A size-dependent model for shear deformable laminated micro-nano plates based on couple stress theory

Abstract

In this paper, a modified first order shear deformation theory is proposed for laminated micro-nano plates. Couple stress components are involved in the differential equations of equilibrium for shear deformable micro-nano plates, which are established by considering the deformation mechanism of an element cut from the micro-nano plate. Based on the mechanics of composite materials and Koiter’s theory, the constitutive equations for laminated micro-nano plates are established to characterize the relationships between Cauchy stress tensor and the strain tensor, the couple-stress tensor and the curvature tensor. Governing equations with size-dependent effect are derived and are expressed in terms of the displacements and the rotation functions. The proposed theory is applied to investigate the eigenvalue buckling and free vibration of laminated micro-nano plates simply supported at the four sides. Matlab program based on the analytical solutions is developed and is testified by comparing the solutions for isotropic micro-nano plates from the proposed theory against those from an available model and from ABAQUS analysis. Various examples are studied to investigate the influence of scale ratio, aspect ratio, length-to-thickness ratio, material properties and stacking sequence on the buckling and free vibration response of the laminated micro-nano plates. The size-dependent effect is captured for all examples, particularly for micro-nano isotropic laminated plate with geometry size similar to its material length scale parameter, where the buckling load may be nearly up to 10 times larger than that given by the FSDT theory. A good agreement among solutions from the Matlab program, the existing literature and ABAQUS analysis demonstrates the efficiency and accuracy of the proposed theory.