General modeling on substrate curvatures and film stresses in bilayer systems

Based on the transform of reference plane of bending from substrate mid-plane to an optimum site, and synchronously involved with the hypothesis of a coupled lattice shared by atoms of two crystals on the both sides of interface between film and substrate, a new closed-form solution with symmetric form in mathematics for residual stress in a film-substrate system is developed. Examples are discussed for the thermal and epitaxial stresses. As the ratio of film thickness to substrate thickness becomes larger, stress given by the new solution differs obviously from those predicted by the previous solutions [For instances: G. G. Stoney, Proc. R. Soc. (London) A82, 172 (1909); L. B. Freund, J. A. Floro, and E. Chason, Appl. Phys. Lett. 74, 1987 (1999); and Q. S. Mu, AIP Adv. 8, 065224 (2018)], which may be considered as an extended modification of this paper to the existing studies.