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

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.

Eshelbian mechanics of novel materials: Quasicrystals

“…(2)-(8) are valid in the framework of generalized incompatible and compatible elasticity theories of quasicrystals. If dislocations are present inside a quasicrystalline material, then the appropriate framework for dislocations is the theory of generalized incompatible elasticity of quasicrystals [Agiasofitou et al, 2010;Lazar and Agiasofitou, 2014;Ding et al, 1995]. The gradient of the phonon displacement vector u i and of the phason displacement vector u ⊥ A can be decomposed into their elastic and plastic parts according to (e.g., Agiasofitou et al [2010]; Lazar and Agiasofitou [2014])…”

“…(16) gives an example of a phason force density caused by the gradient of the plastic fields of dislocations. A dislocation in a quasicrystal can be considered as a hyperdislocation in the hyperlattice by means of a generalized Volterra process, since the hyperlattice is periodic (see, e.g., Wang and Hu [2002]; Feuerbacher [2012]; Lazar and Agiasofitou [2014]). The Burgers vector b = (b i , b ⊥ A ) of the hyperdislocation consists of phonon and phason components which are given by the following surface and line integrals…”

Fundamental elastic field in an infinite medium of two-dimensional hexagonal quasicrystal with a planar crack: 3D exact analysis

“…Owing to his great discovery, Shechtman was awarded the 2011 Nobel Prize in Chemistry and the International Union of Crystallography (1992) changed the official definition of a crystal. In the past three decades, the mechanical behaviors of QCs have stimulated much scientific interest and a variety of topics have been discussed, such as, dislocation (Lazar and Agiasofitou, 2014;Li and Liu, 2012), indentation (Wu et al, 2013;Li et al, 2014b), bending of layered plate , self-action in phason field (Colli and Mariano, 2011;Mariano and Planas, 2013), fundamental elastic fields in infinite and/or half infinite spaces Li et al, 2013;Li et al, 2014a), and so forth. As a class of novel materials, which have a lower friction coefficient, lower adhesion, higher wear resistance and lower porosity in comparison with traditional media, QCs have been found a plenty of potential industrial applications (Kenzari et al, 2012).…”

Renormalized energy for dislocations in quasi-crystals

“…Dislocations are one-dimensional defects in a crystalline-type material, whose presence may greatly affect the elastic and other properties (see [11] and [15]). Dislocation lines of quasi-crystals were observed in experiments soon after Shechtman's discover (see [1], [12], [13], [14]). In a quasi-crystal undergoing a shear deformation, a screw dislocation may be described by a position (x, y) ∈ Ω and a Burger's vector b = b e z .…”