A continuum model for size-dependent deformation of elastic films of nano-scale thickness

Abstract

Ultra-thin elastic films of nano-scale thickness with an arbitrary geometry and edge boundary conditions are analyzed. An analytical model is proposed to study the size-dependent mechanical response of the film based on continuum surface elasticity. By using the transfer-matrix method along with an asymptotic expansion technique of small parameter, closed-form solutions for the mechanical field in the film is presented in terms of the displacements on the mid-plane. The asymptotic expansion terminates after a few terms and exact solutions are obtained. The mid-plane displacements are governed by three two-dimensional equations, and the associated edge boundary conditions can be prescribed on average. Solving the two-dimensional boundary value problem yields the three-dimensional response of the film. The solution is exact throughout the interior of the film with the exception of a thin boundary layer having an order of thickness as the film in accordance with the Saint-Venant’s principle.