Wrinkle wavelength analysis of film-substrate systems with shear deformation

This study investigates the wrinkle wavelength of film-substrate system in stretchable electronics, focusing on the shear deformation commonly overlooked in conventional models but critical for two-dimensional (2D) materials like graphene. Wrinkles are generated by releasing pre-strain in thin films bonded to compliant substrates, enabling stretchability for flexible electronic applications. Traditional theory treats films as continuum plates and ignores shear deformation, which becomes inaccurate for layered 2D materials bonded by weak van der Waals interactions, where interlayer shear deformation is significant. To address this limitation, an improved analytical model is developed based on Timoshenko beam theory to incorporate shear effects into the system's total free energy. The governing equation for critical wrinkle wavelength is derived via variational principles; due to its complexity, the perturbation method is used to obtain a third-order solution with sufficient accuracy. Theoretical predictions are validated against finite element analysis (FEA) with refined interface meshing, showing that wrinkle wavelength increases with film shear modulus and converges to traditional theory as shear modulus approaches infinity. The proposed model outperforms the conventional theory for low-shear-modulus films, thick films, and laminated structures. This work provides an accurate theoretical framework for analyzing wrinkling behaviors in film-substrate system, especially for 2D materials and layered structures where shear deformation cannot be neglected.