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+gamma delta(3))/(1+delta), where gamma designates the ratio of the biaxial moduli and delta 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 delta less than or equal to 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 delta similar or equal to 0.4; and (c) Brenner-Senderoff-type expressions can be very misleading and should be avoided. (C) 2000 American Institute of Physics. [S0021-8979(00)05021-0].