265 / 2019-09-25 18:30:24

Size-dependent penalty finite elements with nodal rotation DOFs based on the modified couple stress theory

modified couple stress theory; unsymmetric FEM; rotation degree of freedom; penalty function

计算力学新方法，新理论 > 微纳米尺度理论与计算

To effectively capture the size-dependent mechanical behaviors of the micro/nano structures, the higher-order continuum theories such as the strain gradient theory and the couple stress theory are widely used. However, as these higher-order theories are more complex than the classical one, a very limited number of problems with simple geometries and boundary conditions can be solved by using the analytical approaches. Hence, the numerical approaches with good efficiency and accuracy are clearly required, in which the finite element method (FEM) is commonly recognized as the most popular and powerful one for solids. In this work, the size-dependent low-order C0 quadrilateral and hexahedral elements with rotation degrees of freedom (DOFs) are developed by applying the shape-free unsymmetric FEM to the modified couple stress theory (MCST). In the present element formulations, the independent nodal rotation DOFs are employed to approximatively design the physical rotation field which is further used for deriving the curvature test function, as well as to enhance the displacement and strain test functions which are formulated at the base of the standard isoparametric interpolations. Meanwhile, the penalty function is included in the virtual work principle for the enforcement of the weak C1 continuity condition. Besides, the stress trial function is formulated based on the equilibrium stress functions. To validate the capabilities of the proposed new elements, several benchmark tests are examined and the results show that the elements can capture the size effect well and show good tolerance to mesh distortion.