FEM Modeling of Phase-Field Fracture in Python

This repository contains the materials used during my research visit to CNRS-LSPM laboratory at Université Sorbonne Paris Nord, kindly hosted by Yann Charles.

The materials include three presentations:

• FullyKineticHydrogen.pdf – the development of the fully kinetic hydrogen transport models in metals.
• GeometricPFF.pdf – a light introduction to the geometric phase-field fracture method.
• PHIMATS.pdf – an overview of the finite element software PHIMATS.

Additionally, two Jupyter notebooks implementing $J_2$ plasticity and the geometric phase-field fracture method of Miehe et al.:

• Homogenous.ipynb – is basically a scalar implementation of coupling $J_2$ plasticity with homogenous phase-field fracture.
• PFF_Plastic.ipynb – a full FEM implementation of a double notched sample with assisting functions in utils_Quad4.py.

These notebooks provide students and researchers with a step-by-step implementation in python of

• Basic nonlinear finite element method.
• Rate-independent plasticity with isotropic hardening.
• A Modular phase-field fracture formulation.

My aim of sharing this repository is to provide an educational tool for understanding the underlying algorithms and/or assist with testing new constitutive relations before moving to a more programming-intensive platforms such as PHIMATS.

---

Features

The Jupyter notebook PFF_Plastic.ipynb provides scripts to:

• Read and preprocess meshes generated by Gmsh via meshio.
• Basic FEM formulation for Quad4 elements (8 DOFs for mechanics and 4 for PFF).
• Implement a step-by-step return-mapping algorithm for $J_2$ plasticity (plane-strain), and assemble the global stiffness matrix.
• Nonlinear global Newton-Raphson solver.
• Geometric phase-field fracture with:
  • Spectral decomposition of elastic strain tensor, ensuring that only tensile energy contributes to crack driving forces.
  • Support for brittle, elastoplastic, and elastoplastic with threshold damage models.
  • Suppression of crack growth under compression via the function ConstrainElPhi with the algorithm proposed by Ambati et al.
• Store simulation outputs in HDF5 format.
• Support visualization through XDMF files compatible with ParaView (v5.9.1)

---

Documentation

The FEM theory behind these implementations can be found in the cited references in the functions' documentations and also summarized in PHIMATS theory manual: Finite Element Theory of PHIMATS

---

Download and installation

• Download the repository via

git clone https://github.com/ahcomat/PhaseFieldFracture.git

• All required Python libraries are included in the phimats Conda environment. To create the environment, run:

conda env create -f environment.yml

💡 Note: If you already have PHIMATS installed, you can use the existing phimats Conda environment directly.

• ParaView v5.9.1 for visualization.

---

Citation

If you use this code, please cite it via:

A. Hussein, Finite Element Theory for PHIMATS, arXiv preprint, 2025. DOI: 10.48550/arXiv.2502.16283

---

License

This code is released under the GNU General Public License v3.0, or later.