AN ENERGY-BASED LOCALIZATION THEORY .1. BASIC FRAMEWORK

The basic framework for an energy-based theory of localization is developed through the analysis of dynamic simple shearing motion of a thermo-viscoplastic solid. The key role of the kinetic energy of the deforming body as far as the characterization of shear band initiation is concerned has been illustrated by Shawki [1988, 1992, 1994a, 1994b]. In Shawki's work, a modified linear stability analysis takes full account of the time dependence of the dynamic simple shear homogeneous solution consistent with constant boundary velocities and adiabatic boundary conditions. The linear stability analysis indicates that the onset of localization is tied to positive rates of change of the kinetic energy of absolute perturbations. Subsequently, Shawki, Sherif, and Cherukuri [1992] illustrated that the fundamental role of the kinetic energy extends far beyond the initiation of shear localization. In this article, we present the general, energy-based framework for localization analysis in which the total kinetic energy serves as a single parameter for the characterization of the full localization history. A characteristic evolution profile of the kinetic energy is shown to correspond to a localizing deformation. The various stages of localization are redefined in view of the foregoing evolution profile. Furthermore, we present a convergence analysis for the finite difference algorithm which benefits significantly from the current characterization of shear localization. We also illustrate that numerical schemes may converge to incorrect late time solutions due to the insufficiency of the classical von Neumann stability constraints.