Refined Nonconforming Finite Element Methods in Couple Stress/Strain Gradient Theory

Chen      Wanji; Ma      Xu; Li      Li

Abstract

This paper reviews the recent development of finite element method for the couple stress/strain gradient theory. Based on the second order couple stress/strain gradient theory, the frame of the C0 continuity and C1 continuity finite elements and convergence criterion as enhanced patch test are discoursed in this paper. The enhanced patch test is used to test convergence of weak C1 continuity element and the capability of C0 continuity penalty element for approaching C1 continuity element. Three couple stress penalty elements including 9-DOF triangular element (CST+BT3) and element (ALT+BT3) and 12-DOF 8-note quadrilateral element (MQ8) are adopted to compare the refined nonconforming 18-DOF triangular element (RCT9+RT9) with weak C1 continuity. Numerical results show that the C0 continuity penalty elements (CST+BT3) and (ALT+BT3) and (MQ8) cannot pass enhanced patch test and are unable to approaching the C1 continuity solution. The C1 weak continuity refined element (RCT9+RT9) can pass the enhanced patch test and possesses higher accuracy. In addition, this paper also includes some information of recent patents on the applications of couple stress/strain gradient theory.

Keywords: Couple stress/strain gradient theory, enhanced patch test, weak C¹ continuity element, C⁰ continuity penalty element, strain independent theory, strain gradient theory, Mindlin plate element, strain, 2D/3D elasticity, linear elasticity, Finite element method, micromachined transducers, Modelling, gradient elasticity