An analysis of strain localization in a shear layer under thermally coupled dynamic conditions. Part 2: Localized thermoplastic models

The analyses presented in Part 1 of this work have shown the ill-posedness of the governing equations for a shear layer governed by a coupled thermomechanical continuum model with strain softening. Finite element solutions of the problem have been shown to exhibit a pathological mesh-size dependence in this case, as a consequence of the aforementioned ill-posedness. The lack of a localized dissipative mechanism has been traced back to the origin of these di culties. We consider in this paper the incorporation of such localized dissipative mechanism in the form of a cohesive law along a discontinuity of the displacements, the so-called strong discontinuity. A coupled thermomechanical model of these discontinuities is formulated involving a continuous temperature eld. The discontinuity of the heat ow (i.e. the derivative of the temperature) accommodates the presence of the localized dissipation associated to the thermomechanical cohesive law. In this context, we obtain the exact solution of a problem of wave propagation in the shear layer when the response of the bulk of the layer is elastic. The physical nature of the solution, in contrast with the solutions obtained in Part 1 for continuum models, illustrates the regularizing e ect of these localized models, showing in particular the continuous dependence of the solution on the material parameters. Furthermore, we obtain the exact solution of the problem involving a regularized discontinuity in a domain of nite length. This analysis allows to gain a full understanding of the properties of nite element approximations of the problem. Several numerical simulations are presented in this context corroborating the conclusions drawn from the analysis