Bifurcation for general curvature variation

The foregoing discussion of bifurcation is based on the assumption of spherical curvature of the substrate midplane prior to bifurcation and on the assumed transverse deflection (2.82) with spatially uniform principal curvatures following bifurcation. It was noted in the discussion of axially symmetric deformation in the previous section that the deformed shape of the substrate midplane can depart significantly from a shape with uniform curvature. Therefore, the bifurcation analysis is repeated in this section, but without a priori assumptions on the deformed shape of the substrate midplane, by means of the numerical finite element method. 

To precipitate stable deformation beyond the point of bifurcation, a slight imperfection in the system was introduced in the form of an anisotropic mismatch strain. Typically, the mismatch strain in the x-direction y-direction) was taken to be 0.01% larger (smaller) than the nominal value Cm- With this level of imperfection, the deformation prior to bifurcation 

For a circular substrate, the elementary model yielded em(bif) = 1-43 as the critical value for bifurcation, while the finite element analysis yielded the estimate em(bif) = 1.54 for a range of values of R/hg. Most calculations were carried out for hg/hf = 100, that is, the dependence of the value of bifurcation strain on variations on hg/hf was not examined systematically. On the basis of a few calculations, it appears that the bifurcation value of Cm is lowered slightly as the value of hg/h is increased. In any case, the magnitude of mismatch strain needed for bifurcation to occur which is implied by ni(bif) = l 4 with R/hg = 100, hg/hi = 100 and Mg/Mf = 1, is roughly 0.01. 

Finally, it is noted that the behavior of thin structures of the kind being discussed here is very sensitive to imperfections in the system. To illustrate the point in the present context, the calculation which led to 

The results obtained for film-substrate systems in Section 2.5 and in the present section provide connections between substrate curvature and film mismatch strain that define boundaries between regimes of behavior. For the case of a very thin film on a relatively thick substrate, it was shown in Section 2.5.1 that the response is linear with spherical curvature for normalized mismatch strain in the range 0 em 0.3. Furthermore, for a circular substrate, the response is geometrically nonlinear but axially symmetric for 

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