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Nonconforming finite element methods for the equations of linear elasticity. (English) Zbl 0747.73044
The article concerns the adaptation of nonconforming finite element methods to the well-known equations of linear elasticity. In considering pure traction boundary conditions, the problem is approximated by use of piecewise linear, quadratic and cubic finite elements, where, for the piecewise quadratic and cubic cases, the usual straightforward extension of nonconforming methods is used while the linear case is based on a modified version in which a local projection is added.
In the foregoing analysis, the main problem arises from the proof that an appropriate version of Korn’s second inequality is valid, combined with an equivalence statement of the displacement formulation of the elasticity equations to a Stokes-like formulation involving displacements and a single stress variable. The contribution of the article is to show that Korn’s second inequality holds for nonconforming piecewise quadratic and cubic elements and that it fails for nonconforming linears. However, for nonconforming linears, a modified method is proposed producing a nonsymmetric approximation to the stress tensor. This method is then shown to be equivalent to a mixed formulation in which the symmetry of the stress tensor is guaranteed by use of a Lagrangian multiplier. A discussion of optimal error estimates, including Poisson’s ratio \(\nu\in[0,1/2)\), is included in the article.
Reviewer: W.Ehlers (Essen)

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74B05 Classical linear elasticity
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