The Computation of One-Parameter Families of Bifurcating Elastic Surfaces

We consider the problem of constructing the middle surface of a deformed elastic shell from its first and second fundamental forms, âαβ and b̂αβ . The undeformed shell is a spherical cap of radius R and thickness h with an angular width 2θ0 where 0 < θ0 < π/2. The cap is subjected to a constant uniform load λ and is simply supported at its edge. We seek to compute the one–parameter families of buckled states which branch from the unbuckled state of the shell. This is accomplished in two steps. First, a finite element method is used to solve the governing shell equations, a pair of fourth–order nonlinear partial differential equations. A solution of this system is a curvature potential w, a stress potential f , and the load λ. Using Liapunov-Schmidt reduction, it can be shown that solutions possessing a variety of symmetries bifurcate from the unbuckled state of the shell. In the work that is presented here, we will numerically continue these local branches. We parametrize solution branches in terms of a pseudo-arc-length parameter ρ (i.e., (λ, f, w) = (λ(ρ), fρ, wρ)), enabling us to track them around turning points. The second step in our solution process is to solve numerically for the parametrization X̂ρ corresponding to the middle surface of the buckled shell Ŝρ. We do so by integrating the partial differential equations of Ŝρ. The coefficients in these differential equations involve the first and second fundamental forms of the deformed shell Ŝρ which can be computed from (λ(ρ), fρ, wρ). A number of bifurcation diagrams corresponding to the first three branch points of a spherical cap of size θ0 = 12.85◦ are presented. For this example, a secondary bifurcation point was found connecting two distinct nonaxisymmetric solution branches. Computer graphics are used to display images of various buckled surfaces which branch from the unbuckled state of the shell.