Locking-Free Finite Elements for Unilateral Crack Problems in Elasticity

Z. Belhachmi; J.-M. Sac-Epée; S. Tahir

Mathematical Modelling of Natural Phenomena (2009)

  • Volume: 4, Issue: 1, page 1-20
  • ISSN: 0973-5348

Abstract

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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.

How to cite

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Belhachmi, Z., Sac-Epée, J.-M., and Tahir, S.. "Locking-Free Finite Elements for Unilateral Crack Problems in Elasticity." Mathematical Modelling of Natural Phenomena 4.1 (2009): 1-20. <http://eudml.org/doc/222437>.

@article{Belhachmi2009,
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.},
author = {Belhachmi, Z., Sac-Epée, J.-M., Tahir, S.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {crack problems; variational inequalities; smooth domain method; mixed finite elements; a priori estimates; mixed finite elements},
language = {eng},
month = {1},
number = {1},
pages = {1-20},
publisher = {EDP Sciences},
title = {Locking-Free Finite Elements for Unilateral Crack Problems in Elasticity},
url = {http://eudml.org/doc/222437},
volume = {4},
year = {2009},
}

TY - JOUR
AU - Belhachmi, Z.
AU - Sac-Epée, J.-M.
AU - Tahir, S.
TI - Locking-Free Finite Elements for Unilateral Crack Problems in Elasticity
JO - Mathematical Modelling of Natural Phenomena
DA - 2009/1//
PB - EDP Sciences
VL - 4
IS - 1
SP - 1
EP - 20
AB - 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.
LA - eng
KW - crack problems; variational inequalities; smooth domain method; mixed finite elements; a priori estimates; mixed finite elements
UR - http://eudml.org/doc/222437
ER -

References

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