Some Consequences of the Distributional Stress Equilibrium Condition

TL;DR

This paper explores the implications of the distributional stress equilibrium condition for piecewise smooth stress fields with singularities, deriving local and global equilibrium conditions using distribution theory.

Contribution

It introduces new local and global equilibrium conditions for singular stress fields, extending the theory to non-contractible domains with disjoint boundaries.

Findings

Derived local equilibrium conditions at bulk and interface

Established necessary and sufficient conditions for stress functions

Demonstrated the use of distribution theory for singular stress analysis

Abstract

We derive two consequences of the distributional form of the stress equilibrium condition while incorporating piecewise smooth stress and body force fields with singular concentrations on an interface. First we obtain the local equilibrium conditions in the bulk and at the interface, the latter including conditions on the interfacial stress and stress dipole. Second we obtain the necessary and the sufficient conditions on the divergence-free non-smooth stress field for there to exist a stress function field such that the equilibrium is trivially satisfied. In doing so we allow the domain to be non-contractible with mutually disjoint connected boundary components. Both derivations illustrate the utility of the theory of distributions in dealing with singular stress fields.