Fracture is one of the most commonly encountered failure modes of engineering materials and structures. Prevention of cracking-induced failure is, therefore, a major concern in structural designs. Computational modeling of fracture constitutes an indispensable tool not only to predict the failure of cracking structures but also to shed insights into understanding the fracture processes of many materials such as concrete, rock, ceramic, metals, and biological soft tissues. This chapter provides an extensive overview of the literature on the so-called phase-field fracture/damage models (PFMs), particularly, for quasi-static and dynamic fracture of brittle and quasi-brittle materials, from the points of view of a computational mechanician. PFMs are the regularized versions of the variational approach to fracture which generalizes Griffith's theory for brittle fracture. They can handle topologically complex fractures such as initiation, intersecting, and branching cracks in both two and three dimensions with a quite straightforward implementation. One of our aims is to justify the gaining popularity of PFMs. To this end, both theoretical and computational aspects are discussed and extensive benchmark problems (for quasi-static and dynamic brittle/cohesive fracture) that are successfully and unsuccessfully solved with PFMs are presented. Unresolved issues for further investigations are also documented.

Phase-field modeling of fracture

Modified Newton methods for solving fully monolithic phase-field quasi-static brittle fracture propagation

Variational phase-field formulation of non-linear ductile fracture

A convergence study of phase-field models for brittle fracture

https://hal.archives-ouvertes.fr/hal-02541808/document

An open-source Abaqus implementation of the phase-field method to study the effect of plasticity on the instantaneous fracture toughness in dynamic crack propagation

A framework for polyconvex large strain phase-field methods to fracture

Summary
In this paper, a mixed variational formulation for the development of energy–momentum consistent (EMC) time-stepping schemes is proposed. The approach accommodates mixed finite elements based on a Hu–Washizu-type variational formulation in terms of displacements, Green–Lagrangian strains, and conjugated stresses. The proposed discretization in time of the mixed variational formulation under consideration yields an EMC scheme in a natural way. The newly developed methodology is applied to a high-performance mixed shell finite element. The previously observed robustness of the mixed finite element formulation in equilibrium iterations extends to the transient regime because of the EMC discretization in time. Copyright © 2016 John Wiley & Sons, Ltd.

An energy–momentum consistent method for transient simulations with mixed finite elements developed in the framework of geometrically exact shells

A mixed variational framework for the design of energy-momentum schemes inspired by the structure of polyconvex stored energy functions

https://cronfa.swan.ac.uk/Record/cronfa39597/Download/0039597-30042018114431.pdf

Alexander Janz

Papers

A framework for polyconvex large strain phase-field methods to fracture

An energy–momentum consistent method for transient simulations with mixed finite elements developed in the framework of geometrically exact shells

A mixed variational framework for the design of energy-momentum schemes inspired by the structure of polyconvex stored energy functions

An energy–momentum time integration scheme based on a convex multi-variable framework for non-linear electro-elastodynamics

An energy momentum consistent integration scheme using a polyconvexity-based framework for nonlinear thermo-elastodynamics