Announcement Detail


Short Course: 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

Dates: August 10 – August 13, 2026
Time: 9:00 AM – 11:00 AM EDT each day
Location: Online (meeting link provided after registration)

Registration Fee:

  • $300 – Non-students

  • $50 – Students

Overview

This short course presents the mathematical foundations and numerical implementation of the phase-field approach to fracture, one of the most significant developments in computational fracture mechanics over the last three decades.

The phase-field approach to fracture emerged from a series of advances by the mathematics and mechanics communities aimed at describing where and when fracture nucleates and propagates in solids under arbitrary loading conditions in a computationally tractable manner. The approach builds upon three key developments:

  1. The formulation of fracture propagation as a variational problem.

  2. The regularization of this variational problem into a system of second-order partial differential equations.

  3. The incorporation of material strength to account for fracture nucleation at large.

Together, these developments provide a powerful framework for predicting fracture nucleation and propagation in brittle materials.

The course will focus on elastic brittle materials such as glass, ceramics, and elastomers, where energy dissipation occurs exclusively through the creation of new fracture surfaces. Participants will learn how fracture toughness, elasticity, and material strength are incorporated into a comprehensive phase-field theory of fracture.

In addition to covering the underlying theory, the course will provide a detailed treatment of the finite element implementation of the governing equations. Live demonstrations will be conducted using both the FEniCSx and MOOSE computational platforms to solve representative fracture problems involving both linear elastic and hyperelastic materials.

Participants will receive lecture notes covering the theoretical foundations of the method, along with the FEniCSx and MOOSE example codes used during the demonstrations.

The course consists of four live online lectures. Recordings will be made available after each session and provided exclusively to registered participants for a limited time.

Learning Outcomes

By the end of this course, participants will be able to:

  • Understand the experimental observations that motivate modern fracture theories

  • Explain the concept of material strength and its role in fracture nucleation

  • Understand Griffith's variational formulation of fracture propagation

  • Describe the phase-field regularization of fracture mechanics problems

  • Formulate complete phase-field models that account for both fracture nucleation and propagation

  • Derive weak forms and finite element formulations of the governing equations

  • Implement staggered solution schemes for coupled fracture problems

  • Apply phase-field methods to practical engineering fracture simulations

  • Use FEniCSx and MOOSE to solve representative fracture mechanics problems

  • Gain the foundation necessary to implement and extend phase-field fracture models in research and engineering applications

Course Schedule

Days 1–2: The Theory

August 10–11, 2026 | 9:00 AM – 11:00 AM EDT

Topics include:

  • A summary of macroscopic experimental observations of fracture nucleation and propagation in nominally elastic brittle materials

  • The definition of material strength

  • Griffith's postulate for fracture propagation as a variational problem

  • The phase-field regularization of the Griffith variational problem

  • Incorporating strength to construct a complete phase-field theory of fracture nucleation and propagation

Days 3–4: Numerical Implementation and Live Demonstrations

August 12–13, 2026 | 9:00 AM – 11:00 AM EDT

Topics include:

  • Weak form and finite element formulation of the governing PDEs

  • Staggered solution schemes for the resulting discretized equations

  • Practical implementation strategies in FEniCSx and MOOSE

  • Representative demonstration 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

Software and Course Materials

The course will feature live demonstrations using:

  • FEniCSx

  • MOOSE

Participants will receive:

  • Comprehensive lecture notes

  • Example simulation codes used during the demonstrations

  • Access to session recordings for a limited time following each lecture

Selected References

  • Francfort GA, Marigo JJ (1998), Journal of the Mechanics and Physics of Solids, 46:1319–1342

  • Bourdin B, Francfort GA, Marigo JJ (2000), Journal of the Mechanics and Physics of Solids, 48:797–826

  • Kumar A, Francfort GA, Lopez-Pamies O (2018), Journal of the Mechanics and Physics of Solids, 112:523–551

  • Kumar A, Bourdin B, Francfort GA, Lopez-Pamies O (2020), Journal of the Mechanics and Physics of Solids, 142:104027

  • Kumar A, Lopez-Pamies O (2020), Theoretical and Applied Fracture Mechanics, 107:102550

  • Lopez-Pamies O, Kamarei F (2025), Extreme Mechanics Letters, 81:102417

  • Lopez-Pamies O, Dolbow JE, Francfort GA, Larsen CL (2025), Computer Methods in Applied Mechanics and Engineering, 433:117520

  • Kamarei F, Zeng B, Dolbow JE, Lopez-Pamies O (2026), Computer Methods in Applied Mechanics and Engineering, 448:118449

Additional historical references and supporting materials will be provided in the course notes.

Additional Resources

FEniCSx Documentation: https://docs.fenicsproject.org/

MOOSE Framework: https://mooseframework.inl.gov/