Extension of Stoney’s formula to non-uniform temperature distributions in thin film/substrate systems. The case of radial symmetry

Current methodologies used for the inference of thin film stress through curvature measurements are strictly restricted to stress and curvature states which are assumed to remain uniform over the entire film/substrate system. By considering a circular thin film/substrate system subject to non-uniform, but axisymmetric temperature distributions, we derive relations between the film stresses and temperature, and between the plate system’s curvatures and the temperature. These relations featured a ‘‘local’’ part which involves a direct dependence of the stress or curvature components on the temperature at the same point, and a ‘‘non-local’’ part which reflects the effect of temperature of other points on the location of scrutiny. Most notably, we also derive relations between the polar components of the film stress and those of system curvatures which allow for the experimental inference of such stresses from full-field curvature measurements in the presence of arbitrary radial nonuniformities. These relations also feature a ‘‘non-local’’ dependence on curvatures making fullfield measurements of curvature a necessity for the correct inference of stress. Finally, it is shown that the interfacial shear tractions between the film and the substrate are proportional see front matter r 2005 Elsevier Ltd. All rights reserved. .jmps.2005.06.003 nding author. Tel.: +1626 395 4523; fax: +1 626 449 6359. dress: [email protected] (A.J. Rosakis).