Remember Me
Or use your Academic/Social account:


Or use your Academic/Social account:


You have just completed your registration at OpenAire.

Before you can login to the site, you will need to activate your account. An e-mail will be sent to you with the proper instructions.


Please note that this site is currently undergoing Beta testing.
Any new content you create is not guaranteed to be present to the final version of the site upon release.

Thank you for your patience,
OpenAire Dev Team.

Close This Message


Verify Password:
Verify E-mail:
*All Fields Are Required.
Please Verify You Are Human:
fbtwitterlinkedinvimeoflicker grey 14rssslideshare1
Nicks, Rachel; Parry, Gareth P. (2014)
Publisher: Miskolci Egyetemi Kiadó
Languages: English
Types: Article
We shall outline geometrical and algebraic ideas which appear to lie at the foundation of the theory of defective crystals that was introduced by Davini [5] in 1986. The focus of the paper will be on the connection between continuous and discrete models of such crystals, approached by consideration of the symmetries inherent in these models. To begin with, we review briefy the results of analysis of variational problems where relevant functionals have the symmetry of perfect (as opposed to defective) crystals, in order to motivate the subsequent study of symmetry in the case when defects are present. In the body of the paper we indicate how the theory of Lie groups, and their discrete subgroups, relates to this geometrical theory of defects, and discuss types of symmetry that occur.
  • The results below are discovered through our pilot algorithms. Let us know how we are doing!

    • [1] L. Auslander, L. Green and F. Hahn, Flows on homogeneous spaces. Ann. Math. Stud. 53. Princeton, NJ: Princeton University Press, 1963
    • [2] M. Baake and J. A. G. Roberts, \Reversing symmetry group of GL(2; Z) and P GL(2; Z) matrices with connections to cat maps and trace maps", J. Phys. A: Math. Gen., Vol. 30, pp. 1549{73, 1997.
    • [3] P. Cermelli and G. P. Parry, \The structure of uniform discrete defective crystals", Continuum Mechanics and Thermodynamics, Vol. 18, pp. 47{61, 2006.
    • [4] M. Chipot and D. Kinderlehrer, \Equilibrium con gurations of crystals", Arch. Rat. Mech. Anal., Vol. 103, pp. 237{277, 1988.
    • [5] C. Davini, \A proposal for a continuum theory of defective crystals", Arch. Rat. Mech. Anal., Vol. 96, pp. 295{317, 1986.
    • [6] C. Davini and G. P. Parry, \A complete list of invariants for defective crystals", Proc. Roy. Soc. London A Vol 432, pp. 341{365, 1991.
    • [7] C. Davini and G. P. Parry, \On defect preserving deformations in crystals". Int. J. Plasticity. Vol. 5, pp. 337{369, 1989.
    • [8] M. Elzanowski and G. P. Parry, \Material symmetry in a theory of continuously defective crystals", J. Elasticity, Vol. 74, pp. 215{237, 2004.
    • [9] I. Fonseca and G. P. Parry, \Equilibrium con guration of defective crystals", Arch. Rat. Mech. Anal., Vol. 120, pp. 245{283, 1992.
    • [10] V. V. Gorbatsevich, \Lattices in solvable Lie groups and deformations of homogeneous spaces", Math. USSR - Sb, Vol 20, pp.249{266, 1973.
    • [11] W. Magnus, A. Karrass, D. Solitar, Combinatorial Group Theory, New York: Dover, 1976.
    • [12] A. Mal'cev, \On a class of homogeneous spaces", Izv. Akad. Nauk SSSR, Ser. Mat. Vol. 13, pp. 9{32, 1949. Am. Math. Soc. Translation Vol. 39, 1949.
    • [13] R. Nicks and G. P. Parry, \On symmetries of crystals with defects related to a class of solvable groups (S1)", Math. Mech. Solids, Vol. 17 No. 6, pp. 631{651, 2011.
    • [14] R. Nicks and G. P. Parry, \On symmetries of crystals with defects related to a class of solvable groups (S2)", Mathematical Methods in the Applied Sciences, Vol. 35, pp. 1741-1755, 2012.
    • [15] R. Nicks and G. P. Parry, \Group elastic symmetries common to continuum and discrete defective crystals", In preparation.
    • [16] P. J. Olver, Equivalence, invariants, and symmetry, Cambridge: Cambridge University press, 1996.
    • [17] G. P. Parry, \Group properties of defective crystal structures", Math. Mech. Solids, Vol. 8, pp. 515{537, 2003.
    • [18] G. P. Parry, \Rotational Symmetries of Crystals with Defects", J. Elasticity, Vol. 94, pp. 147{166, 2009.
    • [19] G. P. Parry, \Elastic Symmetries of Defective Crystals", J. Elasticity, Vol. 101, 101{120, 2010.
    • [20] G. P. Parry and R. Sigrist, \Reconciliation of Local and Global Symmetries for a Class of Crystals with Defects", J. Elasticity Vol. 107, 81{104, 2012.
    • [21] M. Pitteri and G. Zanzotto, Continuum Models for Phase Transitions and Twinning in Crystals, Boca Raton, London, New York, Washington DC: Chapman and Hall, 2003.
    • [22] L. S. Pontryagin, Toplogical Groups, (2nd Edition) New York: Gordon and breach, 1955.
    • [23] W. Thurston Three Dimensional Geometry and Topology, vol 1, Princeton: Princeton University Press, 1997. School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK. E-mail address:
  • No related research data.
  • No similar publications.

Share - Bookmark

Cite this article