Extension of Stoney's formula to arbitrary temperature distributions in thin film/substrate systems

Y. Huang*, A. J. Rosakis

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

20 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)1225-1233
Number of pages9
JournalJournal of Applied Mechanics, Transactions ASME
Issue number6
StatePublished - Nov 2007


  • Interfacial shears
  • Nonlocal effects
  • Nonuniform film temperatures and stresses
  • Nonuniform substrate curvatures
  • Stress-curvature relations

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering


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