Nonlinear dynamic analysis for a corrugated thin film on a pre-strained finite-thickness bi-layer substrate

To improve the robustness of the film/substrate-type stretchable electronics, an intermediate layer has been adopted in their design in recent years. However, the intermediate layer could significantly influence the static and dynamic behaviours of the tri-layer structure (film/intermediate layer/substrate structure). In this paper, considering the shear stress between the film and intermediate layer and the deformation of the intermediate layer and substrate layer, and accounting for the corrugation shape of the film, an improved theoretical model of the tri-layer structure is established and the buckling behaviour of the tri-layer is studied to obtain static buckling amplitude and wavelength. Then, the governing equation of motion of this structure (using the theory of corrugated beam for the buckled film bonded on a finite-thickness bi-layer substrate) is derived, based on the geometrical homogenization theory and the Lagrange equation. By using the numerical results of buckling analysis and the Jacobi elliptic function, the analytical nonlinear frequency of the tri-layer structure with corrugated film is obtained. Finally, numerical examples are analysed to reveal the influences of the Young's modulus and thickness of the intermediate layer on the nonlinear frequency of the corrugated tri-layer structure. From these results, it is concluded that when the thickness of the intermediate layer is hundreds times greater than that of the stiff film (which is the dimension of the current typical design), with the increase of the intermediate layer's Young's modulus, the nonlinear frequency of the tri-layer structure decreases. However, when the intermediate layer is thinner than the film, the nonlinear frequency is independent of the intermediate layer's Young's modulus, which implies that the tri-layer structure degenerates into a bi-layer (film-substrate) structure. These results are helpful for the design of reliable film/substrate-type stretchable electronic devices.