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Within this work, we develop a phase-field description for simulating fractures in incompressible materials. Standard formulations are subject to volume-locking when the solid is (nearly) incompressible. We propose an approach that builds…

Numerical Analysis · Mathematics 2024-12-20 Katrin Mang , Thomas Wick , Winnifried Wollner

In this work, we consider pressurized phase-field fracture problems in nearly and fully incompressible materials. To this end, a mixed form for the solid equations is proposed. To enhance the accuracy of the spatial discretization, a…

Numerical Analysis · Mathematics 2022-09-21 Seshadri Basava , Katrin Mang , Mirjam Walloth , Thomas Wick , Winnifried Wollner

We present a variational reduced-order model for three-dimensional coplanar propagation of sharp cracks in heterogeneous perfectly brittle solids under mixed-mode I+II+III loading. The approach connects the variational fracture formulation…

Materials Science · Physics 2026-05-14 Serafim Egorov , Antoine Sanner , Jean Sulem , Lars Pastewka , Mathias Lebihain

Variational phase-field methods have been shown powerful for the modeling of complex crack propagation without a priori knowledge of the crack path or ad hoc criteria. However, phase-field models suffer from their energy functional being…

Computational Engineering, Finance, and Science · Computer Science 2021-09-15 O. Lampron , D. Therriault , M. Lévesque

Variational phase-field models of brittle fracture are powerful tools for studying Griffith-type crack propagation in complex scenarios. However, as approximations of Griffith's theory-which does not incorporate a strength criterion-these…

Applied Physics · Physics 2026-01-06 Francesco Vicentini , Jonas Heinzmann , Pietro Carrara , Laura De Lorenzis

One of the state-of-the-art strategies for predicting crack propagation, nucleation, and interaction is the phase-field approach. Despite its reliability and robustness, the phase-field approach suffers from burdensome computational cost,…

Numerical Analysis · Mathematics 2022-11-17 Alena Kopaničáková , Hardik Kothari , Rolf Krause

In this work, we extend a phase-field approach for pressurized fractures to non-isothermal settings. Specifically, the pressure and the temperature are given quantities and the emphasis is on the correct modeling of the interface laws…

Numerical Analysis · Mathematics 2019-05-22 Nima Noii , Thomas Wick

In this work, we examine a numerical phase-field fracture framework in which the crack irreversibility constraint is treated with a primal-dual active set method and a linearization is used in the degradation function to enhance the…

Numerical Analysis · Mathematics 2023-07-05 Leon Maximilian Kolditz , Katrin Mang , Thomas Wick

The favored phase field method (PFM) has encountered challenges in the finite strain fracture modeling of nearly or truly incompressible hyperelastic materials. We identified that the underlying cause lies in the innate contradiction…

Numerical Analysis · Mathematics 2022-05-04 Fucheng Tian , Jun Zeng , Mengnan Zhang , Liangbin Li

We propose a variational phase-field model of fracture capable of accounting for arbitrary closed convex strength domains. Unlike traditional models based on Ambrosio and Tortorelli regularization, the phase-field variable does not affect…

Applied Physics · Physics 2025-07-01 Blaise Bourdin , Jean-Jacques Marigo , Corrado Maurini , Camilla Zolesi

A field theory is presented for predicting damage and fracture in quasi brittle materials incorporating effects of irreversible (plastic) deformation as well as elastic moduli that soften with damage. The new observation made here is that…

Materials Science · Physics 2026-03-17 Hayden Bromley , Robert Lipton

The variational approach to fracture is effective for simulating the nucleation and propagation of complex crack patterns, but is computationally demanding. The model is a strongly nonlinear non-convex variational inequality that demands…

Numerical Analysis · Mathematics 2016-05-18 Patrick E. Farrell , Corrado Maurini

We develop robust solvers for a class of perturbed saddle-point problems arising in the study of a second-order elliptic equation in mixed form (in terms of flux and potential), and of the four-field formulation of Biot's consolidation…

Numerical Analysis · Mathematics 2020-11-11 Wietse M. Boon , Miroslav Kuchta , Kent-Andre Mardal , Ricardo Ruiz-Baier

A moving mesh finite element method is studied for the numerical solution of a phase-field model for brittle fracture. The moving mesh partial differential equation approach is employed to dynamically track crack propagation. Meanwhile, the…

Numerical Analysis · Mathematics 2018-01-17 Fei Zhang , Weizhang Huang , Xianping Li , Shicheng Zhang

This paper presents a macroscopic theory, alongside its numerical implementation, aimed at describing, explaining, and predicting the nucleation and propagation of fracture in viscoelastic materials subjected to quasistatic loading…

Soft Condensed Matter · Physics 2025-06-23 Farhad Kamarei , Evan Breedlove , Oscar Lopez-Pamies

This work presents a new adaptive approach for the numerical simulation of a phase-field model for fractures in nearly incompressible solids. In order to cope with locking effects, we use a recently proposed mixed form where we have a…

Numerical Analysis · Mathematics 2021-06-18 Katrin Mang , Mirjam Walloth , Thomas Wick , Winnifried Wollner

This paper concerns the analysis and implementation of a novel iterative staggered scheme for quasi-static brittle fracture propagation models, where the fracture evolution is tracked by a phase field variable. The model we consider is a…

Numerical Analysis · Mathematics 2020-02-19 Mats Kirkesæther Brun , Thomas Wick , Inga Berre , Jan Martin Nordbotten , Florin Adrian Radu

A preconditioning framework for the coupled problem of frictional contact mechanics and fluid flow in the fracture network is presented. The porous medium is discretized using low-order continuous finite elements, with cell-centered…

Numerical Analysis · Mathematics 2022-05-25 Andrea Franceschini , Laura Gazzola , Massimiliano Ferronato

We present a stable finite element method for incompressible nonlinear elasticity based on a four-field mixed formulation involving the displacement, displacement gradient, first Piola--Kirchhoff stress and pressure. Unlike existing…

Numerical Analysis · Mathematics 2026-03-11 Santiago Badia , Wei Li , Ricardo Ruiz-Baier

In this work, the matrix-free solution of quasi-static phase-field fracture problems is further investigated. More specifically, we consider a quasi-monolithic formulation in which the irreversibility constraint is imposed with a…

Numerical Analysis · Mathematics 2024-08-27 Leon Maximilian Kolditz , Thomas Wick
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