# Development of a new constitutive model for thermoplastic copolyester materials

Pierre Robard, Laurent Gornet, Patrick Rozycki, Gilles Marckmann, Jean-Charles Guldner, Frédéric Maitay

► To cite this version:

Pierre Robard, Laurent Gornet, Patrick Rozycki, Gilles Marckmann, Jean-Charles Guldner, et al. Development of a new constitutive model for thermoplastic copolyester materials. 11th European Conference on Constitutive Models for Rubber (ECCMR 2019), Jun 2019, Nantes, France. pp.305-310, 10.1201/9780429324710-54. hal-05310149

HAL Id: hal-05310149

https://hal.science/hal-05310149v1

Submitted on 14 Oct 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Development of a new constitutive model for thermoplastic copolyester materials

P. Robard, L. Gornet, P. Rozycki & G. Marckmann

GeM Institut de Recherche en Genie Civil et Mécanique, UMR 6183 CNRS Ecole Centrale de Nantes

J.C. Guldner & F. Maitay

Trelleborg Boots France

**ABSTRACT:** The present work is dedicated to a sector market of Trelleborg Boots: the constant velocity joint boots (CVJ). During their use, joints are subjected to severe constraints such as wide temperature range, self-contact, friction between coils or grease. In order to increase their lifespan, Trelleborg Boots is investigating new materials and particularly thermoplastics copolyester-ether (TPE-E). The later exhibits some mechanical properties similar to rubber which includes large deformations, viscoelasticity, incompressibility...... However, an experimental campaign shows that viscoplastic effects must be considered. Consequently, as both rubber-like and metallic-like models of the literature are not suitable for TPE-E, a new one is proposed in order to predict the CVJ behavior by finite element (FE) analysis. Based on a multiplicative decomposition of the deformation gradient tensor, the constitutive model is inspired by the Bergström and Boyce model (Bergström and Boyce 1998) and written in a thermodynamical formalism. Both hyperelastic and damage parts are inspired by the GDM model (Gornet et al. 2012) and extended with a viscoelastic behavior. Finally, the viscoplastic part is based on the work done by (Shutov and Kreißig 2008).

## 1 INTRODUCTION

Due to their remarkable properties halfway from thermoplastics and rubbers, thermoplastic elastomers (TPE) are used in a growing number of specific applications. CVJ are a good example: during their use, they are subjected to cyclic loading (0 to 30 Hz) up to 30% strain under extreme environmental conditions (grease, temperature from -40°C to 140°C). These conditions do not allow the use of thermoplastics in the long run (because of their limited deformation) or elastomers (because of their low resistance to grease and high temperatures). By combining the advantages of these two materials, TPE copolyesters-ethers (also known as TPE-E or TPC) meet the needs. Indeed, TPC polymeric chains are built by the alternance of two monomers. Polyethers (soft segment) is chosen for its ability to bring the needed flexibility to the material. Polyester (hard segment) is used for its resistance to grease and high temperature. The aimed macroscopic behavior is set by adjusting types and proportions of monomers. At microscopic scale, the phases tend to separate and the rigid segment to cristallise, resulting into a semicristalline material.

Thermoplastic elastomers are the subject of many research works since few decades, in particular the

mechanical behaviors (Drobny 2015) (Drobny 2007) (Rinaldi et al. 2011). As a result of the complex microscopic structure, macroscopic mechanical behaviour exhibits some phenomena already known for elastomers and thermoplastics: Mullins effect (Mullins 1969), large hysteresis loops (Lion 1997), time-dependant behavior and large inelastic effects.

Numerous constitutives models have been proposed in the literature to describe large deformation behaviors of polymeric materials over the past several decades. A large range of polymeric material and specific behaviours have been studied: amorphous glassy polymers (Struik 1978), evolution of crystallinity under loading (Huneau 2011), semicrystalline polymers (Bartczak & Galeski 2010). . . Bergström and Boyce Boyce (1998) proposed a physically motivated model that generalized the Standard Linear Solid model (also known as Zener model) to finite strain. With a rheological approach, this model can be seen as an assembly of two sets (or branches) made of rheological elements (spring, damper,...). A first one represents the hyperelastic rubbery network behavior, consequence of the entropy change in the soft domain due to molecular network orientation (see also (Holzapfel 2000)). This behavior can be modeled by both Neo-Hoookean or Arrunda Boyce eight-chains models.
The second branch provides the time dependency and the dissipative effect. The friction between chains during deformation is modeled by a viscoelastic behaviour. For semicristalline polymers, it can be extended with a viscoplastic term that captures the configuration change of hard and soft domains Estes et al. (1971) due to the breaking and reforming process of hydrogen bonds. This model has been tested and improved several time by the authors, for exemple to take into account the influence of the fractions of hard and soft phases (Cho et al. 2017) on the mechanical behaviour. Other works done by Ayoub et al. 2010 on high density polyethylene suggests the use of three branches. As the Bergström-Boyce (BB) model, the first one represents the molecular network resistance, extended to include viscoplastic effects. Two viscoelastic-plastic branches complete the model and capture respectively the intermolecular resistance of crystalline and amorphous phases.

A new formalism of the Zener model based on the definition of internal variables and dissipation potential Sidoroff (1988) has been proposed by Lejeunes et al. (2011) for rubber materials. The behavior of this viscoelastic model can be easily extended to viscoplaticity with the appropriate choice of the dissipation potential. Vandenbroucke et al. (2010) proposes to add several parallel branches to the Zener model to capture both the viscoelastic behavior and the residual hysteresis loop. The model is called HVH for Hyper-Visco-Hysteretic. Zerbe et al. (2017) proposes to add a viscoplatic behavior in serial to represent the long term creep behavior.

Qi & Boyce (2005) and more recently Cho et al. (2017) proposed models for thermoplastic polyurethanes (TPUs), a block copolymer that exhibits a mecanical response close to the material of the present work. The authors extend the BB model by adding damages consideration to the hyperelastic network branch and the viscous effects are now considered as thermally activated viscoplastic effects.

<figure><img src="image_2.png" /><figcaption>Figure 1. Material response during a cyclic traction-relaxation test</figcaption></figure>

To conclude, few works concern the mechanical behaviour of block copolymers in the litterature. Thus we propose here a two-step study. First, we highlight the mechanical behaviour through tensile tests. Then, we propose a model based on the knowledge obtained in existing ones.

## 2 MATERIAL CHARACTERIZATION

### 2.1 Test methods

In this work, all tests were conducted on a bi-column electric testing machine, Instron model 3369 with a maximum testing velocity of 500 mm/min. Displacements are mesured with a long travel extensometer clamped to the specimen. Tensile force is given by 1kN load cell. The sample is made of copolyester-ether thermoplastic elastomer provided by Dupont. Dumbell specimens are obtained with a die cutter from molded plates (2 mm thickness) such as they are oriented along the injection direction. They are 40 mm long ($L_0$) and have an initial section of $S_0= 8 \text{ mm}^2$ for tensile test and of $S_0 = 144 \text{ mm}^2$ for compressive test. All tests are displacement driven.

### 2.2 Cyclic tensile test with relaxation phases

We note $\pi$ as the first Piola-Kirchoff stress tensor.

Figure 1 and 2 show the response of cyclic tensile test at a strain rate of $8.310^{-3} \text{s}^{-1}$ and $4.210^{-5} \text{s}^{-1}$, interrupted with 5 minutes relaxation phases. The material is stretched until 35% strain. Then unloaded and reloaded to observe hysteresis.

Although the ultimate deformation of this material is more than 300%, this work focuses on the range of 0 to 30%.

First of all, the material exhibits a non-linear behaviour observed during the first stretching up to 35%. The unload-reload step reveals a residual

<figure><img src="image_3.png" /><figcaption>Figure 2. Stress response during a cyclic traction-relaxation test as a function of time</figcaption></figure>
inelasticity, a softening of the material and an hysteresis loop. During reload, an inflexion point can be observed with an increase of the stiffness. The phenomenon is well-known for polymer and can be explained by the limit of polymer chains extensibility.

The time dependency of the material behaviour is highlighted by both relaxation of the stress during fixed displacement phases (figure 2) and the difference between the stress values for two tests at different strain rates. During the relaxation process, the stress strongly decreases in the first few seconds and then more slowly.

Finally, for the test at $8.310^{-3} \text{s}^{-1}$, the material response at the end of a relaxation phase seems to coincide with the response of the quasi-static test. However, it is interesting to note that the mechanical behavior exhibits an equilibrium hysteresis as described by Lion (1997) for rubber.

## 2.3 Conclusion

According to the previous part, we propose a constitutive model for large strain deformation of TPC. This model is able to model the following main characteristics: nonlinear large strain elastomeric behaviour, time dependence, softening of the material, residual inelasticity, time dependant/independent hysteresis loops.

# 3 THEORETICAL CONSTITUTIVE LAW

## 3.1 Introduction

The model proposed here (figure 3) can be seen as an extension of the BB (Bergström and Boyce 1998) and the HVH (Vandenbroucke et al. 2010) models. A first Hyperelastic-viscoplastic with damages branch ① is chosen to represent the rubbery network contribution (softening behavior and inelastic behavior). A second branch is proposed to capture the viscoelastic behavior ②. The equilibrium hysteresis loop is obtained by a last viscoelastoplastic branch with isotropic and kinematic hardening ③.

All the above observations can be considered with a rheological point of view. (Shutov and Kreißig 2008) and (Kießling et al. 2016) proposed a set of rheological elements and an assembly method to obtain a finite strain viscoelastic-plastic behavior model with isotropic and kinematic hardening. After adding damages, this model corresponds to the third branch ③ of the figure 3. One can notice that the two other branches can be obtained from the third one with appropriate parameters. In practice, ③ can be adapted and included in an assembly to build a large number of complex models.

In this work, we assume that the material is isotropic. The proposed model is written with a Lagrangian formalism and by defining of a dissipation potential (Lejeunes et al. 2011).

<figure><img src="image_4.png" /><figcaption>Figure 3. Proposed rheological model</figcaption></figure>

Due to quasi-incompressibility, we consider a multiplicative volumetric-deviatoric split of all the deformation gradient tensors (Bonet and Wood 2008), defining the unimodulus tensor as:

$$ \bar{F} = J^{-1/3}F \quad \text{with} \quad J = \det(F) \quad (1) $$

A classical multiplicative decomposition of the deformation tensor into an elastic (designated with “e”) and in inelastic (designated with “p”) parts is done. The last one is also decomposed in the same way.

$$ \bar{F} = F_e F_p \quad \text{and} \quad F_p = F_{pe} F_{pp} \quad (2) $$

The left Green-Langrange tensor can now be computed from each deformation gradient tensor:

$$ C = {}^t FF \quad C_e = {}^t F_e F_e \quad C_p = {}^t F_p F_p \quad (3) $$

And the elastic strain invariants can be defined such as:

$$ I_1^e = \text{tr}(C_e) = \text{tr}(C \cdot C_p^{-1}) \quad (4) $$

$$ I_2^e = 1/2((I_1^e)^2 - \text{tr}(C_e^2)) \quad (5) $$

We choose the following form for free energy:

$$ \Psi = \Psi_{iso} + \Psi_{dev} \quad (6) $$

$$ = U(J) + \bar{\Psi}_i(\bar{C}, \bar{C}_p) + \bar{\Psi}_{kin}(\bar{C}_p, \bar{C}_{pp}) + \Psi_{iso}(r) $$

And the dissipation can be derived as:

$$ \mathcal{D} = 2S : \dot{C} - \bar{\Psi} \geq 0 \quad (7) $$

$$ = 2 \left( S - \frac{\partial \Psi_i}{\partial C} \right) \dot{C} - 2 \left( \frac{\partial \Psi_i}{\partial C_p} + \frac{\partial \Psi_{kin}}{\partial C_p} \right) \dot{C}_p + 2 \left( \frac{\partial \Psi_{kin}}{\partial C_{pp}} \right) \dot{C}_{pp} - \left( \frac{\partial \Psi_{iso}}{\partial r} \right) \dot{r} $$

The dissipation potential $\varphi$ is introduced as:

$$ \mathcal{D} = \dot{\varphi} = \frac{\partial \varphi}{\partial \dot{C}} \dot{C} + \frac{\partial \varphi}{\partial \dot{C}_p} \dot{C}_p + \frac{\partial \varphi}{\partial \dot{C}_{pp}} \dot{C}_{pp} + \frac{\partial \varphi}{\partial \dot{r}} \dot{r} \quad (8) $$

Terms of Eq. (8) can be obtained by Eq. (7): With 7 and 8 we can identify:
$$
\left\{
\begin{array}{lcl}
\displaystyle \frac{\partial \varphi}{\partial C} & = & S - 2 \frac{\partial \Psi}{\partial C} = 0 \\[2ex]
\displaystyle \frac{\partial \varphi}{\partial C_p} & = & -2 \left( \frac{\partial \Psi_{ik}}{\partial C_p} + \frac{\partial \Psi_{kin}}{\partial C_p} \right) = -S_v \\[2ex]
\displaystyle \frac{\partial \varphi}{\partial C_{pp}} & = & -2 \left( \frac{\partial \Psi_{kin}}{\partial C_{pp}} \right) = -S_{pp}^{kin} \\[2ex]
\displaystyle \frac{\partial \varphi}{\partial r} & = & -2 \left( \frac{\partial \Psi_{kin}}{\partial r} \right) = -R
\end{array}
\right.
\qquad (9)
$$

$$
\text{with : } S_v = S_p^i + S_p^{kin} = 2 \frac{\partial \Psi_i}{\partial C_p} + 2 \frac{\partial \Psi_{kin}}{\partial C_p} \quad (10)
$$

Defining a dissipation pseudo-potential $\varphi^*$, one
can show that:

$$
\left\{
\begin{aligned}
\dot{C}_p &= - \frac{\partial \varphi^*}{\partial S_v} \\
\dot{C}_{pp} &= - \frac{\partial \varphi}{\partial S_{pp}^{kin}} \\
\dot{r} &= - \frac{\partial \varphi^*}{\partial R}
\end{aligned}
\right.
\qquad (11)
$$

3.2 Hyperelasticity

Numerous hyperelastic models have been proposed
in the past, a review of them has been done by
Marckmann et al. Marckmann et al. (2012). Assum-
ing that the material is isotropic and in order to rep-
resent the material hardening for high strain, the
Gornet-Desmorat-Marckmann model (Gornet et al.
2012) is choosen for the first branch. The free
energy function is extended for quasi-incompressible
materials and become:

$$
\bar{\Psi}_1 = h_1 \int e^{h_3(I_1^e - 3)^2} dI_1 + 3h_2 \int \frac{1}{\sqrt{I_2^e}} dI_2 \quad (12)
$$

where $h_1$, $h_2$ and $h_3$ are material parameters. For
the branches ② and ③, a Neo-Hookean model is
choosen:

$$
\bar{\Psi}_{2,3} = \frac{\mu}{2} (I_1^e - 3) \quad (13)
$$

3.3 Damage

We propose to model the material softening with a
continuum damage mechanics approach as proposed
by Lemaitre & Chaboche (2009) and adapted to
polymers by Gornet et al. Gornet et al. (2012). Thus,
we here consider a non standard damage model. The
damage evolution is governed by the maximum
value of the first invariant $I_1^e$.

$$
\bar{\Psi}_1 = \tilde{h}_1 \int e^{h_3(I_1^e - 3)^2} dI_1 + 3\tilde{h}_2 \int \frac{1}{\sqrt{I_2^e}} dI_2 \quad (14)
$$

$$
\tilde{h}_{i=1,2,3} = (1 - d_i)h_i \qquad (15)
$$

$$
d_{i=1,2} = d_i^\infty \left( 1 - \exp \left( - \frac{I_i^{\max}}{\eta_i} \right) \right) \quad (16)
$$

$$
\text{with : } I_1^{\max} = \max_{\tau \le t} (I_1^{\max}(t), I_1^{\max}(\tau)) \quad (17)
$$

As specified in Gornet et al. (2012), d₃ is not
independent but constrained by the polymer network
mass conservation. Then d₁ and d₃ are linked by:

$$
d_3 = 1 - F(d_1); F(d_1) = 1 - \frac{b}{h_3(3(a/\tilde{h}_1 - 1)^2)} \qquad (18)
$$

The same approach can be used for Neo-hookean
potential:

$$
\bar{\Psi}_{2,3} = \frac{\tilde{\mu}}{2} (I_1^e - 3)^2 \quad (19)
$$

with

$$
\tilde{\mu} = (1 - d_{\mu})\mu \quad (20)
$$

$$
d_{\mu} = d_{\mu}^{\infty} \left( 1 - \exp \left( - \frac{I_{1}^{\max}}{\eta_{\mu}} \right) \right) \quad (21)
$$

3.4 Viscous effects

In order to represent the viscous effects, several
approach have been proposed in the literature such
as: the integral formulation (Simo and Hughes
1998), the differential formulation (Sidoroff 1988) or
the statistic approach (Drozdov 2000) (Martinez
et al. 2011). Here, the second one is chosen with the
following dissipation pseudo-potential:

$$
\varphi^* = \frac{\eta_p}{\nu_p + 1} \left\langle \frac{F_p}{\eta_p} \right\rangle^{\nu_p + 1} + \frac{\eta_{kin}}{\nu_{kin} + 1} \left\langle \frac{F_{kin}}{\eta_{kin}} \right\rangle^{\nu_{kin} + 1} \quad (22)
$$

with (Simo 1988)

$$
\begin{align}
F_p &= \|\mathrm{DEV}(S_v)\|_L^{C_p} - R \tag{23} \\
&= \sqrt{\mathrm{tr}(C_p \mathrm{DEV}(S_v) C_p \mathrm{DEV}(S_v))} - R \nonumber
\end{align}
$$

$$
F_{kin} = \left\| \mathrm{DEV}(S_{pp}^{kin}) \right\|_{L}^{C_{pp}} \quad (24)
$$

and DEV(A) = [A - 1/3 tr(A · C)C⁻¹], the lagran-
gian deviator. The system 11 becomes:

$$
\left\{
\begin{aligned}
\dot{C}_p &= - \left\langle \frac{F_p}{\eta} \right\rangle^{\nu} \frac{C_p \mathrm{DEV}(S_v)}{\|\mathrm{DEV}(S_v)\|_L^{C_p}} C_p \\
\dot{C}_{pp} &= - \left\langle \frac{F_{kin}}{\eta_{kin}} \right\rangle^{\nu_{kin}} \frac{C_{pp} \mathrm{DEV}(S_{pp}^{kin})}{\|\mathrm{DEV}(S_{pp}^{kin})\|_L^{C_{pp}}} C_{pp} \\
\dot{r} &= \left\langle \frac{F_p}{\eta} \right\rangle^{\nu}
\end{aligned}
\right.
\qquad (25)
$$

Furthermore, viscoelasticity or quasi-static plasti-
city can be obtained by setting R = 0 or η_E < 1
respectively.

3.5 Kinematic hardening

In order to capture equilibrium hysteresis, endochro-
nic theory is proposed (Valanis and Read 1986)(Erli-
cher and Point 2006). The gist of this theory is the
definition of an intrinsic time z with the following
properties:

* intrinsic time increases monotonically with the physical time *t*
* only if the strain rate of the endochronic element is not zero, the intrinsic time will accumulate

* the higher the deformation rate is, the faster the intrinsic time passes

The time $z$ is defined as

$$z(t) = k_t \mathcal{L}_p \quad (26)$$

with $k_t = 1$s and $\mathcal{L}_p$ the strain arc length.

The derivative with respect to intrinsic time is defined such as:

$$\frac{\partial C_{pp}}{\partial z} = \frac{\partial C_{pp}}{\partial t} \frac{\partial t}{\partial z} = \frac{1}{k_t \dot{\mathcal{L}}_p} \dot{C}_{pp} \quad (27)$$

And the equation 25-2 becomes:

$$\dot{C}_{pp} = - \frac{1}{k_t \dot{\mathcal{L}}_p} \left\langle \frac{F_{kin}}{\eta_{kin}} \right\rangle^{\nu_{kin}} \frac{C_{pp} \text{DEV}(S_{pp}^{kin})}{\left\| \text{DEV}(S_{pp}^{kin}) \right\|_L^{C_{pp}}} C_{pp} \quad (28)$$

with $\dot{\mathcal{L}}_p = \sqrt{\frac{1}{6} \| C_p^{-1} \dot{C}_p \|}$.

## 3.6 Integration scheme

The problem is solved by return mapping algorithm (Simo and Hughes 1998). The internal variables are updated at each time-step by solving the system Eq. (25). The initial conditions are:

$$\left\{ \begin{array}{lcl} C_p(t=0) & = & \mathbf{1} \\ C_{pp}(t=0) & = & \mathbf{1} \\ r(t=0) & = & 0 \end{array} \right. \quad (29)$$

with **1** the identity tensor. System Eq. (25) can also be written as

$$\dot{Y} = f(Y)Y \quad (30)$$

<table><caption>Table 1. Model parameters</caption><thead><tr><th>Branch</th><th>Behaviour</th><th>law</th><th>variables</th></tr></thead><tbody><tr><td rowspan="4">1</td><td>Hyperelasticity</td><td>GDM</td><td>$h_1, h_2, h_3$</td></tr><tr><td>Damages</td><td>exponential</td><td>$d_{i=1,2}^{\infty}, \eta_{i=1,2}, a, b$</td></tr><tr><td>Plasticity</td><td>linear</td><td>$R_1 + a_1 r$</td></tr><tr><td>Viscous Effects</td><td>linear</td><td>$\eta_p$</td></tr><tr><td rowspan="4">2</td><td>Hyperelasticity</td><td>Neo</td><td>$\mu^{ve}$</td></tr><tr><td>Damages</td><td>Hookeen</td><td>$d_{i}^{ve,\infty}, \eta_{i}^{ve}$</td></tr><tr><td>Viscous Effects</td><td>linear</td><td>$\eta_{ve}$</td></tr><tr><td>Hyperelasticity</td><td>Neo</td><td>$\mu^{vp}$</td></tr><tr><td rowspan="5">3</td><td>Hyperelasticity</td><td>Hookeen</td><td>$d_{i}^{vp,\infty}, \eta_{i}^{vp}$</td></tr><tr><td>Damages</td><td>exponential</td><td>$R_3 + a_1 r$</td></tr><tr><td>Iso. Plasticity</td><td>linear</td><td></td></tr><tr><td>Iso. Viscous Effects</td><td>linear</td><td></td></tr><tr><td>Kin.</td><td>Neo-</td><td>$\mu^{vp,kin}$</td></tr><tr><td>Hyperelasticity</td><td>Hookean</td><td></td></tr><tr><td>Kin. Viscous Effects</td><td>linear</td><td>$\eta_{kin}$</td></tr></tbody></table>

with $Y = [\dot{C}_p, \dot{C}_{pp}, \dot{r}]^t$. This non-linear equation is solved with an implicit Euler algorithm:

$$Y_{n+1} - Y_n = f(Y_{n+1}) Y_{n+1} \Delta t \quad (31)$$

The tangents modulus of (25) are four order tensors. They are computed using (Itskov 2000). The global tangent modulus, defined as $\frac{\partial f}{\partial Y}$, is computed at the first iteration and reused for the following ones.

The plastic incompressibility condition is enforced by setting both $C_p^{n+1} = \dot{C}_p^{n+1}$ and $C_{pp}^{n+1} = \dot{C}_{pp}^{n+1}$

## 3.7 Parameters identification

All the parameters are summarized in table 1. They are optimized with gradient descent algorithm. The error function is defined such as:

$$\varepsilon = \sqrt{\sum_{i} i (\sigma_{i}^{\text{exp}} - \sigma_{i}^{\text{computed}})^{2}} \quad (32)$$

## 3.8 Results

The experimental data and the model results are showed in figures 4 and 5. There is a good agreement between experimental data in the proposed model.

<figure><img src="image_5.png" /><figcaption>Figure 4. Model-results comparison for a compressive test, strain-stress response</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 5. Model-results comparison for a compressive test, time-stress response</figcaption></figure>
4 CONCLUSION AND PERSPECTIVES

As highlighted in this study, the main characteristics of
the TPC mechanical behaviour are: 1. nonlinear large
strain elastomeric behaviour, 2. time dependence, 3.
softening of the material, 4. residual inelasticity 5. time
dependant/independent hysteresis loops. A model
based on the decomposition of the stress in three rheo-
logical branches have been proposed. The first one
captures the rubber-like hyperelastic response with
damage and plasticity. The second one capture the vis-
cous effects. Finally, a third one uses the endochronic
formalism to model the equilibrium hysteresis effects.
The proposed model exhibit good results for a com-
pressive test on the TPC material. Future works will be
carried out to validate this model in the case of alter-
nating traction-compression and multi-axial loading.
Then, the model will be implemented in a finite elem-
ent software. Finally, a temperature and aging influence
on the model's parameters will be studied.

REFERENCES

Ayoub, G., F. Zaïri, M. Naït-Abdelaziz, & J. M. Gloaguen (2010). Modelling large deformation behaviour under loading-unloading of semicrystalline polymers: Application to a high density polyethylene. *International Journal of Plasticity* 26(3), 329–347.

Bartczak, Z. & A. Galeski (2010, aug). Plasticity of Semi-crystalline Polymers. *Macromolecular Symposia* 294(1), 67–90.

Bergström, J. S. & M. C. Boyce (1998). Constitutive modeling of the large strain time-dependent behavior of elastomers. *Journal of Mechanics and Solids* 46(5), 931–954.

Bonet, J. & R. D. Wood (2008). *Nonlinear Continuum Mechanics for Finite Element Analysis (2nd Edition)*.

Cho, H., S. Mayer, E. Pöselt, M. Susoff, P. J. in 't Veld, G. C. Rutledge, & M. C. Boyce (2017, oct). Deformation mechanisms of thermoplastic elastomers: Stress-strain behavior and constitutive modeling. *Polymer* 128, 87–99.

Drobny, J. G. (2007). *Handbook of Thermoplastic Elastomers*. Wiley-VCH.

Drobny, J. G. (2015). *Specialty Thermoplastics*, Volume 13. Springer-Verlag Berlin Heidelberg.

Drozdov, A. D. (2000). Model of cooperative relaxation in finite viscoelasticity of amorphous polymers. *International Journal of Non-Linear Mechanics* 35(5), 897–909.

Erlicher, S. & N. Point (2006). Endochronic theory, non-linear kinematic hardening rule and generalized plasticity: a new interpretation based on generalized normality assumption. *International Journal of Solids and Structures* 43(14-15), 4175–4200.

Estes, G. M., R. W. Seymour, & S. L. Cooper (1971, jul). Infrared Studies of Segmented Polyurethane Elastomers. II. Infrared Dichroism. *Macromolecules* 4(4), 452–457.

Gornet, L., G. Marckmann, R. Desmorat, & P. Charrier (2012). A new isotropic hyperelastic strain energy function in terms of invariants and its derivation into a pseudo-elastic model for Mullins effect: Application to finite element analysis. *ResearchGate* (May), 265–271.

Holzapfel, G. (2000). *Nonlinear solid mechanics: A continuum approach for engineering*, Volume First Edit.

Huneau, B. (2011, sep). Strain-induced crystallization of natural rubber: a review of X-ray diffraction investigations. *Rubber Chemistry and Technology* 84(3), 425–452.

Itskov, M. (2000). On the theory of fourth-order tensors and their applications in computational mechanics. *Computer Methods in Applied Mechanics and Engineering* 189(2), 419–438.

Kießling, R., R. Landgraf, R. Scherzer, & J. Ihlemann (2016). Introducing the concept of directly connected rheological elements by reviewing rheological models at large strains. *International Journal of Solids and Structures* 97_98, 650–667.

Lejeunes, S., A. Boukamel, & S. Méo (2011). Finite element implementation of nearly-incompressible rheological models based on multiplicative decompositions. *Computers and Structures* 89(3-4), 411–421.

Lemaitre, J. & J.-L. Chaboche (2009). *Mecanique des materiaux solides, 2eme edition*. Dunod.

Lion, A. (1997, nov). On the large deformation behaviour of reinforced rubber at different temperatures. *Journal of the Mechanics and Physics of Solids* 45(11-12), 1805–1834.

Marckmann, G. & E. Verron (2006). Comparison of Hyper-elastic Models for Rubber-Like Materials. http://Dx.Doi.Org/10.5254/1.3547969, 835–858.

Martinez, J., A. Boukamel, S. Méo, & S. Lejeunes (2011, nov). Statistical approach for a hyper-visco-plastic model for filled rubber: Experimental characterization and numerical modeling. *European Journal of Mechanics - A/Solids* 30(6), 1028–1039.

Mullins, L. (1969, mar). Softening of Rubber by Deformation. *Rubber Chemistry and Technology* 42(1), 339–362.

Qi, H. & M. Boyce (2005, aug). Stressstrain behavior of thermoplastic polyurethanes. *Mechanics of Materials* 37 (8), 817–839.

Rinaldi, R. G., M. C. Boyce, S. J. Weigand, D. J. Londono, & M. W. Guise (2011, dec). Microstructure evolution during tensile loading histories of a polyurea. *Journal of Polymer Science Part B: Polymer Physics* 49(23), 1660–1671.

Shutov, A. V. & R. Kreißig (2008). Finite strain viscoplasticity with nonlinear kinematic hardening: Phenomenological modeling and time integration. *Computer Methods in Applied Mechanics and Engineering* 197 (21-24), 2015–2029.

Sidoroff, F. (1988). Internal variables and phenomenological models for metals plasticity. *Revue de Physique Appliquée* 23(4), 649–659.

Simo, J. C. (1988). A Framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition: part I. continuum formulation. 66, 199–219.

Simo, J. C. & T. J. R. Hughes (1998). *Computational inelasticity*. Springer.

Struik, L. C. E. (1978). Physical Aging in Amorphous Glassy Polymers and Other Materials. *Elsevier*, 78–85.

Valanis, K. & H. Read (1986, sep). An endochronic plasticity theory for concrete. *Mechanics of Materials* 5(3), 277–295.

Vandenbroucke, A., H. Laurent, N. Aït Hocine, & G. Rio (2010, may). A Hyperelasto-Visco-Hysteresis model for an elastomeric behaviour: Experimental and numerical investigations. *Computational Materials Science* 48(3), 495–503.

Zerbe, P., B. Schneider, E. Moosbrugger, & M. Kaliske (2017). A viscoelastic-viscoplastic-damage model for creep and recovery of a semicrystalline thermoplastic. *International Journal of Solids and Structures* 110-111, 340–350.