Thixotropic behaviour of rubber under dynamic loading histories: Experiments and theory

Abstract

In the first part of this essay we investigate the thermomechanical behaviour of carbon black-filled rubber under dynamic loading conditions. The loadings consist of static predeformations which are superimposed by small sinusoidal oscillations. The frequency was varied between 0.1 Hz and 100 Hz, the deformation amplitude between 0.006 and 0.06, the temperature between 253 K and 373 K and the storage and dissipation moduli measured. The data show that the frequency dependence of the moduli is of the powerlaw type. They also depend on the deformation amplitude and the temperature. If the temperature is constant, increasing amplitudes lead to decreasing moduli. As discussed by Payne (1965), this behaviour can be interpreted in terms of a thixotropic change. If, on the other hand, the deformation amplitude is kept constant, increasing temperature levels lead to decreasing moduli. In the second part of this work we develop a constitutive theory to represent the material behaviour observed. We consider the dissipation principle of thermodynamics and formulate the model for three-dimensional finite deformations. To describe thermal expansion effects, we decompose the deformation gradient into a thermal and a mechanical part as proposed by Lu and Pister (1975). We prescribe the thermal part by a constitutive function and assume the mechanical part to be the driving force for the stress tensor. As motivated in earlier works we split the total stress into an equilibrium stress and a rate-dependent overstress. We represent the equilibrium stress using a modified Mooney-Rivlin strain energy function and the overstress by a series of Maxwell elements whose springs are of Neo-Hookean type. Both the kinematic tensors and the associated stress measures are defined by using the concept of dual variables proposed by Haupt and Tsakmakis (1989). In order to represent the thixotropic effects, the viscosities depend on the temperature and the deformation history. The history dependence is implied by an internal variable which is a measure for the deformation amplitude and has a relaxation property as discussed by Payne (1965). To determine the material parameters, we linearise the constitutive equations with respect to the static predeformation and take the analytical solution for harmonic strain-controlled loadings into account. Numerical simulations demonstrate that the constitutive theory describes all phenomena experimentally observed with a fairly good approximation. The model is compatible with the second law of thermodynamics in the form of the Clausius Duhem inequality.