On the nonlinear viscoelastic behavior of rubber-like materials: Constitutive description and identification

Abstract The main concern of this paper is the development of a three dimensional viscoelastic model at finite strain to describe nonfactorizable behavior of rubber-like materials. The model is developed within the framework of rational thermodynamics and internal state variable approach such that the second law of thermodynamics in the form of Clausius–Duhem inequality is satisfied. The nonfactorizable aspect of the behavior is introduced via a strain dependent relaxation times. The model is applied to describe the response of the isotropic Pipkin multi-integral viscoelastic model and the Bromobutyl (BIIR) material, several parameters involved are then identified using quasi-static and dynamic experiments thanks to a least-square minimization procedure. The proposed model is able to reproduce quasi-static response and show a good ability to predict the dynamic response of nonfactorizable rubber-like materials (BIIR) and the multi-integral model of Pipkin in a wide range of strain.


Introduction 1
It is well known that rubber-like materials exhibit nonlinear viscoelastic 2 behavior over a wide range of strain and strain rates confronted in several engineering applications such as civil engineering, automotive and aerospace 4 industries. This is due to their capacity to undergo high strain and strain 5 rates without exceeding the elastic range of behavior. Further, the time de-6 pendent properties of these materials, such as shear relaxation modulus and 7 creep compliance, are, in general, functions of the history of the strain or the 8 stress [1]. Therefore, in a wide range of strain, a linear viscoelasticity theory 9 is no longer applicable for such material and new constitutive equations are 10 required to fully depict the behavior of rubber-like materials for quasi-static 11 and dynamic configurations of huge interest in engineering applications. 12 In the literature, several phenomenological models have been developed to 13 describe the nonfactorizable behavior of rubber-like materials, namely the  In this work we shall develop a nonlinear viscoelastic model at finite strain 30 within the framework of rational thermodynamics and the approach of inter- identified using data generated from the multi-integral viscoelastic model of 35 Pipkin [12] and experimental data for bromobutyl (BIIR) from [13]. 36 This paper is organized as follow: in section 2, a one dimensional nonlinear 37 viscoelastic model is developed using a modified Maxwell rheological model. 38 In the section 3, this model is extended to the fully nonlinear formulation us-    In figure 1 it is plotted the logarithm of the shear relaxation module G(t) The total stress σ derive directly from the rheological model of figure 2 as 88 the difference between the elastic equilibrium stress and the non-equilibrium 89 stresses q i .
a( ) is a non-negative strain function called strain shift function. Therefore, 99 the law of evolution of the equation 1 became a linear differential equation The formulation in the nonlinear range is based on the decomposition of the 123 gradient F (X, t) into a volume-preserving and pure dilatational part as its [21] among others as follow: sition is its validity near and far away from the thermodynamic equilibrium 128 [11]. The Cauchy-Green strain tensor associated and the Lagrangian strain 129 tensor associated with the volume-preserving gradient are expressed as I is the metric tensor in the reference configuration. Furthermore, several 131 applications of the chain rule lead to the following I is the fourth order unit tensor i.e ( I ijkl = 1 if (i = j = k = l); else I ijkl = 133 0), the sign ⊗ designates the tensorial product in the reference basis. Hence, 134 we postulated an uncoupled free energy density as its expressed in [9] by a 135 Taylor series in which terms higher than the second order are omitted.
Q is a second order overstress tensor internal variable akin to the second As in the previous section, a C is a function of the invariants of the volume-151 preserving right Cauchy-Green strain tensorC and ξ is referred to as the 8 where β j = β j (I 1 , I 2 , I 3 ) are the elastic response functions. In terms of the free energy density they are given by The free energy density has an alternative form in terms of the principle 178 stretches given by Henceforth, the material is considered incompressible so that the Cauchy 188 stress tensor σ and the Kirchhoff stress tensor τ are the same quantity.
The condition of incompressibility J = 1 leads to the following expression of 201 the deformation gradient tensor 202 For simple extension and (26) For the simple extension and can be obtained from the equilibrium deviatoric elastic stress via: parameters. The objective function is defined by the squared 2-norm The identification procedure turns out into a minimization problem which In the case of uniaxial experiment, the nominal stress which is the measured 232 quantity, actual force over reference area, and the principle stretch are related 233 11 through the free energy Ψ o by the equation For the simple extension and is the linearized strain tensor.
From equations 39 and 40 the shear relaxation modulus follows In this work we adopted the Prony series form of the shear relaxation modulus As mentioned in the previous section, the relaxation times τ i are a-priori fixed The optimization problem of equation 48 is an ill-posed problem [31]. There-298 fore, a Tikhonov [32] regularization method was employed to solve this sys-299 tem. The results of this identification using randomly perturbed simulated 300 and real experimental data are shown in the latter section of this paper.
where c 1 and c 2 are two positive material parameters to be fitted using a 324 nonlinear curve fitting algorithm with Matlab software.

Pipkin isotropic model
Pipkin [12] proposed a third order development of the tensorial response 344 function Q for an isotropic incompressible material. The principle of material 345 indifference requires that the Cauchy stress tensor takes the following form: R is the rotation tensor obtained from the polar decomposition of the trans-347 formation gradient tensor F and p is the indeterminate parameter due to 348 incompressibility. The third functional development of Q reads A crucial choice of the parameters a i , b i and c i enables us to describe the 356 behaviour of the material for any given strain history.
in the case of simple extension and Principle stretch       however, that the identification procedure was performed using results of the 380 5% level of strain. Prony series parameters are reported in Table 1.     [36] and then applied to dynamic data from [13].

456
• Tikhonov regularization method: 457 The linear system arising from the identification of the Prony series 458 parameters from dynamic data is an ill-posed problem [31]. From the 459 original system of equation 48 the following system arise: in which A is the global matrix of the system to be calculated from The proof of 63 and further development of the convergence of the 470 regularized Tikhonov problem are well studied in [37].