How accurate are Stoney’s equation and recent modifications

Stoney’s equation has been—and still is—extensively used to evaluate the macrostress acting in a coating deposited on a thick substrate. In principle, the formula applies only in the “thin-film approximation,” that is, for coatings much thinner than the substrate. The main purpose of this communication is to demonstrate that, based on a general theory of elastic interactions in multilayer laminates, the correct formula for the stress can be expressed in terms of Stoney’s equation and a correction factor equal to (1+γ δ3)/(1+δ), where γ designates the ratio of the biaxial moduli and δ is the ratio of the layer thicknesses. In this light, it is shown that (a) Stoney’s equation does not cause serious errors for thickness ratios δ⩽0.1; (b) Atkinson’s recently proposed modification, which does not require information on the coating’s modulus, yields much improved results for thickness ratios up to δ≃0.4; and (c) Brenner–Senderoff-type expressions can be very misleading and should be avoided.