Force-Balance Analysis of Surface Instability in a Tri-Layer Film/Substrate Structure

Driven by increasing engineering demands, the need for high-performance flexible electronics has surged, accelerating the development of stretchable devices within mechanics. Among multilayer structures, the film/substrate architecture serves as a typical example, and its buckling behavior remains a longstanding focus of mechanical investigation. This work examines how an elastic film bonded to a soft tri-layer substrate loses stability, producing wrinkled surface patterns under compression. We first construct a mechanical model, then derive an analytical expression for the wrinkle amplitude using a force-balance approach, and finally employ finite-element simulations and theoretical comparisons, we systematically explore how the middle layer’s stiffness and thickness jointly govern the onset, wavelength and amplitude of surface buckling, revealing quantitative selection rules that have not previously been reported for tri-layer structures. The results show that the tri-layer film/substrate structure exhibits two instability modes: film-intermediate co-buckling and film-only wrinkling. By simply varying the middle layer’s elastic modulus or its thickness, one can move the structure across the boundary that separates the film-only and bi-layer buckling regimes, providing a direct mechanical selection for on-demand mode. In addition, the wrinkle amplitude increases monotonically with the applied initial strain. Those findings offer a theoretical reference for designing flexible electronics based on film/substrate structures.