# zbMATH — the first resource for mathematics

Some asymptotic results concerning the buckling of a spherical shell of arbitrary thickness. (English) Zbl 1342.74063
Summary: For a spherical shell of arbitrary thickness which is subjected to an external hydrostatic pressure, symmetrical buckling takes place at a value of $$\mu _{1}$$ which depends on $$A_1/A_2$$ and the mode number, where $$A_{1}$$ and $$A_{2}$$ are the undeformed inner and outer radii, and $$\mu _{1}$$ 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 $$\mu _{1}$$ on $$A_1/A_2$$ has a boundary layer structure: it is a constant over almost the entire region of $$0 < A_1/A_2 < 1$$ and decreases sharply from this constant value to unity as $$A_1 /A_2$$ 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.

##### MSC:
 74G60 Bifurcation and buckling 74K25 Shells 74B20 Nonlinear elasticity
##### Keywords:
buckling; finite strain; shell; elastic material
Full Text: