On the Stoney formula for a thin film/substrate system with nonuniform substrate thickness

Current methodologies used for the inference of thin film\ stress through system curvature measurements are strictly restricted to stress and curvature states which are assumed to remain uniform over the entire film/substrate system. Recently Huang, Rosakis, and coworkers [Acta Mech. Sinica, 21, pp. 362-370 (2005); J. Mech. Pays. Solids, 53, 24832500 (2005); Thin Solid Films, 515, pp. 2220-2229 (2006); J. Appl. Mech., in press; J. Mech. Mater Struct., in press] established methods for the film/substrate system subject to nonuniform misfit strain and temperature changes. The film stresses were found to depend nonlocally on system curvatures (i.e., depend on the full-field curvatures). These methods, however all assume uniform substrate thickness, which is sometimes violated in the thin film/substrate system. Using the perturbation analysis, we extend the methods to nonuniform substrate thickness for the thin film/substrate system subject to nonuniform misfit strain.