Influence of substrate stiffness on surface stability

In the foregoing discussion of stability of the flat surface of a stressed solid, it was assumed that the elastic material is homogeneous. It was also assumed that, prior to formation of fluctuations in surface shape, the material was homogeneously stressed, that is, the equi-biaxial stress acted throughout the material. This assumption on stress is not essential. It will be shown in Section 8.9 that the results are unaffected if the initial mismatch stress acts only to some finite depth, provided only that this depth extends beyond the roots of the valleys in surface fluctuation. Otherwise, a discontinuity in the initial stress across some plane y = constant is immaterial. Discontinuities in elastic properties, however, do have an influence on stability. 

Suppose that a film of thickness h is epitaxially bonded to a substrate 

Qualitatively, the role of substrate stiffness is evident. The film material at the initial stress level is characterized by the natural length parameter C defined in (8.60). If h is large compared to this parameter then the surface material is unaware of the substrate and the substrate stiffness is expected to have only a minor influence on the critical wavelength Act in (8.61). On the other hand, if h is very small compared to C, then the film material is inconsequential and the critical wavelength should be given by (8.61) but with Eg replacing E, where the latter modulus is represented by E in that equation. It follows that a relatively stiff substrate enhances stability of the surface if the film is thin compared to the natural length parameter, whereas a relatively compliant substrate tends to destabilize the surface. The point of the analysis to follow is to provide the transition between these extremes in behavior. 

The result of solving this boundary value problem is that the perturbation stress along the wavy surface is again given by (8.52), except that the expression for the component includes an additional factor in the form 

Following the reasoning that led to (8.61), it is found that the critical 

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