# Models of dynamic damage and phase-field fracture, and their various time discretisations

## TL;DR

This paper presents and analyzes various dynamic damage models in viscoelastic materials, focusing on phase-field fracture approximations, and explores different time discretisation methods for these models.

## Contribution

It introduces and analyzes multiple variants of damage models with inertia and viscosity, including phase-field fracture, and proposes various time discretisation schemes.

## Key findings

- Models successfully incorporate inertia and damageable viscosity.
- Galerkin method effectively analyzes damage models.
- Different time discretisation options are proposed and evaluated.

## Abstract

Several variants of models of damage in viscoelastic continua under small strains in the Kelvin-Voigt rheology are presented and analyzed by using the Galerkin method. The particular case, known as a phase-field fracture approximation of cracks, is discussed in detail. All these models are dynamic (i.e. involve inertia to model vibrations or waves possibly emitted during fast damage/fracture or induced by fast varying forcing) and consider viscosity which is also damageable. Then various options for time discretisation are devised. Eventually, extensions to more complex rheologies or a modification for large strains are briefly exposed, too.