Penalty 4-node quadrilateral element formulation for axisymmetric-torsion problems within consistent couple stress theory

In this work, a novel quadrilateral four-node element capable of simulating the axisymmetric-torsion deformation of small-scale solids of revolution is developed based on the consistent couple stress theory (CCST). To establish the element formulation, the C¹ requirement for displacement in the CCST is enforced in weak sense by using the penalty function method and the independent nodal rotation degrees of freedom are introduced into element construction to approximate the mechanical rotation fields. Besides, the stress functions that can satisfy the relevant equilibrium equation of the axisymmetric-torsion deformation are adopted as the basic functions for designing the element’s stress trial function. Several numerical tests are carried out and the results are compared to the solutions obtained using the analytical method or hexahedral solid element from the literature. It is shown that the new element exhibits good accuracy and captures the size dependences efficiently in prediction of the axisymmetric-torsion behavior of small-scale solids.