# Phase-Field Models for Fatigue Crack Growth

## TL;DR

This paper presents a phase-field modeling approach for fatigue crack growth that captures sub-threshold fracture growth under cyclic loads, reproduces the Paris law, and can simulate complex crack geometries.

## Contribution

The paper introduces a phenomenological phase-field model incorporating cyclic degradation to simulate fatigue crack growth, including multiple cracks and complex geometries.

## Key findings

- Reproduces the Paris law with high exponents for brittle fatigue growth.
- The Paris law exponent decreases with Ginzburg-Landau type phase dynamics.
- Model can simulate crack growth in complex geometries in 2D and 3D.

## Abstract

We introduce a class of models based on near crack tip degradation of materials that can account for fracture growth under cyclic loads below the Griffith threshold. We incorporate the gradual degradation due to a cyclic load through a flow equation that decreases spatially varying parameters controlling the fracture toughness in the vicinity of the crack tip, with the phase and displacement fields relaxed to an energy minimum at each time step. Though our approach is phenomenological, it naturally reproduces the Paris law with high exponents that are characteristic of brittle fatigue crack growth. We show that the exponent decreases when the phase field dynamics is of the Ginzburg-Landau type with a relaxation time comparable to the cyclic loading period, or when degradation occurs on a scale larger than the process zone. In addition to reproducing the Paris law, our approach can be used to model the growth of multiple cracks in arbitrarily complex geometries under varied loading conditions as illustrated by a few numerical examples in two and three dimensions.