Some asymptotic results concerning the buckling of a spherical shell of arbitrary thickness

For a spherical shell of arbitrary thickness which is subjected to an external hydrostatic pressure, symmetrical buckling takes place at a value of μ₁ which depends on and the mode number, where A₁ and A₂ are the undeformed inner and outer radii, and μ₁ is the ratio of the deformed inner radius to the undeformed inner radius. In the large mode number limit, we find that the dependence of μ₁ on has a boundary layer structure: it is a constant over almost the entire region of and decreases sharply from this constant value to unity as tends to unity (the thin-shell limit). Simple asymptotic expressions for the bifurcation condition are obtained. The classical result for thin shells is recovered directly from the equations of finite elasticity, and an asymptotic critical neutral curve (which envelops the neutral curves corresponding to different mode numbers) is obtained.