Bifurcation in equilibrium shape

Consider a thin film in which a uniform mismatch strain is imposed, with the magnitude of the strain being increased from an initial value of zero. For relatively small values of the mismatch strain, the deformed shape of the substrate is essentially spherical as long as the response remains within the 

The issue of bifurcation in equilibrium shape is pursued in two steps. First, a simple energy approach is taken. The deformed shape of the substrate midplane is assumed to be ellipsoidal so that the shape is characterized completely by two principal curvatures which need not be the same. Consistent in-plane displacements, which account for midplane stretching, are also assumed. The principle of stationary potential energy is then invoked to determine the relationship of the principal curvatures to system parameters to ensure that the system is in equilibrium. A more detailed examination of bifurcation on the basis of a finite element simulation, without a priori restrictions on deformation beyond the Kirchhoff hypothesis, is described 

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Figure 2.25 shows the results of observations of pre-bifurcation and post-bifurcation curvatures in the central portion of Si wafers with different in-plane dimensions and thicknesses and with different thicknesses of W thin film deposits. The critical values of normalized curvature and normalized mismatch strain at which bifurcation in equilibrium shape is triggered in the experiments are close to those predicted by (2.87) and (2.88). The... 

A sharp bifurcation in equilibrium shape, such as that shown in Figure 2.24, is possible only for symmetric substrate configurations, such as a circle, a square and an equilateral triangle. 

Figure 4.6 illustrates the PSPS and the chemical equilibrium surface. The PSPS has a hyperbola-type shape and passes through all pure component vertices and the stoichiometric pole n. It intersects the isobutene-MeOH edge and the MeOH-MTBE edge at two points, which are nonreactive binary azeotropes. From Fig. 4.6 one can also see that there exists no reactive azeotrope in this system. All the bifurcation branches and the pure component vertices, as discussed by Venimadhavan et al. [7], are located on the PSPS. 

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