# Mechanical MNIST crack path extended version

## Cite this dataset

Mohammadzadeh, Saeed; Lejeune, Emma (2021). Mechanical MNIST crack path extended version [Dataset]. Dryad. https://doi.org/10.5061/dryad.rv15dv486

## Abstract

The Mechanical MNIST Crack Path dataset contains Finite Element simulation results from phase-field models of quasi-static brittle fracture in heterogeneous material domains subjected to prescribed loading and boundary conditions. For all samples, the material domain is a square with a side length of 1. There is an initial crack of fixed length (0.25) on the left edge of each domain. The bottom edge of the domain is fixed in x (horizontal) and y (vertical), the right edge of the domain is fixed in x and free in y, and the left edge is free in both x and y. The top edge is free in x, and in y it is displaced such that, at each step, the displacement increases linearly from zero at the top right corner to the maximum displacement on the top left corner. Maximum displacement starts at 0.0 and increases to 0.02 by increments of 0.0001 (200 simulation steps in total). The heterogeneous material distribution is obtained by adding rigid circular inclusions to the domain using the Fashion MNIST bitmaps as the reference location for the center of the inclusions. Specifically, each center point location is generated randomly inside a square region defined by the corresponding Fashion MNIST pixel when the pixel has an intensity value higher than 10. In addition, a minimum center-to-center distance limit of 0.0525 is applied while generating these center points for each sample. The values of Young’s Modulus (E), Fracture Toughness (G_{f}), and Failure Strength (f_{t}) near each inclusion are increased with respect to the background domain by a variable rigidity ratio r. The background value for E is 210000, the background value for G_{f} is 2.7, and the background value for f_{t} is 2445.42. The rigidity ratio throughout the domain depends on position with respect to all inclusion centers such that the closer a point is to the inclusion center the higher the rigidity ratio will be. We note that the full algorithm for constructing the heterogeneous material property distribution is included in the simulation scripts shared on GitHub. The following information is included in our dataset:

(1) Locations of the center of the inclusions (the script to extract rigidity ratio matrices with the desired resolution is available on GitHub), (2) the displacement and damage fields every ten simulation step reported over a uniform 256×256 grid (3) the full resolution displacements and damage fields at both the final displacement step and the damage initiation state, and (4) the force-displacement curves for each simulation.

All simulations are conducted with the FEniCS computing platform (FEniCS Project). The code to reproduce these simulations is hosted on GitHub (https://github.com/saeedmhz/phase-field).

## Methods

The “Mechanical MNIST Crack Path” dataset is an addition to the Mechanical MNIST collection that captures crack propagation in a heterogeneous material. In the Mechanical MNIST Crack Path dataset, we model quasi-static brittle fracture in a two-dimensional heterogeneous material with rigid inclusions whose positions are dictated by the Fashion MNIST bitmap using the phase-field method, a state-of-the-art approach to fracture modeling. In the Mechanical MNIST Crack Path dataset, we define the material domain as a square with a side length of 1. There is an initial crack of fixed length (0.25) on the left edge of the domain. The bottom edge of the domain is fixed in x (horizontal), and y (vertical), the right edge of the domain is fixed in x and free in y, and the left edge is free in both x and y. The top edge is free in x, and in y it is displaced such that, at each step, the displacement increases linearly from zero at the top right corner to the maximum displacement on the top left corner. Maximum displacement starts at 0.0 and increases to 0.02 by increments of 0.0001 (200 simulation steps in total). The heterogeneous material distribution is obtained by adding rigid circular inclusions to the domain using the Fashion MNIST bitmaps as the reference location for the center of the inclusions. Specifically, each center point location is generated randomly inside a square region (a fraction of the size of the domain) defined by the corresponding Fashion MNIST pixel. The values of Young’s Modulus (E), Fracture Toughness (Gf), and Failure Strength (ft) near each inclusion are increased with respect to the background domain by a variable rigidity ratio. Note that all Finite Element simulations are performed in FEniCS, and all the related information and data generation scripts are available on GitHub. The version of this dataset hosted here on Dryad is the “extended version” of the Mechanical MNIST Crack Path dataset, which contains more information than the “lite” version hosted in the OpenBU Institutional Repository.

## Usage notes

A list of all files included in this dataset and a short explanation of each file is as follows:

1) inclusions_{test/train}.7z: Locations of the center of the inclusions saved as .txt files for all samples. The first column is the "x" coordinate, and the second column is the "y" coordinate. (the script to extract rigidity ratio matrices with the desired resolution is available on https://github.com/saeedmhz/phase-field)

(2) {damage/displacements}_low-res_{test/train}.zip: {damage/displacements} fields reported over a uniform 256×256 grid and recorded every ten simulation steps into text files with .npy format. Each file has 20 rows and 256×256 columns, and each row contains information of one simulation step.**

3) {damage/displacements}_dmg-init_{test/train}.zip*: Full mesh resolution {damage/displacements} fields for all samples recorded at the early stages of damage initiation.**

4) {damage/displacements}_last-step_{test/train}.zip*: Full mesh resolution {damage/displacements} fields for all samples recorded at the final step of the simulation.**

5) force-vs-disp-{test/train}.7z: The reaction force in "y" direction (second column) at each maximum displacement (first column) is recorded in .txt files for all samples.

6) mesh.xml: The mesh used for all simulations.

Please refer to https://github.com/saeedmhz/phase-field for further details and scripts.

*All files are FEniCS Functions saved as .h5 files. All scripts and necessary information on how to loading these files are available on https://github.com/saeedmhz/phase-field.

** A folder with the name XXXX-YYYY contains information related to case #XXXX through case #YYYY.

## Funding

Boston University, Award: David R. Dalton Career Development Professorship

Boston University, Award: Hariri Institute Junior Faculty Fellowship