Group Elastic Symmetries Common to Continuum and Discrete Defective Crystals

Nicks, Rachel; Parry, Gareth P.

Abstract

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.

Citation

Nicks, R., & Parry, G. P. (2014). Group Elastic Symmetries Common to Continuum and Discrete Defective Crystals. Journal of Elasticity, 115(2), 131-156. https://doi.org/10.1007/s10659-013-9450-5

• Journal Article Type: Article
• Online Publication Date: Aug 2, 2013
• Publication Date: 2014-04
• Deposit Date: Oct 17, 2014
• Publicly Available Date: Oct 17, 2014
• Journal: Journal of Elasticity
• Print ISSN: 0374-3535
• Electronic ISSN: 1573-2681
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 115
• Issue: 2
• Pages: 131-156
• DOI: https://doi.org/10.1007/s10659-013-9450-5
• Public URL: https://nottingham-repository.worktribe.com/output/999427
• Publisher URL: http://link.springer.com/article/10.1007/s10659-013-9450-5
• Additional Information: The final publication is available at Springer via http://dx.doi.org/10.1007/s10659-013-9450-5