Mathematical modelling of elastoplasticity at high stress

Abstract

This study describes a simple mathematical model for one-dimensional elastoplastic wave propagation in a metal in the regime where the applied stress greatly exceeds the yield stress. Attention is focused on the increasing ductility that occurs in the overdriven limit when the plastic wave speed approaches the elastic wave speed. Our model predicts that a plastic compression wave is unable to travel faster than the elastic wave speed, and instead splits into a compressive elastoplastic shock followed by a plastic expansion wave.

Longitudinal shock waves in solids: the piston shock analogue

“…It should also be noted that the appearance of a shock in such a field is necessarily involved with an unrealistic jump in cross-sectional area of the bar. Further reduction to the one-dimensional field with no changes in cross-sectional area implies ln a = ε e and by (2.6) we obtain the characteristic velocity 5) which is proportional to tangent modulus and independent of material compressibility. Also, to maintain hyperbolicity of the one-dimensional field (2.7), strain softening (h < 0) is prohibited.…”

“…While the amount of experimental data on shock compression in solids that has accumulated over the years is sufficient to delineate the physical nature of shock processes, the theoretical background has yet to mature. An extensive survey can be found in the recent paper by Howell et al [5] and earlier in Craggs [6] and Morland & Cox [7]. The complexity of the problem, which involves thermomechanical coupling and strain-rate effects at hypervelocities, and stresses well above yield, has held back theoretical investigation.…”

“…The second derivative of the longitudinal stress (3.10), with respect to stretch, is negative throughout the field with no transitions; hence in view of the shock condition (2.9), the compression field will be discontinuous while the tension field will be continuous. Longitudinal shock waves have been studied in a recent paper by Howell et al [5], employing an elastic viscous constitutive equation. That model has a natural time constant, unlike the material model used here.…”

“…That model has a natural time constant, unlike the material model used here. The elastic branch of the material investigated by Howell et al [5] is derived from a strain energy function which, in the absence of viscoplastic effects, generates stress components similar to (3.10) only with ln a replaced by (a − 1). Thus (2.27) will be satisfied and no dissipation will take place.…”

“…Recently, Howell et al [5] presented an analysis of the time-dependent response of an elasticviscoplastic solid, with the viscous branch activated beyond the plastic yield, in a similar longitudinal field. Their numerical simulation shows a different yet rich response with elastic precursor and continuous plastic region separated by a shock.…”