Equilibrium of an Elastic Spherical Cap Pulled at the Rim

Abstract

For thin and shallow caps the title problem is carefully formulated. The outcome is a nonlinear system of two ordinary differential equations of second order; this system is amenable to a variational format through reduction to a single functional equation, which turns out to be the Euler–Lagrange equation of a suitable energy integral depending on a load parameter $\pi _0 $ and a thickness parameter $\kappa _0 $.It is shown that, for all admissible values of the parameters, a global minimizes exists that is unique for sufficiently large outward tractions; moreover, no matter what the cap’s thickness, such a global minimizes tends to a flat pseudoconfiguration when $\pi _0 \to + \infty $. It is also shown that, for $\pi _0 = 0$, in addition to the unstressed reference configuration, a $\kappa _0 $-sequence of local minimizers exists, interpretable as everted stressed configurations of the cap; this sequence, for $\pi _0 \to + \infty $, tends to a pseudoconfiguration that is the reflection with respect to ...