Vol. 2, No. 1, 2014

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Delaminated thin elastic inclusions inside elastic bodies

Alexander M. Khludnev and Günter R. Leugering

Vol. 2 (2014), No. 1, 1–21

We propose a model for a two-dimensional elastic body with a thin elastic inclusion modeled by a beam equation. Moreover, we assume that a delamination of the inclusion may take place resulting in a crack. Nonlinear boundary conditions are imposed at the crack faces to prevent mutual penetration between the faces. Both variational and differential problem formulations are considered, and existence of solutions is established. Furthermore, we study the dependence of the solution on the rigidity of the embedded beam. It is proved that in the limit cases corresponding to infinite and zero rigidity, we obtain a rigid beam inclusion and cracks with nonpenetration conditions, respectively. Anisotropic behavior of the beam is also analyzed.

thin inclusion, nonlinear boundary conditions, nonpenetration, crack, variational inequality
Mathematical Subject Classification 2010
Primary: 74-XX
Received: 24 July 2012
Revised: 19 January 2013
Accepted: 25 February 2013
Published: 4 December 2013

Communicated by Paul Steinmann
Alexander M. Khludnev
Lavrentyev Institute of Hydrodynamics
Russian Academy of Sciences
Günter R. Leugering
Institute of Applied Mathematics II
Friedrich-Alexander-Universität Erlangen-Nürnberg
Cauerstrasse 11
D-91058 Erlangen