F-bar-based linear triangles and tetrahedra for finite strain analysis of nearly incompressible solids. Part I: formulation and benchmarking

This paper proposes a new technique which allows the use of simplex finite elements (linear triangles in 2D and linear tetrahedra in 3D) in the large strain analysis of nearly incompressible solids. The new technique extends the F‐bar method proposed by de Souza Neto et al. (Int. J. Solids and Struct. 1996; 33: 3277–3296) and is conceptually very simple: It relies on the enforcement of (near‐) incompressibility over a patch of simplex elements (rather than the point‐wise enforcement of conventional displacement‐based finite elements). Within the framework of the F‐bar method, this is achieved by assuming, for each element of a mesh, a modified (F‐bar) deformation gradient whose volumetric component is defined as the volume change ratio of a pre‐defined patch of elements. The resulting constraint relaxation effectively overcomes volumetric locking and allows the successful use of simplex elements under finite strain near‐incompressibility. As the original F‐bar procedure, the present methodology preserves the displacement‐based structure of the finite element equations as well as the strain‐driven format of standard algorithms for numerical integration of path‐dependent constitutive equations and can be used regardless of the constitutive model adopted. The new elements are implemented within an implicit quasi‐static environment. In this context, a closed form expression for the exact tangent stiffness of the new elements is derived. This allows the use of the full Newton–Raphson scheme for equilibrium iterations. The performance of the proposed elements is assessed by means of a comprehensive set of benchmarking two‐ and three‐dimensional numerical examples. Copyright © 2005 John Wiley & Sons, Ltd.