# American Institute of Mathematical Sciences

July  2013, 12(4): 1705-1729. doi: 10.3934/cpaa.2013.12.1705

## A cohesive crack propagation model: Mathematical theory and numerical solution

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

Received  February 2011 Revised  November 2011 Published  November 2012

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.
Citation: G. Leugering, Marina Prechtel, Paul Steinmann, Michael Stingl. A cohesive crack propagation model: Mathematical theory and numerical solution. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1705-1729. doi: 10.3934/cpaa.2013.12.1705
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