Buckling of an elastic hemispherical shell with an obstacle

We study the buckling of an axially symmetric elastic hemispherical shell, uniformly compressed, subject to a constraint to the radial shifting of the equatorial circumference. The static equilibrium equations, using tensorial notations, are obtained applying the virtual displacements principle to the energy functional. The presence of a constraint does not modify the field equations with respect to the case of a constraint-free buckling, but only influences the boundary conditions, so that, instead of a boundary value problem, we deal with a problem with complementarity conditions on the boundary. We revisit and improve some previously obtained mathematical results, adapting them for the subsequent numerical treatment. Finally, by suitably using a delicate quasi-static shooting technique, numerical results are obtained, which complete the theoretical analysis and give an interesting insight into the behavior of the bifurcation branches.