A physically based method to represent the thermo-mechanical behaviour of elastomers

Abstract

A geometric nonlinear and thermodynamical consistent constitutive theory is proposed, which allows the representation of the thermomechanical behaviour of carbon black filled rubber. In a recent paper [1] it was shown that the mechanical behaviour of this material is mainly influenced by nonlinear elasticity coupled with some inelastic effects, in particular the Mullins-effect, nonlinear rate dependence and a weak equilibrium hysteresis. In the present paper, the Mullins-effect is not taken into consideration. At first we discuss a uniaxial approach, based on a simple spring dashpot system of viscoplasticity. The essential feature of this model is a decomposition of the total stress into a rate independent equilibrium stress and a nonlinear rate dependent overstress. The equilibrium stress is decomposed into a sum of two terms as well: The first term, the elastic part of the equilibrium stress, is a nonlinear function of the total strain, and the second term, the so-called hysteretic part, depends in a rate independent manner on the strain history. Both the overstress and the hysteretic part of the equilibrium stress are determined by nonlinear elasticity relations which depend on internal variables. These internal variables are inelastic strains, and the corresponding evolution equations are developed in consideration of the second law of thermodynamics. Accordingly, we demonstrate that the principle of non-negative dissipation is satisfied for arbitrary deformation processes. In a further step, we transfer the structure of this model to the three-dimensional and geometric nonlinear case. In a certain sense similar to finite deformation elasto-plasticity, we introduce two multiplicative decompositions of the deformation gradient into elastic and inelastic parts. The first decomposition is defined with respect to the overstress and the second one with respect to the hysteretic part of the equilibrium stress. Consequently, two intermediate configurations are induced, which lead two different decompositions of the Green's strain tensor into elastic and inelastic parts. The latter are the internal variables of the model. For physical reasons, we define the corresponding stress tensors and derivatives in the sense of the concept of dual variables [7], [39]. Theconstitutive equations for the overstress and for the hysteretic part of the equilibrium stress are specified by nonlinear elasticity relations, formulated with respect to the different intermediate configurations. In order to facilitate a separate description of inelastic bulk and distortional effects, we introduce kinematic decompositions of the deformation gradient into volumetric and distortional parts. Numerical simulations demonstrate that the developed theory represents the mechanical behaviour of a tread compound at room temperature very well. Thermomechanical heating effects, which are caused by inelastic deformations are also described by the theory. The method proposed in this paper can be utilised to generalise uniaxial rheological models to three-dimensional finite strain theories, which are admissible in the sense of the second law of thermodynamics.