Alternate minimization in fracture mechanics

In the last decades the use of phase-field models in computational fracture mechanics has been constantly increasing and has found many interesting applications. The phase-field approximation of fracture has proven to be particularly effective in combination with alternate minimization algo- rithms, which consist of an iterative energy minimization procedure and result in the identification of equilibrium states of the system. Despite alternate minimization is the reference method in several applications of computational mechanics, it still raises interesting questions regarding its relationship with the theory of rate-independent systems and the correct implementation of the irreversibility of the fracture process. We aim at studying the ability of alternate minimization to approximate a fracture process. In this respect, we will investigate the convergence of alternate minimization schemes in the framework of phase-field fracture mechanics, focusing on the energetic characterization of time-continuous limit evolutions, in combination with different phase-field irreversibility constraints. Eventually, we will consider the passage from phase-field evolutions to sharp interface fracture processes, with applications to modeling and analysis of brittle fracture on thin shells. Our analysis will face three main issues: (i) the involved energies are nonconvex, (ii) the solutions of the evolution problems exhibit time discontinuities, and (iii) the evolutions fulfill inequality constraints promoting irreversibility. Thanks to the tools of the Calculus of Variations and of the theory of (singularly perturbed) gradient flows we will establish the first global in time convergence analysis of the most common numerical methods for fracture simulation. By studying the convergence of critical points of the phase-field system, we will rigorously show that phase-field fracture delivers a reliable approximation of a fracture process. Our analysis will shed light on pros and cons of the several algorithms and models developed in the context of phase-field approximation of fractures. We will therefore provide a sound theoretical foundation to the applied research performed in recent years, contributing to a better understanding of phase-field fracture mechanics and of the related numerical results. As a consequence, our analysis will pave the way to further optimization of the existing computational methods.