Plane strain curvature change due to film cracks

1 Plane strain curvature change due to film cracks 

Consider a uniform elastic film of thickness hf bonded to the surface of an elastic substrate of thickness hs- The film supports a biaxial tensile mismatch stress cjm, but is constrained to deform in plane strain bending for the time being, so as to render discussion of the general approach as transparent as possible. The film is thin which, from Section 2.2, is understood to mean that Efhf Eshs- If the deflections are small in this case, the uniaxial or cylindrical curvature of the substrate is 

The deformation associated with crack formation can be analyzed by considering a reduced problem which is defined to take advantage of the thinness of the film. It is important to recognize that the configuration of the film-substrate system under consideration is symmetric under reflection with respect to any of the planes at a = 0, p. or any of the planes X =. .. midway between the crack planes, far from the substrate 

The main calculation which must be done to estimate curvature in the presence of cracking is the determination of /, the force which plays the role here that served in the derivation of the Stoney formula. The 

Alternatively, reliable results of broad applicability can often be found by other means. For example, if a solution is known for the configuration of interest but with different boundary conditions, then elastic reciprocity can be used to extract useful results with minimal effort. In any case, once / or A/ is known, the plane strain curvature n, or the change in curvature Ak, is given by 

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