Locking-Free Finite Elements for Unilateral Crack Problems in Elasticity

Issue		Math. Model. Nat. Phenom. Volume 4, Number 1, 2009 Modelling and numerical methods in contact mechanics


Page(s)		1 - 20
DOI		10.1051/mmnp/20094101
Published online		27 January 2009

Math. Model. Nat. Phenom. Vol. 4, No. 1, 2009, pp. 1-20
DOI: 10.1051/mmnp/20094101

Locking-Free Finite Elements for Unilateral Crack Problems in Elasticity

Z. Belhachmi, J.-M. Sac-Epée and S. Tahir

LMAM UMR7122, Université Paul Verlaine de Metz, Ile du Saulcy, 57045 Metz, France

belhach@univ-metz.fr

Published online: 27 January 2009

Abstract
We consider mixed and hybrid variational formulations to the linearized elasticity system in domains with cracks. Inequality type conditions are prescribed at the crack faces which results in unilateral contact problems. The variational formulations are extended to the whole domain including the cracks which yields, for each problem, a smooth domain formulation. Mixed finite element methods such as PEERS or BDM methods are designed to avoid locking for nearly incompressible materials in plane elasticity. We study and implement discretizations based on such mixed finite element methods for the smooth domain formulations to the unilateral crack problems. We obtain convergence rates and optimal error estimates and we present some numerical experiments in agreement with the theoretical results.

Mathematics Subject Classification. 65N30, 74M15, 35J85

Key words: crack problems -- variational inequalities -- smooth domain method -- mixed finite elements -- a priori estimates

© EDP Sciences 2009

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