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

Y. Huang, A. J. Rosakis*

*Corresponding author for this work

Research output: Contribution to journalArticle

55 Citations (Scopus)

Abstract

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 non-uniformities. These relations also feature a "non-local" 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 proportional to the radial gradients of the first curvature invariant and can also be inferred experimentally.

Original languageEnglish (US)
Pages (from-to)2483-2500
Number of pages18
JournalJournal of the Mechanics and Physics of Solids
Volume53
Issue number11
DOIs
StatePublished - Nov 1 2005

Fingerprint

Temperature distribution
temperature distribution
curvature
Thin films
symmetry
Substrates
thin films
inference
Temperature
temperature
traction
nonuniformity
methodology
shear
gradients

Keywords

  • Interfacial shears
  • Non-local effects
  • Non-uniform film temperatures and stresses
  • Non-uniform substrate curvatures
  • Stress-curvature relations

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering

Cite this

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abstract = "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 non-uniformities. These relations also feature a {"}non-local{"} 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 proportional to the radial gradients of the first curvature invariant and can also be inferred experimentally.",
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AB - 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 non-uniformities. These relations also feature a "non-local" 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 proportional to the radial gradients of the first curvature invariant and can also be inferred experimentally.

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