AN ENERGY-BASED LOCALIZATION THEORY .1. BASIC FRAMEWORK

The basic framework for an energy-based theory of localization is deve loped through the analysis of dynamic simple shearing motion of a ther mo-viscoplastic solid. The key role of the kinetic energy of the defor ming body as far as the characterization of shear band initiation is c oncerned has been illustrated by Shawki [1988, 1992, 1994a, 1994b]. In Shawki's work, a modified linear stability analysis takes full accoun t of the time dependence of the dynamic simple shear homogeneous solut ion consistent with constant boundary velocities and adiabatic boundar y conditions. The linear stability analysis indicates that the onset o f localization is tied to positive rates of change of the kinetic ener gy of absolute perturbations. Subsequently, Shawki, Sherif, and Cheruk uri [1992] illustrated that the fundamental role of the kinetic energy extends far beyond the initiation of shear localization. In this arti cle, we present the general, energy-based framework for localization a nalysis in which the total kinetic energy serves as a single parameter for the characterization of the full localization history. A characte ristic evolution profile of the kinetic energy is shown to correspond to a localizing deformation. The various stages of localization are re defined in view of the foregoing evolution profile. Furthermore, we pr esent a convergence analysis for the finite difference algorithm which benefits significantly from the current characterization of shear loc alization. We also illustrate that numerical schemes may converge to i ncorrect late time solutions due to the insufficiency of the classical von Neumann stability constraints.