Analytical solutions of film/substrate structure with film bending under elastic boundary and bifurcation deformation analysis

Abstract

Axisymmetric film/substrate systems are indispensable in many applications due to their unique structural requirements. Stoney’s formula is widely utilized for analyzing such systems. The existing theories have relaxed the limitations of the Stoney’s formula, but mainly assume that the film thickness can be neglected. This work proposes the theoretical solution of axisymmetric film/substrate structure with elastic boundary considering film bending under non-uniform temperature field. Parameterization and method of constant variation are used to decouple and determine the governing equations. Finite element method validates the influence of elastic support on the stress in each layer of the film. It is found that the boundary supports exert an anti-bending effect on the film. The two-stage calibration method is employed, wherein the first step determines the linear relationships between surface deflection and the deflections of both the film and the substrate, while the second step establishes the linear coefficients and their dependence on material properties. Then the stress inverse solutions for arbitrary specified normal distributions of temperature fields are provided. The curvature bifurcation of the surface deformation from an equi-biaxial spherical shape to a non-equi-biaxial elliptical shape is analyzed using the energy method and variational principle. As the thickness ratio of the film to the substrate increases, higher thermal strain energy is required to reach a critical state of curvature. And it is found that the surface deformation would exhibit a hyperbolic shape as the temperature continued to increase.