Substrate curvature due to anisotropic films

The strain component normal to the film-snbstrate interface plane is 

For all of the cases of substrate curvature induced by film stress that were considered in Section 2.1, it was assumed that both the film and substrate materials were isotropic. This provided a basis for a relatively transparent discussion of curvature phenomena and it led to results which have proven to be broadly useful. However, there are situations for which some understanding of the influence of material anisotropy of the film material or the substrate material, or perhaps of both materials, is important to know. Therefore, in this section, representative results on the influence of material anisotropy on substrate curvature are included. Results are established for two particular curvature formulas. In the first case, the film is presumed from the outset to be very thin compared to the substrate. Furthermore, the substrate is assumed to be isotropic and the film is considered to be generally anisotropic. In the second case discussed, no restriction is placed on the thickness of the film relative to the substrate, but both materials are assumed to be anisotropic. However, to obtain expressions for curvature which are not too complex to be interpretable, attention is limited to cases for which both the film and substrate materials are orthotropic, that their axes of orthotropy aligned with each other and that one axis of or-thotropy of each material is normal to the film-substrate interface. There is no connection between the values of orthotropic elastic constants of the two materials. Consideration of these two cases illustrates the most useful approaches for anisotropic materials generalizations for cases of greater complexity are evident. 

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