multiscale characterization and modelling of polyurethane...

UNIVERSIDAD POLITECNICA DE MADRID

ESCUELA TECNICA SUPERIOR DE

INGENIEROS DE CAMINOS, CANALES Y PUERTOS

Multiscale Characterization andModelling of Polyurethane Foams

TESIS DOCTORAL

MOHAMMAD MARVI MASHHADI

Ingeniero de Materiales

2018

Departamento de Ciencia de Materiales

Escuela Tecnica Superior de Ingenieros deCaminos, Canales y Puertos

Universidad Politecnica de Madrid

Multiscale Characterization andModelling of Polyurethane Foams

TESIS DOCTORAL

MOHAMMAD MARVI MASHHADIIngeniero de Materiales

Directores de Tesis

Claudio Saul Faria Lopes

Dr. Ingeniero Aeroespacial

Javier LLorcaDr. Ingeniero de Caminos, Canales y Puertos

2018

Seek knowledge from the cradle to the grave.l-Qasim Ferdowsi Tusi

Acknowledgements

I would first like to thank my thesis advisers Dr. Claudio S. Lopes and Prof.

Javier LLorca of the Composite Materials group and Mechanics of Materials group,

respectively, at IMDEA Materials Institute. The office doors of both Prof. Javier

LLorca office and Dr. Claudio S. Lopes were always open whenever I ran into a

trouble or had a question about my research or writing. They consistently allowed

this PhD research to be my own work, but steered me in the right the direction

whenever he thought I needed it.

I would also like to thank the experts who were involved in this research project:

Dr. Jon Molina, Dr. Javier Segurado, Dr. Federico Sket, Dr. Juan Pedro Fernan-

dez and Dr. Miguel Monclus, all the laboratory technicians of IMDEA Materials

Institute and the research group of Dr. Juraj Kosek (Department of Chemical

Engineering, University of Chemistry and Technology Prague). Many thanks to

all people in Fluid Dynamics and Structural Mechanics department of BASF SE,

Germany, and in particular to Dr. Andreas Daiss, Dr. Wolfgang Gerlinger and

Dr. Berend Eling who provide materials and their knowledge on polyurethanes.

This investigation was supported by the 7th Framework Programme of the Eu-

ropean union within the framework of the MODENA project (MOdelling of mor-

phology DEvelopment of micro- and NAnostructures), contract number 604271.

Finally, I must express my very profound gratitude to my family and my fiancee

for providing me with unfailing support and continuous encouragement throughout

my years of study and through the process of researching and writing this thesis.

This accomplishment would not have been possible without them. Thank you.

Mohammad Marvi Mashhadi

14th, March, 2018

Abstract

A modelling strategy based on multiscale characterization and computational

homogenization has been developed to analyze the mechanical behavior of rigid,

open and closed-cell PU foams taking into account the microstructural features.

A multiscale mechanical characterization (i.e. micro- and macromechanical)

was carried out on four different Polyurethane (PU) foams with different densities

and microstructure. The foam microstructure was studied by means of scanning

electron microscopy and X-ray microtomography to determine the cell size distri-

bution, the cell and the strut shape as well as the cell wall thickness. Moreover, the

elastic modulus of the solid PU within the foam was measured using instrumented

nanoindentation creep tests at ambient temperature using a spherical indenter

while the compressive yield stress of the solid PU within the foam was obtained

by means of standard nanoindentation tests with a Berkovich indenter. Macro-

scopic mechanical tests (uniaxial tension, uniaxial compression and shear) were

performed to assess the influence of the density and structural anisotropy on the

mechanical properties.

The information obtained from the microstructural characterization was used

to build representative volume elements of the foam microstructure. This was

achieved by means of the Laguerre tessellation of the space from a random close-

packed sphere distribution, which followed the cell size distribution of the foam.

The polyhedra obtained from the Laguerre tessellation were isotropic and the struc-

tural anisotropy of the foam was introduced by means of an affine deformation of

the polyhedra. The faces and edges of the polyhedra which stand for the cell

walls and struts of the foam were discretized using shell and beam elements, re-

spectively. Special care was taken to ensure that the key microstructural features

(density, cell size distribution, anisotropy, cell wall thickness, strut shape, etc.)

followed the experimental data.

The mechanical properties of the foam in compression were simulated by means

of the finite element simulation of the representative volume element of the mi-

crostructure. The mechanical properties of the solid PU in the model were ob-

tained from the nanoindentation tests. Most of the numerical simulations were

carried out up to the onset of plastic instability to determine the elastic modulus

and the critical stress at the onset of plastic instability. The simulations were in

good agreement with the experimental data and were able to capture the large

influence of the cell anisotropy on the mechanical properties of the foam. In par-

ticular, foams compressed parallel to the orientation of the longest cell dimension

presented much higher elastic modulus and critical stress at the onset of instability

than those deformed in the perpendicular direction. The analysis of the deforma-

tion and failure micromechanisms in the representative volume elements showed

that the instability was triggered by the localized plastic buckling of the struts in

one section of the foam. Moreover, the differences in the mechanical response in

the anisotropic foams were caused by a change in the dominant deformation mode

of the struts with orientation: from axial deformation when the foam was deformed

parallel to the longer axis of the cells to bending in the perpendicular orientation.

Moreover, the potential of the simulation strategy to capture the mechanical be-

havior of the foams up to very large strains, including the effect of densification,

was demonstrated.

Finally, a parametric study was carried out to assess the influence of the mi-

crostructural factors on the elastic modulus and on the critical stress at the onset

of instability . It was found that the main factors that control the mechanical

properties of the foam were the relative density, the fraction of mass in the struts

and in the cell walls and the cell aspect ratio. This information was used to build

surrogate models that are able to predict the elastic modulus and the stress at the

onset of instability of open and closed-cells foam for relative densities in the range

0.025 to 0.2.

RESUMEN

En esta tesis doctoral se ha desarrollado una estrategia de modelizacion para

estudiar el comportamiento mecanico de espumas rıgidas de Poliuretano (PU)

abiertas y cerradas basada en caracterizacion multiescala y en la homogenizacion

numerica.

La caracterizacion de las propiedades mecanicas a nivel microscopico y macros-

copico se realizo en cuatro espumas rıgidas de PU con diferente densidad y mi-

croestructura. La microestructura de las espumas se estudio mediante microscopıa

electronica de barrido y tomografıa de rayos X para medir la distribucion de

tamanos de las celdas, la forma de las celdas y de la aristas ası como el espesor de

las paredes de las celdas. Ademas, el modulo de elasticidad del PU en la espuma

se determino mediante ensayos de fluencia a temperatura ambiente en un nanoin-

dentador con una punta esferica mientras que el lımite elastico en compresion del

PU dentro de la espuma se midio mediante ensayos de nanoindentacion con una

punta Berkovich. Se llevaron a cabo ensayos mecanicos de compresion, traccion,

cortante sobre probetas macroscopicas para medir la influencia de la densidad y

de la anisotropıa en las propiedades mecanicas.

La informacion obtenida mediante la caracterizacion microestructural se uso

para construir volumenes representativos de la microestructura de las espumas.

Esta tarea se llevo a cabo mediante la teselacion de Laguerre del espacio a partir

de una distribucion compacta de esferas cuya distribucion de tamanos seguıa la

distribucion de tamanos de las celdas en la espuma. Los poliedros resultantes de la

teselaci’on son isotropos y se introdujo la anisotropıa aplicando una deformacion

afın a todas las celdas. Las caras y las aristas de los poliedros, que representan

las paredes y las aristas de las celdas en la espuma, se discretizaron con elemen-

tos de placa y de viga, respectivamente. Se tuvo especial cuidado para que los

volumenes representativos fueran un fiel reflejo de las principales caracterısticas

microestructurales de las espumas (densidad, distribucion del tamano de las cel-

das, anisotropıa, espesor de las paredes, forma de las aristas, etc.).

Las propiedades mecanicas de las espumas en compresion se calcularon me-

diante el analisis por elementos finitos de los volumenes representativos de la mi-

croestructura. Las propiedades mecanicas del PU se obtuvieron de los resultados

de los ensayos de nanoindentacion. La mayorıa de las simulaciones numericas se

realizaron hasta el comienzo de la inestabilidad plastica para determinar el modulo

de elasticidad y la tension crıtica para provocar la instabilidad. Los resultados de

las simulaciones se ajustaron a las valores experimentales and fueron capaces de

predecir la enorme influencia de la anisotropıa de las celdas en las propiedades

mecanicas de las espumas. En particular, las espumas comprimidas en la direccion

de la dimension mayor de las celdas presentaron valores mucho mas elevados del

modulo de elasticidad y de la tension crıtica para la inestabilidad que las espumas

defomadas en la direccion perpendicular. El analisis de los micromecanismos de

deformacion en los volumenes representativos demostro que la inestabilidad plas-

tica se origina como consecuencia del pandeo plastico de la aristas de las celdas en

una seccion localizada de la espuma. Ademas, las diferencias en el comportamiento

mecanico en las direcciones perpendiculares de las espumas anisotropas estan cau-

sadas por un cambio en el mecanismo dominante de deformacion de las aristas de

las celdas: de deformacion axial cuando la espuma se deforma en la direccion del

eje mas largo de las celdas a flexion en la direccion perpendicular. Ademas, se ha

demostrado la capacidad de la estrategia de simulacion para modelizar el compor-

tamiento mecanico de las espumas en compresion hasta grandes deformaciiones,

incluyendo el efecto de la densificacion.

Finalmente, se llevo cabo un estudio parametrico de la influencia de los factores

microestructurales en el modulo de elasticidad y en la tension crıtica de inestabil-

idad durante la deformacion en compresion de las espumas. Los resultados del

estudio demostraron que los parametros microestructurales que controlan el com-

portamiento mecanico de las espumas son la densidad relativa, la fraccion de masa

en las paredes y en las aristas, y el factor de forma de las celdas. Esta informacion

se utilizo para construir modelos analıticos sencillos que son capaces de predecir

el modulo de elasticidad y al tension crıtica de inestabilidad en espumas abiertas

o cerradas con densidades relativas en el intervalo 0.025 a 0.2.

Contents

List of Figures XII

List of Tables XIV

1 Introduction and objectives 1

1.1 Polyurethane foams . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Microstructure of PU foam . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Mechanical behavior of PU foams . . . . . . . . . . . . . . . . . . . 8

1.4 Modelling of the mechanical behaviour of PU foams . . . . . . . . . 15

1.4.1 Phenomenological models . . . . . . . . . . . . . . . . . . . 15

1.4.2 Micromechanical models . . . . . . . . . . . . . . . . . . . . 19

1.5 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2 Materials and Experimental Techniques 27

2.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2.1 Microstructural characterization . . . . . . . . . . . . . . . . 28

2.2.2 Micromechanical characterization . . . . . . . . . . . . . . . 29

2.2.3 Thermomechanical characterization . . . . . . . . . . . . . . 30

I

Contents

2.2.4 Mechanical characterization . . . . . . . . . . . . . . . . . . 31

3 Experimental results 35

3.1 Microstructure of PU foams . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Elastic modulus of the solid PU . . . . . . . . . . . . . . . . . . . . 42

3.3 Compressive yield strength of the solid PU . . . . . . . . . . . . . . 47

3.4 Mechanical behavior of the PU foams . . . . . . . . . . . . . . . . . 49

3.4.1 Storage modulus . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4.2 Compressive deformation . . . . . . . . . . . . . . . . . . . . 49

3.4.3 Tensile deformation . . . . . . . . . . . . . . . . . . . . . . . 57

3.4.4 Shear deformation . . . . . . . . . . . . . . . . . . . . . . . 63

4 Computational homogenization strategy 69

4.1 Random packing of spheres . . . . . . . . . . . . . . . . . . . . . . 71

4.2 Laguerre tessellation . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3 Finite element model . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5 Simulation results and discussion 79

5.1 Minimum RVE and mesh size analysis . . . . . . . . . . . . . . . . 80

5.2 Simulation of the isotropic foam . . . . . . . . . . . . . . . . . . . . 82

5.2.1 Experimental validation . . . . . . . . . . . . . . . . . . . . 82

5.2.2 Deformation and failure micromechanisms . . . . . . . . . . 84

5.3 Simulation of the anisotropic foams . . . . . . . . . . . . . . . . . . 88

5.3.1 Experimental validation . . . . . . . . . . . . . . . . . . . . 88

5.3.2 Deformation and failure micromechanisms . . . . . . . . . . 92

5.4 Numerical simulation up to densification . . . . . . . . . . . . . . . 97

II

Contents

5.5 Parametrical study . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.5.1 Fraction of solid material in the struts . . . . . . . . . . . . 101

5.5.2 Cell size distribution . . . . . . . . . . . . . . . . . . . . . . 102

5.5.3 Strut shape . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.5.4 Cell anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . 104

6 Surrogate models for open- and closed-cell foams 109

7 Conclusions and future work 111

7.1 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Bibliography 122

III

List of Figures

1.1 Urethane formation through reaction of isocyanate with hydroxyl

group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 SEM micrographs of (a) open cell; and (b) closed-cell polyethylene

foam (Mills, 2007). . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 SEM micrograph of an edge of a foam. (b) Cross-section of the edge

along its length (Jang et al., 2008). . . . . . . . . . . . . . . . . . . 4

1.4 SEM micrograph of a vertex of a foam (Jang et al., 2008). . . . . . 4

1.5 SEM micrographs of an anisotropic PVC foam. (a) Cross-section in

the plane 1-3. (b) Cross section in the plane 1-2, perpendicular to

the rising direction 3 (Mills, 2007). . . . . . . . . . . . . . . . . . . 8

1.6 Typical stress-strain curve of a PU foam in compression (Gibson &

Ashby, 1999). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.7 Strut uniaxial stress-strain micromechanical testing stage. . . . . . . 13

1.8 Experimental load-penetration curve obtained from the nanoinden-

tation of the solid material of a polymeric foam (Chen et al., 2015)

and the fit with a viscoelastic compliance function. . . . . . . . . . 14

1.9 Marks after indentation of the solid polymer at different loading

rates. (a) 30 µN/s (b) 400 µN/s (Chen et al., 2015). . . . . . . . . . 14

1.10 Equisided tetrakaidekahedron. . . . . . . . . . . . . . . . . . . . . . 20

V

LIST OF FIGURES

1.11 (a) 3D tomography volume of a PU foam. (b) Finite element mesh

superposed to the 3D XCT volume using tetrahedral finite elements

(Youssef et al., 2005). . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.12 3D closed-cell foam model obtained by Laguerre tessellation (Chen

et al., 2015). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.1 Schematic of the foam sample for the nanoindentation tests. . . . . 30

2.2 Experimental set-up for the mechanical tests. (a) Compression. (b)

Tension. (c) Shear. The blue arrows show the loading direction. . . 32

2.3 Speckle pattern on surface of a specimen. . . . . . . . . . . . . . . . 33

3.1 SEM micrographs: (a) parallel and (b) perpendicular to rising di-

rection of the foam 1-3CPW30; (c) parallel and (d) perpendicular

to rising direction of the foam ACPW35; (e) parallel and (f) per-

pendicular to rising direction of the foam ACPW50; (g) parallel and

(h) perpendicular to rising direction of the foam ACPW73. . . . . . 38

3.2 Scanning electron micrographs of the wall thickness of the PU foams.

(a) 1-3CPW30 foam. (b) ACPW35 foam. (c) ACPW50 foam. (d)

ACPW73 foam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3 Cell size distribution of the PU foams measured by XCT. (a) 1-

3CPW30 foam. (b) ACPW35 foam. (c) ACPW50 foam. (d)

ACPW73 foam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4 XCT of the geometrical features of the struts. (a) Longitudinal and

transverse cross-sections of the strut along its length. (b) Variation

of the strut cross sectional area (normalized by the area of the cen-

tral section, A0) as a function of the distance to the center of the

strut, x (normalized by the strut length, L). (c) Volume of the strut

as a function of the strut length. . . . . . . . . . . . . . . . . . . . . 41

VI

LIST OF FIGURES

3.5 (a) Loading scheme during spherical nanoindentation. Representa-

tive load (P ) - penetration depth (h) curve corresponding to the

loading scheme in (a) for: (b) 1-3CPW30 foam. (c) ACPW73 foam.

The experimental curves were corrected by subtracting the compli-

ance due to the porosity in the foam. . . . . . . . . . . . . . . . . . 43

3.6 Atomic force microscopy image of the indented region after nanoin-

dentation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.7 Generalized Maxwell model for a linear viscoelastic solid including

one elastic spring in series with two Kelvin parallel spring and dash-

pot elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.8 Experimental results of the penetration depth, h3/2, vs. time, t, for

solid PU. (a) 1-3CPW30 foam. (b) ACPW73 foam. The solid black

line corresponds to Eq. (3.6), which was fitted to the experimental

data between tR = 20 s and thold = 140 s. . . . . . . . . . . . . . . . 46

3.9 Representative load (P ) - penetration depth (h) curve obtained with

a Berkovich tip to obtain the compressive flow strength of solid PU.

(a) 1-3CPW30 foam. (b) ACPW73 foam. The parameters of the

model (Pmax, hmax, We and Wp) are indicated in the figure. The

curves were corrected for the extra compliance due to the foam

porosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.10 Evolution of storage modulus with temperature. . . . . . . . . . . . 50

3.11 Compressive engineering stress-engineering strain curves of the PU

foams in orientations parallel and perpendicular to the rising di-

rection at ambient temperature. (a) Foam 1-3CPW30. (b) Foam

ACPW35. (c) Foam ACPW50. (d) Foam ACPW73. . . . . . . . . . 51

3.12 Contour plot of the compressive strain along the (vertical) loading

direction on the surface obtained by means of digital image corre-

lation. (a) Compressive engineering strain of 7%. (b) Compressive

engineering strain of 10%. (c) Compressive engineering strain of 20% 52

VII

LIST OF FIGURES

3.13 Compressive engineering stress-engineering strain curves of the PU

foams in orientations parallel and perpendicular to the rising di-

rection at 90 ◦C. (a) 1-3CPW30 foam. (b) ACPW35 foam. (c)

ACPW50 foam. (d) ACPW73 foam. . . . . . . . . . . . . . . . . . 53

3.14 Evolution of the elastic modulus of the foam, E, (normalized by the

elastic modulus of the solid PU, Es) in compression as a function of

the relative density, ρ/ρs, at room temperature. . . . . . . . . . . . 54


elastic modulus of the solid PU, Es) in compression as a function of

the relative density, ρ/ρs, at 90 ◦C. . . . . . . . . . . . . . . . . . . 55

3.16 Evolution of the plateau stress of the foam at room temperature,

σpl, (normalized by the yield stress of the solid PU at ambient tem-

perature, σy) as a function of the relative density, ρ/ρs. . . . . . . . 56

3.17 Evolution of the plateau stress of the foam at 90 ◦C, σpl, (normalized

by the yield stress of the solid PU at ambient temperature, σys) as

a function of the relative density, ρ/ρs. . . . . . . . . . . . . . . . . 57

3.18 Tensile engineering stress-engineering strain curves of the PU foams

in orientations parallel and perpendicular to the rising direction at

ambient temperature. (a) 1-3CPW30 foam. (b) ACPW35 foam. (c)


3.19 Tensile engineering stress-engineering strain curves of the PU foams


90 ◦C. (a) 1-3CPW30 foam. (b) ACPW35 foam. (c) ACPW50 foam.

(d) ACPW73 foam. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59


elastic modulus of the solid PU, Es) in tension as a function of the

relative density, ρ/ρs, at room temperature. . . . . . . . . . . . . . 60

VIII

LIST OF FIGURES


elastic modulus of the solid PU, Es) in tension as a function of the

relative density, ρ/ρs, at 90 ◦C. . . . . . . . . . . . . . . . . . . . . 61

3.22 Evolution of the tensile strength of the foam at ambient tempera-

ture, σut, (normalized by the yield stress of the solid PU at ambient

temperature, σy) as a function of the relative density, ρ/ρs. . . . . . 62

3.23 Evolution of the tensile strength of the foam at 90 ◦C, σut, (normal-

ized by the yield stress of the solid PU at ambient temperature, σy)

as a function of the relative density, ρ/ρs. . . . . . . . . . . . . . . 62

3.24 Contour plot of the tensile strain along the (vertical) loading direc-

tion on the surface of foam 1-3CPW30, obtained by means of digital

image correlation at the strain of 3%. . . . . . . . . . . . . . . . . . 63

3.25 Shear engineering stress-engineering strain curves of the PU foams


ambient temperature. (a) 1-3CPW30 foam. (b) ACPW35 foam. (c)


3.26 Formation of shear bands during the shear test in the 1-3CPW30

foam. The image was taken at the failure point. . . . . . . . . . . . 65

3.27 Evolution of the shear modulus of the foam, G, (normalized by the

elastic modulus of the solid PU, Es) in shear as a function of the

relative density, ρ/ρs, at room temperature. . . . . . . . . . . . . . 66

4.1 Summary of computational homogenization modelling strategy. . . 70

4.2 Laguerre tessellation of a cubic RVE containing 100 cells. The cell

size distribution follows a Gaussian function with an average cell

size diameter of 216 µm and a standard deviation of 67 µm. . . . . 73

4.3 Finite element discretization of a RVE with 100 cells. The variation

of the cross section of the beam elements along the strut can be

observed in the inset. . . . . . . . . . . . . . . . . . . . . . . . . . . 75

IX

LIST OF FIGURES

4.4 Transformation of an isotropic RVE (a) into an anisotropic RVE (b)

and, finally, to the meshed RVE (c). The inset in (c) shows clearly

the variation of the cross section of the beam elements along the

struts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.1 Influence of the number of cells in the RVE on the elastic modulus

of the PU foam, E, (normalized by the elastic modulus of the solid

PU, Es). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2 Influence of the cell wall thickness on the elastic modulus of the PU

foam. The simulation results correspond to the average value and

the standard deviation of five different cell realizations for each cell

wall thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.3 Experimental results and numerical simulations of the engineering

stress - engineering strain curves in compression of the PU foam.

The experimental data correspond to three tests while the numeri-

cal simulations show the results obtained with three different real-

izations of the cell distribution. . . . . . . . . . . . . . . . . . . . . 84

5.4 Central cross-section of the RVE showing the deformation of the cell

walls. (a) Initial cell wall structure. (b) Cell wall structure after 3%

compressive strain. (c) Cell wall structure after 6% compressive

strain. Compression was applied along the vertical Y axis. . . . . . 86

5.5 Contour plot of the axial force in the strut network of the RVE

containing 100 cells. (a) Initial strut configuration. (b) Compressive

strain 3%. (c) Compressive strain 6%. Compression was applied

along the Y axis. The load in the legend is expressed in mN. . . . . 87

5.6 Numerical predictions of the engineering stress-strain curves of the

PU foams parallel and perpendicular to the rising direction. (a)

ACPW35, (b) ACPW50 and (c) ACPW73. The shaded areas en-

close the experimental results. . . . . . . . . . . . . . . . . . . . . . 89

X

LIST OF FIGURES

5.7 Contour plot of the axial forces in the strut network of the ACPW73

foam. (a) Compression parallel to the rising direction up to 3%

strain (Y axis). (b) Compression perpendicular to the rising direc-

tion up to 3% strain (X axis) The load in the legend is expressed

in mN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.8 Cumulative probability distribution of the axial force in struts for a

applied compressive strain of 2%. (a) ACPW35 foam. (b) ACPW50

foam. (c) ACPW73 foam. The probability distributions correspond-

ing to deformation parallel and perpendicular to the rising direction

are plotted in each figure. . . . . . . . . . . . . . . . . . . . . . . . 94

5.9 Fraction of struts in which the central beam element was buckled as

a function of the stress applied to the RVE. (a) ACPW35 foam. (b)

ACPW50 foam. (c) ACPW73 foam. The results are plotted when

the RVE was loaded in the orientations parallel and perpendicular

to the rising direction. . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.10 Compressive response of the RVEs having 100 and 800 cells up to

densification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.11 Contour plots of the displacement in vertical (loading) direction in

the RVE containing 800 cells. (a) Compressive strain of 4%. (b)

Compressive strain of 10%. (c) Compressive strain of 20%. (d)

Compressive strain of 40%. (e) Compressive strain of 60%. The

displacement in the legend is expressed in mm and the initial height

of the RVE is 2 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.12 Influence of the fraction of solid material in the struts, φ, on the

mechanical properties of isotropic foams for different relative foam

densities, ρ/ρs. (a) Normalized elastic modulus, E/Es. (b) Nor-

malized plateau stress, σpl/σys. The error bars correspond to the

standard deviation of the simulation results for three different real-

izations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

XI

LIST OF FIGURES

5.13 Influence of the width of the cell size distribution (characterized

by the standard deviation SD) on the mechanical properties of

isotropic foams for different relative densities of the foam, ρ/ρs. (a)

Normalized elastic modulus, E/Es. (b) Normalized plateau stress,

σpl/σys. The error bars correspond to the standard deviation of the

simulation results for three different realizations. . . . . . . . . . . . 102

5.14 Strut shapes used in the simulations. They are defined by f(x/L),

eq. (3.1), which expresses the variation of the strut cross sectional

area A (normalized by the area of the central section, A0) as a

function of the distance to the center of the strut, x (normalized by

the strut length, L). . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.15 Influence of the strut shape on the mechanical properties of isotropic

foams for different foam densities. (a) Normalized elastic modulus,

E/Es, for φ = 1. (b) Normalized plateau stress, σpl/σys, for φ = 1.

(c) Normalized elastic modulus, E/Es, for φ = 0.6. (d) Normalized

plateau stress, σpl/σys, for φ = 0.6. . . . . . . . . . . . . . . . . . . 105

5.16 Influence of the cell aspect ratio s on the normalized elastic modulus

of the foam, E/Es. (a) φ = 0.2, (b) φ = 0.4, (c) φ = 0.6, (d) φ = 0.8,

(e) φ = 1.0 (open cell foam). The solid lines correspond to eq. (6.1) 107

5.17 Influence of the cell aspect ratio s on the normalized plateau stress

of the foam, σps/σys. (a) φ = 0.2, (b) φ = 0.4, (c) φ = 0.6, (d)

φ = 0.8, (e) φ = 1 (open cell foam). The solid lines correspond to

eq. (6.2) for close cell foams and to eq. (6.3) for open cell foams. . . 108

XII

List of Tables

2.1 Brief description of PU foam samples . . . . . . . . . . . . . . . . . 28

3.1 Microstructural features of the PU foams. D stands for the average

cell diameter, s for the average cell aspect ratio, t for the average

cell wall thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Parameters of the generalized Maxwell model for PU . . . . . . . . 46

3.3 Mechanical properties of the solid PU in compression at ambient

temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4 Mechanical properties of the solid PU in compression at 90◦C. . . . 57

3.5 Mechanical properties of the solid PU in tension at ambient tem-

perature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.6 Mechanical properties of the solid PU in tension at 90 ◦C. . . . . . 61

3.7 Mechanical properties of the solid PU in shear at ambient temperature. 66

4.1 Mechanical properties of solid PU . . . . . . . . . . . . . . . . . . . 77

5.1 Experimental and simulations results of the elastic modulus E (in

MPa) of the anisotropic PU foams in the orientations parallel (‖)and perpendicular (⊥) to the rising direction. . . . . . . . . . . . . 90

XIII

LIST OF TABLES

5.2 Experimental and simulations results of the plateau stress, σp (in

MPa) of anisotropic PU foams in the orientations parallel (‖) and

perpendicular (⊥) to the rising direction. . . . . . . . . . . . . . . . 91

5.3 Influence of the cell aspect ratio, s, on the anisotropy in the elastic

modulus in the orientations parallel (E‖) and perpendicular (E⊥)

to the rising direction. . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.4 Influence of the cell aspect ratio, s, on the anisotropy in the plateau

stress in the orientations parallel (σp‖) and perpendicular (σp⊥) to

the rising direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.1 Constants of eqs. (6.1), (6.2) and (6.3) to determine the elastic

modulus and the plateau stress of closed-cell and open cell foams. . 110

XIV

Chapter 1

Introduction and objectives

1.1 Polyurethane foams

The term polyurethane (PU) is used to designate polymers produced by the

addition reaction of polyfunctional isocyanates with compounds containing at least

two hydroxyl groups (Fig. 1.1). Amines, carboxylic acids and water, as well as

gaseous products (e.g. CO2) are formed when the isocyanate group (NCO) reacts

with alcohols and the gaseous products act as a blowing agent to start forming

the cellular structure. Surface tension determines the cell shape and the cells

can be elongated if the foam rises during production. Silicone surfactants can be

added to control the opening of cells so that final foam microstructure can include

closed-cells or open-cells (Mills, 2007).

Figure 1.1: Urethane formation through reaction of isocyanate with hydroxylgroup.

1

1.1 Polyurethane foams

PU foams have many applications and can be classified in two groups: rigid

and flexible PU foams. Rigid foams are inside the metal and plastic walls of most

refrigerators and freezers, or behind paper, metals and other surface materials in

the case of thermal insulation panels in the construction sector. Rigid PU foams

are also used for moldings which include door frames, columns, balusters, window

headers, pediments, medallions and rosettes as well as resin-transfer moulding

cores in composite processing. Flexible PU foams affect our lives in many ways

and new applications are rolling out on a regular basis. It is used as cushioning for

a wide variety of consumer and commercial products including furniture, carpet

cushion, transportation, bedding, packaging, textiles and fibers. Furthermore, PU

formulations cover an extremely wide range of stiffness, hardness, and densities,

and this wide range of properties opens the application of PU foams in practically

all industrial sectors (Klempner & Frisch, 1991; Mills, 2007).

PU foam processing techniques are mainly classified into three categories: slab-

stock foam, moulded PU foam and batch foaming. Slabstock is a continuous pro-

cess which is able to produce large foam panels. In this process, it is necessary to

control the exothermic reaction to avoid overheat unless special cooling technol-

ogy is used. Blowing agents can be generated by chemical reactions (e.g. CO2)

or added in form of physical blowing agent (e.g. hydrocarbons, particularly pen-

tane isomers) that can be evaporated during foaming process. Another widely

used method is the use of standard injection moulding machines in which blowing

agents are incorporated into the material (moulded PU). Dry chemical blowing

agents are mixed with the granules of plastic, and the mixture is melted in the

barrel of the machine. The heat of the melted plastic causes the blowing agent

to react, leading to the formation of gas, which in turn foams the plastic. The

moulded method is used for the production of the majority of automotive seat

cushions. Finally, the batch process is used to manufacture slow-recovery foams

used, for example, in wheelchair seating. In this method, batch foam machines are

used and polyol and catalysts are poured into the mixing vessel in the appropri-

ate proportions. Mixing is started in the machine and then water is added. The

2

1.2 Microstructure of PU foam

machine mixes the chemicals and CO2 is released while the mixture is introduced

into the mould (Mills, 2007).


Depending on the type of PU foam (e.g. flexible or rigid), there is a wide

range of PU foam microstructures. The whole spectrum from open- to closed-cell

microstructures or even combination of both can exist (Fig. 1.2) (Mills, 2007;

Gibson & Ashby, 1999). The basic morphological unit of a closed-cell foam is

the gas-filled cavity surrounded by cell walls and struts. Typically, these cavities

(cells) are pentagonal dodecahedral or tetrakaidecahedral. Their average dimen-

sions and statistical distribution are known to influence the physical properties

and performance of the foam (Lewis et al., 1996).

Foam structures mainly contain three main elements: struts (edges), vertices

and cell walls (facets). These three features are shown in Fig. 1.2. Cell edges

are straight. If the surface energy is minimized, three faces meet at each edge,

with an interface angle of 120◦. The edges or struts are usually relatively stubby,

with lengths being only a small multiple of their thickness. The variation in the

thickness along the length was characterized by Gong et al. (2005) (Fig. 1.3). A

scanning electron microscope (SEM) micrograph of a typical strut is shown in Fig.

1.3(a), while a set of cross-sections at different locations along the strut are depicted

in Fig. 1.3(b). The edges are nearly linear and have the characteristic three-cusp

hypocycloid cross-section (Plateau borders). The area of the cross-section increases

as the vertices (nodes) with other edges on either end are approached. An SEM

micrograph of a typical vertex of a PU foam is depicted in Fig. 1.4.

In closed-cell PU foams, the thin sheet of PU material surrounded by several

edges is called the cell wall or facet (Fig. 1.2). The number of cell walls per each

cell can vary from 11 to 17 (Fan et al., 2004). Cell wall centers are thinner than

the outer regions, and the thickness is typically less than 1 µm. They behave as

thin membranes, wrinkling under in-plane compressive forces. Optical reflection

3


Figure 1.2: SEM micrographs of (a) open cell; and (b) closed-cell polyethylenefoam (Mills, 2007).

Figure 1.3: SEM micrograph of an edge of a foam. (b) Cross-section of theedge along its length (Jang et al., 2008).

Figure 1.4: SEM micrograph of a vertex of a foam (Jang et al., 2008).

4


microscopy has been used to obtain interference fringes from the cell walls of PU

foam, hence the thickness distribution. It has been shown that even the thinnest

part is significantly thicker than a soap film face, which can be the length of two

surfactant molecules (Rhodes, 1994).

Cell walls are usually planar in undeformed polymeric foams, since there is no

pressure difference between the cells. In comparison with other polymeric foams,

the cell wall thicknesses are much greater than in PU foams, since the highly

viscous polymer melt resists the effect of the extensional flow (Mills, 2007).

Closed-cell polymeric foams have polygonal cells of variable shape and size.

Each cell is a polyhedron, so its geometry must obey rules related to the space-

filling packing of polyhedral. This includes Euler’s formula for any convex two-

dimensional (2D) or three-dimensional (3D) polyhedron,

V − E + F = 1 (2D) (1.1)

− C + V − E + F = 1 (3D) (1.2)

where F is the number of walls or faces, E the number of edges or sides, and V

the number of vertices of C cells. The topology of a foam is characterized by f ,

the number of cell walls per cell and n, the number of struts per cell wall (Mills,

2007).

A honeycomb with regular hexagonal cells obviously has six edges surrounding

each face. An immediate consequence of Euler’s law is that an irregular three-

connected honeycomb also has six sides per face on average. This means that a

five-sided face can only be introduced if a compensating seven-sided face is also

created somewhere. A four-sided face requires an eight-sided face or two seven-

sided ones and so on. In practice, this means that most cells have six sides and

those which do not are paired. The number of edges which meet at a vertex is

the edge-connectivity (Ze) and the number of faces which meet at an edge is the

face-connectivity (Zf ). Ze is usually three in a honeycomb and four in a foam but

5


it can have other values. Zf is usually three in a foam but it can have other values.

If Ze =3, then E/V = 1.5 (the 2 appears because each edge is shared between two

vertices). If Fn is the number of faces with n sides,

∑ nFn

2= E (1.3)

where the denominator 2 appears because an edge separates two faces. Euler’s law

then gives

6−∑nFn

F=

6

F(1.4)

As the total number of faces F becomes large, the right-hand side of this

equation approaches zero. The term∑nFn/F is just the average number of edges

per face (n), which is given by

n = 6 (2D) (1.5)

and a face with five edges can exist only if there is a complementary seven-edge

face, and so on. These results are for an edge-coordination (Ze) of three but this

value is not universal. For a general edge-coordination (Ze), the same argument

gives

n =2Ze

Ze − 2(2D) (1.6)

which reduces to n = 6 when Ze = 3.

In three dimensions, the problem is less constrained (eq. (1.2) has one more

variable), and the results are less general. For an isolated cell (C = 1), the average

number of edges per face (n) is related to the number of faces f (for an edge

connectivity of three on the surface of the cell) by

6


n = 6

(1− 2

f

)(3D) (1.7)

or, more generally,

n =ZeZf

Ze − 2

(1− 2

f

)(3D) (1.8)

An important consequence follows from these results. In most foams, most cells

have faces with five edges, regardless of the shape of the cell. If the cells are one

average dodecahedra (f = 12), the average face has n = 5 edges according to eq.

(1.7) . If they are tetrakaidecahedra (or Kelvin cells with f=14), n = 5.14. Many

foam-like structures (grains in polycrystalline metals and ceramics, for instance)

have f ≈ 14 and n = 5.1, but both f and n depend on how the foam is made, and

the forces which shape its cells.

Considering the cell geometries of collections of soap bubbles, plant cells, and

metal grains, Williams (1968) concluded the following rules:

• The average number of faces per cell is 14.

• The average number of side per face is 5.14.

• The vertices are tetrahedral, with inter-edge angles close to 109◦.

Another important microstructural parameter is the average aspect ratio of

the cells. Most polymeric foams show an anisotropy in the cell shape (Mills, 2007;

Gibson & Ashby, 1999; Montminy et al., 2004; Shulmeister, 1998a,b) as a result

of the foaming process in the mould. The cells tend be elongated along the rising

direction of the foam within the mould and this anisotropy leads to large differences

in the mechanical properties in the orientations parallel and perpendicular to the

rising direction. The anisotropic microstructure of a PVC foam is depicted in Fig.

1.5, which shows elongated cells in the rising direction 3, while the pattern is nearly

isotropic in the plane 1-2, perpendicular to the rising direction (Tita & Caliri JA,

2012).

7

1.3 Mechanical behavior of PU foams

Figure 1.5: SEM micrographs of an anisotropic PVC foam. (a) Cross-sectionin the plane 1-3. (b) Cross section in the plane 1-2, perpendicular to the risingdirection 3 (Mills, 2007).


Since PU foams are highly useful for absorbing the energy of impacts (e.g.

in packing and crash protection) and light weight structures (e.g. in the cores

of sandwich panels), awareness of their mechanical behavior is crucial for design

purposes.

The schematic of the stress - strain curve of a PU foam in compression is shown

in Figure 1.6 (Gibson & Ashby, 1999). The three typical deformation regions could

be seen: linear elastic, plateau region and densification. Observations using optical

and scanning electron microscopy revealed the deformation mechanism associated

with each of these three regimes. The linear elastic regime is controlled by the strut

bending and by the cell wall stretching if the foam has closed-cells. The initial

slope of the stress - strain curve of PU foam is the elastic modulus (E). If the

compressive load increases, elastic-plastic buckling of struts and cell walls results

in the onset of a plastic instability. Further deformation leads to the propagation

of the buckled section along the microstructure at a constant stress (denominated

plateau stress) up to very large strains (60%). The stress carried by the foam

finally increases due to collapse of the cellular structure and the densification of

8


the foam at the densification strain εD (Gibson & Ashby, 1999; Imeokparia et al.,

2003; Fatima Vaz & Fortes, 1993).

Figure 1.6: Typical stress-strain curve of a PU foam in compression (Gibson& Ashby, 1999).

In tension, PU foams are often brittle and the elastic regime is followed by

fracture after yielding. Fracture is triggered by the propagation of a crack from

weak cell walls or pre-existing flaws. Gibson & Ashby (1999); Brezny & Green.

Similarly, the mechanical behavior of PU foams in shear is fairly brittle.

Mechanical properties of PU foams depend on microstructural features such

as open or closed-cell structure, cell size and shape, etc. In addition, the gas

enclosed within the cells also influences the mechanical properties of closed-cell

foams (Shulmeister, 1998b).

To understand the relation between the morphology and properties of the cel-

lular solids (e.g. PU foams), it is important to consider the critical geometrical

9


features of the foam microstructure. They are the following, according to Berlin

(1980)

• Relative number of open cells.

• Relative foam density.

• Cell size.

• Cell shape, or geometrical anisotropy.

• Cell wall thickness and distribution of solid material between struts and faces.

• Mechanical properties of the solid PU.

In most cases, foams contain closed as well as open cells. A study of closed-cell

foams performed by Berlin (1980) suggests that the fraction of open cells (ϑ) is

a function of the foam relative density. ϑ is approximately equal to 0.3 for low-

density foams with ρ/ρs < 0.06 (where ρ is the foam density and ρs the density of

the solid material), and decreases as the foam density increases. This behavior is

explained because the membranes become very thin and can easily rupture during

or after processing in the low-density foams. The relative content of open cells

in closed-cell foam plays a considerable role in the physical properties of cellular

materials (Berlin, 1980; Yasunaga et al., 1996; Shulmeister, 1998b).

The foam relative density is defined as ρ/ρs where ρ is the foam density and ρs

is the density of the solid material in the foam, which is generally in the range of

900-1200 kg/m3 for PU. Low density foams have ρ/ρs < 0.2, while structural foams

manufactured by injection mouldings have ρ/ρs in the range of 0.4-0.8 (Mills, 2007).

The mechanical, acoustic, electrical and thermal properties of a foam material are

strongly affected by its relative density (Gibson & Ashby, 1999). Numerous studies

of sandwich panels with foam cores have shown that the structural strength of a

panel increases with the core foam density (Issac M et al., 2009; Sharaf et al., 2010).

It is also well known that the yield strength and elastic modulus increase with the

density of the foam (Gibson & Ashby, 1999). The proportionality of elastic and

10


shear moduli with the square of relative density shows the strong influence of this

parameter in the mechanical properties of PU foams. The mass density of solid

polyurethane can also vary, but the range of variation is small when compared

with the effective relative density of typical foam.

The average cell diameter D is one important geometrical characteristic of the

foam, that can influence the mechanical properties of foams (Chen et al., 2015).

Different investigators have reached contradictory conclusions about this issue.

Morgan et al. (1981) found out that the elastic modulus is independent of the

cell size in closed-cell glass foam. The elastic constants of open-cell alumina were

measured as a function of their relative density and cell size by (Hagiwara & Green,

1987). They found that the stiffness increased with the cell size. It was also found

that the foam stiffness is sensitive to the microstructure of the cell struts, and

especially porosity within these struts. Brezny & Green (1990) reported that the

compressive and bending strength of a brittle reticulated vitreous carbon foam were

inversely proportional to the cell size. Image analysis revealed that the change in

compressive strength was due to a change in the strut strength with cell size. The

fracture toughness and elastic modulus were, however, independent of the cell size.

It is obvious from these results that there is no clear consensus on the effect of cell

size on the mechanical properties of foams.

The final foam microstructure is often anisotropic (Mills, 2007) and the degree

of anisotropy in the shape of the cells can be controlled through the processing

parameters. Anisotropy influences the mechanical properties and, for instance, PU

foams can be 20 to 50 times stiffer in the rising direction (Lee & Lakes, 1997). Uni-

axial tension and compression, and simple shear tests on the PU foam containing

elongated cells have shown the large effect of anisotropy in the foam mechanical

properties (Huber & Gibson, 1988; Witkiewicz & Zielinski, 2006).

The solid material in foams can be distributed among three geometrical fea-

tures: cell walls, struts and vertices. It has been noted that a critical cell wall

thickness exists for each specific material, which is the minimum thickness for that

material to allow closed-cell foam formation (Mills, 2007). The distribution of the

11


material between walls and struts in the closed-cell foam will influence the me-

chanical properties of the foam considerably (Shulmeister, 1998b). Variation of

cell wall thickness on the stiffness of closed-cell foam has been studied by Grenest-

edt & Bassinet (2000). They found that the elastic and shear moduli were reduced

by roughly 19% when the thickest cell wall was 19 times thicker than the thinnest

cell wall.

In addition, the mechanical behavior of the PU foams also depends on the

mechanical behavior of the solid material. This relevant information is very difficult

to obtain because characterizing the mechanical behavior of the solid material

within the foam is not straightforward. Moreover, the properties of the material

before foaming differ from those after the foaming process and solidification leads

to chemical or physical changes in the primary molecular structure and composition

of base material during foaming. For example, polymer macromolecules may get

particular orientations in the foamed medium, changing the mechanical properties

of the solid. Furthermore, the presence of stabilizers and nucleating agents may

also influence the microstructure and properties of the solid (Shulmeister, 1998b;

Warburton et al., 1990).

Thus, the properties of the unfoamed solid PU are not the properties of the

material inside the grown foam and it is necessary to determine the properties of

the material inside the foam. This problem is difficult because of the size of the

foam structural elements. The strut length is often in the range 10 to 100 µm and

mechanical tests become challenging. Gong et al. (2005) used a micromechanical

testing stage to test foam edges from reticulated flexible polyester urethane foams

(Fig. 1.7). The overall length of the edges was of the order of 3 mm. The nodes of

the edges were bonded to two small plates, which were then clamped in the device.

Several edges were tested at different displacement rates. The small size of the

ligaments, the non-uniform and special shape of their cross section, the small initial

curvature in the ligaments, difficulties in clamping the ends, and uncertainty about

the exact size of the test section led to uncertainties in the measurements. It should

be noted that polymers exhibit time dependent behavior which plays a significant

role in the deformation and fracture of polymeric foams. Chen et al. (2015) has

12


Figure 1.7: Strut uniaxial stress-strain micromechanical testing stage.

recently used instrumented nanoindentation to measure the elastic modulus of the

solid material in closed-cell polymeric foams considering the viscoelastic effects.

Instead of using the conventional Oliver-Pharr method (Oliver & Pharr, 1992), in

which the elastic modulus is related to the initial slopes of the unloading curves, the

creep compliance of the viscoelastic solid polymer inside the foam was obtained by

fitting the initial elastic part of the loading curves into a compliance function based

on viscoelastic contact analysis (Fig. 1.8) (Lu, 2003). They also found that the

indentation marks left after the removal of the loads were hardly identifiable when

the maximum penetration during the nanoindentation test was below a critical

value and thus the material remained in the viscoelastic regime during indentation

(Fig. 1.9(a)). Nevertheless, permanent deformations were found after indentation

at higher depths (Fig. 1.9(b)).

13


Figure 1.8: Experimental load-penetration curve obtained from the nanoin-dentation of the solid material of a polymeric foam (Chen et al., 2015) and thefit with a viscoelastic compliance function.

Figure 1.9: Marks after indentation of the solid polymer at different loadingrates. (a) 30 µN/s (b) 400 µN/s (Chen et al., 2015).

14

1.4 Modelling of the mechanical behaviour of PU foams

1.4 Modelling of the mechanical behaviour of PU

foams

The mechanical behavior of PU foams has been the subject of numerous analyt-

ical and numerical investigations. The typical stress-strain behavior of a PU foam

in compression was depicted in Fig. 1.6 (Avalle et al., 2001). As indicated above,

the curve exhibits three different regions: linear elastic, plateau and densification.

The behavior is linear elastic at small strains (below 5%), and the slope is given

by the elastic modulus of the foam. As the load increases, the foam cells begin to

collapse by elastic buckling, plastic yielding or brittle crushing, depending on the

mechanical properties of the cell walls. Collapse progresses at roughly constant

load, leading to a stress plateau, until the opposing walls in the cells meet and

touch, and densification leads to an increase in the load carried by the foam. In

this last region, the slope of stress-strain curve is close to that of the solid PU.

During unloading, the stress varies non-linearly with the strain. The specific

energy (energy per unit volume) absorbed by the foam during deformation is given

by the area under the stress-strain curve. Very little energy is stored in the linear

elastic region and most of the energy is absorbed during the plateau region at

nearly constant load.

1.4.1 Phenomenological models

Gibson and Ashby tried to describe these three regions with three equations

according to (Gibson & Ashby, 1999):

σ(ε) = Eε for σ ≤ σy (1.9)

σ(ε) = σy for εy ≤ ε ≤ εD(1−D−1) + εy (1.10)

15


σ(ε) = σy1

D

(εD

εD − ε

)m

for ε > εD(1−D−1) + εy (1.11)

where σ and ε are engineering stress and engineering strain, respectively. This

model has five parameters: E is the elastic modulus of the foam, σy the yield stress,

εD the densification strain, and m and D two constants. This model assumes that

the stress has a constant value at the plateau region and that the stress-strain

curve is not smooth at the boundaries of two regions (Gibson & Ashby, 1999).

The pioneer work to relate the parameters with the microstructure of the foam

was carried out by Gibson & Ashby (1999) based on the analysis of a simple cubic

cell structure. The elastic modulus of closed-cell foams was expressed according

to (Gibson & Ashby, 1999) by

E

Es

= c1φ2

(ρ

ρs

)2

+ c′1(1− φ)

(ρ

ρs

)+

p0(1− 2ν)

Es(1− ρ/ρs)(1.12)

where E is the elastic modulus of the foam, Es the elastic modulus of the solid

material, φ the volume fraction of solid material in the cell edges (struts) (and

1− φ the corresponding volume fraction of solid materials in the cell walls), ρ the

foam density, ρs the density of the solid material, p0 the fluid pressure within the

closed-cell foam cells, ν the Poisson’s ratio of the foam, and c1 and c′1 constants

that depend on the geometrical characteristics of the foam. (Gibson et al., 1982)

proposed c1 = 1 and c′1 = 1 based on various experimental results of foams with

regular periodic cell structure or on average foam parameters if the size and shape

do not change much. If the fluid pressure in the cells is near atmospheric pressure,

p0/Es will be much smaller than E/Es and can be neglected. Then, Eq. (1.10)

can be rewritten as

E

Es

= c1φ2

(ρ

ρs

)2

+ c′1(1− φ)

(ρ

ρs

)(1.13)

16


and the effective modulus of the foam E is determined from φ and the relative

density. (Gibson & Ashby, 1999) indicated that 0.6 ≤ φ ≤ 0.8 for common closed-

cell foams.

The nonlinear deformation in compression begins when the applied stress reaches

the yield stress σy. According to the Gibson-Ashby model for a simple cubic mi-

crostructure, Gibson & Ashby (1999), the yield stress is expressed by

σyσys

= c3

(φρ

ρs

)1.5

+ c4(1− φ)ρ

ρs+p0 − patσys

(1.14)

where σys is the yield stress of the solid material, and pat the atmosphere pressure.

The coefficients c3 and c4 are usually found based on the experimental data.

In the plastic regime, the stress carried by the material remains constant un-

til the onset of densification, which occurs at the densification strain εD. The

densification strain is given by:

εD = 1− 1.4ρ

ρs. (1.15)

The stress carried by the foam increased sharply for ε > εD due to the densifi-

cation process, which finishes when the strain reaches the value εL, given by

εL = 1− ρ

ρs. (1.16)

Most PU foams often show a marked anisotropy in the cell shape (Mills, 2007;

Gibson & Ashby, 1999; Montminy et al., 2004; Shulmeister, 1998a) as a result of

the foaming process in the mould. The cells tend be elongated along the rising

direction of the foam within the mould and this anisotropy leads to large differences

in the mechanical properties in the orientations parallel and perpendicular to the

rising direction. For instance, the stiffness and the plateau stress along the rising

direction of PU foams with an average cell aspect ratio of 2 can be 8 and 2.6

17


times higher, respectively, than those along the perpendicular direction (Gibson &

Ashby, 1999).

Moreover, anisotropy also leads to changes in the shape of the stress-strain

curve under compression. In the case of isotropic foams, the elastic region of the

stress-strain curve (characterized by the elastic modulus) is followed by a plateau

whereby the foam deforms at a constant stress (the plateau stress) by the pro-

gressive collapse of the cells. At larger strains, the stress increases as a result of

densification. However, anisotropic foams present different behaviors when they

are deformed along the rising direction or perpendicular to it. In the former, they

show a large drop in the stress after the onset of plastic collapse, which is followed

by the plateau region in which the foam deforms at constant stress. In the latter,

the stress drop at the onset of plastic collapse is negligible but the stress increases

continuously during deformation afterwards. Finally, the stress-strain curves in

both orientations are equivalent in both orientation in the last stages of deforma-

tion, which are controlled by the densification of the foam (Tu et al., 2001; Gibson

& Ashby, 1999; Amsterdam et al., 2008).

Although the structural anisotropy is very important to tailor the mechanical

properties of the foam, very few investigations have analyzed this behavior in

detail. (Gibson & Ashby, 1999) proposed a simple expression for the influence

of the cell aspect ratio, s, on the anisotropy of the elastic modulus of closed-cell

foams based on a simple cubic cell geometry according to

E‖/E⊥ =2s2[

1 + (1/s)3] + (1− φ)

2s[1 + (1/s)

] (1.17)

where E‖ and E⊥ stand for the foam elastic modulus in the rising direction and in

the perpendicular one, respectively.

More detailed analyses of the influence of the structural anisotropy on the

stiffness of the foams were carried out in the framework of models that idealized

the open cell foam by a space-filling ensemble of elongated Kelvin cells (Sullivan

et al., 2008; Li et al., 2003; Jang et al., 2010; Gong et al., 2005). Nevertheless,

18


these simplified micromechanical models can provided trends of the stiffness but

not quantitative predictions and are not appropriate to analyse the onset of the

damage, which depends on the details of the heterogeneous spatial distribution of

the cells within the foam.

The equations of Gibson and Ashby are based on cellular materials with simi-

lar cell size and periodic cell structure, which leads to c1 = c′1 = 1. Furthermore,

these constants are derived from the fitting of experimental stress-strain curves.

Gibson and Ashby also mentioned that the values of c1 and c′1 might be inaccu-

rate due to the differences in cell morphology (Gibson & Ashby, 1999). In fact,

foams with similar density and porosity show a wide range of cell volumes and the

mechanical properties are greatly different from regular cellular foams. Thus, it is

well accepted that the phenomenological models can only provide a rough approx-

imation to the mechanical properties of foams (Ma et al., 2011). Micromechanical

models are necessary to obtain accurate values of the mechanical properties as

well as information about the actual deformation and failure mechanisms includ-

ing the effect of microstructural features (e.g. cell size and face thickness) on their

mechanical response.

1.4.2 Micromechanical models

Micromechanical models for the mechanical behavior of cellular foams are de-

veloped under the assumption that the foams is made up by an ensemble of

tetrakaidecahedral unit cells (Kelvin cells) (Jang et al., 2010; Subramanian &

Sankar, 2012; Sullivan et al., 2008). The tetrakaidecahedron is a polyhedron with

14 faces, 24 vertices, and 36 edges (Fig. 1.10). It is obtained by truncating the

corners of an octahedron, and hence also referred to as a truncated octahedron.

The equisided tetrakaidecahedral cell, obtained by truncating a cube, is a 14-faced

figure made up of 8 hexagonal faces and 6 square faces. The 36 struts of an eq-

uisided tetrakaidecahedron have the same length. According to the experimental

evidence, the cross section of the strut is an equilateral triangle, whose dimensions

vary along the length (see Fig. 1.3).

19


Figure 1.10: Equisided tetrakaidekahedron.

An analytical model for the elastic properties of a foam made up by equisided

tetrakaidecahedron was developed by Zhu et al. (1997). The model was modified

for foams with elongated tetrakaidecahedral unit cells by Sullivan et al. (2008)

and Li et al. (2003). Sihn & Roy (2004) carried out a similar analysis using the

finite element method to understand the influence of edge and wall thickness on

the macroscopic properties.

An analytical model based on tetrakaidecahedral unit cells was developed by

Jang et al. (2010) to predict elastic properties of open-cell foams. The foam is

idealized to be periodic using the space-filling Kelvin cell including the major

geometric characteristics found in the real microstructure of foams. The cells are

elongated in the rise direction and the struts are assumed to be straight with

Plateau border cross-sections. Moreover, the strut cross-section was not uniform

and changed gradually from the center toward the ends, as shown in Fig. 1.3. This

variation was quantified by:

A(x) = A0f(x/L) = A0

[86

(x

L

)4

+

(x

L

)2

+ 1.00

](1.18)

where A0 is the area of the cross section at the midspan of the strut, L is the length

of the strut, and x is the position along the length of the strut (−L/2 ≤ x ≤ L/2).

The second moment of inertia, I(x), and the polar moment of inertia, J(x), of the

strut cross section vary as a function of the position along the strut according to

20


I(x) = I0f(x/L) = I0

[86

(x

L

)4

+

(x

L

)2

+ 1.00

]2(1.19)

J(x) = J0f(x/L) = J0

[86

(x

L

)4

+

(x

L

)2

+ 1.00

]2(1.20)

where I0 and J0 are the corresponding moments of inertia at the midpoint of the

strut (x = 0). When the strut cross section is an equilateral triangle of side d, the

cross-sectional properties are given by

A =

√3

4d2 , I =

√3

96d4 , J =

A2

5√

3(1.21)

The elastic strain energy stored in the foam can be evaluated in closed form by

treating the ligaments as Bernoulli-Euler beams and including the effect of axial

and shear deformations. The axial force (N), moment (M), shear force (V ), and

torque (T ) in each ligament are established in terms of the applied loads. The

corresponding strain energy is then given by

U =

∫ 0.5

0.5

N2(x)

2EsA(x)ldx+

∫ 0.5

0.5

M2(x)

2EsI(x)ldx+

∫ 0.5

0.5

V 2(x)

2GsA(x)ldx+

∫ 0.5

0.5

T 2(x)

2GsJ(x)ldx

(1.22)

where Es and Gs are the elastic and shear moduli of the solid material. Closed

form expressions have been developed for the moduli by solving the strain energy

integral. Based on their results, the Kelvin cell model developed is capable of

capturing the elastic behavior of open-cell foams based on Kelvin cells. These

results showed that considering the amount of material distributed at the nodes is

essential to obtain results in agreement with experimental measurements.

Micromechanical models for estimating the elastic constants of foams with

tetrakaidecahedral unit cells tend to assume that the struts are of uniform cross

section to reached analytical solutions. However, a finite element analysis tak-

ing into account the varying cross section of the strut was performed to predict

21


the stiffness and strength properties of foams (Subramanian & Sankar, 2012). In

this case, three different approaches were used to determine the equivalent cross-

sectional properties in the uniform strut model: the equivalent density method, the

harmonic averaging and the equivalent bending rigidity. An accurate prediction of

the shear modulus and strength was obtained using the equivalent cross-sectional

properties.

X-ray computed microtomography (XCT) has been recently used as a non-

destructive technique to characterize cellular solids ((Bart-Smith et al., 1998);

(Elliott et al., 2002)). This allows for the study of parameters of the cellular

microstructure and of the deformation and fracture mechanisms, when coupled

with in-situ loading devices. The actual microstructure of a rigid polyurethane

foam obtained by XCT is depicted in Fig. 1.11 together with finite element mesh

superposed to the tomogram (Youssef et al., 2005). The mechanical behavior of

the foam can be computed from the finite element model built on the XCT images

and the mechanical properties of the solid materials. Moreover, the model can

be validated at the microscopic level by comparison between the local deformation

fields observed by XCT and the finite element results. However, the reconstruction

process to create a 3D volume from XCT and the generation of a finite element

mesh on the reconstructed 3D volume are difficult and time consuming tasks.

Random tessellation models combined with finite element analysis are effective

tools to study the influence of microstructural characteristics on the mechanical

properties of the foams. In most cases, Voronoi tessellations generated by ran-

dom dense packings of hard spheres have been used. For instance, (Grenestedt

& Tanaka, 1998) studied the influence of cell shape using randomly perturbed

Voronoi tessellations of BCC packings (flat faced Kelvin cells). More realistic

foams microstructures, including irregularities in distribution of materials inside

the foam, were obtained by means of the Laguerre tessellation method based on

germination sites (cell centers) originated from the dense unimodal (or bimodal)

packing of spheres (Redenbach et al., 2012).

22


Figure 1.11: (a) 3D tomography volume of a PU foam. (b) Finite element meshsuperposed to the 3D XCT volume using tetrahedral finite elements (Youssefet al., 2005).

Micromechanical modelling of closed-cell polymeric foams using Laguerre tes-

sellation and incorporating realistic cell sizes and cell wall thickness distributions

has been performed by (Chen et al., 2015), Fig. 1.12. It was found that the elastic

and shear modulus predicted by these models were in good agreement with the

experimental data. This emphasizes the fact that the integration of realistic cell

wall and cell size variations is important for foam modeling. Subsequent investi-

gation on the effects of cell size and cell wall thickness variations on the stiffness

of closed-cell foams revealed that the elastic and shear modulus decreased with

increasing cell size and cell wall thickness variations.

It should be noted, however, that computational homogenization of realistic

RVEs is expensive and cannot be used to perform fast and accurate assessments

of the effect of microstructure on the mechanical properties of the foams that can

be used during either component design or material optimization processes. These

tasks require surrogate models based on simple analytical expressions that relate

the most relevant microstructural factors with the mechanical properties. The

pioneer work in this direction was developed by Gibson & Ashby (1999) based on

the analysis of a simple cubic cell structure (see Eqs. (1.13) and (1.14)). It should

23


Figure 1.12: 3D closed-cell foam model obtained by Laguerre tessellation(Chen et al., 2015).

be noted, however, that the predictions of this model could only provide general

trends because of the simplicity of the model and the underlying assumptions, and

were not accurate in many cases Gibson & Ashby (1999); Koll & Hallstrom (2016).

More recently, Koll and Hallstrom Koll & Hallstrom (2016) carried out a para-

metric study based on computational homogenization to determine the stiffness

of closed-cell foams as a function of ρ and φ. The RVEs were built from an ini-

tial Voronoi tessellation of the space and the generated Voronoi partitions were

transformed into dry foams with a minimum total surface area. The mass of the

solid material was distributed between the cell walls and the struts, which have a

constant three-cuspid cross sections. The results of the parametrical study were

used to modify eq. (1.13) to get a better agreement with the predictions of the

numerical simulations according to

E

Es

= f(φ)

(ρ

ρs

)2

+ g(φ)ρ

ρs. (1.23)

where

24

1.5 Objectives

f(φ) = aφ2 + bφ+ c

g(φ) = dφ+ e(1.24)

and the values of parameters a, b, c, d and e can be found in (Koll & Hallstrom,

2016). The simulations also indicated that the variation of the cell size distribution

did not influence the stiffness of the foam.

1.5 Objectives

Although foams are widely used in engineering applications as structural ma-

terials, it is obvious from the information presented above that there is not a

comprehensive modeling strategy able to relate the complex microstructure of the

foams with the macroscopic mechanical properties. This is the main objective of

this doctoral thesis, which will be focused in PU foams.

The mechanical properties of the foam are controlled by the properties of the

solid material within the foam and by the microstructural details of the foam.

Thus, the first step to develop and validate the comprehensive modeling strategy

is to carry out a multiscale characterization of the material. This includes the

detailed analysis of the microstructural features using scanning electron microscopy

and X-ray microtomography as well as of the mechanical properties of the solid

material in the foam by means of instrumented nanoindentation. In addition,

the macroscopic mechanical properties will be obtained using standard mechanical

tests.

This information will be used to develop an accurate model of the topology

of the foam (cell size distribution and shape) by means of tessellation strategies.

The topological model will be enriched with the details of the strut shape and

cell wall thickness and used to build up realistic representations of the foam mi-

crostructure. These representative volume elements will be discretized and the

macroscopic mechanical properties of the foam will be obtained by means of the fi-

nite element simulation of these volumes. The whole modeling strategy (including

25

1.5 Objectives

the microscopic characterization and the computational homogenization) will be

validated by comparison with mechanical properties of the foams measured with

the macroscopic mechanical tests.

Once the multiscale modeling strategy has been validated, the next objective of

the thesis will be to use this tool to get a deeper understanding of the micromech-

anisms of deformation and fracture in PU foams under compression. In particular,

the phenomena responsible for the stiffness and the onset of plastic instability

will be studied because they control the two main mechanical properties, namely

the elastic modulus and the plateau stress. Particular emphasis will be paid the

analyze the effect of anisotropy on the mechanical properties of the foams.

It is evident that this multiscale modelling strategy cannot be used to perform

fast and accurate assessments of the effect of microstructure on the mechanical

properties of the foams to be used during either component design or material

optimization processes. These tasks require surrogate models based on simple

analytical expressions that relate the most relevant microstructural factors with

the mechanical properties. The pioneer work in this direction was developed by

Gibson & Ashby (1999). More recently, Koll & Hallstrom (2016) carried out

a parametrical study based on computational homogenization to determine the

stiffness of closed-cell foams as a function of ρ and φ. Nevertheless, there are

not comprehensive surrogate models that can predict the elastic modulus and the

plateau stress of closed-cell and open-cell foams taking into account the influence

of the mass distribution between cell walls and struts and of the cell anisotropy.

This is the last objective of the current thesis.

26

Chapter 2

Materials and Experimental

Techniques

2.1 Materials

Four rigid PU foams with different density were studied. Materials were pro-

vided by BASF Polyurethanes GmbH, Lemfoerde, Germany. They were obtained

from the reaction of a polyol resin (different polyols have been used) with a polyiso-

cyanate. Foam 1-3CPW30 was machine-made and the other foams were produced

following standard laboratory procedures developed by BASF. Table 2.1 presents

the brief description of PU foam samples. Since Lupranat M20S was used as a Iso-

cyanate for all the samples, Table 2.1 presents the differences among all these four

foams are the polyol type, the blowing agent type and the density. Consequently,

Polyol and blowing agent types are used to name the foams, together with the

nominal relative density. For instance, 1-3CPW30 indicates that polyol 1, 2 and

3 were used, together with cyclopentan (CP) and water as blowing agents. The

final nominal relative density was 30.2 kg/m3.

27

2.2 Experimental Techniques

Table 2.1: Brief description of PU foam samples

1-3CPW30 ACPW35 ACPW50 ACPW73Polyol 1,2,3 A A ABlowing agent CP+Water CP+Water CP+Water CP+WaterNominal density (kg/m3) 30.2 35 50 73Relative density, ρ/ρs 0.025 0.029 0.042 0.061

Moreover, It should be noted that samples ACPW35, ACPW50 and ACPW73

come from the same recipe (same PU composition but different density) and 1-

3CPW30 has different chemical composition.


2.2.1 Microstructural characterization

Scanning electron microscopy

Micrographs of the foam microstructure were obtained in a scanning electron

microscope (EVO MA15, Zeiss) in two different orientations (parallel and perpen-

dicular to the rise direction) to investigate on anisotropy of cells and wall thickness.

In order to make the specimens electrically conductive, they were coated with an

ultrathin coating of gold (90 nm) by sputtering. The magnification, acceleration

voltage, and working distance of SEM were x50, 10KV and 8 mm, respectively.

Micrographs were taken from different areas to take into account the statistical

variation in the cell sizes for each sample.

X-ray computed tomography

Parallelepipedic samples of 2 x 2 x 1 mm3 were machined by wire cutting from

the foams and studied by means of X-ray Computed Tomography (XCT). The

28


effective voxel size in the 3D tomograms was 1.25 x 1.25 x 1.25 µm3. The ImageJ

software tool (Schneider et al., 2012) was used to carry out the image processing.

The strut shape was analyzed by means of XCT. Several struts were chosen

from different areas of the PU foam. Each strut in the 3D image is made up by a

sequence of 2D slices, and the area of the cross-section of strut in each slice was

calculated by counting the number of voxels in each slice. The 3D images of struts

included from 32 to 64 slices depending on the strut length.

2.2.2 Micromechanical characterization

The mechanical properties of solid PU inside the foam were characterized by

means of instrumented nanoindentation (TI950, Hysitron) at room temperature.

The mayor problem to carry out nanoindentation in cellular solids is the additional

compliance due to pores beneath the loading area that can be a significant source

of inaccuracy in the final results. To overcome this limitation, two strategies were

implemented during sample preparation:

• Reduce the thickness of samples as much as was possible.

• Reduce the porosity by means of resin infiltration within the open holes on

the surface of samples to create more support beneath these vertices.

The schematic of the sample prepared for the instrumented nanoindentation

test is shown in Figure 2.1. Foam layers of 1 mm in thickness were prepared by

wire cutting machine and resin (Loctite 401) was infiltrated into the surface holes.

The specimen surface was polished in a rotating disc with abrasive papers of 2000

and 4000 grit. Indentation was carried out in triple points (vertices) that appeared

in the surface after polishing. Bulk samples of Loctite 401 were also prepared to

measure its elastic modulus by nanoindentation as well.

Nanoindentation tests were carried out with a spherical tip of 2 µm in diameter.

The indented area was examined after the test in an atomic force microscope (Park

XE150). Similar tests were carried out on the infiltrated resin within the foam

29


Figure 2.1: Schematic of the foam sample for the nanoindentation tests.

pores, and on the bulk resin specimen. The extra compliance induced by the

porosity of the foam was obtained by comparison of the load-penetration curves

obtained from the tests on the infiltrated resin in the foam pores, and from the tests

on the bulk resin. This extra compliance was subtracted from the load-penetration

curves of the PU foam to correct for the influence of the porosity in the foam.

Nanoindentation tests with a Berkovitch indenter were also carried out to de-

termine the compressive flow stress of the solid PU, following the methodology

developed by Rodrıguez et al. (Rodrıguez et al., 2012). The extra compliance in-

duced by the foam porosity was also subtracted from the load-penetration curves.

2.2.3 Thermomechanical characterization

Dynamic Mechanical Analysis (DMA) tests were carried out (Q800, TA In-

struments) to determine the viscoelastic properties of the foams as a function of

temperature. The clamp for dual/single cantilever was used. Specimens of 17.5 x

10 x 3 mm3 were prepared by wire cutting. Different frequencies and displacement

amplitudes were tested to decide the best experimental set-up. Finally, tests were

carried out under heating rate of 3 ◦C/min from room temperature to 100 ◦C, a

frequency of 1Hz and displacement amplitude of 200 µm. Two specimens of each

material were tested.

30


2.2.4 Mechanical characterization

Compression and tensile test were performed on foam cubes of 50 x 50 x 50

mm3 that were machined out of cast foam panels. The test setup followed the

standard ASTM Standard C297-04 (ASTM, 2010). The compression and tensile

tests were carried out in an universal electromechanical testing machine (Intron

3384). The experimental set-ups for the compression and tensile tests are shown

in Figure 2.2(a) and (b), respectively. The load was measured with a 2 KN load

cell. Tests were carried out under displacement control at 2 mm per minute, which

led to an approximate strain rate of 0.04 s−1.

Three samples of each foam were tested in tension and compression in two

different directions: parallel and perpendicular to the rising direction of foam.

Tests were carried out at 23.1 ◦C and 90 ◦C, the latter within an environmental

chamber.

The shear tests were carried out following the standard ASTM-C273-00 (2000)

in an universal electromechanical testing machine (Instron 3384) and load was

also measured with a 2 KN load cell (Figure 2.2(c)). Specimens of 120 x 50 x 10

mm3 were prepared by wire cutting. Tests were carried out under displacement

control at 0.5 mm per minute, which led to an approximate strain rate of 0.01

s−1. Three specimens of each material were tested in the directions parallel and

perpendicular to the rising direction. Shear tests were carried out only in room

temperature because it was not possible to fit the experimental fixture within the

environmental chamber.

It should be noted that the density of all samples tested in compression was

measured before the tests. To this end, each sample was weighted in electronic

balance and the length, height and width of the sample were measured with a

calliper. The density was calculated by dividing the mass by the volume of the

parallelepiped sample.

The displacement field (and, thus the strain field) on one surface of the spec-

imens tested in compression ant tension was measured by means of digital image

31


Figure 2.2: Experimental set-up for the mechanical tests. (a) Compression.(b) Tension. (c) Shear. The blue arrows show the loading direction.

32


Figure 2.3: Speckle pattern on surface of a specimen.

correlation (VIC 3D). To this end, a speckle pattern was applied by hand on the

specimen surface. The surface of the specimens was painted with a thin layer of

white paint and then a black mist of paint (spray paint) was applied to create the

black speckles (Figure 2.3).

33

Chapter 3Experimental results

3.1 Microstructure of PU foams

The microstructure of the four PU foams is shown in the SEM micrographs

in Figure 3.1(a) to (h). All the samples are made up of closed cells, and the

main geometrical features of the cells (diameter and aspect ratio) were determined

from cross-sections parallel (‖) and perpendicular (⊥) to the rising direction of the

foam. To calculate the cell aspect ratio, s, it was assumed that the cells have an

ellipsoidal shape with circular cross section of radius a perpendicular to the rising

direction and that the ellipsoid semi-axis along the rising direction was b. Thus,

the cell aspect ratio, s = b/a, was obtained as the ratio of the average cell area

parallel to the rising direction to the average cell area perpendicular to the rising

direction (i.e. πab/πa2 = b/a). These parameters were measured in approximately

250 cells for each foam and the average values (together with the corresponding

standard deviations) are presented in Table 3.1 for each foam in cross-sections

parallel and perpendicular to the rising direction.

The cell aspect ratio of specimen 1-3CPW30 in cross-sections parallel and per-

pendicular to rising direction was small (≈ 1.3) and very similar and this result

indicated that the cell morphology of this foam was isotropic. However, the aspect

35


Table 3.1: Microstructural features of the PU foams. D stands for the averagecell diameter, s for the average cell aspect ratio, t for the average cell wallthickness.

Foam 1-3CPW30 ACPW35 ACPW50 ACPW73

D (µm) (SEM) ⊥ 255 ± 50 519 ± 55 455 ± 47 410 ± 47D (µm) (SEM) ‖ 250 ± 32 819 ± 64 610 ± 119 707 ± 136s ⊥ 1.3 ± 0.14 1.3 ± 0.2 1.4 ± 0.1 1.2 ± 0.1s (SEM) ‖ 1.3 ± 0.09 2.1 ± 0.3 2.1± 0.4 1.7± 0.3t (µm) (SEM) 0.44 ± 0.25 0.59 ±0.04 0.59 ± 0.07 0.83 ± 0.05D (µm) (XCT) 216 ± 67 490 ±128 412 ±108 354 ±102

ratio along the rising direction was much higher than in the plane perpendicular

to the rising direction in the ACPW35, ACPW50 and ACPW75 foams, whose mi-

crostructure was anisotropic. The presence of elongated cells parallel to the rising

direction has been observed by other researchers in the PU foams (Lee & Lakes,

1997). In addition, the isotropic 1-3CPW30 foam showed the smallest average cell

size (≈ 250 µm) while the average cell size of the anisotropic foams was much

higher (in the range 600 - 800 µm in the rising direction and 400 - 500 µ in the

perpendicular direction).

The average cell wall thickness, t was measured in high resolution scanning

electron micrographs (Figure 3.2) in the middle of the cell walls. The values

reported in Table 3.1 are the average of ≈ 20 measurements for each foam. These

results indicate that the walls were significantly thicker in the foam with higher

density (ACPW73). It should be noted that wrinkles were often found in the

micrographs of the cell walls (Fig. 3.1) and this observation indicates that they

were buckled before loading due to their small thickness and were not likely to

contribute to the stiffness of the foam.

The cell size distribution of the foams was also measured by means of XCT

by (Nistor et al., 2016). In this case, the voxel-based reconstructed 3D image of

PU foams was used to measure the volume of cells by counting the number of

voxels in each cell. The average cell diameter was computed from the cell volume

assuming a spherical shape. The cell size distributions of the four foams obtained

by XCT are plotted in Figure 3.3. The distributions were very wide in all cases.

36


The isotropic foam 1-3CPW30 showed the narrowest cell size distribution and the

diameter of most of the cells was comprised between 100 and 300 µ. The cell size

distribution of the anisotropic foams was much wider.

The average cell diameters obtained from XCT were slightly different from

those measured by SEM and the differences could be attributed to the different

hypotheses to calculate the average cell diameter in each technique. From the

modeling viewpoint, the cell size distributions obtained by XCT were used to

build the representative volume elements of the microstructure. However, XCT

did not provide information about the cell aspect ratio, and the cell aspect ratios

measured by SEM in the cross-sections were used to introduce the anisotropy in

the models.

The struts that define the polyhedral shape of the cells have a three-cusp

hypocycloid cross-section generated during the foaming process (Jang et al., 2008).

Since the foaming process for all the samples was similar, the shape of the struts

was only measured by high resolution XCT in the foam 1-3CPW30. Measurements

were carried in ≈ 60 struts located in different regions of the foam. The area of dif-

ferent cross-sections perpendicular to the strut axis was determined along the strut

from the number of voxels in each section and the results are plotted, Figure 3.4(a).

The cross-sectional area of each slice of strut, normalized by the area at the

mid-span, A0, is plotted as a function of the distance to the center of the strut,

normalized by the strut length, L, in Figure 3.4(b) for various representative struts.

It was found that the experimental results were reasonably fitted by a fourth order

polynomial

A = A0f(x/L) = A0

[5.45

(x

L

)4

+ 2.63

(x

L

)2

+ 1

](3.1)

where x is the distance to the center of the strut.

Finally, the total volume of the strut was computed as a function of the strut

length for several struts and the results are plotted in Figure 3.4(c). Following

these data, it was assumed that the strut volume was proportional to its length.

37


Figure 3.1: SEM micrographs: (a) parallel and (b) perpendicular to risingdirection of the foam 1-3CPW30; (c) parallel and (d) perpendicular to risingdirection of the foam ACPW35; (e) parallel and (f) perpendicular to risingdirection of the foam ACPW50; (g) parallel and (h) perpendicular to risingdirection of the foam ACPW73.

38


Figure 3.2: Scanning electron micrographs of the wall thickness of the PUfoams. (a) 1-3CPW30 foam. (b) ACPW35 foam. (c) ACPW50 foam. (d)ACPW73 foam.

39


Figure 3.3: Cell size distribution of the PU foams measured by XCT. (a)1-3CPW30 foam. (b) ACPW35 foam. (c) ACPW50 foam. (d) ACPW73 foam.

40


(a)

1

1.2

1.4

1.6

1.8

2

-0.4 -0.2 0 0.2 0.4

Equation (1)

A/A 0

x/L

(b)4000

5000

6000

7000

8000

9000

10000

11000

35 40 45 50 55 60 65 70 75

Stru

t vol

ume

(µm

3 )

Strut length (µm)

(c)

Figure 3.4: XCT of the geometrical features of the struts. (a) Longitudinaland transverse cross-sections of the strut along its length. (b) Variation of thestrut cross sectional area (normalized by the area of the central section, A0) asa function of the distance to the center of the strut, x (normalized by the strutlength, L). (c) Volume of the strut as a function of the strut length.

41

3.2 Elastic modulus of the solid PU


The elastic modulus of the solid PU was determined following the methodology

developed by Oyen (2005) for the analysis of time-dependent mechanical behavior

of materials using load-controlled spherical indentation. Since the foams ACPW35,

ACPW50, and ACPW73 come from the same recipe (similar chemical composi-

tion), the elastic modulus was measured in the 1-3CPW30 and ACPW73 foams.

It was assumed that the properties of the solid PU in the ACPW35 and ACPW50

foams were equivalent to those of the ACPW73 foam. Moreover, the ACPW73

foam has the highest density among the ACPW foams and the microstructural

features are the larger and it was easier to perform the nanoindentation experi-

ments.

The specimens were subjected to the loading scheme depicted in Figure 3.5(a)

with an spherical indenter, which includes a loading ramp followed by a hold time

at the maximum load and final unloading. The penetration depth, h, was con-

tinuously monitored during the test and the corresponding load (P ) - penetration

(h) curve was measured, Figure 3.5(b) and (c). The experimental P - h curve was

corrected by subtracting the compliance associated with the porosity of the foam.

As indicated above, this compliance was determined by comparing the P - h curves

on infiltrated resin within the foam pores and on the bulk resin. The specimen

surface was examined by atomic force microscopy after the test to ensure that

there was not residual imprint and that deformation was confined to the elastic

regime, Figure 3.6.

The indentation depth h of an spherical indenter in the elastic regime can be

expressed as a function of the indentation load P according to (Lee & Radok,

1960)

h3/2 =3

8√R

P

2G(3.2)

where R is the spherical tip radius and G the shear modulus. In the case of a

linear viscoelastic solid, this equation is transformed into (Lee & Radok, 1960)

42


Figure 3.5: (a) Loading scheme during spherical nanoindentation. Representa-tive load (P ) - penetration depth (h) curve corresponding to the loading schemein (a) for: (b) 1-3CPW30 foam. (c) ACPW73 foam. The experimental curveswere corrected by subtracting the compliance due to the porosity in the foam.

43


Figure 3.6: Atomic force microscopy image of the indented region after nanoin-dentation.

h3/2 =3

8√R

t∫0

J(t− u)dP

dudu (3.3)

where u is a dummy variable, J(t) the material creep function which defines the

viscoelastic behavior (Desprat et al., 2005) and t is the time. It was assumed that

the behavior of the PU could be described by the standard linear viscoelastic solid

model, which includes one elastic spring in series with two Kelvin parallel spring

and dashpot elements (Figure 3.7), and it is given by

J(t) = C0 − C1 exp

(−tτ1

)− C2 exp

(−tτ2

)(3.4)

where C0, C1 and C2 stand for the compliance of the springs and τ1 and τ2 for the

time constants of the dashpots.

Eq. (3.3) has to be solved separately for the loading and holding time, Fig-

ure 3.5(a), where

44


Figure 3.7: Generalized Maxwell model for a linear viscoelastic solid includingone elastic spring in series with two Kelvin parallel spring and dashpot elements.

P (t) = kt 0 ≤ t ≤ tR

(3.5)

P (t) = Pmax tR ≤ t ≤ thold

where tR is the time corresponding to the end of loading ramp, k the slope of the

loading ramp, thold the time at the onset of unloading and Pmax the maximum

indentation load. Thus,

h3/2(t) =3k

8√R

{C0tR −

2∑i=1

Ciτi exp

(−tτi

)[exp

(tRτi

)− 1

]}tR ≤ t ≤ thold

(3.6)

The parameters of the generalized Maxwell model for the viscoelastic PU were

obtained by fitting the experimental results of h3/2 as a function of time t be-

tween tR and thold obtained in fifteen different nanoidentation experiments on each

sample to the predictions of eq. (3.6) (Figure 3.8). The experimental value of

the penetration depth, h, was corrected as indicated above to eliminate the ex-

tra compliance due to the porosity of the foam. The corresponding values of the

generalized Maxwell model for the PU can be found in Table 3.2.

45


Figure 3.8: Experimental results of the penetration depth, h3/2, vs. time, t,for solid PU. (a) 1-3CPW30 foam. (b) ACPW73 foam. The solid black linecorresponds to Eq. (3.6), which was fitted to the experimental data between tR= 20 s and thold = 140 s.

Table 3.2: Parameters of the generalized Maxwell model for PU

Foam C0 C1 C2 τ1 τ2(Pa−1 10−12) (Pa−1 10−12) (Pa−1 10−12) (s) (s)

1-3CPW30 944 ± 4 195 ± 4 194 ± 4 37.2 ± 5 17.8 ± 2ACPW73 940 ± 5 197 ± 2 199 ± 2 25.7 ± 4 10.6 ± 3

The instantaneous shear modulus can be obtained from equations (3.2) and

(3.6) for t = tR (Johnson, 1985; Oyen, 2005), leading to

G =1

2(C0 − (C1 + C2))(3.7)

and the instantaneous elastic modulus is thus given by

E = 2G(1 + ν) (3.8)

46

3.3 Compressive yield strength of the solid PU

Assuming ν = 0.35 for PU (Oyen, 2005), the elastic modulus of PU obtained

from nanoindentation experiments is 2.4 ± 0.1 GPa for 1-3CPW30 and 2.48 ±0.06 GPa for ACPW73.


A new methodology was developed by Rodrıguez et al. (2012) to determine

the compressive flow stress of amorphous materials (including polymers) from in-

strumented nanoindentation. In this approach, the amorphous solid is assumed to

follow the Drucker-Prager model, where the flow stress depends on the hydrostatic

stress through the friction angle β. Under these conditions, an universal equation

that relates the hardness with the compressive flow strength was derived from a

exhaustive numerical study of indentation using the finite element method and

was validated in a wide range of amorphous materials, from bulk metallic glasses

to polymers (Rodrıguez et al., 2012; Gonzalez et al., 2017).

In this methodology, nanoindentation was carried out using a Berkovich tip up

to a load Pmax and the indentation depth, hmax, the elastic energy stored during

indentation, We and the plastic energy dissipated during indentation, Wp, are

measured from the P - h curve, which is shown in Fig. 3.9 for 1-3CPW30 and

ACPW73 after subtracting the extra compliance due to the porosity of the foam.

From the P - h curve in Fig. 3.9, We/(We + Wp) = 0.70 ± 0.08 for the 1-

3CPW30 foam and 0.67 ± 0.07 for the ACPW73 foam. Following the analysis in

(Rodrıguez et al., 2012), the Oliver and Pharr method (Oliver & Pharr, 1992) can

be used under these conditions to determine the actual contact area, Ac from the

indentation depth h according to

Ac = 25.5h2 + 2825h (3.9)

where h and Ac are expressed in nm and nm2, respectively. Thus, the actual

hardness of the PU foam is given by

47


Figure 3.9: Representative load (P ) - penetration depth (h) curve obtainedwith a Berkovich tip to obtain the compressive flow strength of solid PU. (a)1-3CPW30 foam. (b) ACPW73 foam. The parameters of the model (Pmax,hmax, We and Wp) are indicated in the figure. The curves were corrected forthe extra compliance due to the foam porosity.

H =Pmax

Ac(hmax)(3.10)

where Ac(hmax) = 0.89 ± 0.02 µm2 and 0.86 ± 0.03 µm2 for the 1-3CPW30 and

ACPW73 foams, respectively.

The compressive flow stress of the PU can be obtained from hardness H from

Fig. 8 and equation (14) in Rodrıguez et al. (2012) if the friction angle β is known.

The friction angle of amorphous solid polymers is around 25-30◦ (Rodrıguez et al.,

2012), which coincides with the value of 28◦ measured in a solid PU provided by

BASF with similar chemical composition to that of the foam. Thus, assuming β

= 28◦, the compressive flow stress of the solid PU was 110 ± 2 and 109 ± 3 MPa

for the 1-3CPW30 and ACPW73 foams, respectively.

48

3.4 Mechanical behavior of the PU foams


3.4.1 Storage modulus

The evolution of the storage modulus (i.e. measurement of the stored energy,

representing the elastic portion of deformation of sample during DMA experiment)

with temperature obtained from the DMA experiments is plotted in Fig. 3.10 for

all the foams. The storage modulus decreased as the temperature increased in

all cases but there were significant differences among the materials. PU foams

ACPW35, ACPW50 and ACPW73 showed a marked reduction in the storage

modulus with temperature between 30 ◦C and 100 ◦C while the storage modulus of

the 1-3CPW30 foam was practically constant in this temperature range. ACPW73

shows the highest reduction in storage modulus among ACPW foams which has to

do with the higher relative density of ACPW73. Moreover, despite of having higher

density, the storage modulus of foam ACPW35 was always lower than storage

modulus of foam 1-3CPW30 in Fig. 3.10. The latter could be caused because

the ACPW35 foam was deformed perpendicular to the rising direction and the

anisotropy may influence the results of the DMA test.

3.4.2 Compressive deformation

The engineering stress -engineering strain curves of the foams in compression

at ambient temperature are plotted in Fig. 3.11. Each plot in Fig. 3.11 has two

sets of curves which stand for the mechanical behavior in compression parallel and

perpendicular to the rising direction. The shape of all the curves is typical for PU

foams: after the initial elastic region, there is a plateau region in which the foam

collapses with limited strain hardening and a final transition to a regime with high

strain hardening as a result of densification.

The mechanical properties of the 1-3CPW30 foam in both orientations, Fig.

3.11(a), show that the behavior of the foam was isotropic, in agreement with the

microstructural analysis. However, the mechanical properties of the ACPW foams

49


Figure 3.10: Evolution of storage modulus with temperature.

(Fig. 3.11(b), (c) and (d)), were highly anisotropic. Compression along the rising

direction led to a peak in the stress at the onset of nonlinear deformation (which

was much higher than the one found in the samples tested in the perpendicular

orientation), and it was followed by a stress drop and a flat plateau region. There

was no peak in the stress-strain curve at the onset of nonlinear deformation in

the samples compressed perpendicular to the rising direction, and the specimens

showed continuous strain hardening in the non linear regime. The curves in both

orientation tended to be superposed (within the experimental scatter) in the last

stages of deformation, when densification was the dominant process (compressive

strains ≥ 0.6). It should be noticed that there was a significant scatter in the values

of the peak stress in the samples compressed along the rising direction although all

the samples were extracted from the same batch. This variability seems to indicate

that peak load is very sensitive to the microstructural details and this point will

be further analyzed by the numerical simulations.

The compressive strain along the loading direction on the sample surface was

obtained by means of digital image correlation (Figure 3.12). It was found that

strain localized very rapidly in one section of the specimen after the peak in the

50


Figure 3.11: Compressive engineering stress-engineering strain curves of thePU foams in orientations parallel and perpendicular to the rising direction atambient temperature. (a) Foam 1-3CPW30. (b) Foam ACPW35. (c) FoamACPW50. (d) Foam ACPW73.

stress-strain curve that marked the onset of the plateau region (Figure 3.12a).

Further deformation led to the localization of damage in a wider zone around

the initial section in which damage was concentrated (Figure 3.12b and c). More

detailed examination of the damage in these sections showed evidence of buckling

of the cell walls and struts as well as of fracture of the cell walls. This latter result

51


Figure 3.12: Contour plot of the compressive strain along the (vertical) loadingdirection on the surface obtained by means of digital image correlation. (a)Compressive engineering strain of 7%. (b) Compressive engineering strain of10%. (c) Compressive engineering strain of 20%

is in agreement with the flat shape of the stress-strain curve up to an applied

strain of 40% because the gas pressure within the cells did not contribute to the

strengthening of the foam (Mills et al., 2009).

The engineering stress-strain curves of the same materials at 90 ◦C are plotted

in Figure 3.13. They show similar features to those measured at ambient temper-

ature although the elastic modulus and the yield strength were significantly lower

and the anisotropy between both orientations has been reduced. In addition, the

drop in stress occurred at higher strains and was slightly higher for all specimens.

The evolution of the elastic modulus in compression at room temperature with

the relative density is plotted in Figure 3.14. The elastic modulus in the rising

direction was always higher than in the perpendicular direction with the exception

52


Figure 3.13: Compressive engineering stress-engineering strain curves of thePU foams in orientations parallel and perpendicular to the rising direction at 90◦C. (a) 1-3CPW30 foam. (b) ACPW35 foam. (c) ACPW50 foam. (d) ACPW73foam.

53


Figure 3.14: Evolution of the elastic modulus of the foam, E, (normalized bythe elastic modulus of the solid PU, Es) in compression as a function of therelative density, ρ/ρs, at room temperature.

of the foam 1-3CPW30, which was isotropic. The large effect of anisotropy on

the elastic modulus of foams is clearer in the foams 1-3CPW30 and ACPW35

deformed in the perpendicular direction. Despite of having higher density, the

elastic modulus of the ACPW35 foam was lower than that of the 1-3CPW30 foam

because the cell aspect ratio in perpendicular direction is smaller for the ACPW35

foam. Moreover, the anisotropy increased with the foam density, as shown in

the ACPW foams. Additionally, the experimental scatter also increased with the

density in the rising direction in the ACPW foams.

The elastic moduli of the foams in compression at 90 ◦C are plotted in Figure

3.15 as a function of the relative foam density. The influence of the density and

of the anisotropy on the elastic modulus is equivalent to the one reported at room

temperature although the absolute values are much smaller due to the thermal

softening of the solid PU in the foam. This was not the case, however, for the

1-3CPW30 foam, which presented very similar properties at room temperature

and 90 ◦C, in agreement with the results of the DMA tests (Figure 3.10). Thus, it

can be concluded that the three main factors that control the compressive elastic

54


Figure 3.15: Evolution of the elastic modulus of the foam, E, (normalized bythe elastic modulus of the solid PU, Es) in compression as a function of therelative density, ρ/ρs, at 90 ◦C.

modulus of the PU foams are the density, the anisotropy and the mechanical

properties of the solid PU.

The plateau stress in the compressive stress-strain curves was determined as

the stress at the onset of plastic instability. This stress value was very similar to

the constant stress in the plateau region of the stress-strain curves in the isotropic

1-3CPW30 foam (Fig. fig:CompressionTesta). In the case of the anisotropic foams,

the stress at the onset of plastic instability tended to be higher than the plateau

stress (Figs. 3.11b), c) and d) but the differences were reduced by the larger scatter

in both the stress at the onset of plastic instability and the plateau stress. In the

specimens tested at 90 ◦C (Fig. 3.13), there was not a clear maximum in the

stress-strain curves after the elastic region and the stress in the plateau region was

not constant and increased with the applied strain. Thus, the plateau stress was

taken at the stress at the beginning of the non-linear deformation.

The plateau stresses of the foams in compression, σpl (normalized by the yield

strength of the solid PU at ambient temperature, σy) are plotted as a function

of the relative density in Figs. 3.16 and 3.17) at ambient temperature and 90

55


Figure 3.16: Evolution of the plateau stress of the foam at room temperature,σpl, (normalized by the yield stress of the solid PU at ambient temperature, σy)as a function of the relative density, ρ/ρs.

◦C, respectively. It is worth noting that the influence of density, temperature and

anisotropy on the plateau stress of the foams is very similar to the one reported

above for the elastic modulus, although the deformation micromechanisms that

control both properties may not be the same.

The mechanical properties of the foams in compression at ambient temperature

and 90◦C are summarized in Tables 3.3 and 3.4, respectively. They include the

results obtained in the orientations parallel (‖) and perpendicular (⊥) to the rising

direction.

Table 3.3: Mechanical properties of the solid PU in compression at ambienttemperature.

Foam 1-3CPW30 ACPW35 ACPW50 ACPW73E/Es (x 10−3) ⊥ 1 ± 0.04 0.6 ± 0.2 1.4 ± 0.1 2.3 ± 0.1E/Es (x 10−3) ‖ 1.2 ± 0.04 2.1 ± 0.1 3.4 ± 0.9 6.2 ± 2σpl/σys (x 10−3) ⊥ 1.1 ± 0.1 0.6 ± 0.05 1.5 ± 0.05 2.6 ± 0.1σpl/σys (x 10−3) ‖ 1.2 ± 0.1 2 ± 0.1 2.8 ± 0.5 5 ± 0.9

56


Figure 3.17: Evolution of the plateau stress of the foam at 90 ◦C, σpl, (nor-malized by the yield stress of the solid PU at ambient temperature, σys) as afunction of the relative density, ρ/ρs.

Table 3.4: Mechanical properties of the solid PU in compression at 90◦C.

Foam 1-3CPW30 ACPW35 ACPW50 ACPW73E/Es ⊥ 0.8 ± 0.04 0.5 ± 0.2 1.4 ± 0.2 1.4 ± 0.2E/Es ‖ 1 ± 0.04 1.1 ± 0.2 2.6 ± 0.8 5.6 ± 1.6σpl/σys ⊥ 0.6 ± 0.01 0.3 ± 0.01 0.6 ± 0.2 1.1 ± 0.1σpl/σys ‖ 0.6 ± 0.02 0.7 ± 0.03 1.3 ± 0.2 1.9 ± 0.3

3.4.3 Tensile deformation

Representative engineering stress-strain curves of all the foams deformed in

tension along orientations parallel and perpendicular to the rising direction are

shown in Figures 3.18 and 3.19 at room temperature and 90 ◦C, respectively. The

mechanical behavior of all the foams at ambient temperature was fairly brittle

and the anisotropy in the mechanical response was very clear: the elastic modu-

lus and the failure strength were much higher in the rising direction than in the

perpendicular one. On the contrary, the failure strain was much higher in the

perpendicular direction. The brittle behavior comes about as a consequence of the

brittle nature of the PU foam at ambient temperature. The mechanical response

57


Figure 3.18: Tensile engineering stress-engineering strain curves of the PUfoams in orientations parallel and perpendicular to the rising direction at ambi-ent temperature. (a) 1-3CPW30 foam. (b) ACPW35 foam. (c) ACPW50 foam.(d) ACPW73 foam.

at 90 ◦C showed the same features, although there was a significant non-linear

behavior before failure. Nevertheless, the failure process was brittle in all cases.

The variation of the elastic modulus in tension at room temperature with foam

density is plotted in Figure 3.20. The influence of the density and of the anisotropy

58


Figure 3.19: Tensile engineering stress-engineering strain curves of the PUfoams in orientations parallel and perpendicular to the rising direction at 90 ◦C.(a) 1-3CPW30 foam. (b) ACPW35 foam. (c) ACPW50 foam. (d) ACPW73foam.

59


Figure 3.20: Evolution of the elastic modulus of the foam, E, (normalized bythe elastic modulus of the solid PU, Es) in tension as a function of the relativedensity, ρ/ρs, at room temperature.

was similar to that reported in compression. All the foams were stiffer in the rising

direction with the exception of the 1-3CPW30 foam, which showed an isotropic

behavior. It should be noted that the experimental scatter in the elastic modulus

did not increase with density in the ACPW foams, as opposed to the behavior re-

ported in compression. It should be noted that the elastic modulus of all specimens

in tension was almost twice higher than the compressive one for the same foam

and orientation. This has never been reported by other researchers who worked on

polymeric foams. The variation of the elastic modulus in tension as a function of

the foam density at 90 ◦C is plotted in Figure 3.21. Thermal softening caused the

reduction in elastic modulus of all the foams, with the exception of the 1-3CPW30

material, as in the case of compression.

Similar trends were observed in tensile strength (σut) of the foams, which is

plotted in Figures 3.22 and 3.23 at ambient temperature and 90 ◦C, respectively.

However, the effect of density on the tensile strength was not always so obvious,

very likely because of the brittle behavior of the PU foams in tension.

60


Figure 3.21: Evolution of the elastic modulus of the foam, E, (normalized bythe elastic modulus of the solid PU, Es) in tension as a function of the relativedensity, ρ/ρs, at 90 ◦C.

The mechanical properties of the foams in tension at ambient temperature

and 90◦C are summarized in Tables 3.5 and 3.6, respectively. They include the

results obtained in the orientations parallel (‖) and perpendicular (⊥) to the rising

direction.

Table 3.5: Mechanical properties of the solid PU in tension at ambient tem-perature.

Foam 1-3CPW30 ACPW35 ACPW50 ACPW73E/Es (x 10−3) ⊥ 2.5 ± 0.02 1.5 ± 0.2 2.9 ± 0.2 4.8 ± 0.2E/Es (x 10−3) ‖ 2.8 ± 0.04 6.7 ± 0.7 8 ± 0.1 11.6 ± 0.5σut/σys (x 10−3) ⊥ 1.5 ± 0.1 1.5 ± 0.2 2.5 ± 0.5 3.9 ± 0.3σut/σys (x 10−3) ‖ 1.5 ± 0.1 4.4 ± 0.4 4.2 ± 0.3 6.7 ± 0.6

Table 3.6: Mechanical properties of the solid PU in tension at 90 ◦C.

Foam 1-3CPW30 ACPW35 ACPW50 ACPW73E/Es (x 10−3) ⊥ 1.75 ± 0.04 1.3 ± 0.1 1.6 ± 0.2 2.9 ± 0.2E/Es (x 10−3) ‖ 2.5 ± 0.05 2.7 ± 0.3 4.2 ± 0.2 7 ± 0.5σut/σys (x 10−3) ⊥ 1.2 ± 0.1 0.9 ± 0.1 1.4 ± 0.2 2.4 ± 0.4σut/σys (x 10−3) ‖ 1.2 ± 0.1 1.6 ± 0.2 2.6 ± 0.5 4.0 ± 0.3

61


Figure 3.22: Evolution of the tensile strength of the foam at ambient tempera-ture, σut, (normalized by the yield stress of the solid PU at ambient temperature,σy) as a function of the relative density, ρ/ρs.

Figure 3.23: Evolution of the tensile strength of the foam at 90 ◦C, σut,(normalized by the yield stress of the solid PU at ambient temperature, σy) asa function of the relative density, ρ/ρs.

62


Figure 3.24: Contour plot of the tensile strain along the (vertical) loadingdirection on the surface of foam 1-3CPW30, obtained by means of digital imagecorrelation at the strain of 3%.

Fracture of the foams in tension took place by the sudden propagation of a

crack perpendicular to the loading axis. This is shown in the contour plot of

the vertical strain (parallel to the loading axis) at an engineering strain of 0.03

in the 1-3CPW30 foam tested at ambient temperature (Figure 3.24). The crack

was nucleated at the specimen surface, close to the mobile clamp of tensile test-

ing machine and propagated rapidly along the whole cross-section, leading to the

catastrophic failure of the specimen.

3.4.4 Shear deformation

Representative engineering stress-strain curves of all foams deformed in shear

at ambient temperature in orientations parallel and perpendicular to the rising

direction are shown in Figure 3.25. The mechanical response in shear of all the

foams was similar from a from a qualitative viewpoint: the initial elastic response

finished at the yield point and the foams presented linear hardening in the non-

linear regime until failure. The shear failure strains were in the range 0.2 to 0.45,

much higher than those found in tension, and shear deformation in the non-linear

63


Figure 3.25: Shear engineering stress-engineering strain curves of the PUfoams in orientations parallel and perpendicular to the rising direction at ambi-ent temperature. (a) 1-3CPW30 foam. (b) ACPW35 foam. (c) ACPW50 foam.(d) ACPW73 foam.

regime was accommodated by the formation of parallel shear bands (Fig. 3.26).

This damage mechanism was common to all foams.

The mechanical response of the isotropic 1-3CPW30 foam was equivalent in

both orientations (Fig. 3.25a). In the case of the anisotropic foams, the initial

64


Figure 3.26: Formation of shear bands during the shear test in the 1-3CPW30foam. The image was taken at the failure point.

shear modulus and the yield strength in the perpendicular orientation were smaller

than those in the parallel orientation (Fig. 3.27). Moreover, the shear strain at

failure in the specimens deformed in the perpendicular orientation was much higher

than that measured in the perpendicular orientation (Fig. 3.25b, c and d). The

slope of the linear hardening region was independent of the orientation and the

shear strength depended on the combination of the yield strength and the shear

strain at failure.

The mechanical behavior th PU foams in shear was characterized by the shear

modulus, G, the shear strength, τu and the shear strain at failure, γu. These

values are depicted in Table 3.7 for the different foams deformed in the orientations

parallel (‖) and perpendicular (⊥) to the rising direction.

In summary, the results of the mechanical tests presented above indicate that

the foam density, the cell anisotropy and the mechanical properties of the solid PU

within the foam are the most important microstructural factors that determine the

65


Figure 3.27: Evolution of the shear modulus of the foam, G, (normalized bythe elastic modulus of the solid PU, Es) in shear as a function of the relativedensity, ρ/ρs, at room temperature.

Table 3.7: Mechanical properties of the solid PU in shear at ambient temper-ature.

Foam 1-3CPW30 ACPW35 ACPW50 ACPW73G/Es (x 10−3) ⊥ 0.9 ± 0.1 0.6 ± 0.04 1.2 ± 0.3 1.7 ± 0.1G/Es (x 10−3) ‖ 0.9 ± 0.1 0.9 ± 0.1 1.5 ± 0.4 2.6 ± 0.1τu (MPa) ⊥ 0.11 ± 0.03 0.14 ± 0.01 0.2 ± 0.03 0.4 ± 0.1τu (MPa) ‖ 0.1 ± 0.05 0.1 ± 0.01 0.2 ± 0.01 0.45 ± 0.02γu ⊥ 0.2 ± 0.05 0.4 ± 0.03 0.4 ± 0.04 0.3 ± 0.04γu ‖ 0.23 ± 0.03 0.26 ± 0.01 0.25 ± 0.01 0.2 ± 0.01

mechanical properties of the foam. While the effect of density is well established in

the literature, the influence of the anisotropy and of the solid PU properties have

received very limited attention in the past. Anisotropy plays a critical role in the

elastic modulus and the plateau stress in compression and in the elastic modulus

and the strength in tension, while its influence is important (but more limited)

in shear. Moreover, the influence of the mechanical properties of the solid PU

on the overall mechanical response in tension and compression was very obvious

when the foams were tested at ambient temperature and 90 ◦C. The computational

66


homogenization strategy that will be presented in the next chapter will try to take

rigorously into account the effect of the solid PU properties and of the anisotropy

on the mechanical behavior of the PU foams.

67

Chapter 4Computational homogenization

strategy

Following the standard strategies for heterogeneous solids, the mechanical be-

haviour of the PU foam can be determined by means of the finite element simu-

lation of a representative volume element (RVE) of the microstructure. The first

step is to create an accurate geometrical model of the foam microstructure by

means of the Laguerre tessellation, according to the strategy presented by Chen

et al. (2015). To this end, the algorithm to obtain a random close packing distri-

bution of polydisperse spheres is first discussed. Afterwards, the different entities

of the geometrical model (cell walls and edges) are suitably discretized into finite

elements to carry out the numerical simulation of the mechanical behavior. Fi-

nally, the details of computational homogenization by means of the finite element

method are presented. The overall modeling strategy is summarized in Figure 4.1.

69

Figure 4.1: Summary of computational homogenization modelling strategy.

70

4.1 Random packing of spheres

4.1 Random packing of spheres

The foam microstructure is formed by a 3D arrangement of cells. The cells were

isotropic and the cell sizes were assumed to follow a Gaussian distribution, which is

characterized by the average cell diameter, µ, and the standard distribution of the

cell diameter, σ. Each cell is assimilated to a sphere and the location of the centers

of the cells in the 3D space can be obtained from the positions of the centers of a

3D ensemble of random close-packed spheres with the same diameter distribution

as the foam.

Dense packing of spheres with higher volume fraction can be achieved using

collective rearrangement algorithms (Segurado & Llorca, 2002; Torquato, 2001).

Among them, the force biased algorithm is widely used because of its high efficiency

(Bargie l & Moscinski, 1991). This algorithm starts with an initial distribution of

n0 spheres S(x i, ri) characterized by the position of the center x i and the radius

ri distributed in a parallelepipedic container. In this stage, overlapping of spheres

is possible and allowed. Then, the algorithm attempts to reduce the overlaps be-

tween spheres by pushing apart overlapped spheres while small spheres are pushed

to fill the empty spaces between large ones. After certain number of iterations,

repositioning of overlapped spheres is stopped and the spheres gradually shrank

to reduce the total amount of overlaps below a certain threshold. Finally, the co-

ordinates of the centers of the spheres and their diameter are provided as output.

This algorithm can provides random packed configurations of spheres with a given

size distribution with a volume fraction of 0.6.

4.2 Laguerre tessellation

A tessellation is a locally finite division of the 3D space into polyhedrons which

intersect only in their boundaries. The most popular one is the Voronoi tessellation

which is generated by a set S of points in the three dimensional space by assigning

a volume Vxito the point Pi(xi) ∈ S formed by all points P (y) in the space which

71

4.2 Laguerre tessellation

have Pi as their nearest neighbour (Redenbach, 2009). Cell facets in the Voronoi

construction are, however, always equidistant from the generators of their cells

and the range of cell patterns which can be generated is limited. This is why

weighted generalizations of the Voronoi model are frequently used. One possible

generalization is the Laguerre tessellation in which the volume Vxiin the space

associated to the point Pi ∈ S is formed by the points P (y) that fullfill the

condition

P (y) ∈ Vxiif dL(y,xi) < dL(y,xj), j 6= i and xj ∈ S (4.1)

where dL(y,xi) is the ”Laguerre” distance between points y and xi, which is given

by

dL(y,xi) = ‖xi − y‖2 − r2i (4.2)

where ri (> 0) is the weight associated to point Pi. This definition leads to a

partition of the space into the Laguerre tesselation formed by convex, space-filling

polyhedrons. It can be shown (Lautensack, 2007; Xue et al., 1997) that if S is

chosen as a system of nonoverlapping spheres characterized by the coordinates of

their centers, xi and the corresponding radius, ri, each cell of Laguerre tessellation

completely contains its generating sphere and the volume distribution of Laguerre

cells is almost equal to volume distribution of their generating spheres.

Thus, the information provided by the algorithm for random packing of spheres

was used to create a Laguerre tesselation of a cubic volume of the foam using Quey

et al. (2011). This 3D array of polyhedrons provides the topology of the foam and

the faces and edges will become the cell walls and struts of the foam, respectively.

The cell size distribution of tessellated volume is very close to the experimental

one (Figure 3.3). A cubic RVE containing a Laguerre tessellation of 100 cells is

depicted in Figure 4.2.

72

4.3 Finite element model

Figure 4.2: Laguerre tessellation of a cubic RVE containing 100 cells. The cellsize distribution follows a Gaussian function with an average cell size diameterof 216 µm and a standard deviation of 67 µm.


The Laguerre tessellation in a cubic volume of dimensions L × L × L was

used as a RVE of the foam microstructure (Figure 4.2). The tessellated volume

was exported to the open-source software Gmsh (Geuzaine & Remacle, 2009) to

perform the finite element discretization. Euler-Bernoulli beam elements (type B31

in Abaqus (2016)) and shell elements (type S3R in Abaqus (2016)) were used to

discretize struts and cell walls, respectively. The total volume of the solid material

in the foam was distributed between the struts and the cell walls. The analysis by

XCT of the foam microstructure showed that the volume of the struts increased

linearly with the strut length. Thus, the volume of strut i, Vi, is given

Vi =VstrutsLi

Lstruts

(4.3)

where Vstruts and Lstruts are, respectively the total volume and length of the struts

in the RVE.

73


In addition, the variation of the area of the strut cross-section along its length

was measured by means of XCT in struts on different length and was given by

Ai = A0f(x/Li), see eq. (3.1).

The number of beam elements to model each strut increased with the strut

length and the variation of the strut cross-section along its length was taken into

account in the model by assigning different cross sectional areas to the beam el-

ements. If n is the number of beam elements in the strut i with volume Vi, the

cross sectional area of the beam element j in the strut i, Aji , is given by,

Aji =

∫ −Lin

j

−Lin

(j−1)A0if(x/Li)dx (4.4)

where j = 1, ..., n.

The beam element section was an equilateral triangle (that can be approxi-

mated from the trapezoidal section in Abaqus (Abaqus, 2016)), which is the closest

one to the three-cusp hypocycloid cross-section of struts in the foam microstructure

(Jang et al., 2008).

Faces of the polyhedrons, which represent the cell walls, were discretized with

S3R shell elements of constant thickness, assuming that the thickness of the cell

walls was constant in the foam. It should be noted that shell elements and beam

elements share the nodes on the struts. As a result, the number of shell elements

in the RVE is controlled by the number of beam elements in the struts. Thus,

increasing the number of beam elements in the struts provides a more accurate

description of cross section variation along the struts, but leads to a large increase

in number of shell elements and a compromise has to be reached. A RVE of the

foam with 100 cells and 345000 elements is shown in Figure 4.3. The inset shows

clearly the variation of the cross section along one strut.

The topology of the polyhedrons obtained with above strategy, after tessellation

step, is isotropic and the structural anisotropy of the foam in the rising direction

was included by means of the affine deformation of all the nodes in the RVE by

a factor s, the average aspect ratio of the foam. Of course, the dimensions of the

74


Figure 4.3: Finite element discretization of a RVE with 100 cells. The variationof the cross section of the beam elements along the strut can be observed in theinset.

RVE were also changed by the same amount (Figure 4.4). In this case, the typical

number of beam elements to discretize a strut of average length was around 30-33.

This number was large enough to obtain results that were independent of the finite

element discretization (Marvi-Mashhadi et al., 2017b).

Numerical simulations of the mechanical behavior of the RVE of the foam, with

the initial unstressed state as reference, were carried out using the finite element

software Abaqus (Abaqus, 2016) within the framework of the theory of finite de-

formations and rotations. The solid PU foam was modeled as an isotropic, elastic-

perfectly plastic solid following the J2 theory of plasticity. The elastic constants

and the compressive flow strength were obtained from the nanoindentation tests

and are summarized in Table 4.1. Both implicit (Abaqus/standard) and explicit

(Abaqus/explicit) simulations were carried out, the former to determine the elastic

constants and the latter to obtain the plateau stress. The rotational degrees of

freedom (uR, vR,wR) around the axes contained in each of the 6 boundary faces of

the model were constrained (see Figure 4.3). In addition, the translational degrees

of freedom normal to three of the faces of the RVE were constrained, i.e. planes

75


(a) (b)

(c)

Figure 4.4: Transformation of an isotropic RVE (a) into an anisotropic RVE(b) and, finally, to the meshed RVE (c). The inset in (c) shows clearly thevariation of the cross section of the beam elements along the struts.

76


Table 4.1: Mechanical properties of solid PU

Foam E ν σys(GPa) (MPa)

1-3CPW30 2.40 0.35 110ACPW73 2.48 0.35 109

Y Z (u=0), XZ (v=0), and XY (w=0). The compression load was applied by

means of the relative displacement of rigid surfaces (load-plates) in contact with

the RVE at the faces XZ and X ′Z ′. In this way, the elements of the model were

prevented from overlapping the boundaries of RVE as a result of buckling. The

friction between the rigid surfaces and the RVE was neglected.

77

Chapter 5

Simulation results and discussion

The computational homogenization strategy presented in section 4 was used to

simulate the mechanical behavior of the PU foams. This process was carried out

in fours steps. In the first step, the details of the computational homogenization

strategy (minimum size of the RVE, maximum element size for the discretization

of the RVE, etc.) were adjusted on the model created to simulate the compressive

deformation of the isotropic 1-3CPW30. The simulations were able to predict accu-

rately the elastic modulus and the stress at the onset of instability in compression

for this foam and, thus, the simulation strategy based on multiscale characteriza-

tion and computational homogenization was validated.

The same simulation strategy was then applied to analyze the mechanical be-

havior of the anisotropic foams with different density (ACPW35, ACPW50 and

ACPW73). The effect of the cell anisotropy on the mechanical properties of the

foams in compression was again captured by the modeling strategy, which was

able to reproduce the differences in the elastic modulus and the stress at the onset

of instability in the orientations parallel and perpendicular to the rising direc-

tion. Moreover, the mechanisms of deformation and failure responsible for these

differences were ascertained through the analysis of the deformation of the RVEs.

79

5.1 Minimum RVE and mesh size analysis

The simulations carried out in the first two steps were only focused in the ini-

tial stages of deformation of the foams in compression (up to 10 % strain). This

regime is enough to capture the elastic properties and the plateau stress, which

are the critical design parameters, and does not require to take into account the

densification processes which take place at larger strains. However, the simula-

tion strategy is able to simulate the whole stress-strain curve in compression if the

contacts between the solid elements of the foam are included although the compu-

tational cost is much larger. Thus, the whole compressive stress-strain curve was

simulated in the case of the isotropic foam to fully demonstrate the capabilities of

the simulation strategy.

Finally, the computational homogenization strategy was used to carry out a

parametric study of the influence of the microstructural parameters on the elastic

modulus and the plateau stress of the foams in compression. This information was

used to build a surrogate model of the mechanical behavior of closed- and open-cell

foams in compression.

The foam densities mentioned in section 2 and the solid PU density was 1200

kg/m3. The cell size distribution used in the simulations was Gaussians with the

parameters obtained by XCT for the PU foam (Table 3.1).


A first set of simulations was carried out to assess the minimum RVE size as

well as the minimum element size to obtain accurate results. The parameters of the

isotropic 1-3CPW30 foam were used in these simulations. Thus, the foam density

was 30.2 kg/m3, the solid PU density was 1200 kg/m3 and the cell size distribution

followed the Gaussian distribution in Fig. 3.3(a). The mechanical properties of the

solid PU were obtained from section 3 for the 1-3CPW30 foam.

It was assumed that 20% of the solid material was in the cell walls and 80% in

the struts. The figure of merit was the value of the elastic modulus and simulations

were performed using Abaqus/Standard (Abaqus, 2016). For the determination

80


Figure 5.1: Influence of the number of cells in the RVE on the elastic modulusof the PU foam, E, (normalized by the elastic modulus of the solid PU, Es).

of the minimum RVE size, simulations were carried out with RVEs containing 50,

100, 150 and 200 cells. The cell size distribution was the same in all cases (Figure

3.3) and five different cell realizations were analyzed for each RVE size. All the

simulations were carried out using a minimum element size in the beams elements

of 0.02 mm. With this element size, a strut of average length in the microstructure

was discretized with 21-22 beam elements.

The average value of the elastic modulus of the foam (normalized by the solid

PU elastic modulus) and the standard deviation of the five realizations are plotted

in Figure 5.1 as a function of the number of cells in RVE . While simulations with

50 cells lead to a large scatter, simulations with 100 cells or more led to very similar

results and, thus, RVEs with 100 cells were used in the numerical simulations to

validate the modeling strategy.

In addition to the RVE size, the influence of the finite element discretization

on the simulation results was assessed in an RVE with 100 cells. The preliminary

results revealed that the element size affects the mechanical response of the RVE.

On the one hand, the number of beam sections on the struts is not large enough

to provide a smooth bending of strut during deformation, the buckling of struts

81

5.2 Simulation of the isotropic foam

and, consequently of the walls, is delayed, leading to a stiffer response as well as

to a higher plateau stress. On the other hand, as mentioned above, reducing the

element size in the struts can severely increase the computational cost. As a result,

minimum element sizes of 0.01, 0.02, 0.05 and 0.10 mm were chosen and the elastic

modulus of the RVE was computed. Similar results were obtained from minimum

element sizes of 0.01 mm and 0.02 mm and this latter value was used in all the

remaining analyses.


5.2.1 Experimental validation

RVEs containing 100 cells were used to simulate the mechanical behavior in

compression of the PU foam 1-3CPW30. The foam density was 30.2 kg/m3, the

cell size distribution followed the one depicted in Figure 3.3(a) and the thickness

of the cell walls was changed between 0.15 µm and 0.6 µm, which correspond to

5% and 23% of solid PU in the cell walls. The main reason to study the different

wall thickness was to cover the experimental variation observed in wall thickness

SEM measurements. The mechanical properties of the solid PU were obtained

from section 3 for the 1-3CPW30 foam.

Five different cell realizations were simulated with Abaqus/standard for each

cell wall thickness and the computed values of the average elastic modulus are

shown in Figure 5.2 as a function of the cell wall thickness. These results show

that the elastic modulus increases rapidly as material is transferred from the struts

to the cell walls, for this particular foam density and cell size distribution. In

addition, the numerical predictions of the elastic modulus were in good agreement

with the experimental data of the foam with an average cell wall thickness of 0.44

µm.

Implicit calculations carried out with Abaqus/Standard were not able to reach

the plateau stress because of instabilities (i.e. buckling of shell and beam ele-

82


Figure 5.2: Influence of the cell wall thickness on the elastic modulus of the PUfoam. The simulation results correspond to the average value and the standarddeviation of five different cell realizations for each cell wall thickness.

ments) impaired the convergency. Thus, explicit simulations were performed with

Abaqus/Explicit up to a deformation of 6.5% using the same finite element dis-

cretization.

Three strategies were used to ensure that quasi-static conditions were met

during the explicit analysis. Firstly, it was verified that the kinetic energy was

always less than 2% of the internal energy. Secondly, the reaction forces on the top

and bottom surfaces were monitored when compressive displacements were applied

either on the top or on the bottom plate. Similar reaction forces were observed

from these two analysis showing the validity of the quasi-static condition. Finally,

the reaction forces extracted from reference nodes of top and bottom plates were

equal throughout the analysis.

Three experimental engineering stress-engineering strain curves of the foam

tested in compression are plotted in Figure 5.3 up to 6.5 % strain, together with

numerical simulations of three different RVEs. The numerical results show clearly

83


Figure 5.3: Experimental results and numerical simulations of the engineeringstress - engineering strain curves in compression of the PU foam. The experi-mental data correspond to three tests while the numerical simulations show theresults obtained with three different realizations of the cell distribution.

the different regions in the stress-strain curves. The linear elastic regime is followed

by a small drop in stress at the onset of yielding and continues with a non-linear

deformation at an almost constant plateau stress, in excellent agreement with the

experimental results. Both experimental data and the numerical results showed

very limited scatter in the elastic modulus (which is controlled by the averaged

properties of the foam) and much larger scatter in the plateau stress (which de-

pends on the localization of damage).

5.2.2 Deformation and failure micromechanisms

The analysis of the simulated microstructure provided more information about

the dominant deformation and damage mechanisms of the PU foam during com-

pression. Cross-sections of the foam microstructure at the middle along the Z

axis are depicted in Figures 5.4a to c for different values of the compressive strain,

84


0%, 3%, 5% and 6%, respectively. The straight lines in Figure 5.4a correspond to

the intersection of the plane perpendicular to the Z axis with the cell walls. It is

important to notice that buckling of the cells faces that are oriented to ± 45◦ from

the compression axis occurs very early during the elastic regime, Figure 5.4b, as

indicated by the ripples in the cell walls. The early buckling of faces is not sur-

prising, since the face thickness was 0.44 µm and this extremely thin faces could

be bent and buckled easily. This mechanism was also observed when the applied

strain reached 6%, above the critical strain leading to the onset of the plateau

region but it is evident that cell wall buckling was not responsible for the plateau

region and localization of damage.

The deformation of the struts is depicted in Fig. 5.5, which shows the contour

plot of the axial force in the struts for different values of the applied compressive

strain (0%, 3% and 6%). The load in the elastic regime, Fig. 5.5b, is homogeneously

distributed among the struts in the network. They are loaded in compression

(struts in red and yellow) with the exception of those oriented perpendicularly to

the deformation axis (struts in black). The compressive load in the struts increases

with deformation and, in addition, the stresses carried by the struts show large

differences at an applied strain strain of 6%, Fig. 5.5c. This indicates that the

deformation is no longer homogeneous and the plateau stress is associated with

the localization of damage. In fact, the struts located at the top after 6% strain

have buckled in response to the high compressive load along their axis, as shown

in the inset of Fig. 5.5c.

In summary, the numerical simulations of the mechanical behavior of the foam

in compression were in agreement with the experimental results of the elastic

modulus and of the stress at the onset of instability. The detailed analysis of

the numerical simulations showed that the load applied to the foam was mainly

supported by the struts, while buckling of the very thin cell walls occurred in the

early stages of the elastic deformation. The onset of plastic instability was dictated

by the plastic buckling of the struts in one section of the foam perpendicular to

the applied load, in agreement with the mechanisms observed in the macroscopic

samples by means of digital image correlation.

85


Figure 5.4: Central cross-section of the RVE showing the deformation of thecell walls. (a) Initial cell wall structure. (b) Cell wall structure after 3% com-pressive strain. (c) Cell wall structure after 6% compressive strain. Compressionwas applied along the vertical Y axis.

86


a)

c)

b)

Y

XZ

4

8

12

0

4

Figure 5.5: Contour plot of the axial force in the strut network of the RVEcontaining 100 cells. (a) Initial strut configuration. (b) Compressive strain 3%.(c) Compressive strain 6%. Compression was applied along the Y axis. Theload in the legend is expressed in mN.

87

5.3 Simulation of the anisotropic foams


RVEs with 100 cells of the three foams (ACPW35, ACPW50 and ACPW73)

were created from the experimental values of the relative density, Gaussian cell

size distribution, cell wall thickness and cell aspect ratio in Table 3.1 and Figures

3.3(b), (c) and (d). The mechanical properties of the solid PU were obtained

from section 3 for the ACPW73 foam. Numerical simulations of the mechanical

behavior in compression were carried out for each foam in the rising orientation

and in another perpendicular direction. Three different statistical realizations of

the RVE were simulated in each case.

5.3.1 Experimental validation

The engineering stress-engineering strain curves of the ACPW35, ACPW50

and ACPW73 foams obtained from the numerical simulations are plotted in Figs.

5.6(a), (b), and (c), respectively. Results of three simulations parallel and perpen-

dicular to the rising direction are plotted in each figure, together with the shaded

areas which enclose the scatter of the experimental results. The main objective

of the numerical simulations was to capture the elastic stiffness and the stress

at the onset on linearity of the foams, which are the main design parameters for

these materials. It was not possible, however, to extend the simulations to larger

strains due to the development of numerical instabilities resulting from the exten-

sive buckling of the shell elements and they were stopped at ≈ 3% and ≈ 6-8%,

depending on the orientation.

Overall, the numerical simulations were able to capture the strong effect of

anisotropy in the stiffness and in the plateau stress of the foams as well as the

differences in the stress-strain curves observed experimentally. Deformation along

the rising direction led to a very stiff response and non-linearity in the stress-

strain curve was only observed near the peak stress. Moreover, a sudden drop in

the stress was observed after the peak in several simulations, in agreement with the

experiments (Fig. 3.11). The mechanical response in the direction perpendicular to

88


0 . 0 0 0 . 0 3 0 . 0 6 0 . 0 90 . 0

0 . 2

0 . 4

0 . 6E x p . S i m .

Engin

eerin

g stre

ss (M

Pa)

E n g i n e e r i n g S t r a i n

P a r a l l e l P e r p e n d i c u l a r

( a )

0 . 0 0 0 . 0 3 0 . 0 6 0 . 0 90 . 0

0 . 2

0 . 4

0 . 6

Engin

eerin

g stre

ss (M

Pa)


( b )E x p . S i m .


0 . 0 0 0 . 0 3 0 . 0 6 0 . 0 90 . 0 0

0 . 1 4

0 . 2 8

0 . 4 2

0 . 5 6

0 . 7 0

Engin

eerin

g stre

ss (M

Pa)


( c )

E x p . S i m . P a r a l l e l P e r p e n d i c u l a r

Figure 5.6: Numerical predictions of the engineering stress-strain curves of thePU foams parallel and perpendicular to the rising direction. (a) ACPW35, (b)ACPW50 and (c) ACPW73. The shaded areas enclose the experimental results.

89


Table 5.1: Experimental and simulations results of the elastic modulus E (inMPa) of the anisotropic PU foams in the orientations parallel (‖) and perpen-dicular (⊥) to the rising direction.

Foam ACPW35 ACPW50 ACPW73Exp. (‖) 6.5 ± 0.3 8.7± 3 19.5 ± 2Sim. (‖) 7.5 ± 0.24 10 ± 0.3 23.7 ± 0.6Exp. (⊥) 1.8 ± 0.16 3.6 ± 1 7.4 ± 0.7Sim. (⊥) 1.9 ± 0.04 4.3 ± 0.01 4.86 ± 0.02

the rising direction was more compliant, the non-linearity in the stress-strain curve

was noticed at much smaller stress and the plateau stress was also significantly

reduced. Moreover, no sudden drop in the stress was detected after the plateau

stress was attained.

The average values of the elastic modulus and of the stress at the onset of insta-

bility (together with the corresponding standard deviations) are depicted in Tables

5.1 and 5.2, respectively. They are in good agreement and validate the capabilities

of the simulation strategy but it should be noticed that the experimental scatter

(particularly for the plateau stress) is very large and this is not well captured by

the simulations. This difference, may be due to the limited size of the RVEs in

the simulations, which only contain 100 cells. The plateau stress is attained when

damage is localized in one section of the foam and this process is very dependent

on the microstructural details, including the presence of broken struts and cell

walls, cells with longer or shorter aspect ratio, etc.

The particular of effect of the structural anisotropy on the anisotropy in the

elastic modulus and on the plateau stress is depicted in Tables 5.3 and 5.4, respec-

tively, for the three foams. The experimental values of the anisotropy (E‖/E⊥ and

σp‖σp⊥) are included in these tables, together with the predictions of the simula-

tions and of eq. (1.17). The experimental anisotropies of the stiffness in the case

of the ACPW35 and ACPW50 foams are in good agreement with the numerical

simulations and with the predictions of Gibson & Ashby (1999) based on a simple

cubic cell geometry. It is worth noting that this simple model provided estima-

90


Table 5.2: Experimental and simulations results of the plateau stress, σp (inMPa) of anisotropic PU foams in the orientations parallel (‖) and perpendicular(⊥) to the rising direction.

Foam ACPW35 ACPW50 ACPW73Exp. (‖) 0.23 ± 0.01 0.28 ± 0.1 0.61 ± 0.1Sim. (‖) 0.16 ± 0.01 0.28 ± 0.01 0.55 ± 0.03Exp. (⊥) 0.06 ± 0.01 0.13 ± 0.03 0.28 ± 0.02Sim. (⊥) 0.07 ± 0.01 0.17 ± 0.01 0.21 ± 0.01

Table 5.3: Influence of the cell aspect ratio, s, on the anisotropy in the elasticmodulus in the orientations parallel (E‖) and perpendicular (E⊥) to the risingdirection.

Foam ACPW35 ACPW50 ACPW73s 1.57 ± 0.20 1.34 ± 0.30 1.72 ± 0.38

E‖/E⊥ (Exp.) 3.6±0.4 2.4±1 2.6±0.4E‖/E⊥ (Sim.) 3.93±0.01 2.35±0.03 4.87±0.02

E‖/E⊥ (Eq. (1.17)) 3.66±0.02 2.55±0.03 5.11±0.02

tions of the anisotropy very close to those obtained with numerical simulations.

However, both models overestimated the mechanical anisotropy in the case of the

ACPW73 foam. In particular, the numerical simulations were able to predict cor-

rectly the stiffness and strength in the rising direction but underestimated both of

them in the perpendicular direction (Fig. 5.6c).

Table 5.4: Influence of the cell aspect ratio, s, on the anisotropy in the plateaustress in the orientations parallel (σp‖) and perpendicular (σp⊥) to the risingdirection.

Foam ACPW35 ACPW50 ACPW73s 1.57 ± 0.20 1.34 ± 0.30 1.72 ± 0.38

σp‖/σp⊥ (Exp.) 3.89±0.5 2.10±0.3 2.17±0.2σp‖/σp⊥ (Sim.) 2.30±0.17 1.65±0.04 2.61±0.02

91


5.3.2 Deformation and failure micromechanisms

Although it is well known that structural anisotropy plays a very important role

on the mechanical properties of PU foams (Gibson & Ashby, 1999; Kanakkanatt,

1973), the mechanisms responsible for this behaviour are not well understood. This

objective can be achieved, however, through the analysis of the deformation and

failure micromechanisms within the RVE in the numerical simulations.

The contour plot of the axial forces in the struts in the RVE of the ACPW73

foam are plotted in Figs. 5.7a) and b) for an axial strain of 3% parallel and per-

pendicular to the rising direction, respectively. The effective response of the foam

was linear and the contour plots show that compressive loads are carried out by

the struts parallel to the loading direction in each case, while the struts perpen-

dicular to the deformation axis are loaded in tension (black struts). However, the

compressive axial forces were much higher when the foam was loaded parallel to

the rising direction due to the anisotropy of the cells in the foam. Similar effects

were also found in the RVEs of the ACPW35 and ACPW50 foams, although the

differences between both orientations in these foams were less marked due to the

smaller anisotropy.

A quantitative analysis of the influence of the structural anisotropy on the load

carried by the struts can be found in Figs. 5.8a) to c) which show the cumulative

probability distribution of the axial force in all the beam elements in the struts

in the RVEs of the three foams. The probability distribution was determined

when the axial compressive strain was 2% within the elastic regime. Most of

the struts were loaded in compression and tensile loads were only found in a few

struts oriented perpendicular to the loading axis. The cumulative distributions also

showed that the fraction of struts subjected to high compressive stresses was always

higher when the foam was deformed parallel to the rising direction and that the

differences between the compressive stresses in the struts between loading in both

orientations increased with the anisotropy of the foam. These results indicate that

deformation parallel to the rising direction is progressively dominated by the axial

deformation of the struts as the aspect ratio increases, leading to a large increase

92


(a) (b)

15

15

30

45

60

0

5

5

10

15

20

0

Figure 5.7: Contour plot of the axial forces in the strut network of theACPW73 foam. (a) Compression parallel to the rising direction up to 3% strain(Y axis). (b) Compression perpendicular to the rising direction up to 3% strain(X axis) The load in the legend is expressed in mN.

in stiffness. On the contrary, the axial stresses in the struts are not significantly

modified when the foam in deformed perpendicular to the rising direction, and

this indicates that strut bending (rather than axial deformation) is the dominant

deformation mechanism.

The plateau stress of the foam was controlled by the buckling of the struts

and the development of the buckling in the simulations was tracked in the beam

elements at the center of each strut. This element has the minimum cross-section

and is likely to be the first to buckle in each beam. The fraction of struts in which

the central beam element was buckled is plotted in Figs. 5.9a) to c) as a function

of the applied stress for the three foams loaded in both orientations. The fraction

of buckled beams in the foams deformed along the rising direction remains close

to 0 in the elastic regime, and this result is in agreement with the linearity of the

stress-strain curves (Fig. 5.6). Due to the structural anisotropy of the foam, the

struts that carry higher loads are oriented parallel to the deformation axis and

mainly support axial forces. As a result, the load necessary to initiate buckling is

much higher than when the foam is loaded in the perpendicular direction, leading

to higher stiffness and strength. However, the fraction of buckled beams increased

93


- 3 0 - 2 0 - 1 0 0 1 00

2 0

4 0

6 0

8 0

1 0 0

Prob

abilit

y (%)

F o r c e ( m N )


( a )

- 3 0 - 2 0 - 1 0 0 1 00

2 0

4 0

6 0

8 0

1 0 0


Prob

abilit

y (%)

F o r c e ( m N )

( b )

- 3 0 - 2 0 - 1 0 0 1 00

2 0

4 0

6 0

8 0

1 0 0

Prob

abilit

y (%)

F o r c e ( m N )


( c )

Figure 5.8: Cumulative probability distribution of the axial force in struts fora applied compressive strain of 2%. (a) ACPW35 foam. (b) ACPW50 foam.(c) ACPW73 foam. The probability distributions corresponding to deformationparallel and perpendicular to the rising direction are plotted in each figure.

94


rapidly after the onset of damage and this was responsible for the sudden drop

in the stress carried by the foam after the beginning of the non-linear regime.

This behavior was found in the RVEs of three foams deformed along the rising

direction but it was more marked (higher stress at the onset of damage and faster

localization of damage) in the ACPW73 foam which showed the highest structural

anisotropy. On the contrary, damage by buckling of the struts occurred from the

beginning of the deformation in the foam loaded in the direction perpendicular

to the rising orientation. The rate of buckled beams was progressively accelerated

during deformation and this lead to a strong non-linearity in the stress-strain curve

(Fig. 5.6), as opposed to the behavior parallel to the rising direction. The stress

at which damage localization was dominant was much smaller than in the foams

loaded along the rising direction and this behavior was controlled by the presence

of bending stresses, which facilitated buckling at lower compressive axial stresses.

In summary, the results of the numerical simulations were in good agreement

with the experimental results in terms of the elastic modulus and the plateau

stress of the anisotropic foams. The analysis of the micromechanical fields showed

that the load during deformation along the rising direction in anisotropic foams was

mainly carried by the axial deformation of the struts oriented parallel to the loading

axis. This led to a stiff and linear response up to the onset of instability, which

was triggered by sudden propagation of buckling in the RVE. On the contrary,

the load during deformation perpendicular to the loading axis was carried by axial

loading and bending of the struts. As a result, the stiffness was significantly lower

and buckling of the struts loaded in compression and bending started at much

lower stresses. Moreover, the fraction of buckled struts increased progressively as

a function of the applied strain, leading to a non-linear stress-strain curve until

buckling was dominant.

95


0 . 0 0 0 . 0 4 0 . 0 8 0 . 1 2 0 . 1 60

1 0

2 0

3 0

4 0

5 0

Fracti

on of

buck

led st

ruts (

%)

S t r e s s ( M P a )


( a )

0 . 0 0 . 1 0 . 2 0 . 30

1 0

2 0

3 0


Fracti

on of

buck

led st

ruts (

%)


( b )

0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 60

1 0

2 0

3 0

4 0

5 0

Fracti

on of

buck

led st

ruts (

%)


P a r a l l e lP e r p e n d i c u l a r

( c )

Figure 5.9: Fraction of struts in which the central beam element was buckledas a function of the stress applied to the RVE. (a) ACPW35 foam. (b) ACPW50foam. (c) ACPW73 foam. The results are plotted when the RVE was loaded inthe orientations parallel and perpendicular to the rising direction.

96

5.4 Numerical simulation up to densification

5.4 Numerical simulation up to densification

In order to demonstrate the potential of the simulation strategy to reproduce

the full stress-strain curve of the foam in compression up to densification, selected

simulations were carried out including the effect of contact between struts and cell

walls during deformation. The General Contact option in Abaqus (Abaqus (2016))

was chosen assuming that the friction coefficient was 0.

The simulated engineering stress-strain curve in compression for the 1-3CPW30

foam up to densification is plotted in Figure 5.10 together with experimental

curves. Two simulations, carried out with RVEs containing 100 and 800 cells,

are presented. Both simulations were able to reproduce the behavior of the foam

in the plateau regime and the final densification, although the onset of densification

in the simulations was delayed with respect to the experimental observations. This

difference may be due to the presence of flexible cells, which were not broken during

deformation, and contained air. Thus, it is expected that including the effect of gas

pressure within the cells may improve the accuracy of the simulations. It is also

remarkable that the RVE containing 800 cells results in a smoother compressive

response, denoting a more gradual deformation of the foam microstructure that

more accurately represents the experimentally observed deformation mechanism

(see Figure 3.12).

Cross-sections of the foam microstructure along the loading direction are de-

picted in Figures 5.11a to e for different values of the compressive strain, 4%, 10%,

20%, 40% and 60%, respectively. A homogeneous distribution of buckled cell walls

can be observed in the RVE at 4% strain, while a single layer of collapsed cells

perpendicular to the loading direction was developed at 10% strain. This layer of

cells is dividing the RVE to two different areas of top (mostly in blue) and bottom

(mostly in red), Figures 5.11c to e. The displacements in the top region are higher

and this zone is active in deformation while the cells resist again collapse in the

bottom region up to very high strains.

The gradual increase of the applied compression load leads to the progressive

collapse of the layers of cells adjacent to the initial collapsed layer, Figures 5.11c

97

5.5 Parametrical study

Figure 5.10: Compressive response of the RVEs having 100 and 800 cells upto densification.

to e. Hence the collapse of the cells progresses gradually layer-by-layer throughout

the whole RVE until all cells become crushed. At this stage the slope of the

stress-strain curve approaches the elastic modulus of the bulk PU material.


The influence of a number of microstructural parameters (distribution of solid

material between struts and walls, foam density, cell size distribution, strut shape

as well as cell aspect ratio) on the elastic modulus, E, and the plateau stress, σpl,

was ascertained by means of the numerical simulations. The elastic modulus, Es,

and the yield strength, σys, of the solid material in all simulations were 2.4 GPa

and 110 MPa, respectively. The baseline parameters that define the properties of

the foam in the parametric study were the following. The average foam relative

density, ρ/ρs, was 2.52% and the fraction of solid material in the struts, φ, was

0.6. The cell size followed a Gaussian distribution characterized by an average cell

98


size of 216 µm and a standard deviation of 67 µm. The strut shape was given

by eq. (3.1). These parameters were used in all the simulations if not indicated

otherwise. For each specific configuration of parameters, three different statistical

realizations of the RVE were simulated to check the variability of the results and

the average values were used. The results of the parametric study are presented

below.

99


Figure 5.11: Contour plots of the displacement in vertical (loading) directionin the RVE containing 800 cells. (a) Compressive strain of 4%. (b) Compressivestrain of 10%. (c) Compressive strain of 20%. (d) Compressive strain of 40%.(e) Compressive strain of 60%. The displacement in the legend is expressed inmm and the initial height of the RVE is 2 mm.

100


Figure 5.12: Influence of the fraction of solid material in the struts, φ, on themechanical properties of isotropic foams for different relative foam densities,ρ/ρs. (a) Normalized elastic modulus, E/Es. (b) Normalized plateau stress,σpl/σys. The error bars correspond to the standard deviation of the simulationresults for three different realizations.

5.5.1 Fraction of solid material in the struts

The fraction of solid material contained in the struts, φ, is known to have a

large effect of the mechanical properties of the foam Gibson & Ashby (1999); Koll

& Hallstrom (2016) and this was the first parameter analyzed. Simulations were

carried out for 0.2 ≤ φ ≤ 1 (where φ = 1 stands for an open cell foam) and relative

foam densities in the range 0.025 ≤ ρ/ρs ≤ 0.2. The average results of the elastic

modulus, E, (normalized by Es) and of the plateau stress, σpl, (normalized by σys)

corresponding to three different realizations are plotted in Figs. 5.12a) and b) as

a function of the fraction of solid material in the struts, φ. The simulations show

that both the stiffness and the plateau stress of the foam increase as φ decreases

and material is moved from the struts to the cell walls for a constant value of

the relative density. The cell walls, which are connected to the struts, increase the

bending stiffness and the critical load for plastic buckling of the latter and improve

the overall mechanical properties of the foam.

101


Figure 5.13: Influence of the width of the cell size distribution (characterizedby the standard deviation SD) on the mechanical properties of isotropic foamsfor different relative densities of the foam, ρ/ρs. (a) Normalized elastic modulus,E/Es. (b) Normalized plateau stress, σpl/σys. The error bars correspond to thestandard deviation of the simulation results for three different realizations.

5.5.2 Cell size distribution

The cell size distribution of the foam was characterized by a Gaussian function

with an average cell size of 216 µm and different values of the standard deviation

SD (normalized by the average cell size, µ) 0.001 < SD/µ <0.5. The influence

of the width of the cell size distribution on both the normalized elastic modulus

and the plateau stress is shown in Figs. 5.13a) and b), respectively. In general,

the influence is negligible and only an slight increase in both the elastic modulus

and the plateau stress was found for uniform cell size distributions (SD = 0.001)

which was more noticeable for larger relative foam densities (ρ/ρs ≥ 015). Overall,

the influence of this microstructural factor on the mechanical properties is clearly

minor.

102


Figure 5.14: Strut shapes used in the simulations. They are defined by f(x/L),eq. (3.1), which expresses the variation of the strut cross sectional area A(normalized by the area of the central section, A0) as a function of the distanceto the center of the strut, x (normalized by the strut length, L).

5.5.3 Strut shape

The shape of the struts in the baseline simulations followed the function f(x/L)

in eq. (3.1), which was obtained from experimental data provided by X-ray com-

puted microtomography (Marvi-Mashhadi et al., 2017b). However, it is important

to assess the influence of this factor on the foam mechanical response, as it may

change significantly from one type of foam to other (e.g. polymeric vs. metallic)

or depending on the manufacturing process. Thus, three different strut shapes

(depicted in Fig. 5.14) were used to determine the normalized elastic modulus

and plateau stress of isotropic foams with different densities (0.025 ≤ ρ/ρs ≤ 0.2).

Together with the experimental average strut shape, two other shapes correspond-

ing to more uniform or more curved shapes were used for the simulations (Fig.

5.14). Both shapes were in agreement with the extremes of the experimental re-

sults obtained by X-ray microtomography (Marvi-Mashhadi et al., 2017b). The

effect of the strut shape on the mechanical response was analyzed in closed-cell

foams (φ = 0.6) and open-cell foams (φ = 1).

103


The elastic modulus and the plateau stress of the foams with different strut

shapes are plotted in Figs. 5.15a) and b), respectively, for open cell foams and

c) and d) for closed-cell foams. The influence of the strut shape was negligible in

the range of 0.025 ≤ ρ/ρs ≤ 0.2. It was found that the changes of strut shape

according to Fig. 5.14 led to a variation ≤ 7% in A0. This difference was minimum

and the mechanical response of the foams was not influenced.

5.5.4 Cell anisotropy

Finally, the influence of the cell anisotropy on the mechanical properties of the

foam was analyzed by means of computational homogenization. The anisotropy of

the foam was characterized by the parameter s which stands for the aspect ratio of

the elongated cells defined as the average cell length in the loading direction divided

by the average cell length in the perpendicular direction (Marvi-Mashhadi et al.,

2017a). Typical values of the s are in the range 0.5 ≤ s ≤ 2 (Marvi-Mashhadi et al.,

2017a) and the parametric study covered this range. The normalized values of the

elastic modulus and of the plateaus stress of the foams as plotted as a function of

s in Figs. 5.16 and 5.17, respectively, for different values of φ = 0.2, 0.4, 0.6, 0.8

and 1 and different relative densities (ρ/ρs = 0.025, 0.05, 0.1, 0.15 and 0.2). Each

point in these plots is the average results of three different realizations. The rest

of the parameters of the foam (cell size distribution, strut shape) correspond to

the baseline values indicated above.

The results in both figures indicate the strong influence of the cell aspect ratio

on the mechanical properties of the foam, in agreement with the experimental data

in the literature (Amsterdam et al., 2008; Shulmeister, 1998b; Marvi-Mashhadi

et al., 2017a; Huber & Gibson, 1988). In general, both the elastic modulus and

the plateau stress increased with s, leading to a strong anisotropy in the mechanical

properties of the foam (because both properties simultaneously increase along the

direction with s > 1 and decrease in the perpendicular orientation with s < 1).

The only exception to this behavior is found in the plateau stress of open cell

foams (Fig. 5.17e) for large aspect ratios (s ≥ 1.5) which seems to be fairly

104


Figure 5.15: Influence of the strut shape on the mechanical properties ofisotropic foams for different foam densities. (a) Normalized elastic modulus,E/Es, for φ = 1. (b) Normalized plateau stress, σpl/σys, for φ = 1. (c) Normal-ized elastic modulus, E/Es, for φ = 0.6. (d) Normalized plateau stress, σpl/σys,for φ = 0.6.

105


indenpendent of the aspect ratio. This behavior is associated with the reduction

in the central section of the struts when they have subjected to a large elongation.

As a result, the increase in the buckling stress due to the orientation of the struts

along the loading axis (normal forces dominate over bending forces) is balanced

by the reduction in the central cross-sectional area of the struts, which facilitates

buckling. This behavior is not found in the case of closed-cell foams because the

presence of cell walls compensates the reduction in the cross-sectional area of the

struts.

In summary, the results of the parametric study showed that the mechanical

properties of the foams depended mainly on the density, fraction of material in

the cell walls and struts and the cell shape, and were largely insensitive to the

cell size distribution and the strut shape. Building on these numerical results,

accurate surrogate models will be proposed for the elastic modulus and the plateau

stress of open- and closed-cell foams in the following section. They relate the

microstructural features of the foam to the macroscopic properties and can be

used for the design of foams with optimized properties for particular applications.

106


Figure 5.16: Influence of the cell aspect ratio s on the normalized elasticmodulus of the foam, E/Es. (a) φ = 0.2, (b) φ = 0.4, (c) φ = 0.6, (d) φ = 0.8,(e) φ = 1.0 (open cell foam). The solid lines correspond to eq. (6.1)

107


Figure 5.17: Influence of the cell aspect ratio s on the normalized plateaustress of the foam, σps/σys. (a) φ = 0.2, (b) φ = 0.4, (c) φ = 0.6, (d) φ = 0.8,(e) φ = 1 (open cell foam). The solid lines correspond to eq. (6.2) for close cellfoams and to eq. (6.3) for open cell foams.

108

Chapter 6

Surrogate models for open- and

closed-cell foams

The surrogate models developed by Gibson and Ashby Gibson & Ashby (1999)

for the elastic modulus the plateau stress of foams, eqs. (1.13) and (1.14), respec-

tively, were based on very simple assumptions and could provide rough estimations

but not accurate values (unless the parameters of the model were adjusted by com-

parison with experimental data for a given type of foam). The estimations of Koll

and Hallstrom Koll & Hallstrom (2016), eq. (1.23) for the elastic modulus of

isotropic foams were much more accurate in so far they were obtained from nu-

merical simulations of RVEs of the foam microstructure. However, they did not

include the effect of the anisotropy and surrogate models for the plateau stress are

not available in the literature.

Thus, the results of the numerical simulations plotted in Figs. 5.16 and 5.17

for the elastic modulus and the plateau stress, respectively, were used to develop

surrogate models for these properties. The models depend on the three main

parameters that control the properties of the foam, namely relative density (ρ/ρs),

fraction of solid material in the struts (φ) and aspect ratio (s). In the case of the

elastic modulus, the experimental results were reproduced by the function

109

Table 6.1: Constants of eqs. (6.1), (6.2) and (6.3) to determine the elasticmodulus and the plateau stress of closed-cell and open cell foams.

c1 c′1 c3 c4 c5 c6 c70.158 0.5155 0.156 0.760 1.614 -0.077 0.282

E

Es

= c1s1.2

(φρ

ρs

)1.5

+ c′1s1.1(1− φ)

(ρ

ρs

)1.2

(6.1)

for both open- and closed-cell foams.

The plateau stress for closed-cell foams could be obtained as

σplσys

= c3s0.6φ3.5

(ρ

ρs

)1.5

+ c4s0.8(1− φ)

(ρ

ρs

)1.5

. (6.2)

while it was better reproduced in the case of open-cell foams by

σplσys

=

(ρ

ρs

)c5(c6s

2 + c7s

). (6.3)

The constants in these equations can be found in Table 6.1. The predictions

of eqs. (6.1), (6.2) and (6.3) are compared with the results of the numerical

simulations is Figs. 5.16 and 5.17 and the agreement is very good for the whole

range of densities, cell aspect ratios and anisotropies covered by the simulations.

It is worth noting that the numerical simulations and eq. (6.3) predict a max-

imum value of the plateau stress of open-cell foams (Fig. 5.17e) for a given values

of the anisotropy aspect ratio s. It is believed by authors that the optimum values

of the aspect ratio that maximize the elastic modulus or the plateau stress will be

found for all open-cell foam densities if the numerical results plotted in Figs. 5.16

and 5.17 are extended to the higher values of s.

110

Chapter 7Conclusions and future work

7.1 Concluding remarks

The main conclusions of current study are listed as follows:

• A modeling strategy based on micromechanical characterization and com-

putational homogenization of a representative volume of the microstructure

has been developed to determine the mechanical behavior of rigid, open-

and closed-cell PU foams. The model takes into account the microstructural

features of the PU foams (i.e. cell size distribution, strut shape, cell wall

thickness as well as cell anisotropy) together with mechanical properties of

solid PU, and it has been successfully validated again experimental results

of isotropic and anistropic PU foams in compression.

• The viscoelastic properties of the solid PU within the foam were determined

by means of spherical nanoindentation creep experiments while the compres-

sive flow stress was measured by means of Berkovich indentations at ambient

temperature.

• The deformation mechanisms during compression of PU foams were ascer-

tained through he detailed analysis of the numerical simulations. These

111

7.1 Concluding remarks

analyses showed that the load applied to the foam was mainly supported by

the struts, while buckling of the very thin cell walls occurred in the early

stages of the elastic deformation. The plateau stress was dictated by the

onset of plastic buckling of the struts in one section of the foam perpendic-

ular to the applied load, in agreement with the mechanisms observed in the

macroscopic samples by means of digital image correlation.

• The simulations also explained the large effect of anisotropy in the mechani-

cal response of the PU foams. The load during deformation along the rising

direction in anisotropic foams was mainly carried by the axial deformation

of the struts oriented parallel to the loading axis. This led to a stiff and

linear response up to the onset of instability, which was triggered by sudden

propagation of buckling in the RVE. On the contrary, the load during defor-

mation perpendicular to the rising direction was carried by axial loading and

bending of the struts. As a result, the stiffness was significantly lower and

buckling of the struts loaded in compression and bending started at much

lower stresses.

• The parametric study showed that the main geometrical features which affect

significantly the mechanical response of PU foams are the relative density,

the distribution of solid PU between struts and walls as well as the cell

anisotropy, while the cell size distribution and strut shape have almost no

effect.

• Based on calculated results, new surrogate models have been proposed for

closed- and open-cell foams. They describe the effect of the relative density,

the volume fraction of material in the struts and the anisotropy on the foam

stiffness and the plateau stress in compression. These surrogate models can

be very useful in the design of components and in the optimization of the

microstructure of foams for specific applications.

112

7.2 Future work

7.2 Future work

• The simulations of the closed-cell foams did not take into account the effect of

internal pressure during deformation. This was not necessary for the analysis

of the rigid foams studied in this thesis. Nevertheless, this effect is important

in the case of flexible foams and has to be included the model to reproduce

the mechanical response of flexible foams in compression up to large strain.

• The modeling strategy was only applied to simulate the mechanical behavior

in compression and at ambient temperature. An obvious extension of this

work is to apply the modeling strategy to simulate the mechanical behavior of

the foams in tension and shear and also at high temperature. This will require

to extend the micromechanical characterization of the foam to determine the

tensile and shear strength of the solid PU within the foam and to carry out

the micromechanical tests at high temperature.

113

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