HIA: A Hybrid Integral Approach to model incompressible isotropic hyperelastic materials—Part 1: Theory

Highlights

• Hyperelastic model constructed from the molecular and phenomenological theories.

• The phenomenological integral density takes into account the effect of non-affine deformation.

• The model contains six rheological parameters, connected to the polymer chemistry and to the macroscopic behavior.

• Four sets of experimental data from the literature (Treloar, Yeoh–Fleming, Arruda–Boyce and Nunes–Moreira) are used to identify the rheological parameters and to assess the proposed model.

Abstract

This paper provides a new constitutive model for rubber-like materials. The model adds to the 8-chain density introduced by Arruda and Boyce, two phenomenological components: an original part made of an integral density and an interleaving constraint part represented by a logarithmic function as proposed by Gent and Thomas. The model contains six rheological parameters connected to the polymer chemistry and to the macroscopic behavior. Four sets of experimental data from the literature are used to identify the rheological parameters and to assess the proposed model. The model is able to reproduce with a good accuracy experimental data performed under different loading conditions such as uniaxial and equibiaxial tension, uniaxial compression, pure and simple shear as well as the Mooney plot.

Introduction

Rubbers and elastomers are used in various applications such as seals, tires or vibration mounts. Their chemical and mechanical properties make them good sealing elements against humidity, pressure and temperature. They possess in addition very good properties of energy absorption. Industrial use of rubber requires characterizations that need modeling behavior and a very large variety of models was proposed to this purpose in the literature during the last sixty years [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13].

Several constitutive models are based, for example, on the statistical mechanics as the neo-Hookean strain energy deduced from the Gaussian law [14], [15], [16] or on the strain energy of polymer chains using the Langevin statistics [1], [2], [5], [12], [17], [18].

In addition to the statistical mechanics-based models mentioned above, it exists many phenomenological models for rubber materials in the literature, see for example the reviews of Steinmann et al. [18] and Marckmann and Verron [19] on this topic. According to the models developed by Mooney [4], Yeoh [8], Gent [20], Pucci and Saccomandi [21] and Beda [10], [22], the strain energy of rubber material can be expressed with respect to the invariants of the strain tensor. Alternatively, the models suggested by Valanis and Landel [23], Ogden [6] and Davidson and Goulbourne [12] are expressed according to the principal stretches of the strain tensor. However, these models present difficulties to predict the behavior of the material for large compressive loading, particularly in the case where the Mooney plot is used to represent the experimental data [24].

In this context, we propose a new constitutive model for rubber materials which attempts to reconcile basic concepts from the molecular and phenomenological theories of hyperelasticity. This model is based on the 8-chain energy density introduced by Arruda and Boyce [1] to predict the affine deformation of the molecular chain [2]. It also includes the logarithmic function suggested by Gent and Thomas [3] to model the interleaving of the molecular chains. Finally, it includes two original contributions:

• An integral density allowing an excellent fit of the experimental data for large compressive loading.

• A new accurate computation of the inverse of the Langevin function to well describe the affine part of the rubber behavior [25].

In this way, the new model can be regarded as an hybridization between the molecular and the phenomenological theories. It contains six rheological parameters that need to be determined on the basis of experiments. Once these rheological parameters have been identified, the model was successfully compared to experimental data from the literature [1], [26], [27], [28] providing a remarkable agreement with the compressive part of the Mooney plot.