SC17-011: The Phase-Field Approach to Brittle Fracture: Theory and Numerical Implementation

This short course will present the mathematical formulation and the associated numerical implementation of the phase-field approach to fracture. In a nutshell, the phase-field approach to fracture is the culmination of combined efforts (started at the end of the 1990s) by the mathematics and computational mechanics communities aimed at describing when and how fracture nucleates and propagates in solids under arbitrary mechanical loads in a computationally tractable manner. These efforts comprise three pivotal ideas: (i) the casting of the phenomenon of fracture propagation as a variational problem [1], (ii) its regularization into second-order PDEs [2], and (iii) the generalization of these PDEs to account for fracture nucleation at large [3-5]. The latter two ideas constitute the phase-field approach to fracture.

Specifically, the course will focus on the phase-field approach to fracture in elastic brittle materials like glass, ceramics, and elastomers. In such materials, the energy is dissipated only through the creation of new surfaces and is proportional to the amount of surface area created. Fracture toughness is the proportionality constant and constitutes one of the three material inputs in the theory. The second material input is the stored-energy function describing the elasticity of the material. The third material input is the strength surface.

The course will include a detailed introduction to the three pivotal ideas listed above, and the constitutive choices that are made to develop a general phase-field model. The casting of the model in a finite element formulation will be discussed, and a live demonstration in Python (using FEniCS library [6]) will be given to solve representative boundary-value problems involving fracture nucleation and propagation in both linear elastic and hyperelastic materials. The course material will include lecture notes on the fundamentals of the method in addition to the set of Python codes that will be used for the live demonstration. Helpful references are listed below.