This research project deals with nonlinear evolution problems that arise in the study of the inelastic behaviour of solids, in particular in plasticity and fracture. The project will focus on selected problems, grouped into three main topics, namely: 1. Plasticity with hardening and softening, 2. Quasistatic crack growth, 3. Dynamic fracture mechanics. The analysis of the models of these mechanical problems leads to deep mathematical questions originated by two common features: the energies are not convex and the solutions exhibit discontinuities both with respect to space and time. In addition, plasticity problems often lead to concentration of the strains, whose mathematical description requires singular measures. Most of these problems have a variational structure and are governed by partial differential equations. Therefore, the construction of consistent models and their analysis need advanced mathematical tools from the calculus of variations, from measure theory and geometric measure theory, and also from the theory of nonlinear elliptic and parabolic partial differential equations. The models of dynamic crack growth considered in the project also need results from the theory of linear hyperbolic equations. Our goal is to develop new mathematical tools in these areas for the study of the selected problems. Qua-sistatic evolution problems in plasticity with hardening and softening will be studied through a vanishing viscosity approach, that has been successfully used by the P.I. in the study of the Cam-Clay model in soil mechanics. Quasistatic models of crack growth will be developed under different assumptions on the elastic response of the material and on the mechanisms of crack formation. For the problem of crack growth in the dynamic regime our aim is to develop a model that predicts the crack path as well as the time evolution of the crack along its path, taking into account all inertial effects.
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