Cohesive zone-type delamination in visco-elasticity

Marita Thomas; Chiara Zanini

doi:10.3934/dcdss.2017077

Article Contents

2017, Volume 10, Issue 6: 1487-1517. Doi: 10.3934/dcdss.2017077

This issue Previous Article Optimal control of a rate-independent evolution equation via viscous regularization Next Article Smooth and non-smooth regularizations of the nonlinear diffusion equation

Cohesive zone-type delamination in visco-elasticity

Marita Thomas^1, , and
Chiara Zanini^2,

1.
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39,10117 Berlin, Germany
2.
Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

* Corresponding author: Marita Thomas.
* Corresponding author: Marita Thomas.

Received: August 31, 2016

Revised: November 30, 2016

Published: December 2017

This research has been carried out during several research stays of MT at Politecnico di Torino and of CZ at WIAS, Berlin. The hospitality of the two institutions is gratefully acknowledged. MT also acknowledges the partial financial support by GNAMPA 2014 and by the DFG Project “Finite element approximation of functions of bounded variation and application to models of damage, fracture, and plasticity” within the DFG Priority Programme SPP 1748 “Reliable Simulation Techniques in Solid Mechanics. Development of Non-standard Discretisation Methods, Mechanical and Mathematical Analysis.” CZ acknowledges the partial financial support through the ERC Project No. 267802, “Analysis of Multiscale Systems Driven by Functionals”. CZ is also a member of the Progetto di Ricerca GNAMPA 2016 “Analisi di processi inelastici nella meccanica dei solidi e delle cellule: proprietà fini delle soluzioni” and of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)..

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Abstract

We study a model for the rate-independent evolution of cohesive zone delamination in a visco-elastic solid, also exposed to dynamics effects. The main feature of this model, inspired by [32], is that the surface energy related to the crack opening depends on the history of the crack separation between the two sides of the crack path, and allows for different responses upon loading and unloading.

Due to the presence of multivalued and unbounded operators featuring non-penetration and the 'memory'-constraint in the strong formulation of the problem, we prove existence of a weaker notion of solution, known as semistable energetic solution, pioneered in [41] and refined in [38].

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Mathematics Subject Classification: Primary: 35A15, 35Q74, 74H20, 74C10, 49J53, 49J45; Secondary: 74C05.

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Figure 1. Typical cohesive laws. Solid lines: curves corresponding to maximal loading envelope, dashed lines: loading -unloading rules.

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