An accurate and stable algorithm for high strain-rate finite strain plasticity

Abstract

A computational scheme is introduced for the integration of rate-dependent constitutive relations for the high strain-rate finite inelastic deformation and failure of metallic materials. Fundamental issues of accuracy, stability, and stiffness that are intrinsically related to the evolution of dynamic inelastic deformation and failure modes, such as shear-band formation, are addressed. Due to this evolution, accuracy, and stability characteristics of the constitutive relations change throughout the integration domain. Stiffness, a stability problem, may occur due to widely dispersed solution components that result from material instabilities. An adaptive methodology is introduced to classify these characteristics, and appropriate numerical methods were used. The current deformation configuration was updated by obtaining the current total deformation and the plastic deformation-rate tensors. The total deformation-rate tensor was obtained by an explicit finite element method. A nonlinear initial value problem is derived to update the plastic deformation-rate tensor. A combination of a fifth-order accurate explicit Runge-Kutta variable-step method and the A-stable backward Euler method were used to integrate this initial value system. The explicit method was used in nonstiff domains, where accuracy is required. If a time-step reduction was due to stability, a harbinger of numerical stiffness, the algorithm was automatically switched to the A-stable method. To distinguish a time-step reduction due to stability from one due to accuracy, a stiffness ratio is defined to measure the eigenvalue dispersion of the initial-value system. The adaptability of the proposed algorithm to a wide class of inelastic constitutive relations is illustrated by applications to crystal plasticity. This study underscores the importance of understanding the origin of numerical instabilities, such that these instabilities are not mistaken for inherent material instabilities.