Variational formulation of micropolar elasticity using 3D hexahedral finite-element interpolation with incompatible modes

A three-dimensional micropolar elasticity is cast in terms of the rigorous variational formulation. The discrete approximation is based on hexahedral finite element using the conventional Lagrange interpolation and enhanced with incompatible modes. The proposed element convergence is checked by performing patch tests which are derived specifically for micropolar finite elements. The element enhanced performance is also demonstrated by modelling two boundary value problems with analytical solutions, both exhibiting the size-effect. The analyzed problems involve a cylindrical plate bending and pure torsion of circular cylinders, which were previously used in the experimental determination of the micropolar material parameters. The numerical results are compared against the analytical solution, and additionally against existing experiments on a polymeric foam for the pure torsion problem. The enhancement due to incompatible modes provides the needed improvement of the element performance in the bending test without negative effects in the pure-torsion test where incompatible modes are not needed. It is concluded that the proposed element is highly suitable for the numerical validation of the experimental procedure.