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On the incompressible constraint of the 4-node quadrilateral element. (English) Zbl 0854.73066
An analytical expression for the incompressible constraint of a 4-node quadrilateral finite element of plane stress/strain is derived. It is proved that the volume constraint arises naturally from an element matrix related to the first order terms of the Taylor series expansion of the element shape functions. The same matrix is obtained from a 1-point Gauss integration. The incompressibility condition was formulated in a weak sense, so that the element displacement field is divergence-free when integrated over the element volume. The resulting algebraic constraint is shown to coincide with a particular eigenvector of the constant gradient matrix which is obtained from the first order terms of the Taylor series. It is shown that incompressibility could be enforced without the use of a mixed variational principle and without introduction of the pressure as an independent variable. The analytical result can be used to verify the formulation of elements in the incompressible limit by means of a numerical eigensystem analysis.

##### MSC:
 74S05 Finite element methods applied to problems in solid mechanics
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##### References:
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