Tuesday, June 2, 2026
8:00 AM - 3:00 PM | The Phase-Field Approach to Fracture: Theory and Numerical Implementation
Instructors:
- Aditya Kumar, Georgia Tech ([email protected])
- Abhinav Gupta, Vanderbilt University ([email protected])
- Ravindra Duddu, Vanderbilt University ([email protected])
Course Outline:
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 – the aim was to describe when and where a 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]. These ideas constitute the phase-field approach to fracture.
The literature on the phase field approach and its applications is vast, so the course will focus on the fundamental theoretical principles of the approach from a macroscopic point of view and its application to elastic brittle materials like glass, ceramics, and elastomers (generalization to ductile materials will be briefly discussed). In such materials, energy is dissipated only through the creation of new surfaces and is proportional to the area created. Three key pivotal ideas establishing this approach are: (1) fracture toughness as the proportionality constant between dissipated energy and crack surface area governs fracture nucleation and propagation from large pre-existing cracks; 2) the strength surface governs crack initiation in homogeneous domain under uniform stresses, and (3) the mediation of toughness and strength governs the crack nucleation under non-uniform stress states and weak stress singularities.
This one-day short course will include a detailed introduction to the three pivotal ideas listed above, and the constitutive choices that are made to develop a generalized phase-field model. The casting of the model in a finite element formulation will be discussed along with a live demonstration in Python (using FEniCS library [8, 9]) to solve representative boundary-value problems involving fracture nucleation and propagation in both linear elastic and hyperelastic materials. We will briefly discuss adaptive mesh refinement method in FEniCS to accelerate large computations [6, 7]. The course material will include lecture notes on the fundamentals of the method in addition to the Python codes used for the live demonstration. A Slack discussion group will be created for the workshop attendees. Helpful references are listed below.
References:
1. Francfort GA, Marigo JJ (1998) J Mech Phys Solids 46:1319–1342.
2. Bourdin B, Francfort GA, Marigo JJ (2000) J Mech Phys Solids 48:797–826.
3. Kumar A, Francfort GA, Lopez-Pamies O (2018) J Mech Phys Solids 112:523–551.
4. Kumar A, Bourdin B, Francfort GA, Lopez-Pamies O (2020) J Mech Phys Solids 142:104027.
5. Kumar A, Ravi-Chandar K, Lopez-Pamies O (2022) Int J Fract 237, 83–100.
6. Gupta A, Krishnan UM, Mandal TK, Chowdhury R, Nguyen VP (2022) Comput Methods Appl Mech Eng 399:115347.
7. Gupta A, Nguyen DT, Hirshikesh, Duddu R (2024) Eng Fract Mech 306:110252.
8. FEniCS computing platform, https://fenicsproject.org/.
9. FEniCS workshop, https://abhigupta.io/fenics-workshop/.
Agenda:
Hours 1-2: The Theory
- Griffith idea for fracture
- Fracture propagation as a variational problem
- Regularization of Griffith-type surface energy and introduction to phase-field
- Euler-Lagrange equations of the variational problem
- Ingredients for describing fracture nucleation at large
- Concept of strength surface
- Generalizing the Euler-Lagrange equations of the variational problem to account for the strength surface
- Generalization to ductile fracture
Hours 3-6: The Numerical Implementation
- Weak form and finite element formulation of the PDEs
- Staggered formulation for solving coupled PDEs
- Choice of regularization length scale, mesh size, irreversibility
- Calibration of toughness and strength parameters
- Adaptive mesh refinement
- Representative boundary-value problems:
➢ Double cantilever beam problem for fracture propagation
➢ Nucleation under torsion
➢ Nucleation in Brazilian fracture test
➢ Indentation problem with a cylindrical indenter
➢ Mixed-mode propagation in bending experiments
➢ Dynamic branching test
➢ Nucleation and propagation in an elastomeric specimen
➢ Thermal quenching of a plate