Thermal deformation of initially curved substrates coated by thin inhomogeneous layers

Thermal curvature changes and membrane strains are analyzed for elastic sha llow shell substrates which are coated by thin, generally inelastic, inhomo geneous and anisotropic layers. The analysis is restricted to linear kinema tics. It is shown that the deformation is governed by the corresponding sol ution for a flat substrate and a correction due to the initial curvature. T he correction is determined from a shallow shell problem for the bare subst rate with a loading expressed by the coefficients of thermal curvature for the substrate/layer system. For constant initial curvature, certain analyti c solutions are presented. For situations when the initial deflection of th e substrate is much larger than the substrate thickness, a boundary layer s olution is derived. In the particular case of a circular isotropic substrat e with a spherical initial curvature and a coating of arbitrary anisotropy, the solution is presented in closed form. For nonflat substrates, measured curvatures can generally not be used to extract layer stresses without a p roper compensation for the initial curvature. In this paper, it is explicit ly presented how to accurately compensate for a spherical initial curvature . The results are particularly discussed in relation to curvature measureme nts on Silicon substrates.