Keywords: constitutive model, anisotropic-stress-softening, compressible, Mullins effect.

Rubberlike materials are frequently used in structures. They are often treated as incompressible nonlinear purely elastic solids and isotropic with respect to the undeformed stress-free configuration. However, in reality this is not the case; there are numerous rubberlike materials that are nearly incompressible and behave inelastically. Anisotropic stress softening behaviour is often seen, e.g., in carbon black filled rubber even though the virgin material is isotropic. Quite often, treating a nearly incompressible material as incompressible gives erroneous results. However, most constitutive models developed in the past are isotropic models which cannot describe the anisotropic stress-softening effect, often called the Mullins effect.

In this paper we develop a constitutive equation for compressible rubberlike materials to take into account anisotropic stress softening induced by strain. The virgin material is "isotropic" in the stress free reference state. The quasi-static constitutive equation is purely phenomenological and does not take account the underlying physical structure of the material; hence it can be applied to any compressible material exhibiting anisotropic stress-softening induced by strain.

The proposed constitutive model is based on the work of Shariff [1] who developed a constitutive equation for incompressible anisotropic stress softening rubberlike materials and of Shariff & Parker [2] where they analysed nearly incompressible nonlinear anisotropic elastic materials via a variational principle. In order to develop the theory in close parallel with that described in Shariff [1] for incompressible materials we decompose the right stretch tensor into its isochoric and dilatational parts. The constitutive equation is written explicitly in terms of the dilatational part, the principle values and principal directions of the isochoric part of the right stretch tensor. To facilitate the modeling of our strain induced anisotropic stress softening constitutive equation we introduce the concepts damage function and damage point. The damage function can be considered as a measure of strain intensity when its argument is a principal stretch. The damage point corresponding to a principal stretch depends on the principal direction and on the history of the right stretch tensor. When the argument of the damage function is a damage point, the damage function measures amounts of damage caused by deformation on a line element in a prinicipal direction. The effect of shearing on stress softening materials is described via shear-history parameters. These parameters are the maximum and minimum values of the cosine of the angle between two principal-direction line elements throughout the history of the deformation.

The constitutive equation uses a parametric strain energy function where the parameters are damage and shear-history tensors. Both of these second order tensors are symmetric. The damage tensor contains the damage points and is positive definite. The shear-history tensor contains the shear-history parameters. In this communication, we only consider a class of parametric energy functions that is a subset of a wider class of energy functions introduced in this paper. A specific form of this special class is employed and this form seems to simplify the analysis of the three dimensional model. Energy dissipation is shown via the Clausius-Duhem inequality by treating the damage tensor as an internal variable.

To demonstrate the capabilities of the proposed theory, results are given for hydrostatic compression deformation and simple tension deformation. The theoretical results obtained are consistent with expected behaviour and compare well with the experimental data of Mullin and Tobin [3]

1

M.H.B.M. Shariff, "An Anisotropic Model of the Mullins Effect", Jou. of Engng. Maths, in press. doi:10.1007/s10665-006-9051-4

2

M.H.B.M. Shariff and D.F. Parker "An extension of Key's principle to nonlinear elasticity", Jou. of Engng. Maths., 37, 171-190, 2000. doi:10.1023/A:1004734311626

3

L. Mullins and N.R. Tobin, "Theoretical model for the elastic behaviour of filler-reinforced vulcanized rubbers", Rubber Chem. Technol., 30, 551-571, 1957.

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