Geometrical Structure of Two-Dimensional Crystals with Non-Constant Dislocation Density

Abstract

Abstract:We outline mathematical methods which seem to be necessary in order to discuss crystal structures with non-constant dislocation density tensor(ddt) in some generality. It is known that, if the ddt is constant (in space), then material points can be identified with elements of a certain Lie group, with group operation determined in terms of the ddt -the dimension of the Lie group equals that of the ambient space in which the body resides, in that case. When the ddt is non-constant, there is also a relevant Lie group (given technical assumptions), but the dimension of the group is strictly greater than that of the ambient space. The group acts on the set of material points, and there is a non-trivial isotropy group associated with the group action. We introduce and discuss the requisite mathematical apparatus in the context of Davini's model of defective crystals, and focus on a particular case where the ddt is such that a three dimensional Lie group acts on a two dimensional crystal state -this allows us to construct corresponding discrete structures too.Response to Reviewers: I have amended the manuscript to deal with the suggestions of Reviewer #1. Powered by Editorial Manager® and ProduXion Manager® from Aries Systems CorporationNoname manuscript No.(will be inserted by the editor)Geometrical structure of two-dimensional crystals with non-constant dislocation densityGareth ParryReceived: date / Accepted: date Abstract We outline mathematical methods which seem to be necessary in order to discuss crystal structures with non-constant dislocation density tensor(ddt) in some generality. It is known that, if the ddt is constant (in space), then material points can be identified with elements of a certain Lie group, with group operation determined in terms of the ddt -the dimension of the Lie group equals that of the ambient space in which the body resides, in that case. When the ddt is non-constant, there is also a relevant Lie group (given technical assumptions), but the dimension of the group is strictly greater than that of the ambient space. The group acts on the set of material points, and there is a non-trivial isotropy group associated with the group action. We introduce and discuss the requisite mathematical apparatus in the context of Davini's model of defective crystals, and focus on a particular case where the ddt is such that a three dimensional Lie group acts on a two dimensional crystal state -this allows us to construct corresponding discrete structures too.

Two-Dimensional Defective Crystals with Non-constant Dislocation Density and Unimodular Solvable Group Structure

“…According to [12] there are four different classes: the abelian, nilpotent, solvable and simple classes of Lie algebras. The abelian case leads to traditional crystallography, and we considered the nilpotent case in [1]. The solvable case divides into unimodular and non-unimodular classes-we consider the unimodular case here, and hope to present the non-unimodular solvable and (so-called) simple case in forthcoming works.…”

“…This defines the three-dimensional nilpotent Lie algebra considered in [1], so we do not consider this case any further here. So assume that not both of α and γ are zero in (3.1).…”

“…In the simplest case where the strain energy density depends only on point values of 'lattice vector fields' it B G. Parry gareth.parry@nottingham.ac.uk 1 School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK is commonly taken for granted that there is an associated discrete lattice (with the point values of the vector fields providing a basis of the lattice)-we argue that, if assumptions are formulated appropriately, then it is because the energy is presumed to depend only on the lattice vectors that the corresponding discrete structure is a lattice. In fact we give this argument in generality, in the next subsection, and comment on the simplest version of the argument subsequently.…”

“…2 we recap the apparatus employed in [1], for convenience, then we move on to the extension of [1] to the case of solvable groups of a certain type.…”