Discrete approximation of dynamic phase-field fracture in visco-elastic materials

Marita Thomas; Sven Tornquist

doi:10.3934/dcdss.2021067

Article Contents

2021, Volume 14, Issue 11: 3865-3924. Doi: 10.3934/dcdss.2021067 open access

This issue Previous Article Preface Next Article An experimentally-fitted thermodynamical constitutive model for polycrystalline shape memory alloys

Discrete approximation of dynamic phase-field fracture in visco-elastic materials

Marita Thomas and
Sven Tornquist^,

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany

^* Corresponding author: Sven Tornquist
^* Corresponding author: Sven Tornquist

Received: August 2020

Early access: June 2021

Published: November 2021

The authors gratefully acknowledge the support by the Deutsche Forschungsgemeinschaft in the Priority Program 1748 "Reliable simulation techniques in solid mechanics. Development of non-standard discretization methods, mechanical and mathematical analysis" within the project "Reliability of Efficient Approximation Schemes for Material Discontinuities Described by Functions of Bounded Variation" – Project Number 255461777 (TH 1935/1-2). MT also gratefully acknowledges the partial financial support by the DFG within the Collaborative Research Center 1114 "Scaling Cascades in Complex Systems", Project C09 "Dynamics of rock dehydration on multiple scales"

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Abstract

This contribution deals with the analysis of models for phase-field fracture in visco-elastic materials with dynamic effects. The evolution of damage is handled in two different ways: As a viscous evolution with a quadratic dissipation potential and as a rate-independent law with a positively $ 1 $-homogeneous dissipation potential. Both evolution laws encode a non-smooth constraint that ensures the unidirectionality of damage, so that the material cannot heal. Suitable notions of solutions are introduced in both settings. Existence of solutions is obtained using a discrete approximation scheme both in space and time. Based on the convexity properties of the energy functional and on the regularity of the displacements thanks to their viscous evolution, also improved regularity results with respect to time are obtained for the internal variable: It is shown that the damage variable is continuous in time with values in the state space that guarantees finite values of the energy functional.

Keywords:

Visco-elastodynamic damage,
Ambrosio-Tortorelli model for phase-field fracture,
viscous evolution,
rate-independent limit,
temporal regularity of solutions.

Mathematics Subject Classification: Primary: 74H10, 74H20, 74H30;Secondary: 35M86, 35Q74.

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Figure 1. Qualitative shape of $ w_{ \mathbb{C}}: \mathbb{R}\to[w_{0},w^{*}] $: The function is constant on the intervals $ (-\infty,0]\cup[z^{*},\infty) $, monotonously increasing on $ \mathbb{R} $, and convex on the interval $ (-\infty,z_{*}) $ with $ z_{*}>1 $ but non-convex on $ [z_{*},z^{*}) $. The points $ z_{\ominus}\ll0 $ and $ z_{\oplus}\gg z^{*} $ will play a role later in the proof of Theorem 4.1, Formula (41), when showing that solutions $ z_{\tau}^{k} $ of the space-continuous problem (2b) are bounded in $ [0,1] $

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