Periodic array of parallel film cracks

The discussion up to this point has focused on the relationship between the curvature of an elastic substrate and the stress in a single layer or multilayer film in which the mismatch is invariant under any translation parallel to the interface. The films considered have also been continuous and of uniform thickness over the entire film-substrate interface. Within the range of small deflections, such an equi-biaxial film stress induces a spherical curvature in the substrate midplane, except very near the edge of the substrate. What is the deformation induced in the substrate if such a film does not have uniform thickness or if the mismatch stress varies with position along the interface This question is addressed in this section for the cases when the nonuniformity in mismatch stress or thickness varies periodically along the 

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This estimate is remarkably simple, given the complexity of the underlying boundary value problem. The left sides of the expressions in (3.114) are the curvatures normalized by the effective Stoney curvature based on the average film thickness hfb/p. Furthermore, the right sides of (3.114) do not involve the parameter p. In other words, p enters the estimate for curvature only through the effective Stoney curvature based on the amount of film material involved. When b = p, the response is identical to that for a periodic array of parallel cracks. 

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