On theory and numerics of large viscoplastic deformation

Abstract

A multiplicative theory and corresponding computations of finite elastic-viscoplastic deformations based on unified constitutive equations are presented. Basic features of theory are: (1) The inelastic part of the deformation gradient is understood as a material stretch-type tensor, no need arises for the notion of the intermediate configuration. (2) In the isotropic case, the use of an elastic logarithmic strain tensor is shown to lead to the identification that the elastic strains are given by the quantity C −1 p C with C p being an inelastic Cauchy-Green type tensor. (3) In spite of the fact that the logarithmic strain measure is used, closed forms of the tangent operator within an implicit time integration scheme are given circumventing very involved elaborations when the spectral decomposition is used. (4) Evolution equations of the Bodner and Partom type are employed. A general formalism is developed for the application of the evolution laws of the unified type within the theoretical framework. The formalism can be used to generalize any unified evolution equations formulated for infinetismal strains to the range of finite strains. (5) For the axisymmetric case, in addition to the displacement-based 4-node and 9-node finite elements, an enhanced strain 4-node element is presented. It is shown, anyhow, that in the highly nonlinear regime the enhanced strain formulation may exhibit numerical instabilities. Different numerical examples of finite strain deformations are presented.