This short course will present the mathematical formulation and the associated numerical implementation of the phasefield approach to fracture. In a nutshell, the phasefield 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 secondorder PDEs [2], and (iii) the generalization of these PDEs to account for fracture nucleation at large [35]. The latter two ideas constitute the phasefield approach to fracture.
Specifically, the course will focus on the phasefield 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 storedenergy 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 phasefield 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 boundaryvalue 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.
References:
 Francfort GA, Marigo JJ (1998) J Mech Phys Solids 46:1319–1342.
 Bourdin B, Francfort GA, Marigo JJ (2000) J Mech Phys Solids 48:797–826.
 Kumar A, Francfort GA, LopezPamies O (2018) J Mech Phys Solids 112:523551.
 Kumar A, Bourdin B, Francfort GA, LopezPamies O (2020) J Mech Phys Solids 142:104027.
 Kumar A, RaviChandar K, LopezPamies O (2022) Int J Fract. 237, 83–100.
 FEniCS computing platform, https://fenicsproject.org/.
Syllabus:
Hours 13: The Theory

Griffith idea for fracture

Fracture propagation as a variational problem

Regularization of Griffithtype surface energy and introduction to phasefield

EulerLagrange equations of the variational problem

Ingredients for describing fracture nucleation at large

Concept of strength surface

Generalizing the EulerLagrange equations of the variational problem to account for the strength surface
Hours 46: The Numerical Implementation

Weak form and finite element formulation of the PDEs

Staggered formulation for solving coupled PDEs

Choice of regularization length scale

Calibration of toughness and strength parameters

Representative boundaryvalue problems:

Surfing problem for fracture propagation

Indentation problem with a cylindrical indenter

Nucleation from a Vnotch

Mixedmode propagation in a compact tension test

Nucleation and propagation in an elastomeric pokerchip specimen
