We propose a dynamical version of the three-dimensional translation gauge theory of dislocations. In our approach, we use the notions of dislocation density and dislocation current tensors as the translational field strengths and define corresponding response quantities (pseudomoment stress and dislocation momentum flux). For the equations of motion for dislocations, we derive a closed system of field equations in an elegant quasi-Maxwellian form. The dynamical Peach— Koehler force density is derived in this framework as well. Finally, we discuss the similarities and differences between Maxwell field theory and the dislocation gauge theory developed here.
Abu Al-Rub, RK, and Voyiadis, GZA physically based gradient plasticity theory . International Journal of Plasticity, 22, 654-684 ( 2006).
2.
Frank, FCOn the equations of motion of crystal dislocations. Proceedings of the Physical Society, Section A, 62, 131-134 ( 1949).
3.
Eshelby, JDUniformly moving dislocations. Proceedings of the Physical Society, Section A, 62, 307-314 ( 1949).
4.
Leibfried, G., and Dietze, H.-D.Zur Theorie der Schraubenversetzung . Zeitschrift für Physik, 126, 790-808 ( 1949).
5.
Kröner, E.Die inneren Spannungen und der Inkompatibilitätstensor in der Elastizitätstheorie. Zeitschrift für angewandte Physik, 7, 249-257 (1955).
6.
Kröner, E.Continuum theory of defects, in ed. R. Balian, M. Kléman and J.-P. Poirier, Physics of Defects (Proceedings of the Les Houches Summer School, Session 35), pp. 215-315, North-Holland , Amsterdam, 1981.
7.
Holländer, EFThe basic equations of the dynamics of the continuous distribution of dislocations I; II; III. Czechoslovak Journal of Physics, 10, 409-418; 479-487; 551-560 (1960).
8.
Bovet, D.Continuous theory of dislocations in elastostatics and elastodynamics . Solid Mechanics Archives, 4, 31-96 ( 1979).
9.
Mistura, L.Crystal dislocations and Dirac monopoles. International Journal of Engineering Science, 33, 2149-2159 ( 1995 ).
10.
Golebiewska-Lasota, AADislocations and gauge invariance. International Journal of Engineering Science, 17, 329-333 ( 1979).
11.
Edelen, DGB.A four-dimensional formulation of defect dynamics and some of its consequences . International Journal of Engineering Science, 18, 1095-1116 (1980).
12.
Schaefer, H.Maxwell-Gleichungen, Energiesatz und Lagrange-Dichte in der Kontinuumstheorie der Versetzungen. Acta Mechanica , 10, 59-66 ( 1970).
13.
Kadić, A., and Edelen, DGB.A Gauge Theory of Dislocations and Disclinations , Lecture Notes in Physics, Vol. 174, Springer, Berlin , 1983.
14.
Edelen, DGB , and Lagoudas, DCGauge Theory and Defects in Solids, Mechanics and Physics of Discrete Systems, Vol. 1, North-Holland, Amsterdam, 1988.
15.
Lazar, M.Dislocation theory as a 3-dimensional translation gauge theory. Annalen der Physik (Leipzig), 9, 461-473 ( 2000).
16.
Lazar, M.An elastoplastic theory of dislocations as a physical field theory with torsion . Journal of Physics A: Mathematical and General , 35, 1983-2004 (2002).
17.
Lazar, M. , and Anastassiadis , C.The gauge theory of dislocations: conservation and balance laws. Philosophical Magazine, 88, 1673-1699 (2008).
18.
Lazar, M. , and Anastassiadis , C.The gauge theory of dislocations: static solutions of screw and edge dislocations. Philosophical Magazine, 89, 199-231 (2009).
19.
Lazar, M.The gauge theory of dislocations: a uniformly moving screw dislocation. Proceedings of the Royal Society A, 465, 2505-2520 (2009).
20.
Lazar, M.The gauge theory of dislocations: a nonuniformly moving screw dislocation. Preprint (2010), Physics LettersA374 ( 2010) 3092-3098.
21.
Lazar, M. , and Hehl, FWCartan’s spiral staircase in physics and, in particular, in the gauge theory of dislocations. Foundations of Physics, 2010, DOI:10.1007/s10701-010-9440-4.
22.
Hehl, FW, and Obukhov, YNFoundations of Classical Electrodynamics: Charge, Flux, and Metric , Birkhäuser, Boston, 2003.
23.
Frankel, T.The Geometry of Physics: An Introduction, Cambridge University Press, Cambridge, 1997.
24.
Hehl, FW, McCrea, JD, Mielke, EW, and Ne’eman, Y.Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance . Physics Reports, 258, 1-171 (1995).
25.
Ogden, RWNon-linear Elastic Deformations, Dover, New York, 1997.
26.
Marsden, JE , and Hughes, TJR.Mathematical Foundations of Elasticity, Dover, New York, 1994.
27.
Maugin, GAMaterial Inhomogeneities in Elasticity, Chapman and Hall , London, 1993.
28.
Kossecka, E.Mathematical theory of defects. Part II. Dynamics . Archives of Mechanics, 27, 79-92 ( 1975).
29.
Kosevich, ANCrystal dislocations and the theory of elasticity , in ed. F. R. N. Nabarro, Dislocations in Solids, Vol. 1, pp. 33-165, North-Holland , Amsterdam, 1979 .
30.
Landau, LD, and Lifschitz, EMElastizitätstheorie, Band 7, Lehrbuch der Theoretischen Physik , Akademie-Verlag, Berlin, 1989.
Landau, LD, and Lifschitz, EMStatistische Physik: Teil 1, Band 5, Lehrbuch der Theoretischen Physik, Akademie-Verlag , Berlin, 1987.
33.
Bilby, BA, Gardner, LRT, and Stroh, ANContinuous distributions of dislocations and the theory of plasticity, Extrait des Actes du IXe Congrès International de Méchanique Appliquèe (Bruxelles), pp. 35-44, 1957.
34.
Kröner, E.Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Archive for Rational Mechanics and Analysis , 4, 273-333 (1960).
35.
Lee, EHElastoplastic deformation at finite strains . ASME Journal of Applied Mechanics, 36, 1-6 ( 1969).
36.
Cartan, É.Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion. Comptes Rendus Académie des Sciences Paris, 174, 593-595 ( 1922). English translation: Kerlick , G.D.On a generalization of the notion of Riemann curvature and spaces with torsion, in ed. P. G. Bergmann and V. De Sabbata, Proceedings of the 6th Course of the International School on Cosmology and Gravitation: Spin, Torsion, Rotation, and Supergravity (Erice, 1979), pp. 489-491; with subsequent comments by Trautman, A. Comments on the paper by Élie Cartan: Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion, pp. 493-496, Plenum, New York, 1980.
37.
Cartan, É.On Manifolds with an Affine Connection and the Theory of General Relativity (English translation of the French original of 1922-23), Bibliopolis, Napoli , 1986.
38.
Kondo, K.On the geometrical and physical foundations of the theory of yielding . Proceedings of the 2nd Japan National Congress for Applied Mechanics (Tokyo), pp. 41-47, 1952.
