Variational Formulation and Numerical Solution Techniques for the Phase-field Approach to Fracture

In order to use materials more efficiently, there is a growing tendency to narrow safety margins in structural design by taking non-linear material behavior into account, allowing to subject materials closer to their ultimate strength. A thorough understanding of the behavior of materials at such high load levels requires constitutive models which can accurately describe phenomena such as, for instance, plasticity and fracture. While a significant amount of research has been performed on the computational modeling of solids and structures in the past 50 years, the numerical modeling of fracture remains an intriguing topic and is, due to the availability of increasing computational power, receiving more and more attention. Despite the complexity of the phenomenon, several powerful theories exist to accurately describe fracture in both a qualitative and a quantitative way. The phase-field approach, which is closely related to Griffith's theory of brittle fracture, considers a diffuse approximation of the cracks. A major advantage of this approach is the ease in which crack initiation, propagation, and branching can be modeled, without any ad-hoc criteria.

The aim of this project is to derive several phase-field models in a variationally and thermodynamically consistent setting and to develop rigorous numerical solution techniques for the governing equations. In a variational framework, the problem can be expressed in terms of extremal energy principles including evolution constraints, which require the use of constrained optimization techniques. In this context, particular attention is given to interior-point methods, which allow for a rigorous solution of the system of constrained equations that follows from this approach. In order to prevent brutal crack propagation, path-following techniques are suggested, which, additionally, restore energetic deficiencies in the formulation and enhance the convergence of the solution techniques. Several phase-field models are considered, and considerable attention is paid to ductile fracture first, in which the formation of cracks is accompanied by substantial plastic deformations. Next, several heterogeneous structures with predefined interfaces, such as composite and masonry structures, are considered. The proposed solution techniques are applied to several benchmark problems, and a comparison with other methods is performed. Where possible, the numerical results are compared to experimental results reported in the literature.