Semi-discrete modeling of systems of wedge disclinations and edge dislocations via the Airy stress function method

TL;DR

This paper develops a variational and analytical framework for modeling lattice defects like disclinations and dislocations in elastic materials, showing their energetic equivalence in the limit of vanishing core radius.

Contribution

It introduces a rigorous Airy stress function approach for incompatible elasticity and proves the energetic equivalence of disclination dipoles and edge dislocations asymptotically.

Findings

Disclination dipoles and edge dislocations have asymptotically equivalent energies.

The Airy stress function method effectively models incompatible elastic systems.

Kinematic characterization of edge dislocations by disclination dipoles is energetically exact.

Abstract

We present a variational theory for lattice defects of rotational and translational type. We focus on finite systems of planar wedge disclinations, disclination dipoles, and edge dislocations, which we model as the solutions to minimum problems for isotropic elastic energies under the constraint of kinematic incompatibility. Operating under the assumption of planar linearized kinematics, we formulate the mechanical equilibrium problem in terms of the Airy stress function, for which we introduce a rigorous analytical formulation in the context of incompatible elasticity. Our main result entails the analysis of the energetic equivalence of systems of disclination dipoles and edge dislocations in the asymptotics of their singular limit regimes. By adopting the regularization approach via core radius, we show that, as the core radius vanishes, the asymptotic energy expansion for disclination dipoles coincides with the energy of finite systems of edge dislocations. This proves that Eshelby's kinematic characterization of an edge dislocation in terms of a disclination dipole is exact also from the energetic standpoint.