Biaxial curvature due to lines

Because the film material is not deposited with uniform thickness in this case, the parameter Kst is perhaps less suitable as a normalizing factor for curvature than it was in the discussion of a periodically cracked film. The mean film thickness in this case is hfb/p, so a more suitable normalizing factor for curvature might be bKst/p- Nevertheless, the interpretation of the 

If both p and /if are very small compared to substrate thickness hg, then the configuration can be analyzed within the framework established in considering the Stoney formula (2.7). The main task here is to determine the resultant forces per unit length in coordinate directions that act on symmetry planes of the configuration with the substrate in its undeformed state these resultant forces are again denoted by fx and fy. The calculation can be pursued on the basis of elastic reciprocity (3.94), as in the case of a periodic array of cracks, and only a few steps in the procedure will be included. In the present case, the force resultants can be written as 

The second term on the right side of (3.113) accounts for the fact that the side faces of the stripes are separated by the gap of width p — 6, so this contribution represents an additional relative displacement of these faces due to the roughly uniform strain u /p in this gap region. When the elastic properties of the film and substrate are the same, the factor tanh(fe/. ) in (3.113) can be replaced by the alternative form T(h /b) as defined in (3.99). 

Aside from the task of devising an estimate for 8 in the case of film stripes, the analysis is otherwise identical to that followed in deriving (3.108). The results for principal curvatures of the midplane of a substrate of thickness hs and elastic properties Eg and are 

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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