Reconciliation of Local and Global Symmetries for a Class of Crystals with Defects

Parry, Gareth; Sigrist, Rachel

doi:10.1007/s10659-011-9342-5

Cited by 8 publications

(20 citation statements)

References 17 publications

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“…(11)) determines and is determined by φ(e a ), a = 1, 2, 3. This fact lies at the heart of the traditional procedure (of using symmetries of lattices as material symmetry groups for 'perfect crystal' continua) -it is a rather obvious fact in the case that S = 0, but the analogous result in the case of defective crystals (where S = 0) is not so obvious (see Parry and Sigrist [3], Nicks and Parry [4] for the case S = constant, S = 0).…”

Section: Perfect Crystals N =mentioning

confidence: 99%

“…(We would not be able to make this assumption were the energy function in (16) to depend on derivatives of S). When (14) does hold, then, one can exploit the properties of the Lie group associated with the given ddt S to determine symmetries of corresponding discrete structures and examine whether or not these extend to symmetries of the continuum [3][4][5] .…”

Section: Defective Crystals N =mentioning

confidence: 99%

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

Parry

Zyskin

2016

J Elast

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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.

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Section: Perfect Crystals N =mentioning

confidence: 99%

Section: Defective Crystals N =mentioning

confidence: 99%

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

Parry

Zyskin

2016

J Elast

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“…Furthermore (since we assume the crystal is uniform) the fields ℓ a (·) also satisfy (1.9) for some group composition function ψ on R 3 . The following survey of facts about the Lie group G = (R 3 , ψ) follows that given in [16], [17], [18], [21], [22] and is given here for completeness. The reader who is familiar with this background material may omit section 2 and focus on the subsequent new material.…”

Section: Elements Of Lie Group Theory and Discrete Defective Crystalsmentioning

confidence: 99%

“…For such groups, Parry and Sigrist [22] construct explicitly all sets of generators of a given discrete subgroup. The formulae that connect different sets of generators generalise the perfect crystal case given by (1.2).…”

Section: Introductionmentioning

confidence: 99%

A Classification of the Symmetries of Uniform Discrete Defective Crystals

Nicks

2014

J Elast

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Crystals which have a uniform distribution of defects are endowed with a Lie group description which allows one to construct an associated discrete structure. These structures are in fact the discrete subgroups of the ambient Lie group. The geometrical symmetries of these structures can be computed in terms of the changes of generators of the discrete subgroup which preserve the discrete set of points. Here a classification of the symmetries for the discrete subgroups of a particular class of three-dimensional solvable Lie group is presented. It is a fact that there are only three mathematically distinct types of Lie groups which model uniform defective crystals, and the calculations given here complete the discussion of the symmetries of the corresponding discrete structures. We show that those symmetries corresponding to automorphisms of the discrete subgroups extend uniquely to symmetries of the ambient Lie group and we regard these symmetries as (restrictions of) elastic deformations of the continuous defective crystal. Other symmetries of the discrete structures are classified as 'inelastic' symmetries.

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“…For the nilpotent case, the structure of the discrete subgroup

scriptD

has been studied in Cermelli and Parry , and the symmetries of that structure have been found in Parry and Sigrist . The solvable case falls naturally into two cases, and Auslander, Green and Hahn denote the corresponding groups S 1 and S 2 .…”

Section: Flow Along Right‐invariant Fields Discrete Subgroups Of Liementioning

confidence: 99%

On symmetries of crystals with defects related to a class of solvable groups (S₂)

Nicks

Parry

2012

Math Methods in App Sciences

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Reconciliation of Local and Global Symmetries for a Class of Crystals with Defects

Cited by 8 publications

References 17 publications

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

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

A Classification of the Symmetries of Uniform Discrete Defective Crystals

On symmetries of crystals with defects related to a class of solvable groups (S₂)

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Reconciliation of Local and Global Symmetries for a Class of Crystals with Defects

Cited by 8 publications

References 17 publications

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

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

A Classification of the Symmetries of Uniform Discrete Defective Crystals

On symmetries of crystals with defects related to a class of solvable groups (S2)

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On symmetries of crystals with defects related to a class of solvable groups (S₂)