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

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 secondary buckling of the shell.

Shallow shell theory of the buckling energy barrier: From the Pogorelov state to softening and imperfection sensitivity close to the buckling pressure

“…(48), we also obtain the relation F 2 = 3πJ 0 /2, which shows that the force-indentation relation at p = 0 from Ref. [19],F = F 2z 1/2 + O(1), is exactly identical to the force-indentation relation that has been obtained before in Refs. [21,58].…”

“…For small indentationsz 1, the linear regime withψ,w ∝z is a good approximation up to the barrier, and the typical radial extent of the indentation is ρ = O(1), resulting inĒ B ∝z 3 B according to (12). For deep indentationsz 1, the characteristic behavior of a mirror-inverted Pogorelov dimple is ∂ ρw ∼z 1/2 andψ ∼ z 1/2 over a width ∆ρ = O(1) at ρ ∼z 1/2 (in the absence of pressure) [19], which results inĒ B ∝z 3/2 B according to (12). This means both scaling limits in (31) can be rationalized by nonlinear shallow shell theory.…”

“…which should beψ ∼ −F /2πρ according to (9) or (10). Both (19) and (20) lead to the same dimensionless linear stiffness where τ ≡ −p/p c and with an arcsin branch −π/2 ≤ arcsin x ≤ π/2.…”

“…Following Ref. [19] (where the case p = 0 and F > 0 was considered), we smooth these discontinuities by an ansatz (x ≡ ρ −z…”

Buckling of a Pressurized Hemispherical Shell Subjected to a Probing Force

“…In the limit of small indentation (with respect to the shell thickness, n < t), the force is described by the linear Reissner's rigidity [18,25,26]. At larger values of the indentation, we observe a nonlinear shell deflection as described by Pogorelov [27,28]. When the level of depressurization of the shell is increased above p o /p c > 0.11, the load-indentation behavior becomes nonmonotonic and the force plateau reaches a maximum and decreases with increasing probe displacement.…”