Transition from equatorial to whole-shell buckling in embedded spherical shells under axisymmetric far-field loading

Motivated by the need to understand the compression of syntactic foam composites, we solve the canonical problem of buckling of a thin spherical shell embedded in a medium that is much softer than the shell. Syntactic foams comprise shells that usually have diameters in the micron range and are distributed inside a matrix medium that is typically polymeric. Such foams are commonly employed in a range of applications where high stiffness to density ratios are of interest. This can be tailored via choice of shell thickness and type, and filler volume fraction.

Embedded glass microspheres fracture under sufficiently high loading, leading to a permanent softening of the syntactic foam. Embedded polymeric Expancel microspheres however are thought to buckle because the associated softening of the foam is recoverable. We determine critical buckling pressures in the practical scenarios of hydrostatic and uniaxial compressive loading states by solving a more general uniaxial loading problem. Critically, we investigate the thin-stiff shell limit, which yields very different results from a standard thin-shell limit under the assumption that the shell and matrix have stiffnesses of the same order. We employ nonlinear shell theory, linear stability analysis and rigorous asymptotics. We present numerical results for the critical buckling pressure over a wide range of shell thickness and contrasts in shell/matrix stiffnesses. Results for hydrostatic loading are compared against existing analytical and semi-analytical models for embedded shells. Under uniaxial loading we note that there are two distinct regions of parameter space, corresponding to equatorial and non-equatorial buckling regimes. The two non-dimensional parameters of critical importance are the shell thickness to radius ratio h/R and the shell to matrix shear modulus ratio µ_m/µ_s. By fixing one whilst varying the other we observe and describe the transition between these two regimes.