3 ELEMENT TECHNOLOGY
3.1 Locking and “classical” remedies
The problem of element performance in finite element computations of elasto-plastic media exhibiting incompressible or dilatant behavior has been investigated intensively in recent years with the aim of developing low order elements which do not lock. Early approaches to overcome locking due to incompressibility used reduced selective integration, mean dilatation (Nagtegaal & al. 1974), and later on the B-bar approach (Hughes 1980). More recent approaches adopted mixed formulations, derived from the Hu-Washizu variational principle, and enhanced strain approximations; this led to the so-called Enhanced Assumed Strain approach (EAS) (Simo & Rifai 1990, Taylor & al. 1976). Unfortunately, this method does not apply to many well known finite elements (like linear triangles in 2D or linear tetrahedrons in 3D) and, moreover, it requires the specific design of the enhanced part of the strain field for each element separately. Yet another approach is presented in this paper. It is based on a mixed continuous displacement-pressure formulation, and complements the Galerkin scheme by least-squares terms which enhance its stability.
3.2 B-bar approach
The B-bar approach introduces a modification in the volumetric contribution to the standard B matrix of space derivatives of shape functions Bvol, which is underintegrated or averaged over the element. This formulation has proven to be appropriate to overcome locking due to incompressibility in very general situations, but poor performance was observed in dilatant cases (de Borst & Groen 1995).
3.3 Enhanced assumed strain approach
Several comprehensive studies on EAS elements, based on the Hu-Washizu functional, have recently been done by different authors (Andelfinger & Ramm 1993) (Groen 1994) (de Borst & Groen 1995), mainly in conjunction with simple isotropic elastoplastic models. In these cases, it has been shown that a simple enhancement of normal strains only, is sufficient to overcome the problem of volumetric locking due to dilatant constitutive behavior. In the case of real rock-like layered materials, the need of enhancement for both normal and shear strains is confirmed by a number of numerical tests (Truty & al. 1997). In the case of an enhanced shear strain field, some spurious mechanisms may however be produced and, so far, it has been impossible to overcome locking when a mixture of elements of different types are to be used.
3.4 Mixed stabilized elastoplastic formulation
This new approach follows ideas which were developed and proven successful for fluid mechanics in (Hughes & al., 1986, Franca, 1987) and following papers. A least-squares term is added to the standard Galerkin formulation for stabilization. The following general form of the least square term is discussed in Truty & Zimmermann 1997, Truty 2002, and Commend 2001).
