Plastic Flow Rules with Microstructural Evolution and
Effects on Strain Localization

Haizhen Pan
Ph.D. Candidate
Advisor: Professor John L. Bassani
Mechanical Engineering and Applied Mechanics
University of Pennsylvania

Abstract
In this lecture, we consider a class of elastic-plastic materials that possess local orthotropic symmetry which is naturally represented in terms of second-order orientation tensors that can evolve with deformation.  Applications include textured polycrystals, oriented polymers, and composite materials. At finite strain, the standard multiplicative elastic-plastic decomposition is adopted, and the flow rule is defined in the intermediate configuration.  The spin of the orthotropic axes, i.e. the microstructural spin, is defined to be the difference between the material and plastic spin.  The theory of invariants coupled with representations for tensor-valued functions are utilized to develop phenomenological constitutive relations, including an equation for plastic spin. The classical normality flow rule leads to generators for the plastic part of the rate of deformation that only depend linearly on the stress tensor. In constructing a constitutive equation for plastic spin in the intermediate configuration, we assume similar dependencies on stress tensor in addition to nonlinear dependencies on the invariants of stress and orientation tensors. Comparisons with experimental data for textured polycrystals under uniaxial tension and simple shear loading are promising.  Significant effects of microstructural evolution on limits to ductility are predicted. Sheet necking and shear banding are considered in this paper.  The effects of microstructural evolution can increase or delay the tendency for localization from uniform states of deformation, depending upon parameters in the equation for plastic spin that determine the direction and degree of rotations as well as the initial degree of anisotropy and the orientation of orthotropic axes relative to the loading direction.