ETD

• bordo libero
• disuguaglianza variazionale
• elasto-plastic problem
• free boundary
• problema elasto-plastico
• ridge
• Sobolev
• variational inequality

We are going to discuss the elasto-plastic problem: it is given a cylindrical bar which cross-section satisfies some reasonable hypothesis of regularity, subject to a torsion around the axis. The target is to find a twice differentiable function whose gradient has modulus at most 1, satisfies a Poisson-type equation where equality does not hold and is identically null on the boundary. Existence and uniqueness of a weak solution are granted by some well-known general results; the discussion is, then, ready to evolve into an inquiry about the classical regularity: it is Lipschitz continuous along with its first derivatives. By a theorem of Stampacchia the solution also minimizes a certain quantity related to the Poisson-type equality it satisfies, so it makes sense to consider the variational Lagrangian and start looking for its Lagrange multiplier: it exists uniquely and it is also possible to retrieve an explicit description. The Thesis also provides a glimpse on a practical aspect of this model: the original formulation does not allow a simple numerical analysis, so it is proven equivalent to an obstacle-type problem. Finally, the section of the bar is split into an elastic part and a plastic part, according to the behavior of the absolute value of the gradient of the solution, as we wish to uncover some information about the region of the section that separates the two subsets and the region in which the function “distance from the boundary” is not regular enough.