The role of the sample shape and size on the internal stress induced curvature of thin-film substrate systems

The system considered consists of a rectangular plate coated on one face with a perfectly adherent thin film. Internal stresses in this film induce an observable curvature of the substrate. Under the assumption of thin films, we perform numerical computations by the finite element method (CAST3M code) to study the role of the sample width/length ratio on the deformed shape and on the relationship between the curvature and the internal stress. It is well known that large displacements result in non-linear geometrical effects (decrease of the curvature when compared to the linear case, occurring of a bifurcation point associated with an increasing curvature inhomogeneity). In the case of square plates, our computations are in good agreement with the analytical approaches and numerical results of the literature. In contrast, in the case of slender blades, we put into evidence a strong reduction of the non-linear effects related to the curvature, which is not expected from the analytical approaches but is in good agreement with experiment. In short, it appears that in the case of slender blades, the classical linear approaches (Stoney's for instance) can constitute an approximation valid even in the case of large displacements.