DOI: 10.5281/zenodo.14749906
title: Accompanying dataset for the paper "How to introduce an initial crack in phase field simulations to accurately predict the linear elastic fracture propagation threshold?"

description: Contributions



Author FL did contribute to data curation, software development, and data collection.

Author VL did contribute to the supervision of the project, its funding, and data validation.


Funding sources

The work was supported by:



Agence de l'Innovation de Défense – AID – via Centre Interdisciplinaire d'Etudes pour la Défense et la Sécurité – CIEDS – (projects 2022 - FracAddi),

Agence Nationale de la Recherche (ANR-23-CE51-0054 3FAM).


Data structure and information

Data organization

The data is split into two parts:



the figures data in figures, containing the data and images of the figures in the article,

the raw data and associated scripts in raw_data_and_scripts with the raw output from the simulations and the post-processing scripts.


Figures: Images and Data

This directory includes the processed figure data used in the article.The files are named according to their associated figures and represent specific configurations or time steps (e.g., t0 or tf).Images are organized by figure number and contain detailed information in their filenames:





Figure 1: Illustrations of the crack initialization techniques (initial geometry and phase fields).- Example files:- figures/article-figure1-GEO-T0-NEU_t0.png: Image of the geometry and phase field at time t0 for type GEO-T0-NEU.- figures/article-figure1-GEO-T1-TIP_t0.png: Image of the geometry and phase field at time t0 for type GEO-T1-TIP.




Figure 2: Phase field construction pre- and post-regularization.- Example files:- figures/article-figure2_construction_PHA-T0_pre_reg.png: Image of the pre-regularization phase field construction for PHA-T0.- figures/article-figure2_construction_PHA-T1_post_reg.png: Image of the post-regularization phase field construction for PHA-T1.




Figure 4: Evolution data and phase field for infinitely thin initial cracks (T0). - CSV files are stored in figures/article-figure4_data/, with corresponding post-propagation phase fields such as:- figures/article-figure4-GEO-T0-NEU_tf.png.




Figure 5: Evolution data and phase field for one-element-wide initial cracks (T1). - CSV files are stored in figures/article-figure5_data/, with corresponding post-propagation phase fields such as:- figures/article-figure5-GEO-T1-NEU_tf.png.




Figure 6: Evolution data and phase field for optimal initial cracks wiht an unstructured mesh.- CSV files are stored in figures/article-figure6_data/, with corresponding post-propagation phase fields such as:- figures/article-figure6-GEO-T1-WHL_tf.png.



Figures: Data (CSV Files)

The CSV files are organized by figure number into subdirectories (e.g., article-figure3_data/, article-figure4_data/).Each subdirectory contains numerical data used to generate the corresponding figure, with filenames reflecting the configurations or conditions. The quantities in the CSV files are described in the following table.




Symbol
Quantity
Unit




a
Crack length
m


G
Energy release rate
J/m²


F_imp
Unscaled reaction force
N


u_imp
Unscaled imposed displacement
m


cmod
Crack mouth opening displacement
m


l
Load factor
dimensionless


u
Imposed displacement
m


F
Reaction force
N


fraction_dssipation
Fracture dissipation
J




Example Files:





figures/article-figure3_data/:- GEO-T0-NEU.csv: Processed data for simulation of the initial crack GEO-T0-NEU.- ref_lefm.csv: Reference data from the Linear Elastic Fracture Mechanics (LEFM) sharp crack model.




figures/article-figure6_data/:- GEO-T1-WHL_UNSTRUCTURED.csv: Data for unstructured mesh using the initial crack GEO-T1-WHL.- PHA-T1_UNSTRUCTURED.csv: Data for unstructured mesh using the initial crack PHA-T1.



Raw data: Reference simulation results (raw_data_and_scripts/LEFM_REF)

The raw_data_and_scripts/LEFM_REF directory contains the script and raw data generated using fragma. It contains the elastic simulations directories in results/, which are named after the crack length (in meters). Those directories contain:



the mesh.msh mesh file,

the parameters.toml file that parameterizes the FEM solver,

the log files of the simulations: gmsh.log for the mesh generation and fragma.log for the FEM solver,

the results subdirectory containing the raw simulation results: - Displacement.pvd (along with the associated .pvtu and .vtu files): VTK file containing the displacement field, - probes.csv containing scalar data from the simulations: - Energy release rate in J/m$^2$ computed with the $G(\theta)$ method, - Forces in N: F_1 and F_2 at the bottom and top boundaries, - Energies in J: elastic energy, fracture dissipation, undamaged elastic energy, external work, - Displacement data in m representing the displacement field at various locations in the simulation domain.


The main directory also contains the display_results.py python script. This script depends on numpy, matplotlib, and pandas. Among other things, this script plots the reference force-displacement curves and generates the ref_lefm.csv script used in the figures.

Raw data: Phase-field simulation results (raw_data_and_scripts/GEO*, raw_data_and_scripts/PHA*)

The raw_data_and_scripts directory contains the simulation results in the directories with the codified name employed in the article (see the notes below). The simulation subdirectories contain the following data:



the mesh.geo GSMH file employed for mesh generation and the generated mesh file mesh.msh,

the parameters.toml file that parameterizes the FEM solver,

the log files of the simulations: gmsh.log for the mesh generation and fragma.log for the FEM solver,

the results subdirectory containing the raw simulation results: - CrackPhase.pvd (along with the associated .pvtu and .vtu files): VTK file containing the crack phase field evolution, - Displacement.pvd (along with the associated .pvtu and .vtu files): VTK file containing the displacement field evolution, - probes.csv containing scalar data from the simulations: - Forces in N: F_1 and F_2 at the bottom and top boundaries, - Energies in J: elastic energy, fracture dissipation, undamaged elastic energy, external work, - Displacement data in m representing the displacement field at various locations in the simulation domain.


Script to generate figure data from raw data (raw_data_and_scripts/postprocess)

The figures from the article can be generated from the raw data in raw_data_and_scripts. Those scripts require the following dependencies: numpy, matplotlib, pandas. Some scripts must be run using pvpython from Paraview (see the accompanying makefile). The scripts can be run using the makefile in the raw_data_and_scripts directory.

The following scripts are available in the raw_data_and_scripts directory:



postprocess/display_results.py: displays line plots such as the force-displacement curves for the structured meshes.

postprocess/export_crack_tip_images.py (Paraview): export the images of the crack phase around the crack tip. Note that this script circumvents the Paraview issue described here: https://discourse.paraview.org/t/unsymmetric-contour-rendered-by-symmetric-data/4118.

postprocess/compare_struct_unstruct.py: displays line plots such as the force-displacement curves for comparing structured vs unstructured meshes.

postprocess/export_pha_field_construction_images.py (Paraview): export the images used to illustrate the construction of the initial phase field for the PHA technique.

postprocess/export_PHA-T0_thickening_images.py (Paraview): export the images to illustrate the thickening of the phase field for the PHA-T0 technique.


Before running the scripts, all raw simulation data must be accessible.

Notes





File Naming Conventions:- GEO refers to geometric initial cracks, while PHA refers to phase field initial cracks.- T0 and T1 denote the initial crack thickness (infinitely thin or one element wide).- t0 and tf indicate initial and final states, respectively.




Usage:- These processed data files are sufficient to reproduce the figures in the article.- If using this data in a publication or presentation, please cite the accompanying article appropriately.



Acknowledgment

This dataset contains the final processed results; the raw simulation data are not included due to storage constraints. For further inquiries or specific requests regarding the raw data, please contact the corresponding author.

Paper Description

Variational phase-field fracture models are now widely used to simulate crack propagation in structures. A critical aspect of these simulations is the correct determination of the propagation threshold of pre-existing cracks, as it highly relies on how the initial cracks are implemented. While prior studies briefly discuss initial crack implementation techniques, we present here a systematic investigation. Various techniques to introduce initial cracks in phase field fracture simulations are tested, from the crack explicit meshing to the replacement by a fully damaged phase field, including different variants for the boundary conditions. Our focus here is on phase field models aiming to approximate, in the $\Gamma$-convergence limit, Griffith quasi-static propagation in the framework of Linear Elastic Fracture Mechanics. Therefore, a sharp crack model from classic linear elastic fracture mechanics based on Griffith criterion is the reference in this work. To assess the different techniques to introduce initial cracks, we rely on path-following methods to compute the sharp crack and the phase field smeared crack solutions. The underlying idea is that path-following ensures staying at equilibrium at each instant so that any difference between phase field and sharp crack models can be attributed to numerical artifacts. Thus, by comparing the results from both models, we can provide practical recommendations for reliably incorporating initial cracks in phase field fracture simulations. The comparison shows that an improper initial crack implementation often requires the smeared crack to transition to a one-element-wide phase band to adequately represent a displacement jump along a crack. This transition increases the energy required to propagate the crack, leading to a significant overshoot in the force-displacement response. The take-home message is that to predict the propagation threshold accurately and avoid artificial toughening, the crack must be initialized either setting the phase field to its damage state over a one-element-wide band or meshing the crack explicitly as a one-element-wide slit and imposing the fully cracked state on the crack surface.

This directory contains the images and CSV files used to generate the figures in the accompanying article.These processed data files allow for reproducing the plots and analyses presented in the paper.