Nonlocal plasticity, instability, and well-posedness of the elastic-plastic initial-boundary value problem

Traditional associative plasticity theories have previously been shown to be incompatible with experimental data. For this reason, a nonassociative plasticity theory is commonly adopted. Nonassociative plasticity theories are prone to several forms of instability that result in the governing equations becoming ill-posed. Two of these instabilities are discussed: the localization instability and the Sandler-Rubin instability, which is a nonphysical instability that may occur with any degree of nonassociativity. The primary purpose of this dissertation is to describe how traditional nonassociative plasticity theory can be reformulated to eliminate the Sandler-Rubin instability while maintaining agreement with experimental data. Numerical and analytical techniques are used to investigate the effects of three nontraditional plasticity models on the existence of the Sandler-Rubin instability: viscoplasticity, incrementally non-linear plasticity, and nonlocal plasticity. Of these, it is shown that only incremental nonlinearity eliminates the instability while maintaining agreement with existing experimental data. Standard laboratory tests cannot detect nonlinearity in a material's incremental response. For this reason a new experimental method and a new data analysis technique are presented and used to validate incrementally nonlinear plasticity theory. The new technique uses a cyclic load path and an interpolation scheme to infer the material response to several loading directions at the same material state. This new technique was used to study the incremental response of aluminum 6061-T0. The new technique suggests that there is significant nonlinearity in the incremental response of this material. Though it will be demonstrated that the Sandler-Rubin instability is not eliminated with nonlocal theory, this theory is nevertheless well established at regularizing otherwise ill-posed localization problems. Few efficient numerical schemes exist for solving the equations of nonlocal plasticity. Schemes have previously been developed for solving these equations as part of a finite-element or element-free Galerkin method, but no general convergence criterion had been developed for these methods. A new numerical scheme for solving these equations using the material point method (MPM) is presented. The new scheme uses the MPM background grid for particle-to-particle communication, and results in a simple, matrix-free algorithm. A convergence crite- rion derived for the new method is furthermore shown to be applicable to some of the methods developed by other researchers.