Nonlinear spherical caps and associated plate and membrane problems

Abstract

This paper deals with the existence and multiplicity problem of the equilibrium solutions of an elastic spherical cap within nonlinear strain theory. We pose the problem in the form of a three parameter bifurcation problem, one parameter being related to the load, the others to the geometry. When the geometrical parameters are different from zero, the so-called generic case, we revisit the nonuniqueness results, and explore the solutions in the parameter space. Then we analyze the formal limits as the geometrical parameters tend to zero. When the curvature tends to zero, we obtain from the nonlinear shell a von Karman plate, a new, although natural, result. When the thickness parameter tends to zero, we get a nonlinear membrane problem. A study of the latter gives infinitely many solutions, and we discuss the construction, shapes, and stability in detail.