Non-uniform, axisymmetric misfit strain: in thin films bonded on plate substrates/substrate systems: the relation between non-uniform film stresses and system curvatures

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 misfit strain distributions in the thin film, we derived relations between the film stresses and the misfit strain, and between the plate system’s curvatures and the misfit strain. These relations feature a “local” part which involves a direct dependence of the stress or curvature components on the misfit strain at the same point, and a “non-local” part which reflects the effect of misfit strain of other points on the location of scrutiny. Most notably, we also derived 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 a full-field measurement a necessity. 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.