Analytical and numerical study of shear localization phenomena

To model shear localization phenomena, we consider the one-dimensional unidirectional shearing of a high strength material. This leads to a coupled parabolic-hyperbolic system of nonlinear partial differential equations (PDE's) governing momentum, elasticity and energy. Three separate approaches to the problem are considered. In each case, boundary layer methods are used to take advantage of the extreme thinness of the localized region, commonly referred to as a shear band.;In one approach to the problem, we employ a thermal flux inhomogeneity along the centerline to initiate localization. Equations governing the leading order perturbations of an elastic-stage solution for stress and temperature are derived. By examining the growth of these perturbations, a criterion for shear band formation is deduced, independent of the initially imposed stimulus.;In another approach to the problem, we derive expressions for jump discontinuities in velocity, temperature gradient, and stress gradient across the shear band. This analysis also yields an expression for the temporal evolution of the thermal boundary layer width, which is interpreted as a measure of shear band thickness. The jump discontinuities are converted to boundary conditions for the half-slab problem which represents a simplified version of the original initial boundary value problem. Computations are made for the limiting case of a semi-infinite half-slab. The results reveal a shear band width that initially contracts to a minimum during localization and then later widens under diffusional effects.;In the last approach, a formulation that allows for the edge effects of a finite half-slab is developed. An asymptotic analysis outside the band is performed to determine the residual contribution of the plastic strain rate in the far field. This residual contribution is used together with the above mentioned jump discontinuities to reduce the problem. Numerical solution of the reduced problem shows excellent quantitative agreement with full scale numerical simulations, with the benefit of a greatly reduced computational workload.