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Two-dimensional defective crystals with non-constant dislocation density and unimodular solvable group structure

Parry, Gareth P.; Zyskin, Maxim

Authors

Gareth P. Parry gareth.parry@nottingham.ac.uk

Maxim Zyskin maxim.zyskin@nottingham.ac.uk



Abstract

In Parry and Zyskin [1] we outlined mathematical methods which seemed to be necessary in order to discuss crystal structures with non-constant dislocation density tensor (ddt). This was part of a programme to investigate the geometry of continuously defective crystals and the symmetries of associated discrete structures -- one can think of the programme as an attempt to generalize the use of crystallographic groups as material symmetries in non-linear elasticity theory, for perfect crystals, to deal with the case where defects are present.
The methods used rely on the following fact: when the ddt is non-constant, (given technical assumptions), there is a Lie group that acts on the set of material points, and the dimension of the group is strictly greater than that of the ambient space in which the crystal resides. So there is a non-trivial isotropy group associated with the group action.
We develop ideas, and recap the requisite mathematical apparatus, in the context of Davini's model of defective crystals, then focus on a particular case where the ddt is such that a solvable three dimensional Lie group acts on a two dimensional crystal state. We construct the corresponding discrete structures too.
The paper is an extension of [1], where the analogous group was nilpotent.

Citation

Parry, G. P., & Zyskin, M. (2018). Two-dimensional defective crystals with non-constant dislocation density and unimodular solvable group structure. Journal of Elasticity, 1-39. https://doi.org/10.1007/s10659-018-9680-7

Journal Article Type Article
Acceptance Date May 25, 2018
Online Publication Date Jun 28, 2018
Publication Date Jun 28, 2018
Deposit Date Jun 6, 2018
Publicly Available Date Jun 28, 2018
Journal Journal of Elasticity
Print ISSN 0374-3535
Electronic ISSN 1573-2681
Publisher Springer Verlag
Peer Reviewed Peer Reviewed
Pages 1-39
DOI https://doi.org/10.1007/s10659-018-9680-7
Keywords Crystals; Defects; Lie groups
Public URL http://eprints.nottingham.ac.uk/id/eprint/52041
Publisher URL https://link.springer.com/article/10.1007%2Fs10659-018-9680-7
Copyright Statement Copyright information regarding this work can be found at the following address: http://creativecommons.org/licenses/by/4.0

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://creativecommons.org/licenses/by/4.0





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