The goal of this course is to provide to graduate students, young researchers and engineers a complete and multidisciplinary presentation of the variational phase-field approach to fracture.
This new paradigm for the theoretical and computational treatment of damage and fracture has emerged in the past two decades as a game changer for both theory and applications and has attracted large attention in the broad fracture mechanics community as well as from industry. The lecturers will present phase-field fracture modeling in the framework of the variational Griffith’ theory of fracture mechanics and its variational regularization, of gradient damage models, as well as of phase-field modeling for phase transformations widely used in material science. Overviews of current research topics and open issues will be provided. Lectures on the theoretical and mathematical aspects will be complemented by hands-on tutorials introducing the participants to appropriate numerical methods.
The variational phase-field modeling of fracture can be seen as the regularization of the variational approach to fracture (Francfort and Marigo 1998). The lectures by Blaise Bourdin will introduce the basic concepts of the variational formulation of brittle fracture. Flaviana Iurlano will give a general introduction to the mathematical notions needed to establish the link between the sharp-crack model and its regularized counterpart (Gamma-convergence). The lectures, addressed to an audience of non-mathematicians, will also highlight the most important mathematical aspects in the larger context of free-discontinuity problems.
The variational phase-field model of fracture can also be constructed as a special class of gradient damage models, thus establishing the link with continuum damage mechanics. The lectures of Laura De Lorenzis and Corrado Maurini will be devoted to the variational derivation of the set of equations which govern the quasi-static evolution of damage in a body, their stability, and the discussion of the related crack nucleation criteria. Reference analytical solutions and extensions to elastoplastic couplings will be presented.
Phase-field modeling as a general framework entails the regularization of problems involving moving interfaces and is a very active area in the larger context of computational material science, fluid mechanics and topology optimization. The lectures by Peter Vorhees will provide a broad overview of the phase-field method, starting from its thermodynamics, and will highlight different classes of models involving conserved and unconserved phase-field variables.
A primary advantage of the phase-field framework lies in its flexibility and performance in numerical computations. Ensuring robust computations and obtaining accurate results is not trivial. Numerical aspects will be the focus of the lectures by Jack Hale. He will first introduce basic concepts of linear and non-linear solvers and parallel computing, then focus specifically on non-linear coupled phase-field computations involving incremental constrained minimization, and finally provide hands-on numerical examples using the FEniCS Project finite element framework.