A pseudo-elastic model for loading, partial unloading and reloading of particle-reinforced rubber
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.
Section snippets
Basic equations
For full details of the relevant theory of elasticity summarized in this section the reader is referred to, for example, Ogden, 1984, Ogden, 2001 and Holzapfel (2000).
We consider a rubberlike solid whose deformation is completely described by the deformation gradient tensor . The polar decompositions of the deformation gradient giveswhere is a proper orthogonal tensor and are positive definite and symmetric tensors (the right and left stretch tensors, respectively).
The spectral
Basic equations
In the theory of pseudo-elasticity developed by Ogden and Roxburgh (1999a) the strain-energy function appropriate for elasticity theory is modified by incorporating an additional variable η into the function. Thus, we writeIn the context of the Mullins effect, which is related to material damage, η is referred to as a damage or softening variable. The inclusion of η provides a means of changing the form of the energy function during the deformation process and hence changing the
A model for unloading and reloading
In this section we use a simple form for the pseudo-elastic constitutive law that was used previously by Ogden and Roxburgh (1999a) to model the idealized Mullins effect. Here, however, it is assumed that the Mullins effect is not present (having been removed by pre-conditioning) and we are concerned with modelling the hysteresis associated with loading–unloading cycles, and more particularly with partial unloading and reloading. The material is again taken to be incompressible and isotropic
Experimental data
To assess the inelastic effect during the loading, partial unloading and reloading response of particle-reinforced elastomers, a series of uniaxial extension tests were carried out at a constant temperature. Dumbbell specimens were provided by SEMPERIT (Austria) and were used as received. The compound contains 60 phr of carbon black and is treated as a filled rubber.
The loading, partial unloading and reloading tests were performed at room temperature using a testing machine designed at the
Discussion and conclusions
In this paper the theory of pseudo-elasticity, originally developed by Ogden and Roxburgh (1999a) to account for the Mullins effect, has been modified to develop a constitutive model for quasi-static loading, partial unloading, reloading and subsequent unloading of a rubber material with hysteretic response. The theory uses a deformation-dependent scalar parameter to modify the elastic strain-energy function to account for the change in material properties under large strains. A number of
Acknowledgements
The research was partially supported by the Research Directorates General of the European Commission (through project GRD1-1999-11095). The authors gratefully acknowledge this support.
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