Axisymmetric couple stress elasticity and its finite element formulation with penalty terms

Length scale dependent deformation in polymers has been reported in the literature in different experiments at the micron and submicron length scales. Such length scale dependent deformation behavior can be described using higher order gradient theories. A numerical approach for axisymmetric problems is presented here where a couple stress elasticity theory is employed. For the numerical formulation, a penalty finite element approach is proposed and implemented with C (0) axisymmetric elements. In this approach, rotations are introduced as nodal variables independent of nodal displacements, and the penalty term is used to minimize the difference in rotations determined from nodal displacements and nodal rotations. Numerical simulations are performed on different examples to assess the performance of the suggested approach. In particular, a circular cylinder with spherical inclusions and the interaction of the inclusion with the boundary are studied. It is found that with the length scale parameter the distance with which the free boundary affects the stress state of the inclusion increases as well.