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Locking-Free Finite Elements for Unilateral CrackProblems in Elasticity

Published online by Cambridge University Press:  27 January 2009

Z. Belhachmi,
J.-M. Sac-Epée and
S. Tahir
Z. Belhachmi*
Affiliation:
LMAM UMR7122, Université Paul Verlaine de Metz, Ile du Saulcy, 57045 Metz, France
J.-M. Sac-Epée
Affiliation:
LMAM UMR7122, Université Paul Verlaine de Metz, Ile du Saulcy, 57045 Metz, France
S. Tahir
Affiliation:
LMAM UMR7122, Université Paul Verlaine de Metz, Ile du Saulcy, 57045 Metz, France
*
belhach@univ-metz.fr
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Abstract

We consider mixed and hybrid variational formulations to the linearizedelasticity system in domains with cracks. Inequality type conditions areprescribed at the crack faces which results in unilateral contact problems. Thevariational formulations are extended to the whole domain including the crackswhich yields, for each problem, a smooth domain formulation. Mixedfinite element methods such as PEERS or BDM methods are designed to avoidlocking for nearly incompressible materials in plane elasticity. We study andimplement discretizations based on such mixed finite element methods for thesmooth domain formulations to the unilateral crack problems. We obtainconvergence rates and optimal error estimates and we present some numericalexperiments in agreement with the theoretical results.

Keywords

crack problemsvariational inequalitiessmooth domain methodmixedfinite elementsa priori estimates

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