This course is designed to equip PhD students with advanced mathematical tools for the formulation and solution of complex problems in solid and structural mechanics. Building on the foundations of functional analysis, it introduces variational methods as a powerful framework to extend the notion of solutions beyond the limitations of classical partial differential equation theory. The course emphasizes variational formulations and their role in addressing challenging nonlinear phenomena, with particular focus on material damage and plasticity.
The lectures begin with an introduction to variational methods and the motivation for weak formulations, followed by a detailed treatment of classical elasticity within a variational setting. Key mathematical tools are developed, including functional spaces such as Banach, Hilbert, and Sobolev spaces, along with fundamental embedding and trace theorems. The course then addresses variational inequalities and their applications to nonlinear mechanics, particularly in modeling damage and plastic behavior.
Further topics include existence and uniqueness of solutions, well-posedness, and essential mathematical properties such as coercivity and continuity, together with numerical approximation techniques like the Galerkin method and the theoretical foundations of the finite element method. The course concludes with an analysis of numerical locking phenomena and an introduction to mixed variational methods, including stability conditions such as the inf-sup condition.
21, 22, 23 aprile; 5, 6, 7 maggio