On the numerical implications of multiplicative inelasticity with an anisotropic elastic constitutive law

The statement that theories of inelasticity at finite strains have arrived at a high level of development is only true in conjunction with isotropic material behaviour. From both points of view (theoretical and computational), the extension to anisotropic material behaviour seems to be a complicated task. The statement is especially true when the multiplicative decomposition of the deformation gradient is considered a basis for the formulation. Of special interest are questions related to the mathematical form of the stored energy function or, equivalently, of the constitutive relation for the material stress tensor as the thermodynamical force. This paper deals with the above issues. The anisotropic formulation is accomplished using the notion of structural tensors. Here we suggest that the privileged directions of the material should be transformed in a specific way under the action of the inelastic part of the deformation gradient. The inelastic behaviour is assumed to be governed by evolution equations of the unified type. Numerically, we deal with the full multiplicative structure of the theory. The numerical treatment is developed in full detail. Expressions concerning the local iteration as well as the tangent operator are derived. Various numerical examples with applications to shells are presented which demonstrate the influence of anisotropy and the applicability of the theory. Copyright © 2003 John Wiley & Sons, Ltd.