OVERVIEW

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

Instructors: Oscar Lopez-Pamies, University of Illinois Urbana-Champaign; John E. Dolbow; Duke University

When: August 10 - August 13, 2026, 9:00am-11:00am EDT

Where: Link sent out following registration

Cost: $300/non-students; $50/students

Description: 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 mechanics communities aimed at describing where and when fracture nucleates and propagates in solids under arbitrary mechanical loads in a computationally tractable manner. These efforts comprise three pivotal ideas, in chronological order: (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 [3-6]. The latter two ideas constitute the phase-field approach to fracture.

Specifically, the course will focus on the phase-field approach to 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 FEniCSx library [7]) and in C++ (using MOOSE [8]) will be given to solve representative initial-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 FEniCSx and MOOSE codes that will be used for the live demonstration. Helpful references are listed below. Additional historical references will also be provided in the lecture notes.

This short course will be delivered via four two-hour online lectures. Participants will be allowed to attend each lecture live.  Recordings of the lectures will also be made available after each session and offered exclusively to short course participants for a limited time.

Selected 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, Lopez-Pamies O (2020) Theor Appl Fracture Mech 107:102550.
  6. Lopez-Pamies O, Kamarei F (2025) Extreme Mechanics Letters 81: 102417.
  7. Lopez-Pamies O, Dolbow JE, Francfort GA, Larsen CL (2025) Comput Methods Appl Mech Eng 433:117520.
  8. Kamarei F, Zeng B, Dolbow JE, Lopez-Pamies O (2026) Comput Methods Appl Mech Eng 448:118449.
  9. FEniCSx computing platform, https://docs.fenicsproject.org/.
  10. MOOSE computing platform, https://mooseframework.inl.gov/.

Days 1-2: The Theory

  • A summary of macroscopic experimental observations of fracture nucleation and propagation in nominally elastic brittle materials
  • The definition of strength
  • Griffith postulate for fracture propagation as a variational problem
  • The phase-field regularization of the Griffith variational problem
  • Accounting for strength to construct a complete phase-field theory of fracture nucleation and propagation

Days 3-4: The Numerical Implementation and Live Demonstrations

  • Weak form and finite element formulation of the governing PDEs
  • Staggered scheme to solve the resulting discretized equations
  • Representative initial-boundary-value problems:
  • Nucleation of fracture under uniaxial tension
  • Nucleation of fracture from a V-notch
  • Propagation of fracture in a pure-shear test
  • Indentation of glass with a cylindrical indenter
  • The Brazilian fracture test for mortar
  • The poker-chip experiment for rubber