Finite Element Of Elasticity With Couple-Stress Using The Analogy Between Plane Couple-Stress And Reissner/Mindlin Plate Bending
基于平面偶应力-Reissner/Mindlin板比拟的偶应力有限元

In order to include the effect of microstructure, the theory of elasticity with couple stress considers couple stress which does not appear in the classical elasticity theory. However, there exists a crucial C1 continuity difficulty in the finite element formulation of elasticity with couple stress. The analogy between plane elasticity with couple stress and Reissner/Mindlin plate bending provides an important way to avoid the C1 continuity difficulty. According to the analogy, the C1 continuity difficulty can be avoided naturally by the formulation in the space of stress functions, and the formulation can be analogous to the one of certain Reissner/Mindlin plate bending element in the space of transversal deflection and rotation. The unsettled problem is how to transform the finite element with stress functions as degree of freedom (DOF) into the one with usual planar displacement and rotation as DOF. Using the analogy, the present work provides an effective and rigorous method to deal with this problem. The final finite element has two important characteristics. Firstly, the formulation in space of stress functions avoids C1 continuity difficulty. Secondly, the discrete unknown DOF are usual displacement and rotation. As an application of the present method, a finite element of plane couple stress with 12 DOF is transformed from the eight nodes serendipity Reissner/Mindlin plate bending element. Numerical results of typical problems show that the present element has satisfactory precision and convergence.