Triangular Mindlin microplate element

In this article, based on the most general form of strain gradient theory (MGSGT), a novel extended triangular Mindlin plate element is proposed. To accomplish this aim, first, the quadratic form of energy functional is obtained by vectorizing the higher-order tensors of energy pairs, from which the stiffness and mass matrices of the element are readily derived. In comparison with the standard Mindlin plate element, the new element needs three additional nodal degrees of freedom (DOF) including derivatives of lateral deflection and rotations, which means a total of nine DOFs per node. Also, as compared to the standard Mindlin plate element which requires only C0 shape functions, the present one requires C1 continuous smooth shape functions due to second derivatives of deflection and rotations. Hence, cubic polynomials are used to interpolate the displacement components. The new element can be reduced to that based on the modified strain gradient theory (MSGT) and the modified couple stress theory (MCST). Moreover, the standard Mindlin plate element is recovered when the gradient-based material parameters tend to zero. The Mindlin microplates with different boundary conditions are considered as the problem under study whose free vibration and bending are analyzed. The results are compared with the exact solutions and excellent agreement is achieved.