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Novel strategy for the hyperelastic parameter fitting procedure of polymer foam materials
Accepted Manuscript Novel strategy for the hyperelastic parameter fitting procedure of polymer foam materials Attila Kossa, Szabolcs Berezvai PII:
S0142-9418(16)30333-6
DOI:
10.1016/j.polymertesting.2016.05.014
Reference:
POTE 4659
To appear in:
Polymer Testing
Received Date: 11 April 2016 Accepted Date: 17 May 2016
Please cite this article as: A. Kossa, S. Berezvai, Novel strategy for the hyperelastic parameter fitting procedure of polymer foam materials, Polymer Testing (2016), doi: 10.1016/ j.polymertesting.2016.05.014. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Property Modelling
Novel strategy for the hyperelastic parameter fitting procedure of
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polymer foam materials Attila KOSSA, Szabolcs BEREZVAI
Department of Applied Mechanics, Budapest University of Technology and Economics
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H-1111 Budapest, Muegyetem rkp. 5., Hungary
Abstract
The most widely used approach to model the large strain elastic response of polymer foams in a
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finite element (FE) simulation is the use of the Ogden–Hill compressible hyperelastic material model. This model is implemented and termed as "hyperfoam" material model in the commercial FE software ABAQUS. The hyperfoam model is able to characterize the large compressibility (in volumetric sense) of the foam material. In order to find the material parameters of the model for a particular foam specimen, we need to fit the simulated
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responses to the available experimental data. This task is easier for incompressible hyperelastic materials because we can use the incompressibility constraint to eliminate the transverse stretch from the stress solutions. However, this simplification cannot be used for the hyperfoam model, therefore, in the stress-strain relations, the transverse stretch is included,
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which makes the parameter fitting procedure more complicated. In this paper, a novel strategy is proposed for the parameter fitting task. The performance of the new algorithm is
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demonstrated by presenting fitted material responses for a particular polymer foam material. The major advantage of the new strategy is that it can be used with any third-party optimization solver and there is no need to write our own code.
Keywords: hyperelastic; polymer foam; parameter fitting.
1. Introduction The material behavior of solid polymers is highly nonlinear and various tests have to be
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performed to completely characterize every aspect [6], [7], [9], [3]. Polymer foams have special microstructure, which makes their mechanical behavior more nonlinear [10], [18], [2], [16]. In order to accurately describe the mechanical characteristics of polymer foams, accurate prediction of their large strain elastic behavior is crucial.
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The finite strain elastic behavior of elastomeric polymer foams can be accurately captured using the Ogden-Hill hyperelastic material model ([31], [28], [18]) based on the work of Ogden and Hill ([20], [21], [11]). This model is implemented as the "hyperfoam" hyperelastic model in the commercial finite element software, ABAQUS [1]. The strain energy function W
N
W (λ1 , λ2 , λ3 ) = ∑ where λk
(k = 1,2,3)
(
)
2 µi α i 1 −αi βi α α λ + λ2 i + λ3 i − 3 + J − 1 , 2 1 αi βi
(1)
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i =1
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of the hyperfoam model is written as
represent the principal stretches; J denotes the volume ratio
(determinant of the deformation gradient F ); N is the order (positive integer) of the model; whereas µi , α i and β i are the hyperelastic material parameters 1 . Consequently, 3 N material parameters have to be found for a specific material. It should be noted that ABAQUS
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uses ν i parameters instead of β i . The relation between them is given by
νi =
βi
1 + 2β i
,
βi =
νi
1 − 2ν i
.
(2)
A possible way to find the material parameters of a hyperelastic model is to fit the model
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responses of particular homogeneous deformations to experimental data. This is a widely used procedure for hyperelastic materials [24], [30], [13], [17], [25], [19], [29]. However, the fitting of
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the hyperfoam model is more complicated than fitting of hyperelastic models proposed for incompressible materials. In the incompressible case, the transverse stretches can be easily calculated from the incompressibility constraint J = 1 in uniaxial compression/tension or in other homogeneous deformations. However, for the hyperfoam model the transverse stretch, in general, cannot be obtained from the zero transverse stress constraint, even in uniaxial compression. Therefore, the parameter fitting procedure is not so trivial for this material
1
It should be noted that the
between them is
µ
ABAQUS i
µi
parameters used by ABAQUS is different than the
= αi ⋅ µ
Ogden i
/2
µi
parameters proposed by Ogden. The relation
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model. This paper presents a novel strategy to find the material parameters of the hyperfoam material models. The main advantages of the new method are that it includes the transverse stretch data in the parameter fitting algorithm and, in addition, it enforces the zero transverse
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stress constraints.
The outline of the paper is as follows: Section 1 is the introduction; Section 2 presents the solutions for the most widely used homogeneous deformation modes; the experimental data used for the parameter fitting procedure is provided in Section 3; Section 4 presents the
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new parameter fitting strategy; the obtained solutions are shown in Section 6; Section 7 closes
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the paper with conclusions.
2. Model responses for homogeneous deformations 2.1 Stress solutions
For a given hyperelastic strain energy potential W , the first Piola–Kirchhoff stress P can be obtained as
∂W nk ⊗ Nk , k =1 ∂λk 3
Pk =
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P=∑
∂W , ∂λ k
(3)
where λk are the principal stretches, N k and n k are the unit vectors along the principal directions in the reference and the current configurations, whereas Pk are the principal
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stresses [12], [4]. Substituting (1) into (3) gives the relation for the principal nominal stresses of the hyperfoam model:
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N
Pk = ∑ i =1
2µi
αi
(λ
α i −1 k
− λ−k1 J
−α i β i
),
k = 1,2,3.
(4)
The stress solutions for the most important homogeneous deformation modes are
summarized in the following paragraphs. Uniaxial extension (UA): The stretch along the loading is denoted by λ1 , whereas the lateral (transverse) stretches are λ2 and λ3 . For isotropic material, we have the identity λ2 = λ3 .
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For simplicity of the presentation let λ ≡ λ1 and λT ≡ λ2 = λ3 , where subscript T refers to the transverse direction. The deformation gradient F and the volume ratio J can be written as 0
λT 0
0 0 , λT
J = detF = λλT2 .
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λ F = 0 0
(5)
Consequently, the principal first Piola–Kirchhoff stresses (principal engineering stresses, principal nominal stresses) can be obtained by inserting J = λλT2 into (4): 1
(λ
2µi
N
∑α λ i =1
− (λλT2 )
i
1
N
−α i β i
2µi
∑α
(λ
αi
λT
),
− (λλT2 )
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P2UA (λ , λT ) = P3UA (λ , λT ) = 0 =
αi
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P1UA (λ , λT ) =
i =1
T
−α i β i
(6)
).
(7)
i
For the first-order model (, N = 1) the transverse stretch can be expressed from the zero transverse stress constraint (7) as
−
λT = λ
β
1+ 2 β
= λ−ν .
(8)
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Therefore, the transverse stretch can be eliminated in (6) and the nominal stress along the loading direction can be written as the function of λ as αβ 2 µ α −1 −1+ 2 β −1 P (λ ) = λ −λ . α
(9)
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UA 1
However, for higher-order models ( N ≥ 2 ), equation (7) cannot be solved for λT . Consequently, the stress along the loading direction cannot be expressed as the function of the
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primary stretch λ and, therefore, the transverse stretch is always included. Confined uniaxial extension (CU): If we fix the transverse deformation ( λT = 1 ) in uniaxial extension, then the deformation gradient and the volume ratio reduce to λ 0 0 F = 0 1 0 , 0 0 1
The solutions for the stresses can be simplified to
J = λ.
(10)
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αi
i =1
(λ
α i −1
N
P2CU (λ ) = P3CU (λ ) = ∑
−α i β i −1
−λ
2µi
αi
i =1
),
(11)
(1 − λ ). −α i β i
(12)
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2µi
N
P1CU (λ ) = ∑
Planar extension (PE): In case of planar extension one of the transverse stretches in uniaxial extension is fixed. If we fix λ3 in (5) then we arrive at λ 0 F = 0 λT 0 0
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0 0, 1
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J = λλT .
(13)
For this deformation mode, we have three distinct principal stresses: P1PE (λ , λT ) =
1
2µi
N
∑α λ i =1
P2PE (λ , λT ) = 0 =
N
1
(λ
αi
N
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(λ
2µi
i =1
P3PE (λ , λT ) = ∑
−α i β i
),
(14)
i
∑α
λT
− (λλT )
2µi
αi
αi
− (λλT )
T
−α i β i
),
(15)
i
(1 − (λλ ) ). −α i β i
T
(16)
For N = 1 , the transverse stretch can be obtained from (15) as −
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λT = λ
β
1+ β
−
=λ
ν 1− v
(17)
.
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Therefore, the non-zero stresses can be written as the function of the primary stretch as P1PE (λ ) =
αβ
2 µ α −1 −1+ β −1 λ −λ , α
αβ − 2 µ 1+ β P (λ ) = 1− λ α PE 3
.
(18)
(19)
However, for N ≥ 2 , (15) cannot be solved for λT .
Equibiaxial extension (BA): In equibiaxial extension the deformation gradient and the volume ratio have the form
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λ 0 F = 0 λ 0 0
0 0 , λT
J = λ2 λT .
(20)
Consequently, the engineering stresses can be written as N
1
2µi
∑α λ i =1
P3BA (λ , λT ) = 0 =
N
1
2µi
∑α
λT
i =1
(λ
− λ2 λT
(
)
i
(λ
αi
− λ2 λT
T
i
λT = λ
2β 1+ β
−
=λ
2ν 1− v
.
)
−α i β i
),
−α i β i
).
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For N = 1 , the transverse stretch can be obtained from (22) as −
(
αi
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P1BA (λ , λT ) = P2BA (λ , λT ) =
(21)
(22)
(23)
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Thus, the engineering stresses in equibiaxial extension become
2αβ 2 µ α −1 − 1+ β −1 P (λ ) = P (λ ) = λ −λ . α BA 1
BA 2
(24)
However, for , N ≥ 2 (22) cannot be solved for λT .
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Simple shear (SS): In case of simple shear, the deformation gradient is given by 1 γ 0 F = 0 1 0 , 0 0 1
J = 1.
(25)
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The solution for the principal stretches can be written as
(
) (
)
1 1 2+γ 2 +γ 4+γ 2 = γ + γ 2 + 4 , 2 2
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λ1 =
) (
(
(26)
)
1 1 2+γ 2 −γ 4+γ 2 = −γ + γ 2 + 4 , 2 2
λ2 =
λ3 = 1.
(27) (28)
It should be noted that J = 1 implies that λ2 = 1/λ1 . The solution for the principal stresses are α −1 −1 2 µi γ 1 2 i γ 1 2 P (γ ) = ∑ + γ + 4 − + γ +4 , 2 2 2 i =1 α i 2 N
SS 1
(29)
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α −1 −1 2 µi γ 1 2 γ 1 2 i P (γ ) = ∑ − + γ + 4 −− + γ +4 , 2 2 2 2 i =1 α i
(30)
P3SS = 0.
(31)
N
SS 2
deformation is characterized by λ as λ 0 0 F = 0 λ 0 , 0 0 λ
(32)
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J = λ3 .
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Volumetric deformation (VOL): In this deformation mode, λ1 = λ2 = λ3 . Thus, the whole
The principal stresses are equal in all the three princpial directions:
(λ ) = P (λ ) = P (λ ) = ∑ 2µi (λαi − λ−3α i βi ). λ α
P
VOL 2
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N
VOL 1
1
VOL 3
i =1
2.2 Constraints on the parameters
(33)
i
The material parameters in (1) cannot be arbitrarily chosen. The linearized (about λk = 1 )
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response of the hyperelastic models must reduce to the classical Hooke’s law. The ground-state bulk modulus, shear modulus and Young’s modulus can be obtained by following the work of Ogden [22], [23].
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calculated as [1]:
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The ground-state bulk modulus K and the ground-state shear modulus µ can be
N 1 K = ∑ 2 µi + β i , 3 i =1
N
µ = ∑ µi .
(34)
i =1
Having K and µ in hand, we can easily express the Young’s modulus E as 2
N N N ∑ µi + 3 ∑ µ j ∑ µi β i 9 Kµ i =1 j =1 i =1 E= = 2 . N 3K + µ ∑ µi (1 + 2β i ) i =1
Material parameters µ and K must be positive, therefore, we have the constraints
(35)
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N
∑µ i =1
1
N
i
> 0,
∑ 2µ 3 + β > 0. i
(36)
i
i =1
It clearly follows that, if these constraints are fulfilled, then E is also positive. It should be noted that ABAQUS accept only β i > −1/3 values but it is an unnecessarily strict constraint for
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N ≥2
3. Experimental data
The main goal of this paper is to present a novel parameter fitting strategy for the hyperfoam
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material model. The experimental data used in this report will be uniaxial compression and biaxial compression data obtained for a closed-cell polyethylene foam material.
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The uniaxial compression data was captured using engineering strain rate of ε& = 10−5 1/ s. The captured data points corresponding to the loading and unloading phases cannot be
distinguished, therefore the viscoelastic effect was eliminated. Thus, the experimental data can be considered as the long-term pure hyperelastic response of the material. The specimen was a cube having 50 mm edge dimensions. The average cell size was about 1mm. The transverse
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deformation of the specimen was recorded using optical measurement [14]. For the equibiaxial compression test, the test fixture developed in [15] was used. The specimen was a cube material having 100 mm edge dimensions. The applied engineering strain rate was the same as in the uniaxial compression. Since the biaxial compression fixture hid the
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specimen, we were unable to measure the transverse stretch using side-view optical
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measurement. Therefore, the transverse stretch will be approximated as UA λTBA = 2(λUA T − 1) + 1 = 2λT − 1,
(37)
where λUA represents the transverse stretch in uniaxial compression. The experimental data T are shown in Figure 1 and in Figure 2, whereas the values are reported in Table 1.
Figure 1: Engineering stresses in uniaxial and equibiaxial compressions
Figure 2: Transverse stretch in uniaxial and equibiaxial compressions
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Table 1: Experimental data
1.00570 1.01212 1.01786 1.02311 1.02915 1.03551 1.04217 1.04962 1.05751 1.06596 1.07543 1.08615 1.09849 1.11285 1.12816
0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55
λTsBAexp 1.0114 1.02424 1.03572 1.04622 1.05830 1.07102 1.08434 1.09924 1.11502
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exp λBA s
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0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25
exp λUA Tr
Biaxial compression PsBAexp [kPa] -13.67 -25.99 -36.04 -47.33 -60.36 -75.82 -96.05 -123.13 -166.03
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exp λUA r
Uniaxial compression PrUAexp [MPa] -9.1 -17.3 -26 -33.8 -41.41 -50.36 -61.15 -74.04 -89.57 -108.67 -132.54 -163.49 -204.91 -263.43 -353.57
4.1 Introduction
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4. Parameter fitting procedure
For the parameter fitting procedure of a hyperelastic material, we should use as many distinct
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experimental data (uniaxial compression, eqibiaxial compression ... etc.) as we can, but at least exprimental data corresponding to two distinct deformation modes is required. If only the uniaxial extension solution fit to the experimental data is used then the model solutions for
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other modes may be very inaccurate (see [30] for instance). In this paper, the hyperfoam model will be fitted to the uniaxial and the equibiaxial
compreression experimental data, including the corresponding measured transverse stretch data. The main idea of the fitting algorithm can be easily extendable for cases when experimental data of other homogeneous deformations are also provided. The parameter fitting algorithm of the first-order ( N = 1 ) model is very simple because we have closed-form solution for the stresses in terms of the primary stretches (see (9) and (24)). In addition, we have an explicit expression for the transverse stretch in terms of the
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primary stretch (see (8) and (23)). Therefore, the quality (or error) function to be minimized can be easily constructed by summing the errors between the experimental data and the model responses. However, the first-order model may serve an inaccurate result and we need to use a higher-order model ( N ≥ 2 ) for the characterization. As discussed in Section 2, for the
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higher-order models, the transverse stretch cannot be eliminated and cannot be fitted separately as in the case of using a first-order model. Consequently, another strategy has to be employed.
A possible way to overcome this difficulty is to neglect the transverse stretches, thus
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letting λT = 1 in uniaxial compression for instance. In this approach, it automatically follows that all β i parameters are 0 . However, this approximation can be used only for materials
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(mainly open-cell foams) where the transverse stretch is really negligible ([5,8,27]). If the transverse stretch is significant, we can use the method as follows [26]. In each step of the parameter fitting procedure, we first solve the zero transverse stress constraint (7) for the transverse stretch λT . Then, the obtained λT value is substituted back into (6) to calculate the model response. The drawback of this method is that there is no guarantee that
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we can solve (7) for any value of the primary stretch. In addition, the predicted transverse stretch value may be very inaccurate. However, the method can be improved by modifying the quality function in the optimization routine including the fitting of the transverse stretch prediction to the experimental data. The main drawback is that we have to use our own
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optimization code alongside the numerical solution scheme for λT , and we cannot use a
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third-party software for the optimization routine.
4.2 New method
In the following, a new strategy is presented. If we use the transverse stretch experimental data in (6) and in (21), then the quality function (error function) to be minimized is defined as Φ = Q1 + Q2 ,
(38)
with
(
exp exp PUAexp − P1UA λUA , λUA r Tr Q1 = ∑ r PrUAexp r =1 R
) , 2
(39)
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(
exp BAexp PsBAexp − P1BA λBA , λTs s Q2 = ∑ PsBAexp s =1 S
) , 2
(40)
where Q1 is the normalized error function for the uniaxial compression data, whereas Q2 is
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the normalized error function for the equibiaxial compression experiment. R and S are the number of captured data points in the uniaxial and equibiaxial compression tests. Superscript "exp" refers to the experimental data. However, in this case, the zero transverse stress constrains (7) and (22) are not satisfied, therefore the parameter fitting procedure may lead to
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a very inaccurate result. The novel idea is to enforce the zero transverse stress constraints via the optimization routine, which can be achieved by adding two additional terms to the quality function (38):
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Φ = Q1 + Q2 + Q3 + Q4
where
(
) ,
(42)
(
) .
(43)
exp exp PUA λUA , λUA r Tr Q3 = ∑ 2 PrUAexp r =1 R
BAexp exp P BA λBA , λTs s Q4 = ∑ 3 PrUAexp s =1
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S
(41)
2
2
Q3 and Q4 is the normalized error functions defined between the model predictions ((7) and
(22)) and the zero stress (as experimental data). The experimental data for the transverse stress
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is obviously zero, therefore, in the normalized error expressions (42) and (43), we can use the stress values corresponding to the loading directions. Minimizing the quality function Q in
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(41) serves a parameter set which tries to enforce the zero transverse stress constraint in addition to the fitting of the transverse stretches to the experimental data. It should be noted that the transverse stretch is not fitted explicitly. However, it is fitted in an implicit manner via the zero transverse stress constraint. The main benefit of this method is that we can use any third-party software (or built-in algorithm) for the minimization of Q . There is no need to solve the highly nonlinear expressions (7) and (22) for the transverse stretch. The new method can be used for the first-order model as well.
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5. Results This section presents the fitted material responses using the new method proposed in the preceding section. The global minimization of the quality function Φ in (41) was performed
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using the built-in function „Nminimize” in Wolfram Mathematica [32]. Function „Nminimize” offers four methods for the minimization: Nelder–Mead, Differential Evolution, Simulated Annealing and Random Search. The best fit was achieved using the Random Search method. The fitted material parameters are listed in Table 2. The accuracy of the first and the
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second-order model is demonstrated by the R 2 values (coefficient of determination) corresponding to each experimental data (stress PUA in uniaxial compression, stress P BA in in uniaxial compression, transverse stretch equibiaxial compression, transverse stretch λUA T
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λTBA in equibiaxial compression). These values are reported in Table 3. The model responses are illustrated in Figure 3. The stress solutions for other deformations were also calculated and they are presented in Figure 4 for the second-order model. The Drucker’s stability check ([1,28]) was performed for these parameters and they serve stable material responses for all deformation modes for all stretches.
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As can be clearly concluded, the fitted material response of the second-order model is very accurate, there is no need to use higher order models. However the new method can be used for higher-order models as well. The model prediction for the transverse stretches is very
EP
accurate, however, the fitting to the transverse stretch data is not explicitly included in the minimization algorithm.
Remark: It should be emphasized that, in a FE calculation, the zero stress constraint in
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uniaxial extension is always satisfied, because the corresponding boundary value problem is solved. Consequently, the transverse stretch is obtained from the zero transverse stress constraint. However, in the new parameter fitting strategy the transverse stretch is provided from the experimental data. Therefore, it is obvious that we have non-zero transverse stress if we substitute back the fitted material parameters into Eq. (6) using the experimental transverse stretch data. To illustrate more clearly this phenomenon, consider the parameter set obtained for N = 1 (see Table 2), for instance. Substituting the parameters µ1 , α1 and β1 back into Eq. (6) and (7) along with the experimental transverse stretch data, we obtain the stress
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solutions P%1UA and P%2UA (see Figure 5). P%2UA is obviously non-zero but it is very close to zero, because this constraint is enforced via the quality function Q3 in Eq. (42). Stress solutions P%1UA and P%2UA are used only in the minimization process. If we could completely achieve zero
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transverse stress ( P%2UA = 0 ) via the minimization algorithm, then we would obtain the same results as would be predicted using FE calculation for the obtained material parameter set. The statements above are valid for the other homogeneous deformation modes. Table 2: Obtained material parameters
104.869
α1 [-]
7.10874
β1 [-] µ2 [kPa] α 2 [-]
0.106469
4.26621
50.8857
20.9835
0.153096 0.041169
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β 2 [-]
58.2225
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µ1 [kPa]
N =2
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N =1
Table 3: R 2 values corresponding to the fitted models PUA
N =1
EP
N =1 N = 2
97.69
λTBA
N =2
N =1
N =2
N =1
N =2
99.60
86.90
99.31
64.02
99.58
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R 2 [%] 98.90 99.25
λUA T
P BA
Figure 3: Illustration of the fitted model solutions. Solid curves represent the model responses, whereas empty circles denote the experimental data. Left column corresponds to the first-order model, whereas right column presents the second-order model. Figure 4: Model solutions for the volumetric compression and for the planar extension for the second-order model. Figure 5: Comparison of the real stress solution and the stress solution used in the minimization process in case of uniaxial compression. Gray dashed curves represent P%1UA and P%2UA , whereas the solid black curve shows the real model response, while empty circles denote the
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experimental data.
6 Conclusions A new parameter fitting strategy for the hyperfoam material model was proposed in this article.
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The transverse stretch values of the experimental data are included in the quality function (error function) of the new method and, in addition, the zero transverse stress constraint was also enforced. Therefore, there is no need to solve the highly nonlinear zero transverse stress equation for the transverse stretch, which is the commonly adopted technique for this
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compressible hyperelastic model. Consequently, the new parameter fitting method can be used with any minimization solver to find the material parameters. In addition, experimental data
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were provided for uniaxial and equibiaxial compressions with the corresponding transverse stretch values.
Acknowledgements
This research has been supported by the Hungarian Scientific Research Fund, Hungary (Project Identifier: PD 108691). The research leading to these results has received funding from the The
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Hungarian-American Enterprise Scholarship Fund’s (HAESF). The foam specimens were provided by Furukawa Electric Institute of Technology Ltd., Hungary. These supports are
References
ABAQUS 6.14 Documentation. Dassault Systems, Simulia Corporation, Providence, Rhode
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[1]
EP
gratefully acknowledged.
Island, USA. [2]
Sz. Berezvai, A. Kossa, Effect of the skin layer on the overall behavior of closed-cell
polyethylene foam sheets, Journal of Cellular Plastics 52 (2016), 215–229. [3]
J.S. Bergstrom, Mechanics of Solid Polymers, William Andrew, Elsevier, 2015.
[4]
A.F. Bower, Applied Mechanics of Solids, CRC Press, 2010.
[5]
C. Briody, B. Duignan, S. Jerrams, J. Tiernan, The implementation of a visco-hyperelastic
numerical material model for simulating the behaviour of polymer foam materials, Computational Materials Science 64 (2012), 47–51.
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[6]
R. Brown, Physical Testing of Rubber, Springer, 2006.
[7]
R. Brown, Handbook of Polymer Testing, Rapra Technology Limited, 2002.
[8]
C.G. Fontanella, A. Forestiero, E.L. Carniel, A.N. Natali, Analysis of heel pad tissues
mechanics at the heel strike in bare and shod conditions, Medical Engineering & Physics 35
[9]
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(2013), 441–447 U.W. Gedde, Polymer physics, Springer, 1995.
[10]
L.J. Gibson, M.F. Ashby, Cellular Solids, 2nd ed., Cambridge University Press, 1997.
[11]
R. Hill, Aspects of invariance in solid mechanics, Advances in Applied Mechanics 18
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(1978), 1–75.
G.A. Holzapfel, Nonlinear Solid Mechanics, Wiley, 2000.
[13]
M. Hossain, P. Steinmann, More hyperelastic models for rubber-like materials: consistent
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[12]
tangent operators and comparative study, Journal of the Mechanical Behavior of Materials 22 (2013), 27–50. [14]
A. Kossa, Effect of the modeling of lateral stretch in the parameter identification
algorithm of hyperelastic foam materials, 31st Danubia-Adria Symposium on Advances in Experimental Mechanics, Kempten, Germany, 2014.
A. Kossa, A new biaxial compression fixture for polymeric foams, Polymer Testing 45
(2015), 47–51. [16]
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[15]
A. Kossa, Sz. Berezvai, Visco-hyperelastic characterization of polymeric foam materials,
[17]
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Materials Today: Proceedings 3 (2016), 1003–1008. L. Meunier, G. Chagnon, D. Favier, L. Orgeas, P. Vacher, Mechanical experimental
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characterisation and numerical modelling of an unfilled silicone rubber, Polymer Testing 27 (2008), 765–777. [18] [19]
N. Mills, Polymer Foams Handbook, Elsevier, 2007. D.C. Moreira, L.C.S. Nunes, Comparison of simple and pure shear for an incompressible
isotropic hyperelastic material under large deformation, Polymer Testing 32 (2013), 240–248. [20]
R.W. Ogden, Large deformation isotropic elasticity - on the correlation of theory and
experiment for incompressible rubberlike solids, Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences 326 (1972), 565–584. [21]
R.W. Ogden, Large deformation isotropic elasticity: On the correlation of theory and
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experiment for compressible rubberlike solids, Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences 328 (1972), 567–583. [22]
R.W. Ogden, Elastic deformations of rubberlike solids, Mechanics of Solids, The Rodney
Hill 60th Anniversary Volume (1982), 499–537. R.W. Ogden, Recent advances in the phenomenological theory of rubber elasticity 59
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[23]
(1986), 361–383. [24]
R.W. Ogden, G. Saccomandi, I. Sgura, Fitting hyperelastic models to experimental data,
Computational Mechanics 34 (2004), 484–502.
M. Sasso, G. Palmieri, G. Chiappini, D. Amodio, Characterization of hyperelastic
SC
[25]
rubber-like materials by biaxial and uniaxial stretching tests based on optical methods, Polymer
[26]
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Testing 27 (2008), 995–1004.
M. Schrodt, G. Benderoth, A. Kühhorn, G. Silber, Hyperelastic Description of Polymer Soft
Foams at Finite Deformations, Technische Mechanik 25 (2005), 162–173 [27]
M.R. Shariatmadari, R. English, G. Rothwell, Effects of temperature on the material
characteristics of midsole and insole footwear foams subject to quasi-static compressive and shear force loading, Materials and Design 37 (2012), 543–559
G. Silber, C. Then, C., Preventive Biomechanics, Springer, 2013.
[29]
S.P. Soe, N. Martindale, C. Constantinou, M. Robinson, Mechanical characterisation of
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[28]
Duraform Flex for FEA hyperelastic material modelling, Polymer Testing 34 (2014), 103–112. P. Steinmann, M. Hossain, G. Possart, Hyperelastic models for rubber-like materials:
EP
[30]
consistent tangent operators and suitability for Treloar’s data, Archive of Applied Mechanics 82
[31]
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(2012), 1183–1217.
B. Storå kers, On material representation and constitutive branching in finite
compressible elasticity, Journal of the Mechanics and Physics of Solids 34 (1986),125–145. [32]
Wolfram Research, Inc., Mathematica, Version 10.1, Champaign, IL, 2016.
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S0142-9418(16)30333-6
DOI:
10.1016/j.polymertesting.2016.05.014
Reference:
POTE 4659
To appear in:
Polymer Testing
Received Date: 11 April 2016 Accepted Date: 17 May 2016
Please cite this article as: A. Kossa, S. Berezvai, Novel strategy for the hyperelastic parameter fitting procedure of polymer foam materials, Polymer Testing (2016), doi: 10.1016/ j.polymertesting.2016.05.014. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Property Modelling
Novel strategy for the hyperelastic parameter fitting procedure of
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polymer foam materials Attila KOSSA, Szabolcs BEREZVAI
Department of Applied Mechanics, Budapest University of Technology and Economics
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H-1111 Budapest, Muegyetem rkp. 5., Hungary
Abstract
The most widely used approach to model the large strain elastic response of polymer foams in a
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finite element (FE) simulation is the use of the Ogden–Hill compressible hyperelastic material model. This model is implemented and termed as "hyperfoam" material model in the commercial FE software ABAQUS. The hyperfoam model is able to characterize the large compressibility (in volumetric sense) of the foam material. In order to find the material parameters of the model for a particular foam specimen, we need to fit the simulated
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responses to the available experimental data. This task is easier for incompressible hyperelastic materials because we can use the incompressibility constraint to eliminate the transverse stretch from the stress solutions. However, this simplification cannot be used for the hyperfoam model, therefore, in the stress-strain relations, the transverse stretch is included,
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which makes the parameter fitting procedure more complicated. In this paper, a novel strategy is proposed for the parameter fitting task. The performance of the new algorithm is
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demonstrated by presenting fitted material responses for a particular polymer foam material. The major advantage of the new strategy is that it can be used with any third-party optimization solver and there is no need to write our own code.
Keywords: hyperelastic; polymer foam; parameter fitting.
1. Introduction The material behavior of solid polymers is highly nonlinear and various tests have to be
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performed to completely characterize every aspect [6], [7], [9], [3]. Polymer foams have special microstructure, which makes their mechanical behavior more nonlinear [10], [18], [2], [16]. In order to accurately describe the mechanical characteristics of polymer foams, accurate prediction of their large strain elastic behavior is crucial.
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The finite strain elastic behavior of elastomeric polymer foams can be accurately captured using the Ogden-Hill hyperelastic material model ([31], [28], [18]) based on the work of Ogden and Hill ([20], [21], [11]). This model is implemented as the "hyperfoam" hyperelastic model in the commercial finite element software, ABAQUS [1]. The strain energy function W
N
W (λ1 , λ2 , λ3 ) = ∑ where λk
(k = 1,2,3)
(
)
2 µi α i 1 −αi βi α α λ + λ2 i + λ3 i − 3 + J − 1 , 2 1 αi βi
(1)
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i =1
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of the hyperfoam model is written as
represent the principal stretches; J denotes the volume ratio
(determinant of the deformation gradient F ); N is the order (positive integer) of the model; whereas µi , α i and β i are the hyperelastic material parameters 1 . Consequently, 3 N material parameters have to be found for a specific material. It should be noted that ABAQUS
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uses ν i parameters instead of β i . The relation between them is given by
νi =
βi
1 + 2β i
,
βi =
νi
1 − 2ν i
.
(2)
A possible way to find the material parameters of a hyperelastic model is to fit the model
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responses of particular homogeneous deformations to experimental data. This is a widely used procedure for hyperelastic materials [24], [30], [13], [17], [25], [19], [29]. However, the fitting of
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the hyperfoam model is more complicated than fitting of hyperelastic models proposed for incompressible materials. In the incompressible case, the transverse stretches can be easily calculated from the incompressibility constraint J = 1 in uniaxial compression/tension or in other homogeneous deformations. However, for the hyperfoam model the transverse stretch, in general, cannot be obtained from the zero transverse stress constraint, even in uniaxial compression. Therefore, the parameter fitting procedure is not so trivial for this material
1
It should be noted that the
between them is
µ
ABAQUS i
µi
parameters used by ABAQUS is different than the
= αi ⋅ µ
Ogden i
/2
µi
parameters proposed by Ogden. The relation
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model. This paper presents a novel strategy to find the material parameters of the hyperfoam material models. The main advantages of the new method are that it includes the transverse stretch data in the parameter fitting algorithm and, in addition, it enforces the zero transverse
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stress constraints.
The outline of the paper is as follows: Section 1 is the introduction; Section 2 presents the solutions for the most widely used homogeneous deformation modes; the experimental data used for the parameter fitting procedure is provided in Section 3; Section 4 presents the
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new parameter fitting strategy; the obtained solutions are shown in Section 6; Section 7 closes
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the paper with conclusions.
2. Model responses for homogeneous deformations 2.1 Stress solutions
For a given hyperelastic strain energy potential W , the first Piola–Kirchhoff stress P can be obtained as
∂W nk ⊗ Nk , k =1 ∂λk 3
Pk =
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P=∑
∂W , ∂λ k
(3)
where λk are the principal stretches, N k and n k are the unit vectors along the principal directions in the reference and the current configurations, whereas Pk are the principal
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stresses [12], [4]. Substituting (1) into (3) gives the relation for the principal nominal stresses of the hyperfoam model:
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N
Pk = ∑ i =1
2µi
αi
(λ
α i −1 k
− λ−k1 J
−α i β i
),
k = 1,2,3.
(4)
The stress solutions for the most important homogeneous deformation modes are
summarized in the following paragraphs. Uniaxial extension (UA): The stretch along the loading is denoted by λ1 , whereas the lateral (transverse) stretches are λ2 and λ3 . For isotropic material, we have the identity λ2 = λ3 .
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For simplicity of the presentation let λ ≡ λ1 and λT ≡ λ2 = λ3 , where subscript T refers to the transverse direction. The deformation gradient F and the volume ratio J can be written as 0
λT 0
0 0 , λT
J = detF = λλT2 .
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λ F = 0 0
(5)
Consequently, the principal first Piola–Kirchhoff stresses (principal engineering stresses, principal nominal stresses) can be obtained by inserting J = λλT2 into (4): 1
(λ
2µi
N
∑α λ i =1
− (λλT2 )
i
1
N
−α i β i
2µi
∑α
(λ
αi
λT
),
− (λλT2 )
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P2UA (λ , λT ) = P3UA (λ , λT ) = 0 =
αi
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P1UA (λ , λT ) =
i =1
T
−α i β i
(6)
).
(7)
i
For the first-order model (, N = 1) the transverse stretch can be expressed from the zero transverse stress constraint (7) as
−
λT = λ
β
1+ 2 β
= λ−ν .
(8)
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Therefore, the transverse stretch can be eliminated in (6) and the nominal stress along the loading direction can be written as the function of λ as αβ 2 µ α −1 −1+ 2 β −1 P (λ ) = λ −λ . α
(9)
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UA 1
However, for higher-order models ( N ≥ 2 ), equation (7) cannot be solved for λT . Consequently, the stress along the loading direction cannot be expressed as the function of the
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primary stretch λ and, therefore, the transverse stretch is always included. Confined uniaxial extension (CU): If we fix the transverse deformation ( λT = 1 ) in uniaxial extension, then the deformation gradient and the volume ratio reduce to λ 0 0 F = 0 1 0 , 0 0 1
The solutions for the stresses can be simplified to
J = λ.
(10)
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αi
i =1
(λ
α i −1
N
P2CU (λ ) = P3CU (λ ) = ∑
−α i β i −1
−λ
2µi
αi
i =1
),
(11)
(1 − λ ). −α i β i
(12)
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2µi
N
P1CU (λ ) = ∑
Planar extension (PE): In case of planar extension one of the transverse stretches in uniaxial extension is fixed. If we fix λ3 in (5) then we arrive at λ 0 F = 0 λT 0 0
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0 0, 1
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J = λλT .
(13)
For this deformation mode, we have three distinct principal stresses: P1PE (λ , λT ) =
1
2µi
N
∑α λ i =1
P2PE (λ , λT ) = 0 =
N
1
(λ
αi
N
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(λ
2µi
i =1
P3PE (λ , λT ) = ∑
−α i β i
),
(14)
i
∑α
λT
− (λλT )
2µi
αi
αi
− (λλT )
T
−α i β i
),
(15)
i
(1 − (λλ ) ). −α i β i
T
(16)
For N = 1 , the transverse stretch can be obtained from (15) as −
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λT = λ
β
1+ β
−
=λ
ν 1− v
(17)
.
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Therefore, the non-zero stresses can be written as the function of the primary stretch as P1PE (λ ) =
αβ
2 µ α −1 −1+ β −1 λ −λ , α
αβ − 2 µ 1+ β P (λ ) = 1− λ α PE 3
.
(18)
(19)
However, for N ≥ 2 , (15) cannot be solved for λT .
Equibiaxial extension (BA): In equibiaxial extension the deformation gradient and the volume ratio have the form
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λ 0 F = 0 λ 0 0
0 0 , λT
J = λ2 λT .
(20)
Consequently, the engineering stresses can be written as N
1
2µi
∑α λ i =1
P3BA (λ , λT ) = 0 =
N
1
2µi
∑α
λT
i =1
(λ
− λ2 λT
(
)
i
(λ
αi
− λ2 λT
T
i
λT = λ
2β 1+ β
−
=λ
2ν 1− v
.
)
−α i β i
),
−α i β i
).
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For N = 1 , the transverse stretch can be obtained from (22) as −
(
αi
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P1BA (λ , λT ) = P2BA (λ , λT ) =
(21)
(22)
(23)
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Thus, the engineering stresses in equibiaxial extension become
2αβ 2 µ α −1 − 1+ β −1 P (λ ) = P (λ ) = λ −λ . α BA 1
BA 2
(24)
However, for , N ≥ 2 (22) cannot be solved for λT .
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Simple shear (SS): In case of simple shear, the deformation gradient is given by 1 γ 0 F = 0 1 0 , 0 0 1
J = 1.
(25)
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The solution for the principal stretches can be written as
(
) (
)
1 1 2+γ 2 +γ 4+γ 2 = γ + γ 2 + 4 , 2 2
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λ1 =
) (
(
(26)
)
1 1 2+γ 2 −γ 4+γ 2 = −γ + γ 2 + 4 , 2 2
λ2 =
λ3 = 1.
(27) (28)
It should be noted that J = 1 implies that λ2 = 1/λ1 . The solution for the principal stresses are α −1 −1 2 µi γ 1 2 i γ 1 2 P (γ ) = ∑ + γ + 4 − + γ +4 , 2 2 2 i =1 α i 2 N
SS 1
(29)
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α −1 −1 2 µi γ 1 2 γ 1 2 i P (γ ) = ∑ − + γ + 4 −− + γ +4 , 2 2 2 2 i =1 α i
(30)
P3SS = 0.
(31)
N
SS 2
deformation is characterized by λ as λ 0 0 F = 0 λ 0 , 0 0 λ
(32)
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J = λ3 .
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Volumetric deformation (VOL): In this deformation mode, λ1 = λ2 = λ3 . Thus, the whole
The principal stresses are equal in all the three princpial directions:
(λ ) = P (λ ) = P (λ ) = ∑ 2µi (λαi − λ−3α i βi ). λ α
P
VOL 2
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N
VOL 1
1
VOL 3
i =1
2.2 Constraints on the parameters
(33)
i
The material parameters in (1) cannot be arbitrarily chosen. The linearized (about λk = 1 )
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response of the hyperelastic models must reduce to the classical Hooke’s law. The ground-state bulk modulus, shear modulus and Young’s modulus can be obtained by following the work of Ogden [22], [23].
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calculated as [1]:
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The ground-state bulk modulus K and the ground-state shear modulus µ can be
N 1 K = ∑ 2 µi + β i , 3 i =1
N
µ = ∑ µi .
(34)
i =1
Having K and µ in hand, we can easily express the Young’s modulus E as 2
N N N ∑ µi + 3 ∑ µ j ∑ µi β i 9 Kµ i =1 j =1 i =1 E= = 2 . N 3K + µ ∑ µi (1 + 2β i ) i =1
Material parameters µ and K must be positive, therefore, we have the constraints
(35)
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N
∑µ i =1
1
N
i
> 0,
∑ 2µ 3 + β > 0. i
(36)
i
i =1
It clearly follows that, if these constraints are fulfilled, then E is also positive. It should be noted that ABAQUS accept only β i > −1/3 values but it is an unnecessarily strict constraint for
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N ≥2
3. Experimental data
The main goal of this paper is to present a novel parameter fitting strategy for the hyperfoam
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material model. The experimental data used in this report will be uniaxial compression and biaxial compression data obtained for a closed-cell polyethylene foam material.
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The uniaxial compression data was captured using engineering strain rate of ε& = 10−5 1/ s. The captured data points corresponding to the loading and unloading phases cannot be
distinguished, therefore the viscoelastic effect was eliminated. Thus, the experimental data can be considered as the long-term pure hyperelastic response of the material. The specimen was a cube having 50 mm edge dimensions. The average cell size was about 1mm. The transverse
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deformation of the specimen was recorded using optical measurement [14]. For the equibiaxial compression test, the test fixture developed in [15] was used. The specimen was a cube material having 100 mm edge dimensions. The applied engineering strain rate was the same as in the uniaxial compression. Since the biaxial compression fixture hid the
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specimen, we were unable to measure the transverse stretch using side-view optical
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measurement. Therefore, the transverse stretch will be approximated as UA λTBA = 2(λUA T − 1) + 1 = 2λT − 1,
(37)
where λUA represents the transverse stretch in uniaxial compression. The experimental data T are shown in Figure 1 and in Figure 2, whereas the values are reported in Table 1.
Figure 1: Engineering stresses in uniaxial and equibiaxial compressions
Figure 2: Transverse stretch in uniaxial and equibiaxial compressions
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Table 1: Experimental data
1.00570 1.01212 1.01786 1.02311 1.02915 1.03551 1.04217 1.04962 1.05751 1.06596 1.07543 1.08615 1.09849 1.11285 1.12816
0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55
λTsBAexp 1.0114 1.02424 1.03572 1.04622 1.05830 1.07102 1.08434 1.09924 1.11502
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exp λBA s
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0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25
exp λUA Tr
Biaxial compression PsBAexp [kPa] -13.67 -25.99 -36.04 -47.33 -60.36 -75.82 -96.05 -123.13 -166.03
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exp λUA r
Uniaxial compression PrUAexp [MPa] -9.1 -17.3 -26 -33.8 -41.41 -50.36 -61.15 -74.04 -89.57 -108.67 -132.54 -163.49 -204.91 -263.43 -353.57
4.1 Introduction
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4. Parameter fitting procedure
For the parameter fitting procedure of a hyperelastic material, we should use as many distinct
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experimental data (uniaxial compression, eqibiaxial compression ... etc.) as we can, but at least exprimental data corresponding to two distinct deformation modes is required. If only the uniaxial extension solution fit to the experimental data is used then the model solutions for
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other modes may be very inaccurate (see [30] for instance). In this paper, the hyperfoam model will be fitted to the uniaxial and the equibiaxial
compreression experimental data, including the corresponding measured transverse stretch data. The main idea of the fitting algorithm can be easily extendable for cases when experimental data of other homogeneous deformations are also provided. The parameter fitting algorithm of the first-order ( N = 1 ) model is very simple because we have closed-form solution for the stresses in terms of the primary stretches (see (9) and (24)). In addition, we have an explicit expression for the transverse stretch in terms of the
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primary stretch (see (8) and (23)). Therefore, the quality (or error) function to be minimized can be easily constructed by summing the errors between the experimental data and the model responses. However, the first-order model may serve an inaccurate result and we need to use a higher-order model ( N ≥ 2 ) for the characterization. As discussed in Section 2, for the
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higher-order models, the transverse stretch cannot be eliminated and cannot be fitted separately as in the case of using a first-order model. Consequently, another strategy has to be employed.
A possible way to overcome this difficulty is to neglect the transverse stretches, thus
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letting λT = 1 in uniaxial compression for instance. In this approach, it automatically follows that all β i parameters are 0 . However, this approximation can be used only for materials
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(mainly open-cell foams) where the transverse stretch is really negligible ([5,8,27]). If the transverse stretch is significant, we can use the method as follows [26]. In each step of the parameter fitting procedure, we first solve the zero transverse stress constraint (7) for the transverse stretch λT . Then, the obtained λT value is substituted back into (6) to calculate the model response. The drawback of this method is that there is no guarantee that
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we can solve (7) for any value of the primary stretch. In addition, the predicted transverse stretch value may be very inaccurate. However, the method can be improved by modifying the quality function in the optimization routine including the fitting of the transverse stretch prediction to the experimental data. The main drawback is that we have to use our own
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optimization code alongside the numerical solution scheme for λT , and we cannot use a
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third-party software for the optimization routine.
4.2 New method
In the following, a new strategy is presented. If we use the transverse stretch experimental data in (6) and in (21), then the quality function (error function) to be minimized is defined as Φ = Q1 + Q2 ,
(38)
with
(
exp exp PUAexp − P1UA λUA , λUA r Tr Q1 = ∑ r PrUAexp r =1 R
) , 2
(39)
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(
exp BAexp PsBAexp − P1BA λBA , λTs s Q2 = ∑ PsBAexp s =1 S
) , 2
(40)
where Q1 is the normalized error function for the uniaxial compression data, whereas Q2 is
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the normalized error function for the equibiaxial compression experiment. R and S are the number of captured data points in the uniaxial and equibiaxial compression tests. Superscript "exp" refers to the experimental data. However, in this case, the zero transverse stress constrains (7) and (22) are not satisfied, therefore the parameter fitting procedure may lead to
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a very inaccurate result. The novel idea is to enforce the zero transverse stress constraints via the optimization routine, which can be achieved by adding two additional terms to the quality function (38):
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Φ = Q1 + Q2 + Q3 + Q4
where
(
) ,
(42)
(
) .
(43)
exp exp PUA λUA , λUA r Tr Q3 = ∑ 2 PrUAexp r =1 R
BAexp exp P BA λBA , λTs s Q4 = ∑ 3 PrUAexp s =1
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S
(41)
2
2
Q3 and Q4 is the normalized error functions defined between the model predictions ((7) and
(22)) and the zero stress (as experimental data). The experimental data for the transverse stress
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is obviously zero, therefore, in the normalized error expressions (42) and (43), we can use the stress values corresponding to the loading directions. Minimizing the quality function Q in
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(41) serves a parameter set which tries to enforce the zero transverse stress constraint in addition to the fitting of the transverse stretches to the experimental data. It should be noted that the transverse stretch is not fitted explicitly. However, it is fitted in an implicit manner via the zero transverse stress constraint. The main benefit of this method is that we can use any third-party software (or built-in algorithm) for the minimization of Q . There is no need to solve the highly nonlinear expressions (7) and (22) for the transverse stretch. The new method can be used for the first-order model as well.
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5. Results This section presents the fitted material responses using the new method proposed in the preceding section. The global minimization of the quality function Φ in (41) was performed
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using the built-in function „Nminimize” in Wolfram Mathematica [32]. Function „Nminimize” offers four methods for the minimization: Nelder–Mead, Differential Evolution, Simulated Annealing and Random Search. The best fit was achieved using the Random Search method. The fitted material parameters are listed in Table 2. The accuracy of the first and the
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second-order model is demonstrated by the R 2 values (coefficient of determination) corresponding to each experimental data (stress PUA in uniaxial compression, stress P BA in in uniaxial compression, transverse stretch equibiaxial compression, transverse stretch λUA T
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λTBA in equibiaxial compression). These values are reported in Table 3. The model responses are illustrated in Figure 3. The stress solutions for other deformations were also calculated and they are presented in Figure 4 for the second-order model. The Drucker’s stability check ([1,28]) was performed for these parameters and they serve stable material responses for all deformation modes for all stretches.
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As can be clearly concluded, the fitted material response of the second-order model is very accurate, there is no need to use higher order models. However the new method can be used for higher-order models as well. The model prediction for the transverse stretches is very
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accurate, however, the fitting to the transverse stretch data is not explicitly included in the minimization algorithm.
Remark: It should be emphasized that, in a FE calculation, the zero stress constraint in
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uniaxial extension is always satisfied, because the corresponding boundary value problem is solved. Consequently, the transverse stretch is obtained from the zero transverse stress constraint. However, in the new parameter fitting strategy the transverse stretch is provided from the experimental data. Therefore, it is obvious that we have non-zero transverse stress if we substitute back the fitted material parameters into Eq. (6) using the experimental transverse stretch data. To illustrate more clearly this phenomenon, consider the parameter set obtained for N = 1 (see Table 2), for instance. Substituting the parameters µ1 , α1 and β1 back into Eq. (6) and (7) along with the experimental transverse stretch data, we obtain the stress
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solutions P%1UA and P%2UA (see Figure 5). P%2UA is obviously non-zero but it is very close to zero, because this constraint is enforced via the quality function Q3 in Eq. (42). Stress solutions P%1UA and P%2UA are used only in the minimization process. If we could completely achieve zero
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transverse stress ( P%2UA = 0 ) via the minimization algorithm, then we would obtain the same results as would be predicted using FE calculation for the obtained material parameter set. The statements above are valid for the other homogeneous deformation modes. Table 2: Obtained material parameters
104.869
α1 [-]
7.10874
β1 [-] µ2 [kPa] α 2 [-]
0.106469
4.26621
50.8857
20.9835
0.153096 0.041169
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β 2 [-]
58.2225
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µ1 [kPa]
N =2
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N =1
Table 3: R 2 values corresponding to the fitted models PUA
N =1
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N =1 N = 2
97.69
λTBA
N =2
N =1
N =2
N =1
N =2
99.60
86.90
99.31
64.02
99.58
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R 2 [%] 98.90 99.25
λUA T
P BA
Figure 3: Illustration of the fitted model solutions. Solid curves represent the model responses, whereas empty circles denote the experimental data. Left column corresponds to the first-order model, whereas right column presents the second-order model. Figure 4: Model solutions for the volumetric compression and for the planar extension for the second-order model. Figure 5: Comparison of the real stress solution and the stress solution used in the minimization process in case of uniaxial compression. Gray dashed curves represent P%1UA and P%2UA , whereas the solid black curve shows the real model response, while empty circles denote the
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experimental data.
6 Conclusions A new parameter fitting strategy for the hyperfoam material model was proposed in this article.
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The transverse stretch values of the experimental data are included in the quality function (error function) of the new method and, in addition, the zero transverse stress constraint was also enforced. Therefore, there is no need to solve the highly nonlinear zero transverse stress equation for the transverse stretch, which is the commonly adopted technique for this
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compressible hyperelastic model. Consequently, the new parameter fitting method can be used with any minimization solver to find the material parameters. In addition, experimental data
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were provided for uniaxial and equibiaxial compressions with the corresponding transverse stretch values.
Acknowledgements
This research has been supported by the Hungarian Scientific Research Fund, Hungary (Project Identifier: PD 108691). The research leading to these results has received funding from the The
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Hungarian-American Enterprise Scholarship Fund’s (HAESF). The foam specimens were provided by Furukawa Electric Institute of Technology Ltd., Hungary. These supports are
References
ABAQUS 6.14 Documentation. Dassault Systems, Simulia Corporation, Providence, Rhode
AC C
[1]
EP
gratefully acknowledged.
Island, USA. [2]
Sz. Berezvai, A. Kossa, Effect of the skin layer on the overall behavior of closed-cell
polyethylene foam sheets, Journal of Cellular Plastics 52 (2016), 215–229. [3]
J.S. Bergstrom, Mechanics of Solid Polymers, William Andrew, Elsevier, 2015.
[4]
A.F. Bower, Applied Mechanics of Solids, CRC Press, 2010.
[5]
C. Briody, B. Duignan, S. Jerrams, J. Tiernan, The implementation of a visco-hyperelastic
numerical material model for simulating the behaviour of polymer foam materials, Computational Materials Science 64 (2012), 47–51.
ACCEPTED MANUSCRIPT
[6]
R. Brown, Physical Testing of Rubber, Springer, 2006.
[7]
R. Brown, Handbook of Polymer Testing, Rapra Technology Limited, 2002.
[8]
C.G. Fontanella, A. Forestiero, E.L. Carniel, A.N. Natali, Analysis of heel pad tissues
mechanics at the heel strike in bare and shod conditions, Medical Engineering & Physics 35
[9]
RI PT
(2013), 441–447 U.W. Gedde, Polymer physics, Springer, 1995.
[10]
L.J. Gibson, M.F. Ashby, Cellular Solids, 2nd ed., Cambridge University Press, 1997.
[11]
R. Hill, Aspects of invariance in solid mechanics, Advances in Applied Mechanics 18
SC
(1978), 1–75.
G.A. Holzapfel, Nonlinear Solid Mechanics, Wiley, 2000.
[13]
M. Hossain, P. Steinmann, More hyperelastic models for rubber-like materials: consistent
M AN U
[12]
tangent operators and comparative study, Journal of the Mechanical Behavior of Materials 22 (2013), 27–50. [14]
A. Kossa, Effect of the modeling of lateral stretch in the parameter identification
algorithm of hyperelastic foam materials, 31st Danubia-Adria Symposium on Advances in Experimental Mechanics, Kempten, Germany, 2014.
A. Kossa, A new biaxial compression fixture for polymeric foams, Polymer Testing 45
(2015), 47–51. [16]
TE D
[15]
A. Kossa, Sz. Berezvai, Visco-hyperelastic characterization of polymeric foam materials,
[17]
EP
Materials Today: Proceedings 3 (2016), 1003–1008. L. Meunier, G. Chagnon, D. Favier, L. Orgeas, P. Vacher, Mechanical experimental
AC C
characterisation and numerical modelling of an unfilled silicone rubber, Polymer Testing 27 (2008), 765–777. [18] [19]
N. Mills, Polymer Foams Handbook, Elsevier, 2007. D.C. Moreira, L.C.S. Nunes, Comparison of simple and pure shear for an incompressible
isotropic hyperelastic material under large deformation, Polymer Testing 32 (2013), 240–248. [20]
R.W. Ogden, Large deformation isotropic elasticity - on the correlation of theory and
experiment for incompressible rubberlike solids, Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences 326 (1972), 565–584. [21]
R.W. Ogden, Large deformation isotropic elasticity: On the correlation of theory and
ACCEPTED MANUSCRIPT
experiment for compressible rubberlike solids, Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences 328 (1972), 567–583. [22]
R.W. Ogden, Elastic deformations of rubberlike solids, Mechanics of Solids, The Rodney
Hill 60th Anniversary Volume (1982), 499–537. R.W. Ogden, Recent advances in the phenomenological theory of rubber elasticity 59
RI PT
[23]
(1986), 361–383. [24]
R.W. Ogden, G. Saccomandi, I. Sgura, Fitting hyperelastic models to experimental data,
Computational Mechanics 34 (2004), 484–502.
M. Sasso, G. Palmieri, G. Chiappini, D. Amodio, Characterization of hyperelastic
SC
[25]
rubber-like materials by biaxial and uniaxial stretching tests based on optical methods, Polymer
[26]
M AN U
Testing 27 (2008), 995–1004.
M. Schrodt, G. Benderoth, A. Kühhorn, G. Silber, Hyperelastic Description of Polymer Soft
Foams at Finite Deformations, Technische Mechanik 25 (2005), 162–173 [27]
M.R. Shariatmadari, R. English, G. Rothwell, Effects of temperature on the material
characteristics of midsole and insole footwear foams subject to quasi-static compressive and shear force loading, Materials and Design 37 (2012), 543–559
G. Silber, C. Then, C., Preventive Biomechanics, Springer, 2013.
[29]
S.P. Soe, N. Martindale, C. Constantinou, M. Robinson, Mechanical characterisation of
TE D
[28]
Duraform Flex for FEA hyperelastic material modelling, Polymer Testing 34 (2014), 103–112. P. Steinmann, M. Hossain, G. Possart, Hyperelastic models for rubber-like materials:
EP
[30]
consistent tangent operators and suitability for Treloar’s data, Archive of Applied Mechanics 82
[31]
AC C
(2012), 1183–1217.
B. Storå kers, On material representation and constitutive branching in finite
compressible elasticity, Journal of the Mechanics and Physics of Solids 34 (1986),125–145. [32]
Wolfram Research, Inc., Mathematica, Version 10.1, Champaign, IL, 2016.
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