# Fracture Mechanics 2025
*A joint Master course of Ecole Nationale des Ponts et Chaussées, Institut Polytechnique de Paris, Sorbonne Université*
To confirm you enrollement in this class, please fill this form
https://framaforms.org/fracture-mechanics-1727090311
We kindly ask you to create an account on [Github](https://www.github.com) to share materials with you, give assignements, and provide computational tools.
## General informations
### Teachers
- Véronique Lazarus (veronique.lazarus@ensta.fr), ENSTA/IPP
- Corrado Maurini (corrado.maurini@sorbonne-universite.fr), Institut Jean Le Rond d'Alembert, Sorbonne Université
### Schedule and rooms
**Important**:
Starting from 23rd sep:
- English class 54.55.201
- French class 56.66.104 until 10:30 and 56.66.109
The room are posted on the following calendar. Please check before going each day. The room can have last-minute changes.
- [html format](https://calendar.google.com/calendar/embed?src=uarpr9gmur6hj67ekrppomd2bg%40group.calendar.google.com&ctz=Europe%2FParis)
- [ical format](
https://calendar.google.com/calendar/ical/uarpr9gmur6hj67ekrppomd2bg%40group.calendar.google.com/public/basic.ics)
The class will be given in parallel in French and in English (same time, Tuesday 9h-12h15, Jussieu Campus) by Veronique Lazarus (English) and Corrado Maurini (French). The program will be the same for the two versions. There will be a unique final exam.
A plan of the Jussieu campus can be found [here]( https://sciences.sorbonne-universite.fr/vie-de-campus-sciences/accueil-vie-pratique/plan-du-campus)
### Slack channel
We use [slack](https://slack.com) as a communication tool. Please join our workspace by subscribing at the following space:
- https://join.slack.com/t/su-mastersolides/shared_invite/zt-3dmyell4m-l6M61UsNyNTQySTCLh1BAA. *IMPORTANT:* Please, subscribe using your name and not fancy nicknames. We need to be able to identify you.
- Join the channel `m2-2025-fracture` inside the slack workspace.
We will post on the channel `m2-2025-fracture` all the comunications related to the course. You can post your questions related to the course. You can also send us private mesxx§xsages within the slack workspace.
You may wish to join also the channel `m2-2025` to have general information and advertisements about the master in Sorbonne.
### Course Material
Course material will be distributed in the following shared folder:
- https://sdrive.cnrs.fr/s/xBdPKSAecNHHz8j
### Internship offers
[You find here our internship offers at IPP and d'Alembert
](https://sdrive.cnrs.fr/s/ptmW3w2tn3gjqPf)
## References
We suggest two basic references, available in the shared folder :
- J.J. Marigo, Plasticité et Rupture, cours de l'Ecole Polytechnique (in French), Edition 2016
- P.Suquet, Plasticité et Rupture, cours de l'Ecole Polytecnique (in French)
A summary in English of the above references can be found in the course of Laura De Lorenzis at ETH (also available in the shared folder)
Further suggested references are
- J.B Leblond, Mécanique de la rupture fragile et ductile, Hermès - Lavoisier, 2003 (available in the university libraries)
- T.L. Anderson, Fracture Mechanics. Fundamentals and applications, CRC Press, Taylor & Francis Group, 2005
- MEALORII Damage Mechanics and Local Approach to Fracture-Ch. 5 and Ch.12 - https://doi.org/10.5281/zenodo.10125170
The online course at http://www.ltas-cm3.ulg.ac.be/FractureMechanics/?p=welcome can be also an additional freely available reference.
## Syllabus
Preliminary program:
- Lecture 1, 23/09/2025
- Introduction to fracture
- Short review of linear elasticity and its variational formulation
- Stress criteria and defects in linear elasticity
- Lecture 2, 30/09/2025
- Stress singularities (anti-plane case, analytical solutions)
- Stress singularities in plane elasticity, fracture modes
- Lecture 3, 07/10/2025
- Advance laws:Irwin criterion and fracture toughness, Fatigue crack advance and Paris law, damage tolerance approaches
- Brittle/Ductile transition
- Lecture 4, 14/10/2025
- Energetic approach to fracture, Griffith’s theory
- Lecture 5, 21/10/2025
- Energetic approach to fracture, numerical methods
- Lecture 6, 04/11/2025
- Variational approach, quasistatic crack propagation
- Lecture 7, 18/11/2025
- Examples
- Lecture 8, 25/11/2025
- Seminars
Exam: 4/12/2025, Jussieu
Final grade: max (final exam, 80% final exam + 20% homework)
## Computational notebooks
Computational notobooks with applications to fracture mechanics are available at the link below:
- https://newfrac.github.io/fenicsx-fracture/
You can download all the code on your computer: https://github.com/newfrac/fenicsx-fracture/archive/refs/heads/main.zip
In this class we will use in particular:
- Meshing: https://newfrac.github.io/fenicsx-fracture/notebooks/linear-elasticity/00-Mesh.html
- Linear elasticity: https://newfrac.github.io/fenicsx-fracture/notebooks/linear-elasticity/01-LinearElasticity.html
- Linear elastic fracture mechanics : https://newfrac.github.io/fenicsx-fracture/notebooks/linear-elasticity/02-LEFM.html
The notebooks are based on the finite element library FEniCSx (https://fenicsproject.org). A tutorial on FEniCSx is available: https://jsdokken.com/dolfinx-tutorial/
Completely the Howework requires to use the library on your laptop.
We suggested the following method to install the library on your laptop:
- If you are on **linux** or **macosx**, install `https://anaconda.org` and a terminal write
- `git clone https://github.com/newfrac/fenicsx-fracture.git` (Dowload the codes)
- `cd fenicsx-fracture`
- `conda env create -f fenicsx-fracture.yml --yes`(Install the required packages on a python enviroment)
- `conda activate fenicsx-fracture` (Activate the python enviroment)
- `jupyter-lab` (Open a Jupyter notebook on your browser)
- If you are on Windows, use *docker* following the instructions at https://newfrac.github.io/fenicsx-fracture/Index.html
- If other methods fails, try the online services with everthing pre-installed:
- https://colab.research.google.com
```
try:
import dolfinx
except ImportError:
!wget "https://fem-on-colab.github.io/releases/fenicsx-install-release-real.sh" -O "/tmp/fenicsx-install.sh" && bash "/tmp/fenicsx-install.sh"
import dolfinx
try:
import gmsh
except ImportError:
!wget "https://fem-on-colab.github.io/releases/gmsh-install.sh" -O "/tmp/gmsh-install.sh" && bash "/tmp/gmsh-install.sh"
import gmsh
```
- https://mybinder.org/v2/gh/newfrac/fenicsx-fracture/HEAD
- https://codespaces.new/msolides-2024/fenicsx-fracture?quickstart=1
## Howework
- You find the homework subject [here](https://codimd.math.cnrs.fr/p1A1a-dARdKw_lPxK5hrQA)
- You should submit the homework as a pdf of maximum 4 pages in the folder TOADD
- Dateline: End of November?
- One pdf for each group of two students. Name of the file: name1name2.pdf