Localisation of deformations in elasto-plastic and elasto-visco-plastic solids

Exact solutions of the material science problem for three constitutive models of softening solid are given. For the rate insensitive elasto-plastic solid the mathematical problem appears to be ill-posed but actually the process looses stability that leads to strain localisation and well-posed dynamic problem. For the rate sensitive elasto-visco-plastic solid numerical solution with unloading that inevitably arises in the material science problem is also obtained. The third model of abrupt damage interprets softening as transformation of one stable solid into another one. It is shown that depending on the material properties the process may be static or dynamic.