Penalty C0 8-node quadrilateral and 20-node hexahedral elements for consistent couple stress elasticity based on the unsymmetric finite element method

Abstract

In this paper, the penalty unsymmetric finite element framework for the consistent couple stress theory is derived from the virtual work principle. The C1 continuity requirement is satisfied in weak form by using the penalty function method to constrain the independently introduced rotations for approximating the mechanical rotations, enabling the utilization of C° continuous interpolations for designing the element displacement without the loss of convergence property. Within the proposed framework, 8-node quadrilateral element and 20-node hexahedral solid element are constructed for analyzing the size-dependent mechanical responses of consistent couple stress elasticity materials. In these developments, the quadratic serendipity isoparametric shape functions are enriched by the rotation degrees of freedom for determining the test functions, whilst the metric stress functions that are derived from the concerned equilibrium equations are used to design the trial functions. A series of numerical benchmarks are examined for verifying their effectiveness and accuracy. It is shown that the elements can efficiently capture the size dependences, exhibiting good accuracy and low susceptibility to mesh distortion.