# Analysis of Staggered Evolutions for Nonlinear Energies in Phase Field Fracture

@article{Almi2019AnalysisOS, title={Analysis of Staggered Evolutions for Nonlinear Energies in Phase Field Fracture}, author={Stefano Almi and Matteo Negri}, journal={Archive for Rational Mechanics and Analysis}, year={2019}, volume={236}, pages={189-252} }

We consider a class of separately convex phase field energies employed in fracture mechanics, featuring non-interpenetration and a general softening behavior. We analyze the time-discrete evolutions generated by a staggered minimization scheme, where fracture irreversibility is modeled by a monotonicity constraint on the phase field variable. After recasting the staggered scheme by means of gradient flows, we characterize the time-continuous limits of the discrete solutions in terms of balanced…

## 11 Citations

Irreversibility and alternate minimization in phase field fracture: a viscosity approach

- Physics
- 2019

This work is devoted to the analysis of convergence of an alternate (staggered) minimization algorithm in the framework of phase field models of fracture. The energy of the system is characterized by…

A Quasi-Static Model for Craquelure Patterns

- Physics
- 2020

We consider the quasi-static evolution of a brittle layer on a stiff substrate; adhesion between layers is assumed to be elastic. Employing a phase-field approach we obtain the quasi-static evolution…

A variational anisotropic phase-field model of quasi-brittle fracture: Energetic solutions and their computation

- PhysicsArXiv
- 2020

By including the two-sided energetic inequality in the solution method, this work describes, for some of the benchmark problems, an equilibrium path when damage starts to manifest, which is different from the one obtained by solving simply the stationariety conditions of the underlying functional.

Γ-convergence for high order phase field fracture: continuum and isogeometric formulations

- MathematicsComputer Methods in Applied Mechanics and Engineering
- 2020

A Dimension-Reduction Model for Brittle Fractures on Thin Shells with Mesh Adaptivity

- Materials ScienceMathematical Models and Methods in Applied Sciences
- 2020

A new 2D brittle fracture model for thin shells via dimension reduction, where the admissible displacements are only normal to the shell surface, which is successfully assessed on two Riemannian settings and proves not to bias the crack propagation.

Fully discrete approximation of rate-independent damage models with gradient regularization

- Mathematics
- 2020

This work provides a convergence analysis of a time-discrete scheme coupled with a finite-element approximation in space for a model for partial, rate-independent damage featuring a gradient…

Weak solutions for unidirectional gradient flows: existence, uniqueness, and convergence of time discretization schemes

- MathematicsNonlinear Differential Equations and Applications NoDEA
- 2021

We consider the gradient flow of a quadratic non-autonomous energy under monotonicity constraints. First, we provide a notion of weak solution, inspired by the theory of curves of maximal slope, and…

Weak solutions for gradient flows under monotonicity constraints

- Mathematics
- 2019

We consider the gradient flow of a quadratic non-autonomous energy under monotonicity constraint in time and natural regularity assumptions. We provide first a notion of weak solution, inspired by…

Approximation of the Mumford–Shah functional by phase fields of bounded variation

- MathematicsAnalysis and Applications
- 2020

In this paper, we introduce a new phase field approximation of the Mumford–Shah functional similar to the well-known one from Ambrosio and Tortorelli. However, in our setting the phase field is…

## References

SHOWING 1-10 OF 42 REFERENCES

A review on phase-field models of brittle fracture and a new fast hybrid formulation

- Physics
- 2015

In this contribution we address the issue of efficient finite element treatment for phase-field modeling of brittle fracture. We start by providing an overview of the existing quasi-static and…

Phase-field model of mode III dynamic fracture.

- PhysicsPhysical review letters
- 2001

A phenomenological continuum model for the mode III dynamic fracture that is based on the phase-field methodology used extensively to model interfacial pattern formation is introduced and two-dimensional simulations that yield steady-state crack motion in a strip geometry above the Griffith threshold are reported.

Convergence of alternate minimization schemes for phase field

- Mathematics
- 2017

We consider time-discrete evolutions for a phase-field model (for fracture and damage) obtained by alternate minimization schemes. First, we characterize their time-continuous limit in terms of…

Consistent finite-dimensional approximation of phase-field models of fracture

- Computer ScienceAnnali di Matematica Pura ed Applicata (1923 -)
- 2018

It is proved that any limit of a sequence of finite-dimensional evolutions is itself a quasi-static evolution of the phase-field model of fracture, showing for the first time the consistency of a numerical scheme for evolutions of fractures along critical points.

Convergence of discrete and continuous unilateral flows for Ambrosio–Tortorelli energies and application to mechanics

- MathematicsESAIM: Mathematical Modelling and Numerical Analysis
- 2019

We study the convergence of an alternate minimization scheme for a Ginzburg–Landau phase-field model of fracture. This algorithm is characterized by the lack of irreversibility constraints in the…

Numerical implementation of the variational formulation for quasi-static brittle fracture

- Computer Science
- 2007

This paper presents the analysis and implementation of the variational formulation of quasi-static brittle fracture mechanics proposed by G. Francfort and J. Marigo in 1998, and proposes a numerical algorithm based on Alternate Minimizations and proves its convergence under restrictive assumptions.

An Adaptive Finite Element Approximation of a Variational Model of Brittle Fracture

- Computer ScienceSIAM J. Numer. Anal.
- 2010

This work formulate and analyze two adaptive finite element algorithms for the computation of its (local) minimizers and presents two theoretical results which demonstrate convergence of these algorithms to local minimizers of the Ambrosio-Tortorelli functional.

A unilateral L 2 L^{2} -gradient flow and its quasi-static limit in phase-field fracture by an alternate minimizing movement

- PhysicsAdvances in Calculus of Variations
- 2017

Abstract We consider an evolution in phase-field fracture which combines, in a system of PDEs, an irreversible gradient-flow for the phase-field variable with the equilibrium equation for the…