Fundamentals in generalized elasticity and dislocation theory of quasicrystals: Green tensor, dislocation key-formulas and dislocation loops

TL;DR

This paper develops fundamental solutions and key formulas in the generalized elasticity and dislocation theory of quasicrystals, including Green tensors, dislocation equations, and interaction energies, applicable to dislocation loops.

Contribution

It provides the first comprehensive derivation of Green tensors and dislocation key-formulas specifically for quasicrystals within generalized elasticity theory.

Findings

Derived the 3D Green tensor for quasicrystals.

Formulated generalized Burgers and Mura-Willis equations for dislocations.

Extended classical dislocation formulas to quasicrystalline materials.

Abstract

The present work provides fundamental quantities in generalized elasticity and dislocation theory of quasicrystals. In a clear and straightforward manner, the three-dimensional Green tensor of generalized elasticity theory and the extended displacement vector for an arbitrary extended force are derived. Next, in the framework of dislocation theory of quasicrystals, the solutions of the field equations for the extended displacement vector and the extended elastic distortion tensor are given; that is the generalized Burgers equation for arbitrary sources and the generalized Mura-Willis formula, respectively. Moreover, important quantities of the theory of dislocations as the Eshelby stress tensor, Peach-Koehler force, stress function tensor and the interaction energy are derived for general dislocations. The application to dislocation loops gives rise to the generalized Burgers equation, where the displacement vector can be written as a sum of a line integral plus a purely geometric part. Finally, using the Green tensor, all other dislocation key-formulas for loops, known from the theory of anisotropic elasticity, like the Peach-Koehler stress formula, Mura-Willis equation, Volterra equation, stress function tensor and the interaction energy are derived for quasicrystals.