REVIEW

On the polymeric foams: modeling and properties

Vivek Srivastava • Rajeev Srivastava

Received: 30 September 2013 / Accepted: 16 December 2013 / Published online: 3 January 2014

� Springer Science+Business Media New York 2013

Abstract Polymeric foams are now widely used and

researched. The physical properties of polymeric foam can

be related to a set of independent structural parameters or

variables of the foam. Study of these variables and corre-

lation with commercial FE packages is essential for reliable

and faster product development. Some aspects of foam

behavior are widely studied while some are little less, like

correlation of physical unloading behavior. For example, a

lot of work in the area of phenomenological constitutive

modeling of uniaxial loading was done, though research in

areas of unloading–reloading and their correlation still

demands more attention. Increasing number of OEMs and

suppliers are moving to computer simulations in the design

phase to assess their future products. Hence, different

parameters within FE packages play a significant role and

also affect the results. Appropriate use of these parameters

will narrow down error band and automatically reduce the

cycle time and development cost. This brief review is

expected to set the perspective for major research work

done so far in terms of FE modeling correlation and con-

stitutive modeling of polymeric foam vis-a-vis to its

properties.

Introduction

Extensive research work is available for characterization of

metals and alloys while a lot of work is under progress in

the area of polymeric foams. Properties of polymeric foams

are complex and require static and dynamic tests for better

representation. The trend of replacing metal and their

alloys with polymers or plastics in most products is rapidly

increasing. For example, Iron and Steel formed 70 % of

gross average weight of automotive vehicle in 1990, while

it was reduced to 65 % in 2000 and further to 61 % in 2011

[1]. Hence, we can conclude a 9 % decline in Iron and

Steel usage, which is compensated by alterative materials,

largely, plastics. Polymeric foam family is composed of

different materials and their types; therefore, we shall

review the development within the following heads for

better presentation of the subject matter:

A. Mechanical properties and governing parameters

B. Polymeric foam modeling

C. FE modeling of foam deformation

D. Conclusion

Mechanical properties and governing parameters

The important characteristics of polymeric foams are

resilience, lightweightedness, stiffness, high porosity, high

crushability, and good energy absorption capacity [2].

Foam material used for packaging and automobile padding

applications is mainly subjected to compressive loads.

Polymeric foam possesses unique physical, mechanical,

and thermal properties, which can be attributed to polymer

matrix. This matrix is formed by cellular structure and gas

composition while its properties are dependent on the

ingredients and their scattering. Properties of gaseous

mixtures are predominantly governed by volumetric char-

acteristics of constituents. Thermodynamic properties, such

as specific heat, equilibrium constant, etc., are governed by

the molecular weight of individual participating compo-

nents [3]. Mechanical properties of foams are broadly

V. Srivastava (&) � R. Srivastava

Department of Mechanical Engineering, Motilal Nehru National

Institute of Technology, Allahabad 211 004, India

e-mail: reachtovivek@gmail.com

123

J Mater Sci (2014) 49:2681–2692

DOI 10.1007/s10853-013-7974-5

controlled by the gas pressure, base material properties,

cell geometry, and manufacturing methods. Foams can be

differentiated as open-cell foam and closed-cell foam,

based on cell type, as shown in Fig. 1. Structure–property

relationship is discussed in the following two sections:

a Modulus and effect of foam density

b Geometrical constitution of foam and its effect

Modulus and effect of foam density

Like any other design parameter, foam density also needs

to be optimized as per the targeted application. Gibson and

Ashby [4] explained that at lower density, densification

zone is reached quickly which results in very high force

prior to full energy dissipation. On the other hand, in case

of high-density foam, force exceeds the critical value way

before adequate energy absorption, resulting in partially

utilized compressive strains. Higher density foams of the

same type experience higher permanent deformation,

whereas compression stresses are lower than that of the

indentation stresses. This behavior suggests the role of

shear and tensile strength in providing additional resistance

to the deformation [5].

Internal structure plays a vital role in defining foam’s

mechanical properties. For any given amount of energy, low-

density foam sustains a large deformation due to the quick

densification and low value of plateau. On the other hand,

high-density foam sustains low deformation for the same

amount of energy [6], which clearly points that ideal foam

should be of intermediate density. Young’s Modulus and yield

stress follow power law dependence with respect to density of

foam shown in Eq. (1). Density exponent is found to be 1.5 and

4 for yield stress and Young’s Modulus, respectively, though

bulk modulus varies linearly with density [7–9]

E ¼ ðqÞn; ð1Þ

where E, q, and n are the modulus, density, and density

exponent of foam. It is well established that yield is controlled

by the maximum principal stress for polystyrene and

polyurethane foams [10, 11]. Effect of density, filler size,

impregnation, cell structure, cell orientation, testing

temperature, and crush behavior has been studied in detail

[12–21]. Constitutive and yield behavior of polyvinylchloride,

along with tensile, shear and compression, and multiaxial yield

response has also been described [22, 23]. Polymeric foams

have always exhibited lower values of maximum force when

compared with solid material block of the same material [6].

At higher strains, even at 80 %, stress increases very rapidly.

At strains as high as 95 %, foam density is not equal to that

of solid foam (polystyrene) and becomes very stiff beyond

2.5 MN m-2 [24]. Higher density foams have higher yield

strength, breaking strength, and Young’s modulus, where

stress and strain follow a linear relationship within the range

of 20 % of tensile strain. Resilient foams can sustain strains

up to 50 % before fracture, while crushable foams can take

only 20 % strain before fracture [5]. In most of the cases,

dynamic stress levels were higher than their static

counterpart at the same strain level. Foams can absorb

more energy in dynamic loading. DYTHERM foam, with

unit volume of 48.08 kg m-3, resulted in 31 % difference in

energy absorption during dynamic and static compression

[5]. At larger compressions, compressive response is

considerably different for similar density foams

(polyethylene), when there is substantial variation in cell

size [25]. Gibson and Ashby [26, 27] derived the modulus of

a closed-cell foam which contained three terms that reflect

the effect of strut bending, membrane (cell face) stretching

[28, 29], and the internal gas pressure of closed cells. The

stiffness of closed-cell foam can be attributed to these three

contributions as in Eq. (2).

E�

Es

� /2 q�

qs

� �2

þ 1� /ð Þ q�

qs

þ qo

Es

� �1� 2t�ð Þ

1� q�

qs

� � ; ð2Þ

Fig. 1 Open-cell and closed-cell foam

2682 J Mater Sci (2014) 49:2681–2692

123

where E* and Es are the modulus of foam and solid

polymer, respectively; q* and qs are the density of foam

and solid polymer; and n is the density exponent. On the

other hand, Goods [7] simplified Gibson and Ashby’s

model [26, 27] by modeling foam as an array of cubic cells

of length ‘‘l, and struts of thickness t, as shown in Fig. 2

(open-cell foam) and Fig. 3 (closed-cell foam). The cells

were staggered in such a way so that corners of one cell rest

upon the midpoint of adjacent cells. Such structure neither

matched to the actual geometric characteristics of real

foam nor could be reproduced to fill the space. It is

supposed to capture the critical physical processes of

deformation and structural stability of a cellular structure.

This unit cell assumption resulted in Eq. (3), which

describes density dependence of the elastic modulus of

open-cell foam, valid only at small displacements [25, 26]

E�

Es

� t4

l4

� �� q�

qs

� �2

ð3Þ

Geometrical constitution of foam and its effect

Substantial amount of effort has been put to relate the

mechanical response of foams and cell deformation

mechanics. Zhu et al. [8] derived a relationship for Young’s

modulus, shear modulus, and Poisson’s ratio as a function of

edge cross section and foam density. A significant amount of

work on cellular solids was based on incorrect assumptions.

For instance, bending stiffness is a better representative of

elastic behavior of foams instead of axial extensions of cell

walls [26, 30, 31]. Initially, cell wall bent was attributed to

elastic behavior of foams which was later not accepted [32,

33]. Later, Menges and Knipschild [34] found the correct

mechanism, which was cell wall bending, and explained that

open- and closed-cell foams have similar stiffness. Cell wall

struts are responsible for a major part of load and contribute

significantly to foam stiffness rather than thin cell wall. Ko

[35] worked out the relationship between cell wall bending

and modulus and so did Patel and Finnie [28]. They

emphasized greatly on the geometrical requirements essen-

tial to fill the space for different three-dimensional structures

and further explained that on foam expansion, participating

bubbles tend to form polyhedral. The only structure that

meets all geometric requirements is the ‘‘minimum area

tetrakaidecahedron’’ having 109.47� angle between adjoin-

ing pairs of cell edges and having hexagonal faces of double

curvature.

Multiple variables are known to affect properties of

foam like: structure of the foam, matrix material of the

foam, density of the foam, cell orientation, and testing

temperature [21].There is a well-established direct rela-

tionship between the internal structural traits (e.g., poros-

ity, cell shape and size, strut length and thickness) and the

mechanical characteristics [26, 36] that are governed by

cell wall material [37].The structural variables, necessary

to describe plastic foams, are density, cell size, cell

geometry, polymer phase composition, and gas-phase

composition [38]. It has been suggested that cell structure

can be assumed oval and with some approximation it can

be modeled by ratio of radius, area, and thickness of cell

wall [39]. It has also been represented by many complex

shapes, and various mechanical properties have been pre-

dicted by these approximate shapes [40–42]. Most of the

work for plastic deformation of foams deal with cell cor-

poreal like strut shape, size, and their effect on properties

Fig. 2 Unit cell of open-cell foam of cubic symmetry [25, 26]

Fig. 3 Unit cell of closed-cell foam of cubic symmetry [25, 26]

J Mater Sci (2014) 49:2681–2692 2683

123

of periodic arrangement of cells, isotropy (effect of disor-

der), and cellular interaction at microscopic scale [43].

Most of the foam material is distributed, either in cell walls

or cell ribs, and affects compression behavior of cellular

foam. Energy dissipation mechanism is likely through cell

bending, buckling, or fracture, but stress is generally lim-

ited by the long and flat plateau of the stress–strain curve

38]. Properties of foam are closely linked to its structural

parameters. Berlin and Shutov [44] listed the following

important parameters affecting the properties of foam:

I. Relative number of open cells

II. Relative foam density

III. Cell size

IV. Cell shape or geometrical anisotropy

V. Cell walls thickness and distribution of solid between

struts and faces

VI. Geometry of a foam cell and its constituents

Relative number of open cells

Practically, foams are a mix of open-cell and closed-cell

matrix and it is obvious that the relative content of open

cells in closed-cell foam governs the resultant physical

behavior of cellular materials. In a study [45] of closed-cell

PUR foam, cell walls are classified as closed, pin holes,

partially open (less than 50 % open), and open. The

effective fraction of open cells is estimated, then, by the

Eq. (4) as under:

t ¼Nopen þ 1

2

� Npart

Nopen þ Npart þ Npin þ Nclosed

; ð4Þ

where ‘‘N’’ corresponds to the number of walls of various

classes.

Relative foam density

Relative foam density qf=qsð Þ defines many mechanical

properties by an empirical relation as in Eq. (5)

Uf

Us

¼ Cqf

qs

� �p

; ð5Þ

where Uf and Us are mechanical properties of foam and

solid phase; correspondingly, C and p are the coefficients

obtained from the experiments. From Gibson and Ashby’s

model [26, 27], Goods et al. [7] gave a simplified relation

of relative density with cell dimensions as in Eq. (6).

q�

qs

/ V�

Vc

/ t

l

� �2

ð6Þ

Also, from Eq. (2) for U ¼ 1 (open-cell foam), negli-

gible gas pressure, small qo, and high relative density,

modulus was found to be varying as the square of the

density; while at low relative densities, the modulus was

linearly dependent on the density.

Cell size

Due to irregular cell structure and variation in size, it has

been observed that during compression, extruded foam

does not yield uniformly [46, 47]. Cell size is a critical

parameter in identification of foam properties, though the

base material may change the behavior completely. For

example, cell size is not a governing parameter for

Young’s modulus [48], whereas increasing cell size makes

closed-cell foam stiffer [45]. Foam tends to align as per the

extrusion direction during compression [49]; and, on the

other hand, microscopic inspection reveals a uniform cell

distribution and orientation for all the examined material

and this infers that mechanical behavior is independent of

the direction of the loading [6].

Cell shape or geometrical anisotropy

Foam properties are direction dependent and mostly

depend on the manufacturing process. Based on this pro-

duction process, mostly three main directions have been

determined in foam with different cell dimensions in these

directions. Mechanical characteristics of foam were found

better in the directions where foam cells have greater

dimensions [50].

Cell walls thickness and distribution of solid between struts

and faces

Major portion of base material was supposed to be found in

walls, struts, and vertices of the cell. It is assumed that the

solid-phase material is contained by walls/struts and struts,

respectively, for closed-cell foam and open-cell foam. The

presence of material in the vertices is assumed negligible

and can be ignored [50].

Geometry of a foam cell and its constituents

Vladimir [50] has compiled the following set of conditions

for foam to be in equilibrium:

• A strut is an intersection of three walls (Plateau’s law

[51]).

• A knot point is an intersection of 4 struts and an

intersection of 6 walls (Plateau’s law [51]).

• Struts are straight in the undeformed state.

• Several struts, belonging to the same cell wall and

connected to each other, lie in one plane.

2684 J Mater Sci (2014) 49:2681–2692

123

• As a result of the above points, walls are flat in

undeformed state.

• During the production process of foam (growing), a

dihedral angle (angle between faces) is equal to 120�.

• Struts intersect in a vertex under the bond angle equal

to 109�2801600.• These laws are based on the principles of the

minimizing surface energy during foam growth.

Cell wall thickness is neither constant nor uniform

within the wall and the same is also applicable for the struts

as well. They can vary, and distribution of the wall thick-

ness and strut cross section may be large and nonuniform.

The use of mean values in the modeling can lead to the

introduction of errors. That is why it is important to mea-

sure a considerable amount of struts and walls and to pay

attention onto the distributions. Foams are found to be very

sensitive to strain rate during loading; therefore, static tests

are of partial use in forecasting cushion performance [52].

During compression, the cells of extruded foam do not

yield uniformly due to cell size and shape variations [47].

Hysteresis caused by strain rate leads to conversion of part

of kinetic energy into heat energy during cyclic loading and

unloading of closed-cell cushions [53]. Orientation of the

cushion during compression has significant bearing due to

affinity of cells getting aligned during extrusion [49]. Two-

parameter Mooney–Rivlin strain energy potential model

describes the hyperelasticity of the solid foam [54]. Ini-

tially, for the strain less than 0.05, cell wall bending

dominates and foam deformation was found to be primarily

in linear elastic region [5]. Very little energy was absorbed

in the linear elastic region. After cell wall bending, cell

edges or walls were formed due to elastic buckling of

plates. This stage was generally characterized as a plateau

of deformation at almost constant stress ,and long plateau

of the stress–strain curve is the cause of large energy

absorption at almost constant load [6].

Despite large amount of research work in the area of

geometric visualization and parameterization, nothing

substantial has been achieved in terms of contribution and

modeling of geometric constitution of polymeric foam.

This is due to some basic flaws in the approach like

adaptation of previous mathematical models relying on

static test data for development of material models for

essentially a rate-dependent material, and ignoring the

error creeping into the model due to averaging of highly

random material structure.

Polymeric foam modeling

Micromechanics of foam is different for open- and closed-

cell foam. Gibson and Ashby [4, 26, 27] derived Eq. (2) for

closed-cell foam and the same is applicable to open-cell

foam when U ¼ 1. The differentiating factor between these

two is the absence of the membrane in the open-cell foam.

In most of the cases, constitutive foam structure consists of

closed as well as open cells. In practice, it is difficult to

make foam only with closed or only with open cells. Open-

cell foams contain mostly open cells (more than 90 %) and

vice versa.

For open-cell foams, existing micromechanics models

can be broadly classified as regular models (which include

Gibson and Ashby [26] model and Kelvin [55] model),

and irregular models (which include Van der Burg et al.

[56] model and Zhu’s Voronoi tessellation model [57]).

However, they were only building blocks and none of

them was the actual representation of the physical phe-

nomenon. The important aspects of polymeric foam’s

mechanics are uniform/nonuniform edge, polyhedral cells

(with uniform edges), full edge and vertex geometry,

irregular cell shapes, anisotropic cell shapes, and variable

cell sizes [58]. Manufacturing parameters, such as rise

direction and blowing agents, play a greater role in the

mechanics of the foam. Foam compression in rise direc-

tion displays higher force compared to the compression

perpendicular to rise direction. This behavior was

observed till 20 % of strain and afterward, force rose

monotonically for compression, perpendicular to rise

direction. This force rose rapidly when the strain exceeds

50 % and edges touched [59, 60]. Open-cell foams have

slender edges with Plateau border cross sections [58]. In

this case, variation in bending and torsional stiffness due

to variation in width along the edge should be considered,

while most of the models assumed edges of constant

width for simplification of analysis. Dement’ev and Ta-

rakanov [61] calculated Young’s modulus of the Kelvin

foam for the [001] compression direction which is rep-

resented below in Eq. (7).

E100 ¼ffiffiffi2p

EP

b=Lð Þ4

2þ b=L; ð7Þ

where E100 is the polymer Young’s modulus, b is the

breadth of the edges with a square cross section, and ‘‘L’’ is

their length. The denominator contains a correction for the

vertex volume. Zhu et al. [8] examined the prediction of

tensile and compressive Young’s Modulus E100 of BCC

lattice of tetrakaidecahedral cells [61] and found erroneous

compressive results. Isotropy of foam could be not justified

due to the lack of computation of shear modulus and not

taking account of torsional edge deformation mechanism.

Assumption of uniform stress and uniform strain results in

lower average modulus and higher average results,

respectively. The predicted Young’s modulus of the

Warren and Kraynik [62] model is about 50 % higher

J Mater Sci (2014) 49:2681–2692 2685

123

than the modulus of a polycrystalline BCC model, in which

the grains have random orientations. Direction of

mechanical axis (rise direction and extrusion direction)

and applied stress primarily governs the stress–strain

response of anisotropic foam. Anisotropy in the foam can

either be introduced by manufacturing process (elongated

cells) or by application of high strain. The foam elastic

anisotropy can be expressed using Zener’s anisotropy

factor ‘‘A*’’ and is defined by

A� ¼ 2 1þ t12ð ÞG12

E100

; ð8Þ

where ‘‘G12’’ is shear modulus and E100 is Young’s

modulus. For low relative density (R), anisotropy factor for

the Kelvin foam is

A� ¼ 3

2

5EI þ GJ

8EI + GJ

� �ð9Þ

For Kelvin foam, the Zener’s anisotropy factor was

found to be close to unity and which was the reason of

preferring Kelvins model over other lattice models for

open-cell foams [58]. The nonuniformity of edge’s cross

section affects the performance of foam. There are many

possible ways to calculate normalized Young’s modulus to

assess the impact of nonuniformity. Two of them could be

by measuring the edge width profile and another by solid

element FEA, though both of them gave similar results [55,

63]. Randomization of foam model was tried to assess the

effect on Young’s modulus. This was achieved by ran-

domly moving the nuclei from BCC positions or some-

times by Voronoi tessellation of randomly placed nuclei in

a cube. In either case, randomization affected Young’s

modulus severely, when compared to the regular Kelvin

foam model. An increase of 46–50 % was observed after

introducing the randomness into model [56–64]. The same

response could, however, not be reproduced for edges with

circular cross sections even after introducing the foam

irregularity on the same model [65], which could be

attributed to the fact: despite being an irregular foam

model, it was not the Voronoi model. Therefore, effect of

generation of foam irregularities and the corresponding

effect of the same on mechanical properties are not clearly

known. Mechanical properties of closed-cell foam are

dominated by polymer base material and gas inside the

cells. Cell faces, air, air–polymer interactions, polyhedral

cells, irregular cell shapes, and bead boundaries are criti-

cally important aspects of foam mechanics [58]. Sko-

chdopole and Ruben’s [38] qualitative model suggested

that when gas and polymer act in parallel during com-

pression, they both undergo the same strain as the cell air

and polymer. Contribution of cell gas [66] during uniaxial

compression of closed-cell foam can be related as under in

Eq. (10)

rG ¼paef

1� e� R; ð10Þ

where pa is atmospheric pressure (the assumed gas pressure

in the undeformed foam cells), e is the applied compressive

strain, ‘‘f’’ is the fraction of closed cells in the foam, and

R is the foam relative density. Among different models

available for closed-cell foam, models can be again broadly

divided into regular foam models (which includes Gibson

and Ashby’s [26] model and Kelvin [55] foam model) and

irregular foam models (which includes Shulmeister [50]

and Voronoi model of Roberts and Garboczi [67]). Mod-

eling of closed-cell foam is complex compared to open-cell

foam, due to the presence of air-filled cells. Young’s

modulus [26] of foam can be represented as a fraction of

solid material [Eq.(2)] which results in the excellent cor-

relation of experimental data to the proposed elastic–plastic

model of Kelvin foam [68]; whereas Voronoi model of

edges with circular cross section predicts Young’s modulus

10 % higher than the actual [67]. The plastic hinge pattern

in compressed Polystyrene foam is found similar to Alu-

minum foam model of Santoza and Wierzibicki’s [69] for

the crushing of closed-cell foam. Another space filling

polyhedral model is tetrakaidecahedron. This model cell

was found to be closest to the real foam cell [70]. As in

regular foam models, not much is known for irregular foam

models for closed-cell foam. Available foam models are

schematically shown in Fig. 4.

Unrealistic geometry modeling is the downside of most

of the available foam models. In contrast to random cell

structure of foam matrix, in practical scenario, most of the

models are based on regular unit cells. Initiation of random

irregularities in the FE models and their effects are not yet

clearly known. Knowledge of this aspect may very posi-

tively affect the FE modeling and simulations of polymeric

foams. A model based on constitutive equations or gen-

eralized voxel data, representing the irregularity generation

algorithm, must be developed to take the polymeric foam

modeling to the next level.

Finite element modeling of foam deformation

Finite Element Modeling of foam is still in its early stage

of development unlike metals. Some of the metal’s con-

stitutive equations are adapted for foam, though they are

not able to produce very accurate results. Finite element

method (FEM), apart from other things, is used to study the

microstructure dependence of open-cell solids [43]. Models

are based on Voronoi tessellations, level-cut Gaussian

random fields, and the nearest neighbor node-bond rules.

Polymeric foams require different types of material model

as they can experience very large volumetric deformations.

2686 J Mater Sci (2014) 49:2681–2692

123

Ongoing practice is to superimpose stress–strain history on

phenomenological model, which is completely tabulated

input instead of parametric input.

Rusch [66] gave a shape function for empirical curve fit

of uniaxial compression stress ‘‘r’’ versus strain ‘‘e’’ data,

using Eq. (11). Also, strain softening of cellular materials

deformation was to be found occurring in confined regions/

bands. These bands show discontinuity due to a sudden

change in amount of deformation [71, 72].

FðeÞ � rEe¼ aep þ beq; ð11Þ

where ‘‘E’’ is the Young’s modulus and ‘‘a,’’ ‘‘b,’’ ‘‘p,’’

‘‘q’’ are constants. These five independent constants are

found to be sufficient to make a reasonable fit to most data.

The curve-fitting parameters are not found linked to

deformation mechanisms, so cannot be compared with

foam microstructural variables. It is not clear that how

some foam material models in LS-DYNA, which use curve

fitting, predict the foam response under complex strain

states. Commercial softwares like LS-Dyna, RADIOS and

PAM-CRASH do not disclose much about how their

material model functions. Various material models avail-

able for foam within LS-DYNA are listed in Table 1.

There are many ways to obtain test data of materials.

Impact tester, Hopkinson bar system, conventional screw

drive load frame, servo-hydraulic system, and high-rate

servo-hydraulic system are the five testing systems mostly

used to obtain rate-dependent material data [73]. Under

impact, stress state of foam was found to be dominated by

compression and shear [74]. Polymeric foams have very

high strain rate dependency when compared to solid

metallic materials [6], which is primarily due to intrinsic

mechanical characteristics of material. The presence of

fluid (air) also adds to this behavior. Foam deformation

causes inside air compression, which results in air pushing

the cell walls. Fluid flow is found to be essentially affected

by compression rate; hence, the compression behavior is an

important aspect of foam properties. Uniaxial compression,

uniaxial tension, and simple shear tests are the primary

tests, which help understand mechanical properties of

foam. These tests have provided interesting insights about

fracture of foams and two-dimensional response of crush-

able foam under low velocity impact [72, 75]. Some rep-

resentative data/parameters are tabulated in Table 2.

Dynamic tests characterize dynamic behavior and energy

absorption capabilities of foams. Several models and

experimental methods were developed for hydrostatic

compression of foam to study the foam behavior in

numerical simulations [76–78]. Mechanical characteristics

of foams are widely governed by temperature as well.

Uniaxial compression test with varying temperature in the

range of -20 to 80 �C resulted in a considerable decrease

of compressive stress with increase in temperature [79, 80].

An accurate constitutive model, representing true stress–

strain behavior, is the prerequisite for better correlation of

simulations. The actual deformation is molecular structure

dependent and guided by polymer processing. As per the

available literature, elastic plastic theory-based foam mate-

rial model was developed by Kreig [81] in 1972 for Sandia

National laboratories. This study was originally meant for

soil and concrete though it was majorly used for foams under

compression loadings due to similarity in uniaxial com-

pressive stress–strain nature. Kreig model included strain

Foam Models

Open Cell Foam

Regular Model

Kelvin's Model

[55]

Gibson & Ashby Model

[26]

Irregular Model

Vander Burg

Model

[56]

Zhu's V. Tessellation Model

[57]

Closed Cell Foam

Regular Model

Gibson & Ashby Model

[26]

Kelvin's Model

[55]

Irregular Model

Roberts-Garboczi

Model

[67]

Shulmeister

Model

[50]

Fig. 4 Schematic representation of available foam models

J Mater Sci (2014) 49:2681–2692 2687

123

Table 1 Available foam material models for LS- Dyna [94]

MAT

No

Keyword Strain

rate

Fail THERM ANISO DAM TENS

05 MAT_SOIL_AND_FOAM Y

14 MAT_SOIL_AND_FOAM_FAILURE Y Y

38 MAT_BLATZ_KO_FOAM

53 MAT_CLOSED_CELL_FOAM

57 MAT_LOW_DENSITY_FOAM Y Y Y

62 MAT_VISCOUS_FOAM Y

63 MAT_CRUSHABLE_FOAM Y

73 MAT_LOW_DENSITY_VISCOUS_FOAM Y Y Y

75 MAT_BILKHU/DUBIOS_FOAM Y

83 MAT_FU-CHANG_FOAM Y Y Y Y

142 MAT_TRANSVERSLEY_ANISOTROPIC_CRUSHABLE_FOAM Y Y

144 MAT_PITZER_CRUSHABLE_FOAM Y Y

154 MAT_DESHPANDE_FLECK_FOAM Y

163 MAT_MODIFIED_CRUSHABLE_FOAM Y Y

177 MAT_HILL_FOAM Y

178 MAT_VISCOELASTIC_HILL_FOAM Y Y

179 MAT_LOW_DENSITY_SYNTETIC_FOAM Y Y Y Y Y Y

180 MAT_LOW_DENSITY_SYNTETIC_FOAM_ORTHO Y Y Y Y Y

181

183

MAT_SIMPLIFIED_RUBBER/FOAM_(WITH_FAILURE)/

_WITH_DAMAGE

Y Y Y Y

Table 2 Compilation of representative physical test parameters of foam block

Foam block

dimension

Minimum strain

rate

Pressure Shape Densities

studied

(g L-1)

Foam Test type Articles

Maximum strain

rate

Drop tower

height

Initial velocity

25.4 9 Ø19.4 1.7 9 10-5 s-1 – Cylinder 120–610 PU Tension [7]

50.8 9 Ø28.7 1.7 9 10-2 s-1 Compression

50.8 9 Ø28.7 Impact

142.24 9 Ø69.3 – 6.8-213.7 kPa Cylinder 32–80 PU Hydrostatic

triaxial

[83]

152.4 9 152.4 9 50.8 – Rectangular

block

30–35 EPP Drop tower

impact

[52]

609.6–914.4 mm

100 9 100 9 50 0.14 s-1 – Rectangular

block

65–94 PU Compression [105]

14 s-1

50 9 50 9 50 1.6 9 10-3 – Rectangular

block

3.0–9.6 PP,PS,PU Compression [106]

Tension

88 s-1 Hydrostatic

Shear

127 9 127 9 50.8 – Multiple

parameters

Rectangular

block

20–63 PU,EPP Compression [5]

Tension

Indentation4.47 mm msec-1

50 9 50 9 50 0.02 s-1 – Rectangular

block

31–100 EPP,PUR,NORYL Compression [6]

2688 J Mater Sci (2014) 49:2681–2692

123

rate in constitutive equations. This model could, however,

not resolve the issue of coupling of longitudinal loading and

transverse deformation. In continuation of development,

linear elastic constitutive model accurately defined the foam

behavior in the linear elastic region [82]. Further, Neilsen

et al. [83] suggested an improved model, focusing on post-

yield behavior, which incorporated volumetric and devia-

toric part of foam behavior.

Most of the commercial softwares are built-in with foam

material models like ABAQUS Hyperfoam, Crushable

Foam Plasticity Model and for LS-DYNA, MATL 57 (Low

Density Foam), MATL 83 (Fu Chang Foam), and PAM-

CRASH material model 21. These models are based on the

assumption that strain energy function can be determined

by principal stretches with respect to the homogeneous

stress-free natural state [84]. The strain energy density [85]

of the Hyperfoam model is expressed by Eq. (12).

U ¼XN

i¼1

2li

a2i

½kai

1 þ kai

2 þ kai

3 � 3 þ ððJelÞ�aibi � 1Þ�; ð12Þ

where U is the strain energy density and N, li, ai, and ‘‘bi’’

are material parameters. Earlier in the development of

constitutive equations three types of models, viz (1) Uni-

axial crush model, (2) Conventional deviatoric plasticity

model, and (3) Soils model which combines volumetric

plasticity with pressure-dependent deviatoric plasticity,

have been mostly used to model the foam behavior [86].

Although, these three models are not able to accurately

represent the actual physical behavior. Various empirical

nonlinear stress–strain relations exist to describe foam

material characteristics under static loading. All these

models and empirical constitutive equations are found

valid only for uniaxial compression [87–90]. All these

models were without strain effect which was a major

shortcoming; though later, Sherwood and Frost [79] pro-

posed an empirical equation, containing strain rate effect.

This model was applicable only to uniaxial compression

loading and deformation and hence was not suitable for

3-D real world scenarios like crash loadings. Unified

equations are a better representation when compared to

constitutive equations which model the viscoplasticity

behavior. These are the ‘‘unified’’ set of constitutive

equations without a yield criterion or loading/unloading

conditions. All aspects of inelastic deformation, including

plasticity, creep, and stress relaxation, are represented by a

single inelastic strain rate term [91]. Fu Chang et al. [92]

added another nonlinear term in unified equations to

emulate bottoming out of the foam. Most of the foams have

random internal cell structure while most of the available

material models are based on the hypothesis of idealized

unit cell structure and, hence, are not able to reflect the

actual mechanical response of the foam [68]. 2-D models

with circular cells in random fashion were used to under-

stand static (uniaxial tension, compression, and pure shear)

and dynamic performance of foam [93]. Circular cell

model consisting of two-dimensional regular and random

cells used to simulate the microstructure of polymer foams.

This method coupled with FEM when simulated large

deformation of foam resulted in very little strain rate sen-

sitivity. This was essentially due to rate-independent con-

stitutive model [54]. Appropriate material model within FE

package is an essential requirement for better correlation.

For example, material type 57 in LS-DYNA- 3D [94] is

suitable for static loading while MAT 83 is good for rate-

sensitive dynamic analysis [5, 95]. This conclusion also

suggested the need for selecting the appropriate material

models for simulating static and dynamic impact problems.

Conclusion

Organized approach is required to properly understand the

polymeric foams. Most of the work done was adapted for

foam without considering the finer implications on the

property and modeling of polymeric foams. A lot of work

done on honeycomb-like materials is replicated for poly-

meric foams. Honeycomb has a periodic structure and

similar models are developed for foam assuming periodic

properties for idealized unit cell though that is not the case,

foams are mostly very random in structural arrangements

[96]. Many empirical structural property equations,

including foam density and property equations, were

developed based on physics, chemistry, and processing

conditions relationship [97–99]. There is a low volume of

work relating microstructure with macroscopic physical

properties and there are few good structure characterization

methods available [100]. Most of the models are based on

assumptions like monodispersity, rhombic dodecahedral

[101] cell shapes, tetrahedral cell intersections, and Pla-

teau’s laws [9, 30, 35, 51, 98]; and later simplified foam

shape is described using cubic unit cells to represent foam

behavior [7]. These relationship models result in fair

engineering correlation, though these are the generic

models which may not represent polymeric foam [100].

Product design requires repetitive numerical simulation

using finite element program such as LS-DYNA3D [94] for

cost-effective product evaluation. However, Finite element

analysis and material modeling correlations always have

their limitations in terms of fundamental assumptions; for

example, volume and density as volume changes during the

testing, which ideally should be constant, and elastic wave

velocity depends on density which again does not remain

constant along with oscillation issues in test setups and use

of numerical filters. It is necessary to study the

J Mater Sci (2014) 49:2681–2692 2689

123

performance of such material under dynamic loadings. Lots

of experiments were performed on polymer foams to

characterize mechanical properties, whereas few numerical

investigation studies are carried out [102]. Even the

existing numerical studies further required fine tuning in

terms of input parameters. The current worth of the global

polymer foams market is $ 82.6 billion and is estimated to

reach $131.1 billion by 2018, growing at a CAGR of 7.7 %

from 2013 to 2018. The high demand across industries such

as automotive, building and construction, and packaging

will increase the overall polymer foam consumption [103].

There is a scope of further research in the area of corre-

lation of physical behavior with finite element results in

general and unloading behavior in particular. Authors have

identified some possible areas of future research as

following:

a) All available constitutive models are found dependent

on processing history and hence not applicable to any

other polymeric foam or the same foam with different

processing parameters. There is an immediate need to

develop such models to incorporate the mentioned

parameters.

b) Modeling of anisotropic and isotropic closed-cell foam

is not very much explored and requires focus to

research further.

c) There is no accurate and correlated algorithm to

generate or imitate randomness in the FE model of

polymeric foam matrix. Development of such models

will deeply help the modeling of polymeric foam.

d) Most of the work available was performed on ABA-

QUS while the industry mostly prefer LS-DYNA,

PAM-CRASH, or RADIOS for similarity of the

platform across different types of product development

and testing. Hence, lot of cross correlation among

softwares is also required for a unified approach.

e) The constitutive and phenomenological models avail-

able are mostly applicable to limited number of

loadings and on top of all, unloading behavior is not

properly incorporated and validated in constitutive

equations and modeling.

f) There is a lot of work done in the area of physical to

FE correlation, yet effect of different parameters on the

same analysis are still unknown. One analysis can be

performed in various ways with different input param-

eters on software platform and have dissimilar results.

For example, authors have conducted a study to

evaluate different material types for polymeric foam

(expanded polypropylene) in LS-DYNA and found that

there was a difference in energy absorption of 27.83 %

for MAT57 and MAT83 for the same analysis [104]. A

standard set of input parameters is required for

integrated product development. Authors are currently

working in developing a structured framework for

polymeric foams within LS-DYNA.

Correlation studies are very common, though it is not

easy to differentiate between the error due to wrong

selection of simulation parameters and error due to

experimental noise. This is largely due to lack of set pro-

cedures for characterization, which may lead to pragmatic

choices rather than the factual scientific decision based on

material and its behavioral characteristics. Future studies

may focus on a DOE-based study of simulation input

parameters for different FE solvers like LS-DYNA,

ABAQUS, PAM-CRASH, and RADIOS. This may further

lead to correlation with physical results and cross correla-

tion among the selected commercial FE solvers for devel-

opment of unified product development approach. The

future work in assessing foam behavior for loading/

unloading/reloading in finite element environment may

provide a deeper insight. This will surely reduce time and

effort in estimating newer designs and predicating the life

use of foam, overcoming the hassles of repeated physical

testing and correlation.

Acknowledgements Authors would like to acknowledge and thank

George Jacob, Deputy Manager, Chrysler Corporation and Anurag

Verma, Principal, Karyon Consulting for material information and

insights. The authors sincerely thank Dr. Anshul Chandra for effective

help in manuscript preparation.

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