B-bar FEMs for anisotropic elasticity

Abstract

Anisotropic elastic materials, such as homogenized model of fiber-reinforced matrix, can display near rigidity under certain applied stress– the resulting strains are small compared to the strains that would occur for other stresses of comparable magnitude. The anisotropic material could be rigid under hydrostatic pressure if the material were incompressible, as in isotropic elasticity, but also for other stresses. Some commonly used finite element techniques are effective in dealing with incompressibility, but are ill-equipped to handle anisotropic material that lock under stress states that are not mostly hydrostatic (e.g. uniformly reduced serendipity and Q1/Q0 B-bar hexahedra). The failure of the classic B-bar method is attributed to the assumption that the mode of deformation to be relieved is one of near incompressibility. The remedy proposed here is based on the spectral decomposition of the compliance matrix of the material. The spectrum can be interpreted to separate nearly-rigid and flexible modes of stress and strain which leads naturally to a generalized selective reduced integration. Furthermore, the spectral decomposition also enables a three-field elasticity formulation that results in a B-bar method that is effective for general anisotropic materials with an arbitrary nearly-rigid mode of deformation.