Extrapolated fields in the formulation of the assumed strain elements Part II: Three-dimensional problems

Assumed strain eight-node hexahedral elements with significantly extended range of applicability are presented. These elements are formulated using only standard translational displacements at each of their eight nodes and provide accurate solutions for a variety of benchmark problems such as spacial beams, plates, shells as well as general three-dimensional elasticity problems. The elements with such characteristics are particularly useful to model problems of complex geometry, where different regions of the domain might impose entirely different modeling requirements. The formulation starts with introduction of a parallelepiped domain associated with the given eight-node hexahedral element. Then, the assumed strain field is constructed for that associated parallelepiped domain. It is done by identification of various modes of its deformation and by proper modification of the strain field in the constant and linear bending modes. Strain and displacement extrapolation from the associated parallelepiped to the original hexahedral domain is subsequently used to establish the assumed strain field for the given element. Solutions to some popular benchmark problems demonstrate that the proposed assumed strain hexahedral elements exhibit remarkably high accuracy even when severely distorted and high aspect ratio meshes are used. Another advantage of the present elements is that locking for nearly incompressible materials is also mitigated. Unfortunately, the elements pass the patch test only when their shapes are parallelopipeds.