Variational Fracture Mechanics and Phase-Field Models

The course provides a complete and multidisciplinary presentation of the variational phase-field approach to fracture. Emerged over the past two decades, this paradigm has become a game changer for the theoretical and computational treatment of damage and fracture, attracting wide attention in the fracture mechanics community and in industry. It connects Griffith's theory of fracture through the mathematical notion of Gamma-convergence, gradient damage models in continuum mechanics, and the phase-field modelling of phase transformations used in materials science, while offering substantial advantages for computations. The course deliberately takes all of these viewpoints, so that participants gain a presentation of the subject that is both broad and deep.

• From the variational approach to fracture to phase-field modelling: variational formulation of brittle fracture with pre-assigned crack path and in the general case (“sharp-crack” formulation), and its regularization.

• The variational approach to damage and plasticity: local damage models, the variational approach to damage, the transition from local to non-local (gradient) damage models, and the basics of the variational approach to plasticity.

• Phase-field models of brittle fracture as gradient damage models: model ingredients and their consequences, homogeneous and localized solutions, nucleation as loss of stability of the homogeneous solution, the role of energy decompositions under multi-axial stress states, numerical aspects.

• Phase-field models of cohesive and ductile fracture: a formulation in which damage degrades material strength rather than stiffness, allowing arbitrary convex stress-based failure criteria within a consistent variational structure; the equivalent cohesive fracture law; convex programming tools for stable and efficient solvers.

• The mathematical link between the sharp-crack model and its regularized version: Gamma-convergence explained to non-mathematicians, and the key mathematical aspects of phase-field models for fracture and for free-discontinuity problems.

• High-performance phase-field fracture computations: linear and non-linear solvers, parallel computing, incremental constrained minimization for coupled phase-field problems, and hands-on examples in FEniCS.

A short student presentation session on Wednesday afternoonallows participants to introduce themselves and present their current research.

The course addresses a broad public: doctoral students, young researchers and practicing engineers. Since most of it is devoted to theoretical modelling and numerical computation, an inclination for mathematics is advised. No specific prerequisite in damage mechanics or phase-field modelling is required, but knowledge of the basic concepts of continuum mechanics, elasticity and fracture mechanics is recommended. Preliminary knowledge of Python is useful for active participation in the numerical tutorials; participants should bring their own laptops.