A deviatoric couple stress Mindlin plate model and its degeneration

Abstract

Based on the classical couple stress theory, the deviatoric couple stress theory (DCST), where both couple stress and curvature tensors are traceless, is developed. The deviatoric couple stress Mindlin plate is established based on a new hypothesis of the vanishing curvature of the transverse normal, and the governing equations and associated boundary conditions of the simplified size-dependent Mindlin plate are respectively derived by the principle of virtual work. In the absence of couple stresses, a simplified Mindlin plate model between the classical Kirchhoff plate model and classical Mindlin plate model is derived. To illustrate the simplified DCST-based Mindlin plate model, the static bending and free vibration of rectangular plates are studied. The dependence of the length scale parameter and curvature ratio on the deflection, rotation and natural frequency is analyzed. The numerical results show that the consideration of couple stresses and curvatures hardens the stiffness of the microstructure, resulting in a decrease in its deflection and rotation compared with that of the macroscopic structure under the same conditions, while the natural frequency increases, which is consistent with the size effect observed in experiments.