Finite element Lagrange multiplier formulation for size-dependent skew-symmetric couple-stress planar elasticity

We develop a variational principle based on recent advances in couple-stress theory and the introduction of an engineering mean curvature vector as energy conjugate to the couple stresses. This new variational formulation provides a base for developing a couple-stress finite element approach. By considering the total potential energy functional to be not only a function of displacement, but of an independent rotation as well, we avoid the necessity to maintain C¹ continuity in the finite element method that we develop here. The result is a mixed formulation, which uses Lagrange multipliers to constrain the rotation field to be compatible with the displacement field. Interestingly, this formulation has the noteworthy advantage that the Lagrange multipliers can be shown to be equal to the skew-symmetric part of the force-stress, which otherwise would be cumbersome to calculate. Creating a new consistent couple-stress finite element formulation from this variational principle is then a matter of discretizing the variational statement and using appropriate mixed isoparametric elements to represent the domain of interest. Finally, problems of a hole in a plate with finite dimensions, the planar deformation of a ring, and the transverse deflection of a cantilever are explored using this finite element formulation to show some of the interesting effects of couple stress. Where possible, results are compared to existing solutions to validate the formulation developed here.