A third-order plate model with surface effect based on the Gurtin–Murdoch surface elasticity

Abstract

Thin films with micro/nano scale thickness possessing ultra-high specific surface area have been widely used in opto-, thermo-, and electro-mechanical actuators for ultra-sensitive detection of physical signal change and few molecular absorptions. However, there still lacks a rigorous plate model capable of describing the large deflection and, in the meanwhile, the complex surface effects of those films, especially the coupling between surface shear stress and bulk shear deformation. In this paper, via the satisfaction of the surface balance relation in the Gurtin–Murdoch theory, we refine Reddy’s third-order plate theory to accurately characterize the size-dependent deformation and uncover the influences of the surface layer on the second and third order shear deformations of micro/nano plates. It is revealed that the modified displacement field of our plate model can engender changes in local behaviors including the through-thickness distributions of bulk shear strain and shear stress, which bring about the change in global stiffness and stability of the plates, especially those with smaller length-to-thickness ratio. When the magnitude of residual surface stress increases, the stiffness and stability of the plate are enhanced, due to the amplification of bulk shear stress. Our model also indicates that miniature structures do not share entirely the same displacement field with macroscopic ones, and the method of deriving size-dependent displacement field incorporating surface effect can be extended to more complex cases. This work provides a strategy for tuning the static and dynamic behaviors of thin film devices by modifying the surface elastic constants and residual stress.