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

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.

ACOUSTIC SCATTERING FROM AN ELASTIC SPHERICAL SHELL IN AN OCEANIC WAVEGUIDE WITH A PENETRABLE BOTTOM

The paper describes the theory and implementation issues of modeling of the acoustic field scattered by an air-filled spherical elastic shell immersed in a shallow-water waveguide over a homogeneous, fluid half-space. The normal mode evaluation is applied to the source contribution and to the scattering coefficients. The arising branch cut integrals are simplified and expressed via the probability integral. Two cases are analyzed: when a source frequency differs from the critical frequency of a normal mode and when they coincide. The formalism is applied to evaluate the effect of coupling between propagation and scattering on transmission loss.

ACOUSTIC SCATTERING BY AN ELASTIC SPHERICAL SHELL NEAR THE SEABED

The paper describes the theory and implementation issues of modeling of the backscattered field from a thin air-filled spherical elastic shell immersed in water close to the seabed or to the air/water interface. Computational results obtained for the full multiple scattering solution are compared with the model utilizing the single-scatter approximation in a wide-frequency range 0 < k0a ≤ 55. In this frequency range for a thin air-filled spherical shell the main elastic contribution to scattering is due to the lowest-order compressional wave which is the generalization of the Lamb symmetric wave of a flat plate and due to the subsonic mode of the first antisymmetric Lamb wave. Strong resonance peaks produced by these waves in the backscattered form functions have been identified in numerical modeling. It has been shown that when the object is close to the interface in addition to geometrical reflections between the shell and the interface, strong interactions due to these resonances can be observed.

Dynamics of a Hyperelastic Gas-Filled Spherical Shell in a Viscous Fluid

The dynamical response of a gas-filled, spherical elastic shell immersed in a viscous fluid is of interest in a number of different scientific and technological contexts. In this article, this problem is formulated and studied numerically, within a purely mechanical setting. For spherically symmetric motions, a neo-Hookean shell material, and an incompressible surrounding fluid, the equation of motion can be obtained through an integration in the radial coordinate. The resulting nonlinear initial-value problem must be integrated numerically. An interesting feature of the system response is the possibility of a departure from bounded oscillation for large-amplitude far-field forcing. The amplitude at which this departure occurs is found to be highly dependent on the forcing frequency. A stability map in the forcing frequency/amplitude plane is an important result of this study.