×

zbMATH — the first resource for mathematics

Connection and curvature in crystals with non-constant dislocation density. (English) Zbl 1425.74121
Summary: Given a smooth defective solid crystalline structure defined by linearly independent ‘lattice’ vector fields, the Burgers vector construction characterizes some aspect of the ‘defectiveness’ of the crystal by virtue of its interpretation in terms of the closure failure of appropriately defined paths in the material and this construction partly determines the distribution of dislocations in the crystal. In the case that the topology of the body manifold \(M\) is trivial (e.g., a smooth crystal defined on an open set in \(\mathbb{R}^2)\), it would seem at first glance that there is no corresponding construction that leads to the notion of a distribution of disclinations, that is, defects with some kind of ‘rotational’ closure failure, even though the existence of such discrete defects seems to be accepted in the physical literature. For if one chooses to parallel transport a vector, given at some point \(P\) in the crystal, by requiring that the components of the transported vector on the lattice vector fields are constant, there is no change in the vector after parallel transport along any circuit based at \(P\). So the corresponding curvature is zero. However, we show that one can define a certain (generally non-zero) curvature in this context, in a natural way. In fact, we show (subject to some technical assumptions) that given a smooth solid crystalline structure, there is a Lie group acting on the body manifold \(M\) that has dimension greater or equal to that of \(M\). When the dislocation density is non-constant in \(M\) the group generally has a non-trivial topology, and so there may be an associated curvature. Using standard geometric methods in this context, we show that there is a linear connection invariant with respect to the said Lie group, and give examples of structures where the corresponding torsion and curvature may be non-zero even when the topology of \(M\) is trivial. So we show that there is a ‘rotational’ closure failure associated with the group structure – however, we do not claim, as yet, that this leads to the notion of a distribution of disclinations in the material, since we do not provide a physical interpretation of these ideas. We hope to provide a convincing interpretation in future work. The theory of fibre bundles, in particular the theory of homogeneous spaces, is central to the discussion.

MSC:
74E15 Crystalline structure
53Z05 Applications of differential geometry to physics
22E70 Applications of Lie groups to the sciences; explicit representations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] [1] Davini, C. A proposal for a continuum theory of defective crystals. Arch Ration Mech Anal 1986; 96: 295-317. · Zbl 0623.73002
[2] [2] Elżanowski, M, Preston, S. On continuously defective elastic crystals. Miskolc Math Notes 2013; 14(2): 659-670. · Zbl 1299.53161
[3] [3] Kobayashi, S, Nomizu, K. Foundations of differential geometry, vols. I & II. New York: John Wiley & Sons, Inc., 1996.
[4] [4] Davini, C, Parry, GP. A complete list of invariants for defective crystals. Proc R Soc London, Ser A 1991; 432: 341-365. · Zbl 0726.73032
[5] [5] Elżanowski, M, Parry, GP. Material symmetry in a theory of continuously defective crystals. J Elast 2004; 74: 215-237. · Zbl 1058.74028
[6] [6] Olver, PJ. Equivalence, invariants, and symmetry. Cambridge: Cambridge University Press, 1995.
[7] [7] Parry, GP. Discrete structures in continuum descriptions of defective crystals. Philos Trans R Soc London, Ser A 2016; 374(2066): DOI: 10.1098/rsta.20150172. · Zbl 1353.82073
[8] [8] Parry, GP, Silhavý, M. Elastic invariants in the theory of defective crystals. Proc R Soc London, Ser A 1999; 455: 4333-4346. DOI: 10.1098/rspa.19990503. · Zbl 0954.74013
[9] [9] Gorbatsevich, VV, Onishchik, AL, Vinberg, EB. Lie groups and Lie algebras I (Encyclopedia of Mathematical Sciences, vol. 20). Berlin: Springer-Verlag, 1993.
[10] [10] Palais, RS. A global formulation of the Lie theory of transformation groups (Memoirs of the American Mathematical Society, vol. 22). Providence, RI: American Mathematical Society, 1957.
[11] [11] Parry, GP. Group properties of defective crystal structures. Math Mech Solids 2003; 8: 515-538. · Zbl 1072.74016
[12] [12] Poor, WA. Differential geometric structures. New York: McGraw-Hill, 1981.
[13] [13] Čap, A, Slovàk, J. Parabolic geometries I (Mathematical Surveys and Monographs, vol. 154) Providence, RI: American Mathematical Society, 2008.
[14] [14] Baker, A. Matrix groups. London, Springer-Verlag, 2003.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.