Principle of stationary potential

The strategy adopted is to select plausible parametric forms for u r) and w r), and then to invoke the principle of stationary potential energy to determine optimal values of the parameters involved to minimize potential energy. For small deflections, the radial deformation and the out-of-plane or transverse deformation are uncoupled and a choice for the midplane displacement is... 

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... 

---