A pseudo-elastic model for loading, partial unloading and reloading of particle-reinforced rubber

Abstract

Particle-reinforced rubbers exhibit a marked stress softening during unloading after loading in uniaxial tension tests, i.e. the stress on unloading is significantly less than that on loading at the same stretch. This hysteretic behaviour is not accounted for when the mechanical properties are represented in terms of a strain-energy function, i.e. if the material is modelled as hyperelastic. In this paper a theory of pseudo-elasticity is used to model loading, partial or complete unloading and the subsequent reloading and unloading of reinforced rubber. The basis of the model is the inclusion in the energy function of a variable that enables the energy function to be changed as the deformation path changes between loading, partial unloading, reloading and any further unloading. The dissipation of energy, i.e. the difference between the energy input during loading and the energy returned on unloading is accounted for in the model by the use of a dissipation function, the form of which changes between unloading, reloading and subsequent unloading.

Introduction

During the last decade there has been a considerable growth in interest in modelling the mechanical response of rubbery polymers, and more particularly of particle-filled rubbers. This interest has been stimulated by numerous industrial applications of rubbers (vibration isolators, vehicle tyres, seals and shock absorbers, for example) and the availability of computational facilities suitable for running complex models in finite element software.

Many contributions have aimed to model inelastic behaviour of rubber such as the stress softening associated with the Mullins effect. These are mainly rate and time-independent models based on the use of damage theory. Representative examples of these works are the papers by Govindjee and Simo, 1991, Govindjee and Simo, 1992a, Govindjee and Simo, 1992b, extended to allow for viscoelasticity (1992b), Johnson and Beatty, 1993, Johnson and Beatty, 1995, Lion (1996), Huntley et al. (1997), Kaliske and Rothert (1998), Ogden and Roxburgh, 1999a, Ogden and Roxburgh, 1999b, Beatty and Krishnaswamy (2000) and Dorfmann et al. (2002). There are also many contributions dealing with time and/or rate effects and stress–strain cycling involving hysteresis. Representative works from a lengthy list include Johnson et al. (1995), Drozdov (1996), Drozdov and Dorfmann (2001), Ha and Schapery (1998), Reese and Govindjee (1998), Bergström and Boyce, 1998, Bergström and Boyce, 2000, Miehe and Keck (2000), Wu and Liechti (2000) and Yang et al. (2000). The collections of papers contained in the proceedings of the first two European conferences on constitutive models for rubber are also valuable sources of reference: Dorfmann and Muhr (1999) and Besdo et al. (2001). Both phenomenological and micro-structural models are represented in these contributions and for more detailed references we refer to the above-cited works.

The present paper focuses on the quasi-static modelling of the inelastic response of particle-reinforced rubber. In particular, we are concerned with the hysteretic cycles associated with partial unloading and reloading (at constant temperature) following loading after appropriate pre-conditioning aimed at eliminating the Mullins effect. Our starting point is the pseudo-elasticity theory of Ogden and Roxburgh (1999a), which was used to model the Mullins effect. It is adapted so as to model the hysteretic cycles mentioned above. While the theory is applicable to three-dimensional deformations, the details are described primarily for the simple tension specialization. Simple tension experiments on a 60 phr carbon black-filled rubber have been performed for loading, partial unloading, reloading and subsequent unloading in order to test the theory.

The paper is organized as follows. In Section 2 we summarize the required equations of non-linear elasticity, first for three dimensions and then for the appropriate homogeneous uniaxial specialization. In Section 3 the corresponding theory of pseudo-elasticity is outlined. In Section 4, first the specific model of Ogden and Roxburgh (1999a), with some modification, is reviewed and then adapted so as to capture the partial unloading–reloading–unloading response. Section 5.1 contains a brief discussion of the experimental results that are used as the basis for fitting the model. The elastic strain-energy function employed for describing the loading response (after pre-conditioning) is given in Section 5.2, and then the theory of Section 4 is used to fit the actual data.