EFFECT OF INTEGRATION METHODS ON THE SOLUTION OF AN ADIABATIC SHEAR BANDING PROBLEM

Equations governing the thermomechanical deformations of a block made of a viscoplastic material, and undergoing overall simple shearing deformations, are stiff in the sense that time scales associated with the heat conduction, viscous effects, strain hardening and strain‐rate hardening may differ by several orders of magnitude as localization of the deformation initiates and proceeds. Because of the presence of high spatial gradients within the region of localization of the deformation, we also consider a viscoplasticity theory in which the strain gradient is taken as a kinematic variable. The Galerkin approximation of the pertinent initial‐boundary‐value problem yields an initial‐value problem involving a set of coupled non‐linear ordinary differential equations. The solution of these equations by the Crank–Nicolson, Adams–Moulton and the Gear methods reveals that the three methods give qualitatively similar results. However, quantitatively the results by the three methods differ somewhat, the difference being more for the dipolar theory. Copyright © 1990 John Wiley & Sons, Ltd