Elasticity of spheres with buckled surfaces

There has been longstanding interest in the buckling instabilities of core-shell systems, comprising an interior elastic sphere, attached to an exterior shell. Although idealized, such a model has been proposed to underlie myriad biological morphologies. To fully discuss such systems, it is important to properly understand the elasticity of the spherical core. Here, by exploiting well-known properties of the solid harmonics, we present a simple, direct method for solving the elastic problem of spheres and spherical voids with surface deformations, described by a real spherical harmonic. We calculate the corresponding bulk elastic energies, providing closed-form expressions for any values of the spherical harmonic degree (l), Poisson ratio, and shear modulus. We find that the elastic energies are independent of the spherical harmonic index (m). Using these results, we revisit the buckling instability experienced by a core-shell system comprising an elastic sphere, attached within a membrane of fixed area, that occurs when the area of the membrane sufficiently exceeds the area of the unstrained sphere. We determine the phase diagram of the core-shell sphere's shape, specifying what value of l is realized as a function of the area mismatch and the core-shell elasticity. We also determine the shape phase diagram for a spherical void bounded by a fixed-area membrane.