Extension of Stoney’s Formula to Arbitrary Temperature Distributions in Thin Film/Substrate Systems

[+] Author and Article Information
Y. Huang

Department of Mechanical and Industrial Engineering, University of Illinois, Urbana, IL 61801

A. J. Rosakis

Graduate Aeronautical Laboratory, California Institute of Technology, Pasadena, CA 91125

J. Appl. Mech 74(6), 1225-1233 (Feb 09, 2006) (9 pages) doi:10.1115/1.2744035 History: Received June 23, 2005; Revised February 09, 2006

Current methodologies used for the inference of thin film stress through curvature measurements are strictly restricted to stress and curvature states that are assumed to remain uniform over the entire film/substrate system. By considering a circular thin film/substrate system subject to nonuniform and nonaxisymmetric 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 that involves a direct dependence of the stress or curvature components on the temperature at the same point, and a “nonlocal” part that 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 nonuniformities. These relations also feature a “nonlocal” dependence on curvatures making full-field 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 related to the gradients of the first curvature invariant and can also be inferred experimentally.

Copyright © 2007 by American Society of Mechanical Engineers
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Grahic Jump Location
Figure 1

A schematic diagram of the thin film/substrate system, showing the cylindrical coordinates (r,θ,z)



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