Volume averaged stress in terms of curvature

As was noted in Section 3.7, the value of any stress component field Oij averaged over the region contained within a material volume V is denoted by (Tij) and is defined as 

Once the identification (3.119) is made, the means of deducing values of (Txx) and (Tyy) from curvature measurements is straightforward. Consider the relationships (3.73), and associate the coordinate direction x transverse to the line direction with x and associate the coordinate direction y parallel to the lines with X2- Then, solving (3.73) for fx and fy, and invoking (3.119), it follows that the volume averaged stress components in the lines transverse to and along the line directions are given in terms of substrate curvature by 

These expressions represent a reinterpretation of (3.73) based on the identification of the stress resultants fx, fy with average stress values for a thin 

The effect of patterning the continuous film into periodic arrays of parallel stripes on stress evolution can be determined by combining (3.114) and (3.120) to obtain the volume averaged stress components in the lines in terms of the mismatch stress in the original continuous film from which the 

Here am is the mismatch stress that would be present in a continuous film of uniform thickness h under the same conditions. As noted before, when the elastic properties of the patterned film and the substrate are comparable, the factor tanh (b/ ) in (3.121) can be replaced by the alternative form T b/hf) defined in (3.99). The resulting values of average stress components in the latter case are 

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