Penalty 4-Node Quadrilateral Element Formulation for Axisymmetric Couple Stress Problems

To address the issue of size effects in axisymmetric deformation of small-scale solids, this work proposes a 4-node 12-DOF element for axisymmetric problems based on the consistent couple stress theory (CCST), following the framework of the unsymmetric finite element method. With the use of the penalty function method, an independently assumed rotational field is introduced into the virtual work principle to approximate the physical rotation, ensuring the satisfaction of the <i>C</i>¹ continuity requirement of the CCST in a weak form. As a benefit, the enriched <i>C</i>⁰ isoparametric-based interpolation is employed to construct the test functions for displacement and rotation. Furthermore, the force-stress field that satisfies the equilibrium equations related to axisymmetric deformation is employed as the element's force-stress trial function. In order to circumvent locking issues, reduced integration is employed in the penalty stiffness integration process. The numerical results demonstrate that the new element exhibits high computational accuracy and convergence rate in both static and modal analysis problems, effectively capturing size-dependent phenomena.