Modelling of the strain softening and localization in dynamical problems

Abstract
The strain localization problems in softening solids under dynamic loading are considered. Two different models are accepted. One is elastoviscoplastic model with a strain softening diagram and another is the second order gradient elastoplastic model. By the asymptotic method of matching rapidly and slowly changing solutions for the partially differential equations with a small parameter, developed by the author [1], the close form solution for a one-dimensional dynamic localization problem was obtained for the both models. A bar loaded instantaneously by forcing both ends to move with a constant opposite velocity of the magnitude V0 is considered. After the collision of generated elastic waves at the middle point of a bar the plastic strain is appeared. The plastic strains ε>ε^* propagate along a bar and the strains concentrate at the middle point of the bar, ε^* is the strain related to the maximum stress. The zone of these large strains forms a band of strain localization with the width slowly growing in time. The solution describing the structure of the bands is obtained in an analytical form for both models. The profile of the elastoviscoplastic solution is monotone, while for the second gradient theory it is oscillating, but the effective width of the band for the both models is similar and is growing in time as t^1/2. Note, that the linear analysis predicts constant width of a localization band and the solution [2] obtained for strain rate-insensitive elastoplastic material shows that localization takes place only at one point.