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 ba sed on closed-form expressions of the eigenvalues and eigenvectors of a rec tangular element under an axial stress in plane strain. These analyses allo w to identify the locking properties of the finite elements in different si tuations (namely, incompressible and shear locking), as well the appearance of numerical instabilities. In particular, we identify the presence of mat erial instabilities, that is, negative stiffness in the constant strain res ponse of the material land so exact and independent of the particular finit e element under consideration) as the cause for the appearance of numerical instabilities in the form of hourglassing. This is particularly the case i n the original enhanced elements in plane strain problems. This situation i s 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 al ternative regularization of the volumetric stiffness contributions to the h ourglass modes. This observation allows for the identification of a new ass umed/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 constan t scaling of the original enhanced deformation gradient with only two enhan ced modes, maintaining the full locking-free response of the element and th e important strain driven structure of the finite element formulation. Full details of the numerical implementation of these ideas are presented, as w ell as several numerical simulations illustrating the performance of the ne w formulation. (C) 2000 Elsevier Science Ltd. All rights reserved.