Less than 500 lines self-contained Python finite element implementation of the phase-field method for fracture mechanics

This paper presents a simple self-contained finite element implementation of the phase-field method for fracture mechanics. The implementations are done in Python, and they only use the standard NumPy and SciPy libraries for basic matrix operations and to solve the resulting systems of equations. The AT2 phase-field model is adopted and the additive decomposition of the energy density is employed to prevent fracture propagation under compressive stresses. The alternate minimization algorithm is adopted for solving the nonlinear system of equations. The implementation is verified using three examples: a bar under tension, a notched plate under tension, and a three-point bending test. The results display good agreement with analytical solutions and solutions from other authors. Each example is less than 500 lines long, and they are available on GitHub at https://github.com/nathanshauer/phasefield-jr-py and as supplementary data to this article. These Python scripts are intended for educational purposes and to provide a simple starting point for those interested in the phase-field method for fracture mechanics.