Size effect on the static, dynamic and buckling analysis of orthotropic Kirchhoff-type skew micro-plates based on a modified couple stress theory: comparison with the nonlocal elasticity theory

The size effect on orthotropic Kirchhoff-type skew micro-plates is investigated based on a modified couple stress theory. For a three-dimensional orthotropic body, three additional material length scale parameters should be involved in the modified couple stress theory (with respect to the three shear moduli). However, in this study and without restricting the generality, we assume that the 2D couple stress state of the orthotropic micro-plate is described solely by only one material length scale parameter in accordance with the in-plane shear modulus. Furthermore, this reasonable assumption allows us to compare qualitatively the results with those obtained by the nonlocal elasticity theory, which also uses only one material length scale parameter to capture the size effect. Using Hamilton's principle, the governing equilibrium equation of the micro-plate and the associated general boundary conditions are derived in terms of the deflection. The resulting initial boundary value problem is of fourth order, and it is solved employing the analog equation method. Example problems are presented for orthotropic skew micro-plates, and useful conclusions are drawn from the investigation of their micron-scale response. Some of the findings detected in studying the microstructure vibratory response of orthotropic skew micro-plates, based on the modified couple stress theory, are also verified by those obtained by the nonlocal elasticity theory. Nevertheless, a new important finding is that both the frequency and critical load parameters increase by increasing the material length scale parameter of the modified couple stress theory, which is in direct contradiction to that of the nonlocal elasticity theory where these parameters decrease by increasing the nonlocal parameter.