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.

为了提高薄膜/基板型可拉伸电子设备的稳健性，近年来在其设计中采用了中间层。然而，中间层会显着影响三层结构（薄膜/中间层/基材结构）的静态和动态行为。本文考虑薄膜与中间层之间的剪切应力和中间层与基材层的变形，并考虑薄膜的波纹形状，建立了三层结构的改进理论模型，屈曲研究三层的行为以获得静态屈曲幅度和波长。然后，基于几何均匀化理论和拉格朗日方程，推导出了该结构的运动控制方程（使用波纹梁理论用于粘合在有限厚度双层基板上的屈曲薄膜）。利用屈曲分析的数值结果和Jacobi椭圆函数，得到了波纹薄膜三层结构的非线性解析频率。最后通过算例分析揭示了杨氏模量和中间层厚度对波纹三层结构非线性频率的影响。从这些结果可以得出结论，当中间层的厚度比硬膜的厚度（这是当前典型设计的尺寸）大数百倍时，随着中间层杨氏模量的增加，三层结构的非线性频率降低。然而，当中间层比薄膜薄时，非线性频率与中间层的杨氏模量无关，这意味着三层结构退化为双层（薄膜-基材）结构。这些结果有助于设计可靠的薄膜/基板型可拉伸电子设备。