On the locking and stability of finite elements in finite deformation plane strain problems

We present in this paper a discussion of the properties of different mixed and enhanced finite element formulations in the finite deformation range based on closed-form expressions of the eigenvalues and eigenvectors of a rectangular element under an axial stress in plane strain. These analyses allow to identify the locking properties of the finite elements in different situations (namely, incompressible and shear locking), as well the appearance of numerical instabilities. In particular, we identify the presence of material instabilities, that is, negative stiffness in the constant strain response of the material (and so exact and independent of the particular finite element under consideration) as the cause for the appearance of numerical instabilities in the form of hourglassing. This is particularly the case in the original enhanced elements in plane strain problems. This situation is to be contrasted with the standard mixed Q1/P0 element, which is shown to avoid these numerical instabilities in the plane strain case through an alternative regularization of the volumetric stiffness contributions to the hourglass modes. This observation allows for the identification of a new assumed/enhanced formulation that avoids the observed numerical instabilities in the plane strain, without resorting to stabilization techniques relying on user-defined parameters. The new formulation simply involves the constant scaling of the original enhanced deformation gradient with only two enhanced modes, maintaining the full locking-free response of the element and the important strain driven structure of the finite element formulation. Full details of the numerical implementation of these ideas are presented, as well as several numerical simulations illustrating the performance of the new formulation.