Eight-node hexahedral elements for gradient elasticity analysis

Computational analysis of gradient elasticity often requires the trial solution to be C-1 yet constructing C-1 finite elements is not trivial. Further to the recent success of 4-node quadrilateral and 4-node tetrahedral elements which employ only displacement and displacement gradient as the nodal dofs, this paper develops 8-node hexahedral elements for gradient elasticity analyses by the generalized discrete Kirchhoff and the relaxed hybrid-stress methods. Both methods require a C-0 displacement interpolation which is quadratic complete in the Cartesian coordinates. Starting from the 8-node hexahedron with only displacement and displacement gradient as the nodal dofs, it is noted that another 6 mid-face nodes and a condensable bubble node with displacement dofs are required for the quadratic completeness. The dofs of the mid-face node are then constrained to those of the corner nodes defining the same element face. The C-0 displacement of the resultant 8-node hexahedron is quadratic complete only when all element faces are flat. Though the quadratic completeness of the two 8-node elements has been partially compromised, the relaxed hybrid-stress element model is marginally more accurate than the previously devised 4-node tetrahedral elements.