ON THE INCOMPRESSIBLE CONSTRAINT OF THE 4-NODE QUADRILATERAL ELEMENT

The standard plane 4-node element is written as the summation of a con stant gradient matrix, usually obtained from underintegration, and a s tabilization matrix. The split is based on a Taylor series expansion o f element basis functions. In the incompressible limit, the 'locking'- effect of the quadrilateral is traced back to the stabilization matrix which reflects the incomplete higher-order term in the Taylor series. The incompressibility condition is formulated in a weak sense so that the element displacement held is divergence-free when integrated over the element volume. The resulting algebraic constraint is shown to co incide with a particular eigenvector of the constant gradient matrix w hich is obtained from the first-order terms of the Taylor series, The corresponding eigenvalue enforces incompressibility implicitly by mean s of a penalty-constraint. Analytical expressions for that constant-di latation eigenpair are derived for arbitrary element geometries. It is shown how the incompressible constraint carries over to the element s tiffness matrix if the element stabilization is performed in a particu lar manner. For several classical and recent elements, the eigensystem s are analysed numerically. It is shown that most of the formulations reflect the incompressible constraint identically. In the incompressib le limit, the numerical accuracies of the elements are compared.