On a numerical implementation of a formulation of anisotropic continuum elastoplasticity at finite strains

In a recent theoretical study [see C. Sansour, I. Karšaj, J. Sorić, A formulation of anisotropic continuum elastoplasticity at finite strains. Part I: Modelling, International Journal of Plasticity 22 (2006) 2346–2365], a constitutive model for anisotropic elastoplasticity at finite strains has been developed. The model is based on the multiplicative decomposition of the deformation gradient. The stored energy function as well as the flow rule has been considered as quadratic functions of their arguments. In both cases, the list of arguments is extended to include structural tensors which describe the anisotropy of the material response at hand. Non-linear isotropic hardening is considered as well. In this paper, the integration of the constitutive law is presented. The associative flow rule is integrated using the exponential map which preserves the plastic incompressibility condition. The numerical treatment of the problem is fully developed and expressions related to the local iteration and the consistent tangent operator are considered in detail. It is shown that while the consistent linearisation of the model is quite complicated, it still can be achieved if various intriguing implicit dependencies are identified and correctly dealt with. Various numerical examples of three-dimensional deformations of whole structural components are presented. The examples clearly illustrate the influence of anisotropy on finite elastoplastic deformations.