Equilibrium of an elastic spherical cap pulled at the rim. (English) Zbl 0724.73101

Summary: 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 minimizer exists that is unique for sufficiently large outward tractions; moreover, no matter what the cap’s thickness, such a global minimizer 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 \(\kappa_ 0\to +\infty\), tends to a pseudoconfiguration that is the reflection with respect to the horizontal plane of the middle surface of the cap in its reference configuration.


74G60 Bifurcation and buckling
34B15 Nonlinear boundary value problems for ordinary differential equations
74K15 Membranes
74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
49J15 Existence theories for optimal control problems involving ordinary differential equations
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