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On singular behaviour in a plane linear elastostatics problem

Heiko Gimperlein, Michael Grinfeld, Robin J. Knops, Marshall Slemrod

TL;DR

The paper analyzes a planar elastostatic problem with a geometric singularity: a vector field $u(x)$ defined as tangent to a one-parameter family of circles that fill a lens-shaped region $\Omega$ and meet at a double cusp at the origin. Using a semi-inverse method with an Airy stress function, the authors realize $u$ as the displacement field of a nonhomogeneous compressible isotropic plane elastic body whose inner and outer boundaries are rigidly rotated, illustrating how rupture-like behavior can arise from geometric singularities. They identify a family of strongly elliptic Lamé parameters (via a parameter $k>0$) yielding finite yet singular energy near the origin, compute the total force and the total couple on $\Omega$ (notably a zero net force and a finite couple $\Gamma=4\pi(R^{2}-1)$), and show the displacement has non-unique limiting behavior at the origin. The work links geometric singularities to elastostatic boundary-value problems and demonstrates the viability of the semi-inverse Airy approach for constructing explicit nonhomogeneous elastic solutions with controlled singular behavior, while proposing several generalizations and open questions.

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 $Ω$ 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 $Ω$ 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 Lamé parameters strongly-elliptic, but the non-unique limiting behaviour of the displacement is preserved. Other properties of the vector field are also established.

On singular behaviour in a plane linear elastostatics problem

TL;DR

The paper analyzes a planar elastostatic problem with a geometric singularity: a vector field defined as tangent to a one-parameter family of circles that fill a lens-shaped region and meet at a double cusp at the origin. Using a semi-inverse method with an Airy stress function, the authors realize as the displacement field of a nonhomogeneous compressible isotropic plane elastic body whose inner and outer boundaries are rigidly rotated, illustrating how rupture-like behavior can arise from geometric singularities. They identify a family of strongly elliptic Lamé parameters (via a parameter ) yielding finite yet singular energy near the origin, compute the total force and the total couple on (notably a zero net force and a finite couple ), and show the displacement has non-unique limiting behavior at the origin. The work links geometric singularities to elastostatic boundary-value problems and demonstrates the viability of the semi-inverse Airy approach for constructing explicit nonhomogeneous elastic solutions with controlled singular behavior, while proposing several generalizations and open questions.

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 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 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 Lamé parameters strongly-elliptic, but the non-unique limiting behaviour of the displacement is preserved. Other properties of the vector field are also established.
Paper Structure (13 sections, 2 theorems, 119 equations, 2 figures, 1 table)

This paper contains 13 sections, 2 theorems, 119 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation, basic theory and other preliminaries
  3. Unit tangent vector field
  4. Properties of the vector field
  5. Airy stress function
  6. Total force and couple
  7. Geometry
  8. Traction and couple
  9. The strain energy
  10. Further remarks
  11. Acknowledgement.
  12. Airy function. Derivation
  13. Generalised $J(z,\bar{z})$

Key Result

Proposition 4.1

For points on a given circle phidef, the limit of the component $u_{2}(x)$ specified by u222 as the point $(x_{1},\,x_{2})$ tends to the origin depends along which member of the family phidef the origin is approached; that is where the constant $c$ can be arbitrarily chosen in the range $1\le c \le R$. Moreover, the component $u_{2}(x)$ when given by u221 is indeterminate at the origin and is not

Figures (2)

  • Figure 1: The region $\Omega(c_{2},\,c_{1})$.
  • Figure 2: $\partial \Omega(c_2,\,c_1) \cap \partial B(0,a)$.

Theorems & Definitions (4)

  • Proposition 4.1: Limiting behaviour
  • Remark 5.1: Generalisation
  • Remark 5.2
  • Proposition 6.1