-similar regime.

The self-similar regime ends when the sharp interface reaches one of the boundaries of the

specimen. Depending on the aspect ratio of the specimen, the sharp interface will first

64

0.0 0.2 0.4 0.6 0.80.0

0.4

0.8

1.2

0.0 0.2 0.4 0.60.0

0.4

0.8

1.2

F/Fp

d/H d/H

F

d

F

dH H

F/Fp

(a) (b)

Figure 4.9: Scaled force–penetration curves for (a) tall (cubic) and (b) short (brick-shaped)EOC foam specimens penetrated by a conical punch.

reach either the lower boundary or a lateral boundary. If the lower, fixed (and therefore

stiff) boundary is reached first, the mechanical response will display a sizable increase in

stiffness. This prediction is in accord with the experimental results for the short (brick-

shaped) specimens (Fig. 4.3b and Fig. 4.4b). If a lateral, unsupported (and therefore soft)

boundary is reached first, the mechanical response will display a momentary loss of stiffness.

This prediction is in accord with the experimental results for the tall (cubic) specimens

(Fig. 4.3a and Fig. 4.4a).

4.4 Computational simulations

4.4.1 Wedge-shaped punch

We use two-dimensional plane-strain finite elements for the foam specimens. The finite

element meshes are shown in Fig. 4.10. We model the punch and the loading plate using

ABAQUS’ “analytical rigid surface” [66]. The frictional coefficient is 0.3 between the punch

and the foam specimen and also between the foam specimen and the loading plate.

65

d

Fd

F

(a) (b)

Punch

Symmetry

plane

Punch

Figure 4.10: Finite element meshes used in the computational simulations with a wedge-shaped punch. (a) Tall (cubic) specimens and (b) short (brick-shaped) specimens.

None of the computational results to be discussed here have been fitted to the experi-

mental results.

Tall specimens. The force–penetration curves are shown in Fig. 4.11. The computa-

tional results are in good accord with the experimental results. In particular, the punching

force increases linearly with the penetration up to a penetration of about 40% of the height

of the specimen. Then, the response becomes nonlinear, and there is a momentary loss of

stiffness.

To verify the existence of a sharp interface between the low-deformation configurational

phase and the high-deformation configurational phase, we plot the vertical component of the

local stretch over the domain of the specimen (in the undeformed geometry). The result

is shown in Fig. 4.12, where R denotes the vertical distance (measured in the undeformed

geometry) from the tip of the punch to the sharp interface.

The sharp interface sweeps through the specimen as the penetration is increased (Fig. 4.13a).

Further, for as long as the mechanical response is linear, R remains proportional to d

(Fig. 4.13b), in accord with the expected behavior in the self-similar regime.

66

0 0.2 0.40

100

200

300

d/H

F (

N)

Simulated

Experimental

ρ = 0.03

ρ = 0.038

ρ = 0.046d

F

Punch

H

Figure 4.11: Force–penetration curves from simulations with a wedge-shaped punch and tallEOC foam specimens of relative density ρ. As we have seen in chapter 3, the oscillations ofthe force–penetration curves must be ascribed to the discreteness of the finite element mesh.In fact, the amplitude of these oscillations will lessen if the finite element mesh is made moredense.

When the penetration attains a value of about 40% of the height of the specimen, the

sharp interface reaches the lateral, unsupported boundary of the specimen (Fig. 4.14). This

event signals the breakdown of the self-similar regime, and it takes place concurrently with

a momentary loss of stiffness (Fig. 4.14). These computational results are in accord with the

theoretical predictions of Dai.

Short specimens. The force–penetration curves are shown in Fig. 4.15. The computa-

tional results are in good accord with the experimental results. In particular, the punching

force increases linearly with the penetration up to a penetration of about 50% of the height

of the specimen. Then, the response becomes nonlinear, and there is a sizable increase in

stiffness.

When the penetration attains a value of about 50% of the specimen height the sharp

67

Y

(a)

1.0

40

0

Y (

mm

)

λ

(b)

High

deformation

Low

deformation

Symmetry

plane

R80

0.7 0.4

0.45

1.00

λ

Figure 4.12: (a) Contour plot of the local stretch λ for a tall EOC foam specimen of relativedensity ρ = 0.03 under a wedge-shaped punch. The penetration d = 29.1 mm. The plot isin the undeformed geometry. (b) Plot of the local stretch λ along a line of nodes below thetip of the punch. The discontinuity in λ marks the sharp interface.

(a)

R

0

40

80

0.8 0.6 0.41.0

λ

Y (

mm

)

10 20 3010

30

50

70

d (mm)

R (

mm

)

Linear fit

(b)

40

Figure 4.13: (a) Plots of λ along a line of nodes below the tip of the punch for d = 15.0 mm,17.4 mm, 21.1 mm, 24.9 mm, 27.1 mm, 31.1 mm ,33.0 mm and 37.2 mm. (b) Plot of R vs. d.

68

0 0.1 0.2 0.3 0.4 0.50

40

80

120

160

200

d/H

F

(N)

A

B

C

DE

F

G

d

F

λ

0.45

1.00

H

Figure 4.14: Simulated force–penetration curve and attendant contour plots of the stretch λfor a tall EOC foam specimen of relative density 0.03 under a wedge-shaped punch. PointsA through F correspond to d = 10.0 mm, 16.3 mm, 24.1 mm, 32.2 mm, 38.0 mm, 41.0 mmand 51.0 mm, respectively.

interface reaches the lower, fixed boundary of the specimen (Fig. 4.16). This event signals

the breakdown of the self-similar regime, and it takes place concurrently with a sizable

increase in stiffness (Fig. 4.16). These computational results are in accord with the theoretical

predictions of Dai.

Scaled results.

We scale the punching force by the Maxwell force FM obtained from simulations of

EOC foams under compression along the rise direction. (These simulations were discussed

in chapter 3, section 3.3.1). We compare the scaled simulated curves against the scaled

experimental curves in Fig. 4.17. The scaled force–penetration curves collapse on a single

curve in the self-similar regime, in accord with theoretical predictions, and compare well

with the scaled experimental curves.

69

0 0.2 0.4 0.60

50

100

150

200

250

d/H

F (

N)

Simulated

Experimentalρ = 0.046

ρ = 0.038

ρ = 0.030

d

F

H

Figure 4.15: Force–penetration curves for short EOC foam specimens of relative density ρunder a wedge-shaped punch.

4.4.2 Conical punch

We model a quarter of the foam specimen bound by two symmetry planes using three-

dimensional finite elements. The finite element meshes are shown in Fig. 4.18. We model the

punch and the loading plate using ABAQUS’ “analytical rigid surface” [66]. The frictional

coefficient is 0.3 between the punch and the foam specimen and also between the foam

specimen and the loading plate.

None of the computational results to be discussed here have been fitted to the experi-

mental results.

Tall specimens. The force–penetration curves are shown in Fig. 4.19a. We plot the

same curves in log-log scale in Fig. 4.19b. The computational results are in good accord with

the experimental results. In particular, the punching force increases quadratically with the

penetration up to a penetration of about 45% of the height of the specimen. Then, there is

70

0 0.2 0.4 0.6

50

100

150

d/H

F

(N)

λ

0.45

1.00

d

F

A

B

CD

E

F

H

Figure 4.16: Simulated force–penetration curve and attendant contour plots of the stretch λfor a short EOC foam specimen of relative density 0.03 under a wedge-shaped punch. PointsA through F correspond to d = 10.3 mm, 13.5 mm, 16.3 mm, 20.0 mm, 24.3 mm, and 28.1 mmrespectively.

0.0 0.2 0.4 0.60.0

0.4

0.8

1.2

1.6

d/H

F/FM

0.0 0.2 0.4 0.60.0

0.4

0.8

1.2

d/H

F/FM

Simulated

Experimental

d

F

Simulated

Experimental

d

F

H

H

(a) (b)

Figure 4.17: Scaled force–penetration curves for (a) tall and (b) short specimen under awedge-shaped punch. The force is normalized by the plateau force FM

71

Symmetry

planes

Fixed base

(a) (b)

Figure 4.18: Finite element meshes used in the computational simulations with a conicalpunch. (a) Tall (cubic) specimens and (b) short (brick-shaped) specimens.

0 0.2 0.4 0.60

100

200

300

F

d

d/H

F (

N)

Experimental

Simulated

ρ = 0.030

ρ = 0.046

ρ = 0.038

10-2 10-1

101

102

F

d

Experimental

Simulated

1

d/H

F (

N)

2

1

(a) (b)

HH

Figure 4.19: Force–penetration curves from simulations with a conical punch and tall EOCfoam specimens of relative density ρ, in (a) linear-linear scale and (b) log-log scale.

72

(a)

R

0

40

80

0.8 0.6 0.41.0

λ

Y (

mm

)

d (mm)

R (

mm

)

Linear fit

(b)

10 30 5010

20

30

40

Figure 4.20: (a) Plots of λ along a line of nodes below the tip of the punch for d = 15.0 mm,20.0 mm, 25.1 mm, 30.2 mm, 40.1 mm, 45.0 mm and 50.0 mm. (b) Plot of R vs. d.

a momentary loss of stiffness.

We trace the interface radius R as the sharp interface sweeps through the specimen with

increasing penetration and find that as long as the force increases quadratically with the

penetration, R increases in proportion to d (Fig. 4.20).

When the penetration attains a value of about 43% of the height of the specimen, the

sharp interface reaches the lateral, unsupported boundary of the specimen (Fig. 4.21). This

event signals the breakdown of the self-similar regime, and it takes place concurrently with

a momentary loss of stiffness (Fig. 4.21) in accord with theoretical predictions of Dai.

Short specimens. We compare the simulated force–penetration curves to experiments

in Fig. 4.22. The simulated results compare well against experiments. In particular, the

punching force increases quadratically with the penetration up to a penetration of about

60% of the height of the specimen. Then, the response becomes nonlinear, and there is a

sizable increase in stiffness.

When the penetration attains a value of about 60% of the height of the specimen, the

sharp interface reaches the lower, fixed boundary of the specimen (Fig. 4.23). This event

73

0

40

80

120

160

F (N

)

200

0 0.1 0.2 0.3 0.4 0.5 0.6

d/H

A

B

C

D

E F

G

1.00

0.36

λ

F

dH

Figure 4.21: Force–penetration curve and the attendant contour plots of the stretch λ for acubic EOC foam specimen of relative density 0.03 under a conical punch. Points A throughF correspond to d = 15.14 mm, 22.0 mm, 29.1 mm, 36.1 mm, 43.5 mm, 48.6 mm and 57.4 mm,respectively.

signals the breakdown of the self-similar regime, and it takes place concurrently with a sizable

increase in stiffness (Fig. 4.23). These results are in accord with theoretical predictions of

Dai.

Scaled results.

We compare the scaled simulated force–penetration curves and experimental force–penetration

curves in Fig. 4.24. The scaled force-penetration curves obtained from the simulations for

low-density specimens collapse on top of each other as predicted by Dai and the scaled

simulated curves compare well with experiments.

74

40

80

120

160

200

0 0.2 0.4 0.60

d/H0.8

ρ = 0.030

ρ = 0.046

ρ = 0.038

F

d

Experimental

SimulatedF

(N

)

(b)(a)

H

101

102

1

d/H

F (

N)

2

1

F

d

Experimental

Simulated

H

10-1 100

Figure 4.22: Force–penetration curves from simulations with a conical punch and short EOCfoam specimens of relative density ρ, in (a) linear-linear scale and (b) log-log scale.

F

(N)

d/H0 0.2 0.4 0.6 0.80

40

80

120

160

AB

C

D

E

F

1.00

0.47

λ

d H

F

Figure 4.23: Simulated force–penetration curve and attendant contour plots of the stretchλ for a short EOC foam specimen of relative density 0.03 under a conical punch. Points Athrough F correspond to d = 12.2 mm, 17.5 mm, 20.7 mm, 25.6 mm, 29.3 mm, and 32.4 mmrespectively.

75

0.0 0.2 0.4 0.6 0.80.0

0.4

0.8

1..2

Simulated

Experimental

0.0 0.2. 0.4 0.60.0

0.4

0.8

1.2

1.6

Simulated

Experimental

d/H

F/FM

d/H

F/FM

(a) (b)

F

dH

d H

F

Figure 4.24: Scaled force–penetration curves for (a) tall and (b) tall specimens under aconical punch. The force is normalized by the plateau force FM

4.5 Discussion

In this chapter, we have assessed the capacity of our unit-cell model of EOC foams to yield

reliable predictions of the mechanical response of elastic polyether polyurethane foams in

problems where the geometry is complex, the stress fields are highly spatially heterogeneous,

the deformations are very large, and the use of finite-element simulations is indispensable.

We have chosen to focus on punching, which is utterly common in applications of polyether

polyurethane foams.

We have run finite-element simulations of the numerous experiments carried out by Dai.

In these experiments, tall and short specimens of foams of three different relative densities

were penetrated by a wedge-shaped punch and a conical punch. For each experiment we

have run a simulation and found the simulated force–penetration curve to be in good accord

with the corresponding experimental force–penetration curve. The values of the parameters

of the model were fixed a priori and equal to those of the calibration that we carried out in

chapter 3, a calibration that did not involve any experimental results on punching.

76

Wedge-shaped punches and conical punches lack an intrinsic characteristic length, a fact

that was exploited by Dai to formulate a simple theory to explain some curious features of

the force–penetration curves. Dai was able to use this simple theory to explain the linear

force–penetration curves and the quadratic force–penetration curves that he had measured in

his experiments with a wedge-shaped punch and a conical punch, respectively. In addition,

Dai reasoned that the break down of the linear and quadratic force–penetration curves must

occur when the expanding sharp interface reaches one of the boundaries of the specimen and

cannot continue to expand self-similarly. Our computational simulations have allowed us to

test several aspects of Dai’s theory that remained inaccessible to experimental observation.

For example, we have been able to document a sharp interface expanding self-similarly under

both a wedge-shaped punch and a conical punch, and to verify that the breakdown of the

linear or quadratic relation between force and penetration occurs just as the sharp interface

reaches one of the faces of the specimen.

Our results on the punching of elastic polyether polyurethane foams serve to underscore

the importance of two-phase stretch fields in the mechanics of EOC foams. A number of

models of EOC foams might account for the stress–stretch curves of tests 1 to 5 of Dai et

al., but the force–penetration curves appear to be so inextricably linked to the prevalence

of two-phase stretch fields that no model is likely account for these force–penetration curves

unless it allows for the occurrence of phase transitions.

77

Chapter 5

Summary, discussion and concludingremarks

We have formulated a new unit-cell model of elastic open-cell foams (EOC foams). In this

model the microstructure of a foam consists of a three-dimensional, regular skeleton (or

network of bars) and a set of thin-walled bubbles. The skeleton is made up of four-bar

tetrahedra arranged in the hexagonal diamond structure Lonsdaleite. A single bubble is

fitted tightly within each cavity of the skeleton.

The thin-walled bubbles substitute for the membranes that span the openings of the

skeleton as seen in polyether polyurethane EOC foams under a scanning-electron microscope.

It has long been known that these membranes contribute significantly to the elastic modulus

of a foam, and as part of this project we have shown that the membranes have a marked

affect on the mechanical response at large deformations (that is, the type of deformation

that prevails under service conditions in most applications of EOC foams). And yet, to our

knowledge, the membranes have so far been ignored in models of EOC foams.

The model can be fully described using 9 parameters. Each one of these parameters has a

clear geometrical or mechanical significance, and the value of each parameter may be readily

estimated for any given foam. Our experience with a commercial set of of elastic polyether

polyurethane foams has been that 8 of the parameters of the model may be chosen and

ascribed to the entire set of foams. The remaining parameter is the relative density, which

is unique to each foam in the set.

As in common applications of EOC foams the deformations are large and the mechanical

response of a foam is dominated by nonlinear geometric effects, we have used a linear elastic

78

constitutive relation for the skeleton and bubbles. Thus, the mechanical parameters include

the Young’s modulus and the Poisson’s ratio of the skeleton and the bubbles.

All of the geometrical parameters of the model are dimensionless, a reflection of the fact,

which we have proven theoretically, that the mechanical response predicted using the model

is invariant to isomorphic changes in the dimensions of the microstructure of the foam. Thus,

the model may be said to be scale invariant.

Where a unit cell is cut off from the microstructure and subjected to a given displacement

gradient, the skeleton and the set of bubbles are coupled in parallel, and contribute additively

to the strain energy density, and therefore to the stress. We have adopted the Lagrangian

purview and carried out all calculations in the undeformed configuration.

We have implemented the model as a user-defined material subroutine in the general-

purpose finite-element code ABAQUS. To our knowledge, this is the first unit-cell model of

solid foams to be implemented in a general-purpose finite-element code.

To calibrate the model, we have carried out fully nonlinear, three-dimensional finite-

element computational simulations of the experiments of Dai et al. In these experiments, a

set of polyether-polyurethane EOC foams was tested under five loading conditions, namely

compression along the rise direction (test 1), compression along a transverse direction (test 2),

tension along the rise direction (test 3), simple shear combined with compression along the

rise direction (Test 4), and hydrostatic pressure combined with compression along the rise

direction (test 5). The set of polyether polyurethane EOC foams consisted of five foams and

covered the range of commercially available relative densities, from 0.03 to 0.086.

We have compared the experimental stress–stretch curves with the simulated stress–

stretch curves and verified that, with a suitable choice of the values of the parameters of

the model, the simulated stress–stretch curves reproduce all of the major trends seen in the

experiments. Thus we have been able to verify, for example, that both the simulated and

the experimental stress–stretch curves have stress plateaus for low-density foams (ρ < 0.05)

79

tested under compression along the rise direction (test 1). As the relative density of the

foam increases, the stress plateau becomes less broad, and the plateau stress—that is, the

stress level of the stress plateau—increases; for relative densities in excess of about 0.05,

a stress plateau is not present, and the stress remains a monotonic function of the stretch

throughout a test. These trends are shared by the simulated stress–stretch curves and the

experimental stress–stretch curves for compression along the rise direction.

Hardly any model of EOC foams has ever been calibrated on the basis of such an extensive

set of experimental stress–stretch curves. And yet, stress–stretch curves are but one aspect

of the mechanical behavior of EPP foams, and widely disparate models might yield similar

stress–stretch curves. To assess thoroughly the goodness of our model, we have taken into

consideration the nature of the stretch fields that accompanied any given stress–stretch curve.

Of special interest to us have been the two-phase stretch fields associated with a phase

transition. These two-phase stretch fields consist of mixtures of two configurational phases

of the foam, a high-deformation phase and a low-deformation phase.

We have found that for any given simulation the nature of the stretch fields can be

ascertained by inspecting the stress–stretch curve. Where the simulated stress–stretch curve

displays serrations, the attendant stretch fields are invariably two-phase stretch fields.

We have shown that these serrations reflect an increase in the volume fraction of the

high-deformation phase at the expense of the volume fraction of the low-deformation phase;

with space partitioned in discrete finite elements, the volume fractions can alter only in

discrete jumps, and to each jump there corresponds a serration in the stress–stretch curve.

Although the serrations are mere artifacts of the finite-element discretization, they signalize

the presence of two configurational phases in the foam, and may be conveniently used to

diagnose the occurrence of a phase transition.

From our simulations of the experiments of Dai et al., we have concluded that the model

yields two-phase stretch fields only for low-density foams (ρ < 0.05), and only for compres-

80

sion along the rise direction (test 1), simple shear combined with compression along the

rise direction (test 4), and hydrostatic pressure combined with compression along the rise

direction (test 5). These tests 1, 4, and 5 can be characterized as tests in which there is a

component of compression along the rise direction.

Dai et al. carried out experimental measurements of the stretch fields in test 1, and

we have found the experimental measurements of Dai et al. are well reproduced by our

computational simulations. In both experiments and computations, the high-deformation

phase nucleates on bands that span the entire width of a specimen, from a free boundary to

the centerline of the specimen to the other free boundary. Further, in test 1 the two-phase

stretch fields occur always in conjunction with a stress plateau.

We have shown that a plateau stress may be interpreted in light of our model as the

Maxwell stress associated with the process whereby a cell of a low-density foam snaps dynam-

ically from one stable equilibrium configuration (which corresponds to the low-deformation

phase) to a second stable equilibrium configuration of differing stretch (which corresponds

to the high-deformation phase). For test 1, the finite-element simulations automatically give

stress–stretch curves with stress plateaus, and there has been no need for us to calculate

Maxwell stresses by imposing the Erdmann equilibrium condition. The stress plateaus are

straddled by serrations (discussed above), and we have verified that the amplitude of the

serrations, as well as their wavelength, scales with the size of the finite elements.

We have demonstrated that the simulated stress–stretch curves of test 1 lose their stress

plateaus if the bubbles be excluded from the model. To explain the effect of excluding the

bubbles, we have shown that the bubbles act as hoops which provide lateral confinement to

the cells of a foam. Without the lateral confinement provided by the bubbles, the cells do

not snap, the foam does not undergo a phase transition, and the simulated stress–stretch

curves remain monotonic, even for relative densities as low as 0.03. As the experimental

stress–stretch curves do have a stress plateau for relative densities less that 0.05, we have

81

been able to conclude that the bubbles play a crucial role at large deformations, at least

under certain loading conditions.

Dai et al. also carried out experimental measurements of the stretch fields in test 4 and

reached the conclusion that the stretch fields showed no evidence of the presence of two

configurational phases of the foams, regardless of relative density. This conclusion appears

to be at odds with our computational simulations of test 4.

We have noted, however, that in our simulations of test 4 the high-deformation phase

nucleates on the lateral faces of a specimen and spreads towards the centerline of the specimen

only gradually, as the test proceeds. From this observation, we have argued that it would be

hard to detect evidence of a phase transition except close to the lateral faces of a specimen.

As Dai et al. plotted the distribution of local stretch only along the centerline of the specimen,

they might have missed the evidence.

To test this possibility, we have carried out an experiment of our own. We have subjected

a foam specimen of relative density 0.03 to simple shear combined with compression along

the rise direction, measured the stretch fields in the foam via a digital image correlation

technique, and found evidence of a phase transition—but only close to the lateral faces of

the specimen, just as we expected on the basis of our computational simulations. All of the

major features of the stretch fields measured in the experiment have turned out to be well

reproduced in the computational simulations.

We have therefore concluded that the presence of two configurational phase of the foam

in the computational simulations of test 4 is supported by the experimental evidence.

In the last part of this thesis, we have assessed the capacity of our model of EOC foams

to yield reliable predictions of the mechanical response of elastic polyether polyurethane

foams in problems where the geometry is complex, the stress fields are highly spatially

heterogeneous, the deformations are very large, and the use of finite-element simulations is

indispensable. We have chosen to focus on punching, which is utterly common in applications

82

of polyether polyurethane foams.

We have run finite-element simulations of the numerous punching experiments carried

out by Dai. In these experiments, tall and short specimens of foams of three different

relative densities were penetrated by a wedge-shaped punch and a conical punch. For each

experiment we have run a simulation and found the simulated force–penetration curve to be

in good accord with the corresponding experimental force–penetration curve. The values of

the parameters of the model were fixed a priori and equal to those of the calibration that

we had carried out previously, a calibration that did not involve any experimental results on

punching.

Wedge-shaped punches and conical punches lack an intrinsic characteristic length, a fact

that was exploited by Dai to formulate a simple theory to explain some curious features of

the force–penetration curves:

For a wedge-shaped punch, the force–penetration curves remain linear up to a penetration

of about 50% of the height of the specimens, a result that seems hard to reconcile with the

strong geometric and constitutive nonlinearities that prevail near the tip of a wedge-shape

punch. Further, the form in which the linear relation between force and penetration breaks

down depends on the geometry of the specimen: if the specimen be tall, there is a sudden

lessening in stiffness, and the force–penetration curve becomes momentarily almost flat; if

the specimen be short, there is a sudden increase in stiffness, and the force–penetration curve

shoots up steeply into a nonlinear, increasingly hardening stage.

For a conical punch, the force–penetration curves remain quadratic up to a penetration

of about 50% of the height of the specimens. The quadratic relation between force and

penetration breaks down with a sudden lessening in stiffness for tall specimens and a sudden

increase in stiffness for short specimens—the same trends as for a wedge-shaped punch.

In his theory, Dai associated the linear or quadratic relation between force and penetration

with the expansion of a sharp interface that circles the tip of the punch and marks the

83

boundary between two regions in the specimen: an inner region in which there obtains the

high-deformation configurational phase of the foam, and an outer region in which there

obtains the low-deformation configurational phase of the foam. As the two configurational

phases coexist along the sharp interface, the stress along the sharp interface must equal the

Maxwell stress of the foam. For a wedge-shaped punch, the sharp interface is cylindrical; for

a conical punch, the sharp interface is spherical.

Dai was able to use this simple theory to explain the linear force–penetration curves

and the quadratic force–penetration curves that he had measured in his experiments with

a wedge-shaped punch and a conical punch, respectively. In addition, Dai reasoned that

the break down of the linear and quadratic force–penetration curves must occur when the

expanding sharp interface reaches one of the boundaries of the specimen and cannot continue

to expand self-similarly. In a tall specimen the lateral faces of the specimen are reached first.

As the lateral faces are free boundaries, Dai predicted that for a tall specimen there should

be a sudden softening of the mechanical response. In contrast, in a short specimen the lower

face of the specimen is reached first. As the lower face lies on a loading plate and is supported

by it, Dai predicted that for a short specimen there should be a sudden hardening of the

mechanical response. Thus, the predictions of Dai were in agreement with his experimental

results.

Our computational simulations have allowed us to test several aspects of Dai’s theory that

remained inaccessible to experimental observation. Thus, for example, we have been able to

document a sharp interface expanding self-similarly under both a wedge-shaped punch and a

conical punch. From the computational simulations, the breakdown of the linear or quadratic

relation between force and penetration may be said to occur just as the sharp interface

reaches one of the faces of the specimen, but we have found a subtle difference between short

specimens and tall specimens. In a short specimen the sharp interface expands self-similarly

until it makes contact with the lower surface. This is the exact scenario envisioned by Dai

84

in his theory. In a tall specimen the sharp interface ceases to expand self-similarly just

before it makes contact with the lateral faces of the specimen, whereupon a narrow band of

localized deformation shoots out from the sharp interface to emerge on the lateral faces of

the specimen. This scenario differs from the one envisioned by Dai in his theory but is still

consistent with all of the experimental measurements.

Our results on punching have very broad implications that go beyond a narrow interest

in punching tests. In fact, our results serve to underscore the importance of two-phase

stretch fields in the mechanics of EOC foams. The force–penetration curves of the punching

experiments of Dai display a number of striking features, and we have been able to match

each one of these features to the presence of two configurational phases of the foam. Feature

by feature the match is so specific and intricate that we have been lead to conclude that

the force–penetration curves could be taken by themselves as proof of the occurrence of

phase transitions in the foams, even in the absence of any direct experimental evidence on

the stretch fields. A number of models of EOC foams might account for the stress–stretch

curves of tests 1 to 5 of Dai et al., but the force–penetration curves are inextricably linked

to the prevalence of two-phase stretch fields, and it is likely that no model can account for

these curves unless it allows for the occurrence of phase transitions.

To the best of our knowledge, we are the first to have compared experimental and the-

oretical stress-stretch curves for a varied set of loading conditions and the full range of

commercially available foam densities. This practice must become widespread if we are to

provide a reliable simulation tool for engineering applications. In this respect, our model

would benefit from further work to improve, for example, the predictions at small deforma-

tions through the introduction of a suitable nonlinear constitutive relation for the polyether

polyurethane.

In closing, we would like to highlight two practices which must become widespread if mi-

cromechanical models of EOC foams are to provide a reliable simulation tool for engineering

85

applications. First, models should be calibrated not only by comparing experimental and

theoretical stress–stretch curves for a varied set of loading conditions and the full range of

commercially available foam densities, but also by taking into account the relation between

the stress-stretch curve measured in an experiment and the nature of the attendant stretch

fields. And yet, as we have noted, the stretch fields have largely been ignored in experimental

work on EOC foams, and for most models a few, if any, complete stress–stretch curves have

been compared with experimental measurements, for the most part on compression along

the rise direction.

Second, a model should be implemented in a general purpose finite-elements code. In this

respect, we have noted that it would have been impossible for us to judge the performance

of our model if we had not implemented the model in a general purpose finite-element

code. This point can hardly be missed in connection with our work on punching, where the

stress fields are highly spatially heterogeneous. But even in the case of a test where the

stress field is far less complex than in a punching test, and where in principle it would be

possible to compute a force–displacement curve by applying a suitably chosen deformation

gradient to a single unit cell, such a computation would invariably couple the presence of

two configurational phases of the foam to a force–displacement curve with a force plateau by

imposing the Erdmann equilibrium condition. And yet, as we have shown for test 4 of Dai

et al., for example, the presence of two configurational phases of the foam may be actually

coupled to a monotonic force–displacement curve.

86

Appendix A

UMAT implementation

We implement our unit-cell model as a constitutive relationship in the commercial FE code

ABAQUS through its user material subroutine UMAT. The subroutine is called at each

integration point in the finite element mesh. The deformation gradient tensor F at an

integration point is provided as input to the subroutine and is used to update the spatial

stress tensor σ and spatial elasticity tensor c to their values at the end of the increment

for which the subroutine is called. The consistent spatial elasticity tensor for total-form

constitutive relations is defined in ABAQUS as c = ∂τ/∂d where τ is the Kirchoff stress

tensor and d is the rate of deformation tensor [66]. Here we derive expressions to calculate

σ and c for the unit-cell model given the deformation gradient F.

A.1 Stress and elasticity tensor for the skeleton.

In the discussion to follow we use the following notation: the displacement vector at a tip k

is uk and the corresponding force vector is f k. Displacements and forces at interior nodes

are stored in vectors v and R respectively. We move the bar tips in accord with the local

deformation gradient F provided as input to the UMAT subroutine (chapter 2, section 2.3.1),

uki = FijXkj −Xk

j (A.1)

where X k is the position vector of tip k in the undeformed configuration. To determine

87

the equilibrium configuration of the skeleton subject to a given F we minimize the strain

energy density of the skeleton W s(F, v) with respect to the interior displacements v ,

∂W s

∂v

∣∣∣∣F

= R = 0 (A.2)

We solve the nonlinear system of equations (A.2) using a Newton-Raphson scheme to

obtain v . We calculate the first Piola-Kirchoff stress tensor Ps by taking derivatives of the

strain energy density W s at equilibrium,

P sij =

∂W s

∂Fij

∣∣∣∣v

=

Np∑p=1

∂W s

∂upm

∂upm∂Fij

=1

Vo

Np∑p=1

fpm∂upm∂Fij

(A.3)

where Np is the number of bar tips. From (A.1), ∂upm/∂Fij = δimXpj and the first Piola-

Kirchoff stress tensor Ps can be expressed as,

P sij =

1

Vo

Np∑p=1

fpi Xpj (A.4)

The corresponding cauchy stress tensor follows from σs = (1/J)FPsT , where J is the

jacobian of the deformation gradient tensor, thus,

σsij =1

V

Np∑p=1

fpi xpj (A.5)

where V is the volume of the deformed unit cell and x k is the position vector of tip k in the

deformed configuration.

The lagrangian elasticity tensor for the skeleton, Cs, is obtained by taking derivatives of

the stress tensor Ps at equilibrium,

Csijkl =

dP sij

dFkl=∂P s

ij

∂Fkl

∣∣∣∣v

+∂P s

ij

∂vm

∣∣∣∣F

dvmdFkl

(A.6)

88

To evaluate (A.6) we need dv/dF which we obtain from the minimization condition (A.2)

as,

dvmdFkl

= −(∂Rn

∂vm

)−1∂Rn

∂Fkl(A.7)

substituting (A.7) in (A.6) we get,

Csijkl =

∂P sij

∂Fkl−∂P s

ij

∂vm

(∂Rn

∂vm

)−1∂Rn

∂Fkl(A.8)

Using equations (A.2) and (A.3) we can express Ps and R as derivatives of W s and

obtain,

Csijkl =

[∂2W s

∂Fij∂Fkl− ∂2W s

∂Fij∂vm

(∂2W s

∂v∂v

)−1

mn

∂2W s

∂vn∂Fkl

](A.9)

To obtain derivatives of the strain energy density W s we turn to the equilibrium equations

for the skeleton which can be expressed in matrix form as,

Kvv Kvu

Kuv Kuu

v

u

=

R

f

where Kvv,Kvu,Kuv and Kuu are components of the global stiffness matrix for the

skeleton, u is the vector of tip displacements and f is the vector of forces at the tips. We

use equation (A.1) to correlate derivatives of W s to the stiffness matrices as follows,

∂2W s

∂Fij∂Fkl=

Np∑p=1

Np∑q=1

∂2W s

∂upi ∂uqk

XpjX

ql =

1

Vo

Np∑p=1

Np∑q=1

XpjKuu

pqikX

ql (A.10)

89

∂2W s

∂Fij∂vm=

Np∑p=1

Xpj

∂2W s

∂upi ∂vm=

1

Vo

Np∑p=1

XpjKuv

pim (A.11)

∂2W s

∂vm∂vn=

1

VoKvvmn (A.12)

∂2W s

∂vn∂Fkl=

Np∑q=1

Xql

∂2W s

∂vn∂uqk

=

Nq∑q=1

1

VoKvuqnkX

ql (A.13)

We substitute equations (A.10) to (A.13) in (A.9) to obtain an expression for the la-

grangian elasticity tensor Cs in terms of the stiffness matrices of the skeleton,

Csijkl =

1

Vo

[ Np∑p=1

Np∑q=1

Xpj

(Kuupqik −Kuv

pimKvv

−1mnKvu

qnk

)Xql

](A.14)

Since R = 0 at equilibrium, we can define a condensed stiffness matrix K∗ such that,

K∗ = Kuu−KuvKvv−1Kvu (A.15)

Cs can now be expressed in a convenient form,

Csijkl =

1

Vo

[ Np∑p=1

Np∑q=1

XpjK∗pqik X

ql

](A.16)

Finally, the spatial elasticity tensor cs is obtained by pushing forward relation (A.16) to

the current configuration [76],

csijkl = FjaFlbCsiakb − δikτjl =

1

Vo

[ Np∑p=1

Np∑q=1

xpjK∗pqik x

ql

]− δikτjl (A.17)

90

A.2 Stress and elasticity tensor for the bubble

The skeleton and the bubble act in parallel and share the same F. Subject to F the spherical

bubble deforms into an ellipsoid under the action of principal stretches λ1, λ2 and λ3 and

the attendant strain energy density can be calculated (chapter 2, equation (2.13)). The

corresponding cauchy stress tensor σb and spatial elasticity tensor cb along the principal

directions n1, n2 and n3 are [77],

σbαα =λαJ

∂W b

∂λαnα ⊗ nβ (A.18)

cbααββ =3∑

α,β=1

∂2W b

∂lnλα∂lnλβnα ⊗ nα ⊗ nβ ⊗ nβ −

3∑α=1

τ bααnα ⊗ nα ⊗ nα ⊗ nα

+3∑

α,β=1α 6=β

τ bααλ2β − τ bββλ2

α

λ2α − λ2

β

nα ⊗ nβ ⊗ nα ⊗ nβ (A.19)

where J = λ1λ2λ3. For λα = λβ, the third term in equation (A.19) follows from an

application of L’Hopital’s rule as,

limλα→λβ

τ bααλ2β − τ bββλ2

α

λ2α − λ2

β

=

[∂2W b

∂lnλβ∂lnλβ− ∂2W b

∂lnλα∂lnλβ

]− τββ (A.20)

We transform these relations to a cartesian coordinate system using nα =∑3

j=1 Tαjej

where T is the transformation matrix [77],

σbij =3∑

α=1

σααTαiTαj (A.21)

91

cbijkl =3∑

α,β=1

∂2W b

∂lnλα∂lnλβTαiTαjTβkTβl −

3∑α=1

τ bααTαiTαjTαkTαl (A.22)

+3∑

α,β=1α 6=β

τ bααλ2β − τ bββλ2

α

λ2α − λ2

β

TαiTβjTαkTβl

For λα = λβ, the third term in equation (A.22) follows from an application of L’Hopital’s

rule as,

limλα→λβ

τ bααλ2β − τ bββλ2

α

λ2α − λ2

β

=

[∂2W b

∂lnλβ∂lnλβ− ∂2W b

∂lnλα∂lnλβ

]− τββ (A.23)

A.3 Summary

The local deformation gradient F is provided as input to the UMAT subroutine. We subject

the skeleton to tip displacements in accord with F and obtain the deformed configuration

by solving the nonlinear system of equations (A.2). The cauchy stress tensor σs and spatial

elasticity tensor cs are than obtained using equations (A.5) and (A.17) respectively. For

the bubble, we obtain the stress tensor σb and spatial elasticity tensor cb from equations

(A.21) and (A.22) respectively. The stress and elasticity tensor for the unit-cell model that

are required as output from the UMAT subroutine are simply σ = σc + σb and c = cs + cb

respectively.

92

Appendix B

Digital Image Correlation

Digital image correlation is a non-contact, full-field, optical method that is used to measure

surface deformation. In this method, two digital images are taken, one before and one after a

deformation increment. The digital image taken before the deformation increment is mapped

onto the digital image taken after the deformation increment, and a correlation function is

used to quantify the goodness of the mapping. Then, the displacement field is calculated by

determining the mapping that maximizes goodness. The strain field may be computed from

the displacement field by taking derivatives.

We carry out digital image correlation (DIC) on the foam specimen using a Matlab-based

code developed at the John Hopkins university [78]. The code can be freely downloaded

at (http://www.mathworks.com/matlabcentral/fileexchange/12413) and has been used in

several experimental studies [79–83]. The code implements a local DIC method and divides

Rise

Figure B.1: Markers used for DIC on a lateral surface of a low-density foam (ρ = 0.03).

93

the region of interest into correlation windows that cover the entire region of interest. The

image correlation is than successively performed over each correlation window.

We show a lateral surface of the foam on which we used DIC in figure B.1. The images

used for DIC are taken using a 10 megapixel Canon A1000IS ccd digital camera. The

region of interest over which the DIC is performed (the region covered by the markers in

figure B.1) is 8.6 cm x 8.9 cm (1212 pixels by 1242 pixels). The markers are spaced at 10

pixels. The correlation windows are centered at the markers and measure 25 pixels by 25

pixels (This results in an overlap between successive correlation windows which helps smooth

the resultant displacement and strain field). The displacement increment between any two

successive images is 1 mm. Gradients of the displacement fields obtained using DIC are

calculated using the inbuilt matlab function “gradient”.

94

References

[1] S. Gschmeissner. Coloured scanning electron micrograph (sem) of cancellous (spongy)bone. image no. f0013050. http://www.visualphotos.com/image/2x4141968/bone_

tissue_coloured_scanning_electron_micrograph, July 2011.

[2] A bee’s honeycomb. http://www.tcd.ie/Physics/Foams/solid.php, July 2011.

[3] D. Curran. An aluminium foam produced by the shinko wire company (trade namealporas) scanned from a sectioned aluminum foam specimen. http://en.wikipedia.

org/wiki/File:Closed_cell_metal_foam_with_large_cell_size.JPG, July 2011.

[4] Polyurethane foam. http://www.microvisionlabs.com/bus-card-html/bus_foam.

html, July 2011.

[5] S. Perkowitz. Exploring the Science of Nature’s Most Mysterious Substance. Anchorbooks, New York, 2001.

[6] O. Bayer. Das di-isocyanat-polyadditionsverfahren (polyurethane). AngewandteChemie, 59:257–272, 1947.

[7] Global Industry Analysts Inc. Foamed plastics (polyurethane): A global strategic busi-ness report. CODE: MCP-2069, 2011.

[8] L. D. Artavia and C. W. Macosko. Polyurethane flexible foam formation, in Low densitycellular plastics: Physical basis of behavior . Chapman and Hall, London, 1994.

[9] G. Gioia, Y. Wang, and A. M. Cuitino. The energetics of heterogenous deformation inopen-cell solid foams. Proceedings of The Royal Society, 457:1079–1096, 2001.

[10] N. R. Patel and I. Finnie. Structural features and mechanical properties of rigid cellularplastics. Journal of Materials, 5:909–932, 1970.

[11] L. J. Gibson, M. F. Ashby, G. S. Schajer, and C. I. Robertson. The mechanics oftwo-dimensional cellular materials. Proceedings of the Royal Society, A382:25–42, 1982.

[12] W. E. Warren and A. M. Kraynik. Foam mechanics: the linear elastic response oftwo-dimensional spatially periodic cellular materials. Mechanics of Materials, 6:27–37,1987.

95

[13] C. Chen, T. J. Lu, and N. A. Fleck. Effect of imperfections on the yielding of two-dimensional foams. J. Mech. Phys. Solids, 47:2235–2272, 1999.

[14] P. R. Onck, E. W. Andrews, and L. J. Gibson. Size effects in ductile cellular solids.part i: modeling. J. Mech. Phys. Solids, 47:2235–2272, 1999.

[15] H. X. Zhu, J. F. Knott, and N. J. Mills. Analysis of the elastic properties of open-cellfoams with tetrakaidecahedral cells. Journal of the Mechanics and Physics of Solids,45:717–733, 1997.

[16] M. Laroussi, K. Sab, and A. Alaoui. Foam mechanics: nonlinear response of an elastic3d-periodic microstructure. International Journal of Solids and Structures, 39:3599–3623, 2002.

[17] L. Gong, S. Kyrikides, and W. Y. Jang. Compressive response of open-cell foams. parti: Morphology and elastic properties. International Journal for Solids and Structures,42:1355–1379, 2005.

[18] Lord Kelvin. On the division of space with minimum partitional area. PhilosophicalMagazine, 24:503, 1887.

[19] D. Weaire and R. Phelan. A counterexample to kelvin’s conjecture on minimal surfaces.Philosophical Magazine Letters, 69:107–110, 1994.

[20] E. B. Matzke. The three-dimensional shape of bubbles in foam-an analysis of the role ofsurface forces in three-dimensional shape determination. American Journal of Botany,33:58–80, 1946.

[21] M. D. Montminy, A. R. Tannenbaum, and C. W. Macosko. The 3d structure of realpolymer foams. Journal of Colloid and Interface Science, 280:202–211, 2004.

[22] A. N. Gent and A. G. Thomas. Mechanics of foamed elastic materials. Rubber Chemistryand Technology, 36(1):597–610, 1963.

[23] L. J. Gibson and M. F. Ashby. The mechanics of three-dimensional cellular materials.Proceedings of the Royal Society, A382:43–59, 1982.

[24] W. E. Warren and A. M. Kraynik. The linear elastic properties of open foams. Journalof Applied Mechanics, 55:341–346, 1988.

[25] Y. Wang and A. M. Cuitino. Three-dimensional nonlinear open-cell foams with largedeformations. Journal of the Mechanics and Physics of Solids, 48:961–988, 2000.

[26] X. Dai., T. Sabuwala, and G. Gioia. Lonsdaleite model of open-cell elastic foams: theoryand calibration. Journal of elasticity, 104:143–161, 2011.

96

[27] V. Shulmeister, M. W. D. Van der Burg, E. Van der Giessen, and R. Marissen. A numer-ical study of large deformations of low density elastomeric open-cell foams. Mechanicsof materials, 30:125–140, 1998.

[28] I. Finnie and M.R. Patel. Structural features and mechanical properties of rigid cellularplastics. Journal of Materials, 5:909–932, 1970.

[29] G. Menges and F. Knipschild. Estimation of mechanical properties rigid polyurethanefoams. Polymer Engineering and Science, 15:623–647, 1975.

[30] T. T. Huu, M. Lacroix, C. P. Huu, D. Schweich, and D. Edouarda. Towards a more re-alistic modeling of solid foam: Use of the pentagonal dodecahedron geometry. ChemicalEngineering Science, 64:5131–5142, 2009.

[31] L. Gong and S. Kyrikides. Compressive response of open-cell foams. part ii: Initiationand evolution of crushing. International Journal for Solids and Structures, 42:1381–1399, 2005.

[32] K. Li, X. L. Gao, and A. K. Roy. Micromechanical modeling of three-dimensional open-cell foams using the matrix method for spatial frames. Composites: Part B, 36:249–262,2005.

[33] W. Jang, A. Kraynik, and S. Kyriakides. On the microstructure of open-cell foamsand its effect on elastic properties. International Journal of Solids and Structures, 45:1845–1875, 2008.

[34] R. M. Sullivan, L. J. Ghosn, and B. A. Lerch. A general tetrakaidecahedron model foropen-celled foams. Int. J. Solid. Struct., 45:1754–1765, 2008.

[35] P. Thiyagasundaram, J. Wang, B. V. Sankar, and N. K. Arakere. Fracture toughnessof foams with tetrakaidecahedral unit cells using finite element based micromechanics.Engineering Fracture Mechanics, 78:1277–1288, 2011.

[36] L. J. Gibson and M. F. Ashby. Cellular Solids: Structures and Properties. CambridgeUniversity Press, London, 1997.

[37] W. E. Warren and A. M. Kraynik. Linear elastic behavior of a low-density kelvin foamwith open cells. Journal of Applied Mechanics, 64:787–794, 1997.

[38] H. X. Zhu, J. F. Knott, and N. J. Mills. Analysis of the high strain compression ofopen-cell foams. Journal of the Mechanics and Physics of Solids, 45:1875–1899, 1997.

[39] M. C. Shaw and T. Sata. The plastic behavior of cellular materials. InternationalJournal of Mechanical Sciences, 8:469–472, 1996.

[40] M. F. Vaz and M. A. Fortes. Characterization of deformation bands in the compressionof cellular materials. Journal of Materials Science Letters, 12:1408–1410, 1993.

97

[41] X. Dai., T. Sabuwala, and G. Gioia. Experiments on elastic polyether polyurethanefoams under multiaxial loading: mechanical response and strain fields. Journal of Ap-plied Mechanics, 78:031018.1–031018.11, 2011.

[42] M. Zaslawsky. Multiaxial-stress studies on rigid polyurethane foam. Exp. Mech., 13:70–76, 1973.

[43] A. K. Maji, H. L. Schreyer, S. Donald, Q. Zuo, and D. Satpathi. Mechanical propertiesof polyurethane foam impact limiters. J. Eng. Mech., 121:528–540, 1995.

[44] N. Triantafyllidis and M. W. Schraad. Onset of failure in aluminum honeycombs undergeneral in-plane loading. J. Mech. Phys. Solids, 46:1089–1124, 1989.

[45] J. Zhang, Z. Lin, A. Wong, N. Kikuchi, V. C. Li, A. F. Yee, and G. S. Nusholtz.Constitutive modeling and material characterization of polymeric foams. Trans. ASMEJ. Eng. Maters. Tech., 119:284–291, 1997.

[46] V. S. Deshpande and N. A. Fleck. Multi-axial yield behaviour of polymer foams. Actamater., 49:1859–1866, 2001.

[47] E. E. Gdoutos, I. M. Daniel, and K. A. Wang. Failure of cellular foams under multiaxialloading. Compos. A: Appl. Sci. Manuf., 33:163–176, 2002.

[48] P. Viot. Hydrostatic compression on polypropylene foam. Int. J. Impact Eng., 36:975–989, 2009.

[49] E. A. Meinecke and R. C. Clark. Mechanical properties of polymeric foams. TechnomicPub. Co., 1973.

[50] N. C. Hilyard and A. Cunningham. Low density cellular plastics. Chapman and Hall(London, New York), 1994.

[51] D. L. Weaire and S. Hutzler. The physics of foams. Oxford University Press, New York,1999.

[52] Y. Wang. The mechanics of open-cell solid foams: modeling, simulation and experiment.PhD Thesis, Rutgers, The State University of New Jersey, New Brunswick, NJ, August2001.

[53] T. Daxner. Multi-Scale Modeling and Simulation of Metallic Foams. PhD Thesis, In-stitute of Lightweight Design and Structural Biomechanics, Vienna University of Tech-nology, 2003.

[54] L. Gong. The compressive response of open-cell foams. PhD Thesis, The University ofTexas at Austin, May 2005.

98

[55] A. N. Gent and A. G. Thomas. The deformation of foamed elastic materials. Journalof Applied Polymer Science, 1:107–113, 1959.

[56] W. L. Ko. Deformation of foamed elastomers. Journal of Cellular Plastics, 1:45–50,1965.

[57] R. M. Christensen. Mechanics of low density materials. J. Mech. Phys. Solids, 34:563–578, 1986.

[58] K. Li, X. L. Gao, and G. Subhash. Effects of cell shape and strut cross-sectional areavariations on the elastic properties of three-dimensional open-cell foams. Journal of theMechanics and Physics of Solids, 54:783–806, 2006.

[59] P. Thiyagasundaram, B. V. Sankar, and N. K. Arakere. Elastic properties of open-cellfoams with tetrakaidecahedral cells using finite element analysis. AIAA Journal, 48:818–828, 2010.

[60] S.D. Papka and S. Kyriakides. Experiments and full-scale numerical simulations ofin-plane crushing of a honeycomb. Acta Materiala, 46:27652776, 1998.

[61] T. Suminokura, T. Sasaki, and N. Aoki. Effect of cell membrane on mechanical prop-erties of flexible polyurethane foams. Japanese Journal of Applied Physics, 7:330–334,1968.

[62] K. Yasunaga, R. A. Neff, X. D. Zhang, and C. W. Macosko. Study of cell opening inflexible polyurethane foam. Journal of Cellular Plastics, 32:427–448, 1996.

[63] C. Frondel and U. B. Marvin. Lonsdaleite, a hexagonal polymorph of diamond. Nature,214:587–589, 1967.

[64] H. Langhaar. Energy methods in applied mechanics. Krieger, 1989.

[65] G. I. Barenblatt. Scaling, self-similarity, and intermediate asymptotics. CambridgeUniversity Press, 1986.

[66] Hibbit, Karlsson, and Sorensen. ABAQUS/standard user’s manual, version 6.5. Hibbit,Karlsson, & Sorensen, Pawtucket, R.I., 2005.

[67] F. Pierron. Identification of poisson’s ratios of standard and auxetic low-density poly-meric foams from full-field measurements. The Journal of Strain Analysis for Engineer-ing Design, 45:233–253, 2010.

[68] Crash test dummy. http://www.nacinc.com/applications/automotive/, July 2011.

[69] N. Mullins and K. E. Evans. Indentation resilience of conventional and auxetic foams.Journal of Cellular Plastics, 34:231–260, 1998.

99

[70] W. Yu, Y. Li, Y. P. Zheng, N. Y. Lim, M. H. Lu, and J. Fan. Softness measurements foropen-cell foam materials and human soft tissue. Measurement Science And Technology,17:1785–1791, 2006.

[71] N. J. Mills and A. Gilchrist. Modeling the indentation of low-density polymer foams.Cellular Polymers, 19:389–412, 2000.

[72] N. J. Mills and G. Lyn. Design of foam padding for rugby posts. Materials and Sciencein Sports Symposium, pages 105–117, April 2001.

[73] G. Lyn and N. J. Mills. Design of foam crash mats for head impact protection. SportsEngineering, 4:153–163, 2001.

[74] M. Schrodt, G. Benderoth, A. Kuhhorn, and G. Silber. Hyperelastic description ofpolymer soft foams at finite deformations. Technische Mechanik, 25:162–173, 2005.

[75] X. Dai. Mechanical response of polyether polyurethane foams under multiaxial stressand the initial yielding of ultrathin films. PhD Thesis, University of Illinois at Urbana-Champaign, 2010.

[76] J. C. Simo and T. J. R. Hughes. Computational inelasticity. Springer-Verlag, New York,1998.

[77] J. Bonet and R. D. Wood. Nonlinear Continuum Mechanics for Finite Element Analysis.Cambridge University Press, 1997.

[78] C. Eberl, D. S. Gianola, and R. Thompson. Digital image correlation and tracking.Mathworks file exchange server, File ID: 12413, 2006.

[79] W. Sharpe, J. Pulskamp, D. Gianola, C. Eberl, R. Polcawich, and R. Thompson. Strainmeasurements of silicon dioxide microspecimens by digital imaging processing. Experi-mental Mechanics, 47:649–658, 2007.

[80] D.S. Gianola, C. Eberl, X.M. Cheng, and K.J. Hemker. Stress-driven surface topographyevolution in nanocrystalline al thin films. Advanced materials, 20:303–308, 2008.

[81] T. Balk, C. Eberl, Y. Sun, K. Hemker, and D. Gianola. Tensile and compressive mi-crospecimen testing of bulk nanoporous gold. JOM Journal of the Minerals, Metals andMaterials Society, 61:26–31, 2009.

[82] C. Eberl, D. Gianola, and K. Hemker. Mechanical characterization of coatings usingmicrobeam bending and digital image correlation techniques. Experimental Mechanics,50:85–97, 2010.

[83] C. Eberl, D.S. Gianola, X. Wang, M.Y. He, A.G. Evans, and K.J. Hemker. A methodfor in situ measurement of the elastic behavior of a columnar thermal barrier coating.Acta Materialia, 59:3612 – 3620, 2011.

100