A general solution for two-dimensional stress distributions in thin films

We present closed-form solutions for stresses in a thin film resulting from a purely dilatational stress-free strain that can vary arbitrarily within the film. The solutions are specific to a two-dimensional thin film on a thick substrate geometry and are presented for both a welded and a perfectly slipping film/substrate interface. Variation of the stress-free strain through the thickness of the film is considered to be either arbitrary or represented by a Fourier integral, and solutions are presented in the form of a Fourier series in the lateral dimension x. The Fourier coefficients can be calculated rapidly using Fast Fourier Transforms. The method is applied to determine the stresses in the film and substrate for three cases: (a) where the stress-free strain is a sinusoidal modulation in x, (b) where the stress-free strain varies only through the thickness, and (c) where a rectangular inclusion is embedded within the film, and the calculated stresses match accurately with the exact solutions for these cases.