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Journal ArticleDOI

Parameter estimation of a hyperelastic constitutive model for the description of polyurethane foam in large deformation

01 Jan 2013-Cellular Polymers (SAGE PublicationsSage UK: London, England)-Vol. 32, Iss: 1, pp 21-40
TL;DR: In this article, the nonlinear elastic mechanical behaviour of compressible polyurethane foam during the loading and unloading quasi-static compression tests can be described by applying Ogden's modified model.
Abstract: Polyurethane foam is widely used in such areas as the automotive industry and sport, and in the field of packaging because of its low stiffness, high compressibility and its ability to absorb strain energy. The quasi-static behaviours of polyurethane foam are highly nonlinear and elastic. This paper demonstrates that the nonlinear elastic mechanical behaviour of compressible polyurethane foam during the loading and unloading quasi-static compression tests can be described by applying Ogden's modified model. The experimental data from a uniaxial compression of three types of polyurethane foam in three different strain rates are used for parameter identification. A nonlinear optimization method helps to ensure that the parameters are satisfied with stability conditions. Thanks to the optimized parameter results, the numerical simulations agree with the experimental data. Finally, the errors between the model results and the experimental results are analyzed and the unloading phases are discussed in detail.

Summary (3 min read)

1. Introduction

  • Today, polymeric foam materials, such as polystyrene (PS), expanded polypropylene (EPP) and polyurethane (PU), are widely used in numerous industrial applications in engineering, sport, medical care.
  • Polyurethane foams are cellular materials characterized by the spectrum of mechanical properties [1] such as: low stiffness, low Poisson rate, low density (less than 80kg· m -3 for flexible foam), the ability to absorb the strain energy, high compressibility and slow recovery rate.
  • In the literature, there are numerous models designed to fit experimental results for hyperelastic materials.
  • Conventionally, the determination of material parameters is based on the use of test samples with a standardized geometry under a simplified strain state.

2. Experimental

  • The three types of polyurethane foams, designated by foam Type A, Type B and Type C, have characteristics similar to those of automotive seat foam.
  • This device includes a basis frame and an upper block which moves vertically.
  • Before starting the tests, the top plate had to move down slightly for full contact with the material because the top and bottom of each foam samples were not exactly parallel.
  • All the test conditions including the strain rate, the maximum compression level, the number of cycles, the sampling period, and the test mechanical parameters exported were conducted using the BLUEHILL software configuration window.
  • Each specimen had been quasi-statically loaded and then unloaded with a constant speed during the test process.

3.1. Constitutive Theory

  • According to the empirical results, polyurethane foams show large strains, highly non-linear elastic and some inelastic properties.
  • This study is restricted to the elastic properties of polyurethane foam and their stress-strain relationship can be characterized by a strain energy function which is related to the principal stretches.
  • Hyperelastic constitutive models are adapted for this description [25] .
  • If the principal stretches are denoted by i , then the most common quoted triad of invariants are given by: (5) A strain energy function can represent the stress-strain behaviour of hyperelastic materials and the stress tensor can be generated by the derivation of the strain energy function with regard to the strain tensor.

3.2. Ogden’s model

  • From a phenomenological standpoint several attempts have been made to obtain a realistic mathematical explanation of the mechanical behaviour of highly elastic materials.
  • In the present paper, this model is used to determine the mechanical behaviour of polyurethane foams.

3.3. Parameter Optimization

  • To identify the parameters, the optimization methods are used as basic tools.
  • The value of stressstrain experimental data can be obtained from the uniaxial compression test described in Section 2.
  • Then the Ogden model (12) helps to calculate the model results which are compared with the experimental data.
  • The trust region reflective, Levenberg-Marquardt and Gradient methods are three examples of the deterministic methods which are effective when the objective function (function to optimize) changes rapidly.
  • In order to find the minimum results, the authors used the optimization tool in MATLAB with the solver FMINCON (Constrained nonlinear minimization).

4.1. Experimental results

  • As can be seen in figure 2, polyurethane foam deformation in uniaxial compression presents three stages: initial elastic deformation, collapse deformation and compaction deformation.
  • In the third stage, a region of densification occurs, where the cell walls crush together, resulting in a rapid increase of compressive stress.
  • It can also be seen from the figure that there are large differences in the stress corresponding to the same strain level under the loading and the unloading processes.
  • Figure 3 indicates the stress-strain curves of Foam A in three different strain rates (test 1, test 2 and test 3).
  • This means that the model parameters for loading and unloading phases should be calculated separately.

4.2. Model results

  • The first step was to determine the material parameters which describe the loading process for polyurethane foam using the loading experimental curves of three foams.
  • The loading and unloading parameters were determined using Ogden's model in equation (12), with a three-term expression.
  • The parameter results for three foams in the three tests are given in Table 4.
  • It can be seen in Table 4 that the values of parameters are different between the loading and unloading phases, which corresponds to the experimental results.
  • Figure 4 shows the best set of loading-unloading data for three foams in test 1.

4.3. Discussion

  • In Figures 4 and 5, it appears that there is a good correspondence between the Ogden model and the experimental results, especially in the high compression phase.
  • These two parameters are obtained from the identification results of the preliminary test with 15 test samples.
  • The Student law is used for all parameters in this paper.
  • So the conclusion is that the unloading phase is more sensitive than the loading phase because of the residual stress which has a great effect at the end of the test.
  • After a sufficiently large period, the foams will return to the original configuration.

5. Conclusion

  • This paper has presented numerous experiments with three different polyurethane foams in three different strain rates for loading-unloading uniaxial compression tests.
  • The model consists of an incompressibility component and a compressibility component and the stress has been derived from the model in terms of principal stretches.
  • The stability conditions of the Ogden model have been proposed.
  • The results show that the Ogden model can predict the quasi-static mechanical behaviour of polyurethane foam under large strain compression.
  • The model results agree with the experimental results and a detail of the unloading phase in the compression test has been discussed.

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Parameter estimation of a hyperelastic constitutive model for the
description of polyurethane foam in large deformation
M.L. JU
a
, S. MEZGHANI
a,b
, H. JMAL
a
, R. DUPUIS
a,
*, E. AUBRY
a
a
Laboratoire MIPS, Université de Haute Alsace, 12 rue des frères Lumière, 68093 Mulhouse, FRANCE.
b
Research Unit U2MP, National Engineering School of Sfax (ENIS), B.P 1173. 3038-Sfax- TUNISIA.
Abstract
Polyurethane foam is widely used in such areas as the automotive industry and sport, and in the field of packaging
because of its low stiffness, high compressibility and its ability to absorb strain energy. The quasi-static behaviours of
polyurethane foam are highly nonlinear and elastic. This paper demonstrates that the nonlinear elastic mechanical
behaviour of compressible polyurethane foam during the loading and unloading quasi-static compression tests can be
described by applying Ogden’s modified model. The experimental data from a uniaxial compression of three types of
polyurethane foam in three different strain rates are used for parameter identification. A nonlinear optimization method
helps to ensure that the parameters are satisfied with stability conditions. Thanks to the optimized parameter results, the
numerical simulations agree with the experimental data. Finally, the errors between the model results and the
experimental results are analyzed and the unloading phases are discussed in detail.
Keywords: Polyurethane foam; Quasi-static behaviour; Ogden’s model; Uniaxial compression; Parameter identification.
________________________
*Corresponding author. Tel.: +33 3 89 33 69 25; fax: +33 3 89 42 32 82
E-mail address: raphael.dupuis@uha.fr

Nomenclature
Symbols
Units
Definitions
B
Left Cauchy-Green deformation tensor
C
Right Cauchy-Green deformation tensor
Ceil
Round to the nearest integer
D
(sec
-1
)
Strain rate tensor
F
Deformation gradient tensor
1,2,3
i
i
I
Invariants of the right Cauchy-Green deformation
tensor
J
Determinant of the deformation gradient
(MPa)
Initial bulk modulus
L
w
h
(mm
mm
mm)
Initial dimensions of polyurethane foam samples
N
Minimum number of test samples
cyc
N
Number of cycles in quasi-static tests
R
A proper orthogonal tensor
S
(MPa)
Cauchy stress tensor
T
(sec)
Test period
ech
T
(sec)
Sampling period
U
Right stretch tensor
V
Left stretch tensor
W
Strain energy function
1,2,3
i
i
Parameters of Ogden’s model
1,2,3
i
i
Parameters of Ogden’s model
(mm mm
-1
)
Strain
0
(mm mm
-1
)
Initial strain
max
(mm mm
-1
)
Maximum strain
(sec
-1
)
Strain rate
1,2,3
i
i
Principal stretches
1,2,3
i
i
(MPa)
Parameters of Ogden’s model
0
(MPa)
Initial shear modulus
0
(kg/m
3
)
Density of material in reference configuration
(kg/m
3
)
Density of material in deformed configuration
(MPa)
Stress

1
1. Introduction
Today, polymeric foam materials, such as polystyrene (PS), expanded polypropylene (EPP) and
polyurethane (PU), are widely used in numerous industrial applications in engineering, sport,
medical care. Polyurethane foams are cellular materials characterized by the spectrum of
mechanical properties
[1]
such as: low stiffness, low Poisson rate, low density (less than 80kg· m
-3
for flexible foam), the ability to absorb the strain energy, high compressibility and slow recovery
rate. These properties help to improve the comfort of car seats
[2]
.
Polyurethane foams can be categorized as open or closed cell materials depending on the shape and
connectivity of the cells. If foams allow fluids to flow through the cellular structure, these foams are
called open-cell foams
[3]
. There is a great variety of open-cell foams; however, their mechanical
behavior is not fully understood. Therefore, the present study aims to investigate open-cell foams
thoroughly.
There are several standards for foam characterization, such as the American standard D3574-95 of
the American Society for Testing Material
[1]
, and for methods and tests for the assessment of foam
properties. The tests include those of the indentation force deflection, the ball rebound, the
compression force deflection and the dynamic such as transmissivity and impact. There are also
four types of foam studies
[1]
: static behaviour, quasi-static behaviour, dynamic behaviour and
fatigue behaviour. A more detailed description of foam behaviour can be found in
[4, 5]
.
The polyurethane foam stress-strain response shows very large strains with a strongly nonlinear
behaviour which can be described by a number of hyperelastic models based on the definition of
different strain energy functions. In the literature, there are numerous models designed to fit
experimental results for hyperelastic materials. Mooney proposed a model with two parameters
[6]
.
The Neo-Hookean model described by Trelor has only one material parameter
[7]
, but this model
was proved to be a special case of the Mooney model. In 1950, Rivlin modified the Mooney model
and obtained a general expression so-called Mooney-Rivlin model
[8, 9]
. Yang and Shim
[10]
proposed
a visco-hyperelastic model for foams under strain rates to capture the three-dimensional large
compression behaviour. The Blatz-Ko model
[11]
is also used to describe the properties of
hyperelastic materials. In 1972, Ogden
[12, 13]
proposed a strain energy function expressed in terms
of principal stretches, which is a very general expression for describing hyperelastic materials.
There is an excellent correspondence between the Ogden model data and Treloars experimental
data. Other models include those of Yeoh
[14]
, Beatty
[15]
, Arruda-Boyce
[16]
, Bischoff et al.
[17]
, and
Attard
[18]
. Numerous studies have been carried out to solve non-linear problems with the finite
element method. A very detailed review on finite element formulation for non-linear analysis has
been provided by Sussman and Bathe
[19]
. Many models are now available in commercial FEM
software, such as ANSYS and ABAQUS.
Conventionally, the determination of material parameters is based on the use of test samples with a
standardized geometry under a simplified strain state. Then the unknown model parameters are
obtained using curve fittings from experimental data. For polyurethane foams, a wide range of tests
have been used (e.g. compression, shear, and volumetric tests) in the literature to predict these
parameters. These methods normally require large numbers of tests and samples with well-defined
geometries. Seat polyurethane foams are mainly loaded through a compressive force. For these two
reasons, numerous compression tests were performed to obtain a sufficient number of experimental
results for the parameter identification of the polyurethane foams presented here. There are few
articles on the mechanical behaviour of polyurethane foam, especially on the analysis of the

2
unloading phase. Smardzewski et al.
[20]
used a hyperelastic model to determine the elastic
properties of polyurethane foams in compression tests with a loading phase. Zhang et al.
[21]
used a
pseudo elastic model to model the polymeric foam mechanical properties in the loading and
unloading phases.
This paper describes both the loading and unloading phases and explains in detail the unloading
phase affected by residual stress. The purpose of this paper is to only model the elastic behaviour of
polyurethane foam with high compressive deformation and to estimate the model parameters to
allow a good correlation between the model and the experimental results. The corresponding
identification errors are also taken into account to analyze the results. This paper is organized as
follows: the experimental conditions are presented in Section2 which is followed by a description of
the experimental details. The Ogden model is described in Section 3; this model is then used to
estimate the mechanical behaviour of polyurethane foam with the experimental data. Finally, the
experimental data and the model results are discussed and compared in Section 4 and conclusions
are summarized in Section 5.
2. Experimental
In order to investigate the stress-strain relation, a series of loading-unloading uniaxial compression
experiments were carried out at a constant temperature of 25°C. The three types of polyurethane
foams, designated by foam Type A, Type B and Type C, have characteristics similar to those of
automotive seat foam. The properties of the three type foams are summarized in Table 1. Test
specimens of polyurethane foam were cut from a block of foam (2000mm×1200mm×75mm)
obtained through expansion in a free open mould. All specimens had the same mechanical and
environmental histories. They are original specimens and each specimen was compressed only one
time. Contrary to Belouettar et al.
[22]
and White et al.
[23]
who used foam cubes cut from bolsters of
car seat cushions, the type of foam chosen here helps to provide a substantially homogeneous
material and isotropic, repeatable specimens
[24]
.
Table 1. Chemical and morphological foam characteristics
Type A
Type B
Type C
Foam type
Flexible polyurethane
foam
Flexible polyurethane
foam
Flexible polyurethane
foam
Isocynate
Toluene diisocynate TDI
Toluene diisocynate TDI
Toluene diisocynate TDI
Polyol
Polyether
Polyether
Polyether
Expansion gas
2
CO
2
CO
2
CO
Fabrication process
Free rise
Free rise
Free rise
Density
28 kg/m
3
40 kg/m
3
50 kg/m
3
Average cell size
828 μm
941 μm
633 μm
Samples shape
cubic
cubic
cubic
Dimensions
(L×w×h)
75 mm ×75 mm×75 mm
75 mm×75 mm ×75 mm
75 mm ×75 mm×75 mm
Cell type
open
open
open
All the tests were performed on a usual compression-tension testing device INSTRON 33R4204
driven with BLUEHILL software (Figure 1). This device includes a basis frame and an upper block

3
which moves vertically. Two 150 mm diameter compression plates were installed: one on the base
of the machine and the other on the force sensor of the crosshead. The two plates were checked to
be strictly parallel. Before starting the tests, the top plate had to move down slightly for full contact
with the material because the top and bottom of each foam samples were not exactly parallel. In
order to establish a homogeneous deformation field, the shear stresses between the two plates on the
top and bottom sides and the test specimen at the uniaxial compression test had to be eliminated
[25]
.
To minimize the noise contribution, the maximum experimental response force of foam had to be
slightly less than the load cell maximum capacity. All the test conditions including the strain rate,
the maximum compression level, the number of cycles, the sampling period, and the test
mechanical parameters exported were conducted using the BLUEHILL software configuration
window.
Fig. 1. Compression test device
The three types of polyurethane foam were initially put between the upper frame and the basis
frame of the machine. The test started when the upper frame affected the foams and then the upper
block moved down to compress the samples to the final position. The final compression ratio was
80% of the original thickness. At the end of the loading phase, the upper block changed direction
and returned to the initial level. Each specimen had been quasi-statically loaded and then unloaded
with a constant speed during the test process. The conditions of all tests are given in Table 2.
Table 2. Quasi-static compression test conditions
N
cyc
1
sec
0
%
max
%
T
(sec)
ech
T
(sec)
Foam
A B C
Test 1
1
1.06 10
-2
0
80
150
0.0625
Test 2
1
5.33 10
-3
0
80
300
0.125
Test 3
1
6.66 10
-4
0
80
2400
2
At the start, 15 test samples of each strain rate for each foam were taken for the preliminary test.
Then, the minimum numbers of test samples which were determined to ensure the statistical quality
of the parameters were calculated using equation (20) and they are summarized in Table 3. More
details are given in the discussion section.

Citations
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Abstract: Large Elastic Deformations and Non-Linear Continuum Mechanics By Prof. A. E. Green and J. E. Adkins. Pp. xiii + 348. (Oxford: Clarendon Press; London: Oxford University Press, 1960.) 55s. net.

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References
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TL;DR: The linear elasticity of anisotropic cellular solids is studied in this article. But the authors focus on the design of sandwich panels with foam cores and do not consider the properties of the materials.
Abstract: 1. Introduction 2. The structure of cellular solids 3. Material properties 4. The mechanics of honeycombs 5. The mechanics of foams: basic results 6. The mechanics of foams refinements 7. Thermal, electrical and acoustic properties of foams 8. Energy absorption in cellular materials 9. The design of sandwich panels with foam cores 10. Wood 11. Cancellous bone 12. Cork 13. Sources, suppliers and property data Appendix: the linear-elasticity of anisotropic cellular solids.

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TL;DR: In this paper, it was deduced that the general strain energy function, W, has the form W=G4 ∑ i=13(λi−1λi)2+H 4 ∑ t=13 (λi2−1 ε)2 + H 4, where the λi's are the principal stretches, G is the modulus of rigidity, and H is a new elastic constant not found in previous theories.
Abstract: It is postulated that (A) the material is isotropic, (B) the volume change and hysteresis are negligible, and (C) the shear is proportional to the traction in simple shear in a plane previously deformed, if at all, only by uniform dilatation or contraction. It is deduced that the general strain‐energy function, W, has the form W=G4 ∑ i=13(λi−1λi)2+H4 ∑ t=13(λi2−1λi2), where the λi's are the principal stretches (1+principal extension), G is the modulus of rigidity, and H is a new elastic constant not found in previous theories. The differences between the principal stresses are σi[minus]σi=λi∂ W/∂λi[minus]λi∂ W/∂λi.Calculated forces agree closely with experimental data on soft rubber from 400 percent elongation to 50 percent compression.

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TL;DR: In this article, an eight-chain representation of the underlying macromolecular network structure of the rubber and the non-Gaussian behavior of the individual chains in the proposed network is proposed.
Abstract: Aconstitutive model is proposed for the deformation of rubber materials which is shown to represent successfully the response of these materials in uniaxial extension, biaxial extension, uniaxial compression, plane strain compression and pure shear. The developed constitutive relation is based on an eight chain representation of the underlying macromolecular network structure of the rubber and the non-Gaussian behavior of the individual chains in the proposed network. The eight chain model accurately captures the cooperative nature of network deformation while requiring only two material parameters, an initial modulus and a limiting chain extensibility. Since these two parameters are mechanistically linked to the physics of molecular chain orientation involved in the deformation of rubber, the proposed model represents a simple and accurate constitutive model of rubber deformation. The chain extension in this network model reduces to a function of the root-mean-square of the principal applied stretches as a result of effectively sampling eight orientations of principal stretch space. The results of the proposed eight chain model as well as those of several prominent models are compared with experimental data of Treloar (1944, Trans. Faraday Soc. 40, 59) illustrating the superiority, simplicity and predictive ability of the proposed model. Additionally, a new set of experiments which captures the state of deformation dependence of rubber is described and conducted on three rubber materials. The eight chain model is found to model and predict accurately the behavior of the three tested materials further confirming its superiority and effectiveness over earlier models.

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TL;DR: In this article, the correlation of theory and experiment for incompressible isotropic elastic solids under finite strain was extended to incorporate the effects of compressibility (under isothermal conditions) with the result that experimental data on the compressibility of rubberlike materials are adequately accounted for.
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Frequently Asked Questions (2)
Q1. What are the contributions mentioned in the paper "Parameter estimation of a hyperelastic constitutive model for the description of polyurethane foam in large deformation" ?

This paper demonstrates that the nonlinear elastic mechanical behaviour of compressible polyurethane foam during the loading and unloading quasi-static compression tests can be described by applying Ogden ’ s modified model. Finally, the errors between the model results and the experimental results are analyzed and the unloading phases are discussed in detail. 

For further studies, other models will be used and compared with Ogden ’ s model for polyurethane foams and the viscoelasticity behaviour will be presented in a future paper.