A cohesive crack propagation model: Mathematical theory and numerical solution

G. Leugering; Marina Prechtel; Paul Steinmann; Michael Stingl

doi:10.3934/cpaa.2013.12.1705

Article Contents

2013, Volume 12, Issue 4: 1705-1729. Doi: 10.3934/cpaa.2013.12.1705

This issue Previous Article On the backgrounds of the theory of m-Hessian equations Next Article Estimates of solutions for the parabolic $p$-Laplacian equation with measure via parabolic nonlinear potentials

A cohesive crack propagation model: Mathematical theory and numerical solution

1.
Applied Mathematics II, Martensstr. 3, D-91054 Erlangen
2.
Chair of Applied Mechanics, Egerlandstr. 5, D-91058 Erlangen
3.
Applied Mathematics II, Martensstr. 3, D-91058 Erlangen

Received: February 2011

Revised: November 2011

Published online: November 2012

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Abstract

We investigate the propagation of cracks in 2-d elastic domains, which are subjected to quasi-static loading scenarios. As we take cohesive effects along the crack path into account and impose a non-penetration condition, inequalities appear in the constitutive equations describing the elastic behavior of a domain with crack. In contrast to existing approaches, we consider cohesive effects arising from crack opening in normal as well as in tangential direction. We establish a constrained energy minimization problem and show that the solution of this problem satisfies the set of constitutive equations. In order to solve the energy minimization problem numerically, we apply a finite element discretization using a combination of standard continuous finite elements with so-called cohesive elements. A particular strength of our method is that the crack path is a result of the minimization process. We conclude the article by numerical experiments and compare our results to results given in the literature.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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