Group elastic symmetries common to continuum and discrete defective crystals

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

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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.

Item Type: Article
Additional Information: The final publication is available at Springer via
Schools/Departments: University of Nottingham UK Campus > Faculty of Science > School of Mathematical Sciences
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Depositing User: Parry, Dr Gareth
Date Deposited: 17 Oct 2014 14:42
Last Modified: 16 Sep 2016 11:30

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