Non-linear deformation mechanism of circular thin film/substrate systems under film stress

The standard practice of assuming spherical deformation to relate film stress and deformation in thin film/substrate systems proves increasingly inaccurate with larger deformations. This study proposes a paradigm shift, adopting a model that captures the intricate nuances of large deformations through a quadratic curvature function. Based on this, the deformation mechanism of circular thin film/substrate systems is studied and it was found that when the deformation is large, the circumferential compression caused by the radial displacement cannot be ignored, resulting in the loss of radial force balance. The film on the substrate continues to bend until the forces reach equilibrium. This results in a progressive increase in the bending curvature, gradually growing outwards from the center towards the edges. The relationship between stress and deformation is derived through the theory of elasticity and the substrate stress changes significantly along the radial direction. The solved deformation and the stress states agree well with those obtained by finite element method. This novel method boasts its applicability across both linear and non-linear deformation regimes within circular film/substrate systems. It seamlessly transitions back to the classical Stoney formula in the linear limit, demonstrating its compatibility with established theoretical frameworks.