A penalty finite element approach for couple stress elasticity

There is mounting evidence for size dependent elastic deformation at micron and submicron length scales. Material formulations incorporating higher order gradients in displacements have been successful in modeling such size dependent deformation behavior. A couple stress theory without micro-rotation is considered here as micro-rotations increase complexity and necessitate parameters that are difficult to determine. Higher order gradient theories require continuity in both displacements and their derivatives and direct approaches with both displacements and their derivatives as nodal variables results in a large number of degrees of freedom. Here nodal rotations are applied along with nodal displacements to obtain a simpler finite element formulation with fewer degrees of freedom. The difference in rotation gradients determined with nodal displacements and rotations are minimized by a penalty term. To assess the suggested approach simulations are presented and discussed, where the material parameters have been obtained from experiments of epoxy microbeams in the literature.