Thermal Deformation of Initially Curved Substrates Coated by Thin Inhomogeneous Layers

Thermal curvature changes and membrane strains are analyzed for elastic shallow shell substrates which are coated by thin, generally inelastic, inhomogeneous and anisotropic layers. The analysis is restricted to linear kinematics. It is shown that the deformation is governed by the corresponding solution for a flat substrate and a correction due to the initial curvature. The correction is determined from a shallow shell problem for the bare substrate with a loading expressed by the coefficients of thermal curvature for the substrate/layer system. For constant initial curvatures, certain analytic solutions are presented. For situations when the initial deflection of the substrate is much larger than the substrate thickness, a boundary layer solution is derived. In the particular case of a circular isotropic substrate 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 proper compensation for the initial curvature. In the paper, it is explicitly presented how to accurately compensate for a spherical initial curvature. The results are particularly discussed in relation to curvature measurements on Silicon substrates.