Group elastic symmetries common to continuum and discrete defective crystals
The Lie group structure of crystals which have uniform continuous distributions of dislocations allows one to construct associated discrete structures—these are discrete subgroups of the corresponding Lie group, just as the perfect lattices of crystallography are discrete subgroups of R 3 , with a...
| Main Authors: | , |
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| Format: | Article |
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Springer
2014
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| Online Access: | https://eprints.nottingham.ac.uk/2054/ |
| _version_ | 1848790712784519168 |
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| author | Nicks, Rachel Parry, Gareth P. |
| author_facet | Nicks, Rachel Parry, Gareth P. |
| author_sort | Nicks, Rachel |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | The Lie group structure of crystals which have uniform continuous distributions of dislocations allows one to construct associated discrete structures—these are discrete subgroups of the corresponding Lie group, just as the perfect lattices of crystallography are discrete subgroups of R 3 , with addition as group operation. We consider whether or not the symmetries of these discrete subgroups extend to symmetries of (particular) ambient Lie groups. It turns out that those symmetries which correspond to automorphisms of the discrete structures do extend to (continuous) symmetries of the ambient Lie group (just as the symmetries of a perfect lattice may be embedded in ‘homogeneous elastic’ deformations). Other types of symmetry must be regarded as ‘inelastic’. We show, following Kamber and Tondeur, that the corresponding continuous automorphisms preserve the Cartan torsion, and we characterize the discrete automorphisms by a commutativity condition, (6.14), that relates (via the matrix exponential) to the dislocation density tensor. This shows that periodicity properties of corresponding energy densities are determined by the dislocation density. |
| first_indexed | 2025-11-14T18:16:59Z |
| format | Article |
| id | nottingham-2054 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T18:16:59Z |
| publishDate | 2014 |
| publisher | Springer |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-20542020-05-04T20:17:24Z https://eprints.nottingham.ac.uk/2054/ Group elastic symmetries common to continuum and discrete defective crystals Nicks, Rachel Parry, Gareth P. The Lie group structure of crystals which have uniform continuous distributions of dislocations allows one to construct associated discrete structures—these are discrete subgroups of the corresponding Lie group, just as the perfect lattices of crystallography are discrete subgroups of R 3 , with addition as group operation. We consider whether or not the symmetries of these discrete subgroups extend to symmetries of (particular) ambient Lie groups. It turns out that those symmetries which correspond to automorphisms of the discrete structures do extend to (continuous) symmetries of the ambient Lie group (just as the symmetries of a perfect lattice may be embedded in ‘homogeneous elastic’ deformations). Other types of symmetry must be regarded as ‘inelastic’. We show, following Kamber and Tondeur, that the corresponding continuous automorphisms preserve the Cartan torsion, and we characterize the discrete automorphisms by a commutativity condition, (6.14), that relates (via the matrix exponential) to the dislocation density tensor. This shows that periodicity properties of corresponding energy densities are determined by the dislocation density. Springer 2014 Article PeerReviewed Nicks, Rachel and Parry, Gareth P. (2014) Group elastic symmetries common to continuum and discrete defective crystals. Journal of Elasticity, 15 (2). pp. 131-156. ISSN 0374-3535 http://link.springer.com/article/10.1007/s10659-013-9450-5 doi:10.1007/s10659-013-9450-5 doi:10.1007/s10659-013-9450-5 |
| spellingShingle | Nicks, Rachel Parry, Gareth P. Group elastic symmetries common to continuum and discrete defective crystals |
| title | Group elastic symmetries common to continuum and discrete defective crystals |
| title_full | Group elastic symmetries common to continuum and discrete defective crystals |
| title_fullStr | Group elastic symmetries common to continuum and discrete defective crystals |
| title_full_unstemmed | Group elastic symmetries common to continuum and discrete defective crystals |
| title_short | Group elastic symmetries common to continuum and discrete defective crystals |
| title_sort | group elastic symmetries common to continuum and discrete defective crystals |
| url | https://eprints.nottingham.ac.uk/2054/ https://eprints.nottingham.ac.uk/2054/ https://eprints.nottingham.ac.uk/2054/ |