The Computation of One-Parameter Families of Bifurcating Elastic Surfaces

Abstract

The problem of constructing the middle surface of a deformed elastic shell from its first and second fundamental forms $\hat{a} _{\alpha \beta } $ and $\hat{b} _{\alpha \beta } $ is considered. The undeformed shell is a spherical cap of radius R and thickness h with an angular width $2\theta _0 $, where$0 < \theta _0 < \pi/2$. The cap is subjected to a constant uniform load $\lambda $ and is simply supported at its edge. The authors seek to compute the one-parameter families of buckled states that 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 (PDEs). A solution of this system is a curvature potential w, a stress potential f, and the load $\lambda $. 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 presented here, these local branches will be numerically continued. Solution branches are parameterized in terms of a pseudo-arc-length parameter $\rho $ (i.e., $( \lambda , f, w) = ( \lambda ( \rho ), f_\rho , w_\rho ) $), enabling them to be tracked around turning points. The second step in the solution process is to solve numerically for the parameterization $\hat{\bf X} _\rho $ corresponding to the middle surface of the buckled shell $\hat{\mathcal{S}} _\rho $. This is done by integrating the PDEs of $\hat{\mathcal{S}} _\rho $. The coefficients in these differential equations involve the first and second fundamental forms of the deformed shell SP, which can be computed from $( \lambda ( \rho ), f_\rho ,w_\rho )$. A number of bifurcation diagrams corresponding to the first three branch points of a spherical cap of size $\theta _0 = 12.85^\circ $ 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 that branch from the unbuckled state of the shell.