A Review of Modified/Consistent Couple Stress and Strain Gradient Theories for Analyzing Static and Dynamic Behaviors of Functionally Graded Microscale Plates and Shells

This paper provides an overview of various size-dependent theories based on modified/consistent couple stress and strain gradient theories (CSTs and SGTs), highlighting the development of two-dimensional (2D) refined and advanced shear deformation theories (SDTs) and three-dimensional (3D) pure analytical and semi-analytical numerical methods, including their applications, for analyzing the static and dynamic behaviors of microscale plates and shells made from advanced materials such as fiber-reinforced composites, functionally graded (FG) materials, and carbon nanotube/graphene platelet-reinforced composite materials. The strong and weak formulations of the 3D consistent CST, along with their corresponding boundary conditions for FG microplates, are derived and presented for illustration. A comparison study is provided to show the differences in the results of a simply supported FG microplate’s central deflection, stress, and lowest natural frequency obtained using various 2D size-dependent SDTs and 3D analytical and numerical methods based on the consistent CST. A parametric study is conducted to examine how primary factors, such as the effects of dilatational and deviatoric strain gradients and couple stress, impact the static bending and free vibration behaviors of a simply supported FG microplate using a size-dependent local Petrov–Galerkin meshless method based on the consistent SGT. Influences such as the inhomogeneity index and length-to-thickness ratio are considered. It is shown that the significance of the impact of various material length-scale parameters on the central deflection and its lowest natural frequency (in the flexural mode) of the FG microplate is ranked, from greatest to least, as follows: the couple stress effect, the deviatoric strain gradient effect, and finally the dilatational strain gradient effect. Additionally, when the microplate’s thickness is less than 10⁻⁷ m, the couple stress effect on its static and dynamic behaviors becomes saturated. Conversely, the impact of the dilatational and deviatoric strain gradients consistently influences the microplate’s static and dynamic behaviors.