Point Singularities in Incompatible Elasticity

TL;DR

This paper investigates the mathematical behavior of stress and strain fields with point singularities in elastic materials, establishing conditions for solution uniqueness and deriving general solutions for defect-related problems.

Contribution

It provides a rigorous analysis of stress and strain fields with point singularities, proving solution uniqueness and developing a generalized force concept in incompatible elasticity.

Findings

Uniqueness of stress solutions with singularities is established.

A general stress solution for point-supported forces and defects is derived.

A generalized notion of force acting on defects is introduced.

Abstract

The equations of stress equilibrium and strain compatibility/incompatibility are discussed for fields with point singularities in a planar domain. The sufficiency (or insufficiency) of the smooth maps, obtained by restricting the singular fields to the domain away from the singularities, in completely characterizing the equations of equilibrium and compatibility/incompatibility over the entire domain, is established and illustrated with examples. The uniqueness of the solution to the stress problem of incompatible linear elasticity, allowing for singular fields, is proved. The uniqueness fails when the problem is considered solely in terms of the restricted maps. As applications of our framework, a general stress solution, in response to point supported body force and defect fields, is derived and a generalized notion of the force acting on a defect is developed.