Mixed variational principle and finite element formulation for couple stress elastostatics

Skew-symmetric couple stress theory may permit the exploration of solids and fluids at the finest scales for which continuum mechanics applies. This theory is founded upon a true continuum representation. As a result, for solids, the rotation equals one half the curl of the displacement field. In addition, the couple-stress tensor reduces to skew-symmetric form and can be written as a polar couple-stress vector, which is energy conjugate to the mean-curvature polar vector field. The resulting theory is fully self-consistent and parsimonious, requiring only a single extra material parameter, a length scale, for the linear elastic isotropic and cubic single crystal cases. Previous finite element formulations for this theory have required either Lagrange multipliers or penalty parameters to enforce rotation-displacement compatibility, while maintaining C⁰ continuity. Here, we introduce a novel mixed C⁰ variational principle written in terms of displacement and couple-stress polar vectors that avoids any extraneous contributions to the energy functional. This stationary principle, in turn, provides the basis for a robust finite element method, which is developed in this paper for planar quasistatic size-dependent response of linear elastic isotropic media, and then generalized to consider a cubic single crystal example. The finite element implementation uses standard linear three-node triangles for the displacements and tangential edge elements on the triangles to represent the divergence-free couple-stresses. The formulation, however, is quite comprehensive in nature and thus can be extended to examine a broad range of problems, such as those associated with the dynamic response of non-centrosymmetric anisotropic bodies in three-dimensions.