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Journal of Elasticity

, Volume 115, Issue 2, pp 131–156 | Cite as

Group Elastic Symmetries Common to Continuum and Discrete Defective Crystals

  • Rachel Nicks
  • Gareth Parry
Article

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 \(\mathbb{R}^{3}\), 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.

Keywords

Crystals Defects Lie Groups 

Mathematics Subject Classification (2010)

74A20 74E25 

Notes

Acknowledgements

We acknowledge the support of the UK Engineering and Physical Sciences Research Council through grant EP/G047162/1.

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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.School of MathematicsUniversity of BirminghamBirminghamUK
  2. 2.School of Mathematical SciencesUniversity of NottinghamNottinghamUK

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