On singular behaviour in a plane linear elastostatics problem

TL;DR

This paper investigates a geometric singularity in a plane elastostatic problem, revealing non-unique displacement limits at a cusp point in a lens-shaped domain with rotating boundaries.

Contribution

It constructs a vector field with singular behavior in a specific elastostatic setting, demonstrating non-uniqueness and rupture potential at a geometric cusp.

Findings

Vector field exhibits non-unique limits at the cusp.

Solutions exist with strongly-elliptic Lamé parameters.

Displacement behavior indicates possible rupture at the origin.

Abstract

A vector field similar to those separately introduced by Artstein and Dafermos is constructed from the tangent to a monotone increasing one-parameter family of non-concentric circles that touch at the common point of intersection taken as the origin. The circles define and space-fill a lens shaped region $\Omega$ whose outer and inner boundaries are the greatest and least circles. The double cusp at the origin creates a geometric singularity at which the vector field is indeterminate and has non-unique limiting behaviour. A semi-inverse method that involves the Airy stress function then shows that the vector field corresponds to the displacement vector field for a linear plane compressible non-homogeneous isotropic elastostatic equilibrium problem in $\Omega$ whose boundaries are rigidly rotated relative to each other, possibly causing rupture or tearing at the origin. A sequence of solutions is found for which not only are the Lam\'{e} parameters strongly-elliptic, but the non-unique limiting behaviour of the displacement is preserved. Other properties of the vector field are also established.