Analysis of theoretical models and computational methods for sharp crack and phase field fracture

Lectures will be in Aula Beltrami (at the Department of Mathematics) on Tuesday and Thursday from 11 to 13, starting from April 17, 2018.

This course aims to present a modern mathematical theory for fracture. It will be ideally divided in two parts.

• In the first part we will present a simple model for the quasi-static propagation of straight crack, making reference to ASTM compact tension specimens. In this setting, we will introduce the basic ingredients of the classical theory: energy release rate (with its many representations) and Griffith's criterion. We will then see some mathematical characterizations of quasi-static evolutions and their discretizations both in space and time.
• In the second part we will present instead the phase-field approach to fracture, developed in the last twenty years. First we will see how phase field energies are approximations of sharp crack energies, both in Sobolev and Finite Element spaces. We will prove this relationship by means of Γ- convergence, at least in a technically simple case. Then we will turn to evolutions and in particular to the alternate minimization algorithm. We will study the quasi-static evolutions obtained by this scheme, discussing their thermodynamical consistency and the relationship with Griffith's criterion.

A mathematical background and a basic knowledge of Sobolev spaces is required, however, necessary mathematical results from Functional Analysis will be always recalled and explained. Supplementary lectures on technical details, of interest for mathematicians, will be given according to the interest of the audience.