Bifurcation analysis with uniform curvature

As described in Section 2.1, the basic idea is to adopt a parametric family of deformed shapes for the substrate midplane, along with a consistent strain distribution which incorporates the mismatch strain, and then to determine values of the parameters which represent stationary points of the total potential energy. Even though the undeformed substrate is a circular 

As before, the mismatch strain is uniform throughout the film and it is identically zero in the substrate. This deformation is identical to that rep- 

For the present case, the equilibrium conditions can be obtained in relatively simple form for the case when the elastic properties of the film and substrate materials are the same (Masters and Sal am on (1993) Freund (2000)). In particular, for a given mismatch strain and geometric parameters, it is possible to express equilibrium values of Kx and Hy in terms of Cm-If Cm is then eliminated, a relationship between Kx and Ky is obtained which represents the locus of equilibrium states for the system in the plane of k,x versus Ky, namely, 

A number of additional observations on film—substrate deformation in the geometrically nonlinear range can be made on the basis of modeling of the kind introduced here. Among these are  

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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. 

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