Abstract:
In order to include the effect of microstructure, thetheory of elasticity with couple stress considers couple stress which doesnot appear in the classical elasticity theory. However, there exists acrucial C$^{1}$ continuity difficulty in the finite element formulation ofelasticity with couple stress.The analogy between plane elasticity with couple stress and Reissner/Mindlinplate bending provides an important way to avoid the C$^{1}$ continuitydifficulty. According to the analogy, the C$^{1}$ continuity difficulty canbe avoided naturally by the formulation in the space of stress functions,and the formulation can be analogous to the one of certain Reissner/Mindlinplate bending element in the space of transversal deflection and rotation.The unsettled problem is how to transform the finite element with stressfunctions as degree of freedom (DOF) into the one with usual planardisplacement and rotation as DOF. Using the analogy, the present workprovides an effective and rigorous method to deal with this problem. Thefinal finite element has two important characteristics. Firstly, theformulation in space of stress functions avoids C$^{1}$ continuitydifficulty. Secondly, the discrete unknown DOF are usual displacement androtation.As an application of the present method, a finite element of plane couplestress with 12 DOF is transformed from the eight nodes serendipityReissner/Mindlin plate bending element. Numerical results of typicalproblems show that the present element has satisfactory precision andconvergence.
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