The shallow shell approach to Pogorelov's problem and the breakdown of `mirror buckling'

TL;DR

This paper provides a detailed asymptotic analysis of spherical shell indentation, revealing that the classical mirror buckling assumption breaks down near the indenter due to singular stresses, leading to a more complex deformation structure.

Contribution

It introduces a new asymptotic framework that accounts for bending stiffness effects near the indenter, challenging the traditional three-region model of mirror buckling.

Findings

Stress profile becomes singular near the indenter.

Mirror buckling must be modified to include bending stiffness.

Shell exhibits seven distinct spatial regions during indentation.

Abstract

We present a detailed asymptotic analysis of the point indentation of an unpressurized, spherical elastic shell. Previous analyses of this classic problem have assumed that for sufficiently large indentation depths, such a shell deforms by `mirror buckling' --- a portion of the shell inverts to become a spherical cap with equal but opposite curvature to the undeformed shell. The energy of deformation is then localized in a ridge in which the deformed and undeformed portions of the shell join together, commonly referred to as Pogorelov's ridge. Rather than using an energy formulation, we revisit this problem from the point of view of the shallow shell equations and perform an asymptotic analysis that exploits the largeness of the indentation depth. This reveals first that the stress profile associated with mirror buckling is singular as the indenter is approached. This consequence of point indentation means that mirror buckling must be modified to incorporate the shell's bending stiffness close to the indenter and gives rise to an intricate asymptotic structure with seven different spatial regions. This is in contrast with the three regions (mirror-buckled, ridge and undeformed) that are usually assumed and yields new insight into the large compressive hoop stress that ultimately causes the shell to buckle azimuthally.