How do you model crack branching and interaction in brittle materials?

1 Finite element method

The finite element method (FEM) is a numerical technique that can solve complex problems involving stress, strain, displacement, and fracture in solid materials. FEM divides the material domain into small elements connected by nodes, and applies the governing equations and boundary conditions to each element. FEM can capture the nonlinear behavior and large deformations of brittle materials, as well as the crack initiation, propagation, and branching. However, FEM requires a fine mesh around the crack tip to accurately represent the stress singularity and the crack path, which can increase the computational cost and time. Moreover, FEM needs special techniques, such as remeshing, cohesive elements, or extended finite elements, to handle the discontinuity and topology changes caused by crack branching and interaction.

2 Phase field method

The phase field method (PFM) is a continuum approach that can model fracture as a diffuse process, where the crack is represented by a scalar variable that varies smoothly from 0 (intact material) to 1 (fully cracked material). PFM introduces a phase field equation that describes the evolution of the crack variable as a function of the elastic energy and the fracture energy of the material. PFM can simulate complex crack patterns, such as branching, coalescence, and kinking, without explicitly tracking the crack tip or updating the mesh. However, PFM also has some drawbacks, such as the need to choose appropriate parameters for the phase field equation, the difficulty to incorporate plasticity and other material effects, and the high computational demand for fine resolution.

3 Discrete element method

The discrete element method (DEM) is a particle-based technique that can model fracture as a discrete process, where the material is composed of discrete elements that interact with each other through contact forces and bonds. DEM can simulate the initiation, propagation, and branching of cracks by breaking the bonds between elements according to certain failure criteria. DEM can capture the heterogeneity and randomness of brittle materials, as well as the fragmentation and size effects of fracture. However, DEM also has some limitations, such as the need to calibrate the bond properties and failure criteria, the difficulty to represent complex geometries and boundary conditions, and the low efficiency for large-scale problems.

4 Lattice method

The lattice method (LM) is a hybrid technique that can model fracture as a discrete process on a predefined lattice structure that approximates the material microstructure. LM assigns nodes and springs to the lattice sites and links, and applies the constitutive laws and failure criteria to the springs. LM can simulate the crack initiation, propagation, and branching by breaking the springs according to the stress or strain level. LM can incorporate the material anisotropy and heterogeneity, as well as the crack deflection and interaction. However, LM also has some challenges, such as the need to generate suitable lattice models for different materials, the sensitivity to the lattice orientation and spacing, and the difficulty to couple with other methods.

5 Molecular dynamics method

The molecular dynamics method (MD) is a microscopic technique that can model fracture as a discrete process at the atomic scale. MD simulates the motion and interaction of atoms or molecules based on the interatomic potentials and the Newton's laws of motion. MD can capture the fundamental mechanisms and processes of fracture, such as bond breaking, atomic rearrangement, dislocation emission, and crack branching. However, MD also has some limitations, such as the high computational cost and time for large systems, the uncertainty and variability of the interatomic potentials, and the difficulty to scale up to the macroscopic level.

6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?