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 TMF, 2003, Volume 135, Number 2, Pages 338–352 (Mi tmf183)

Wedge Dislocation in the Geometric Theory of Defects

M. O. Katanaev

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We consider a wedge dislocation in the framework of elasticity theory and the geometric theory of defects. We show that the geometric theory quantitatively reproduces all the results of elasticity theory in the linear approximation. The coincidence is achieved by introducing a postulate that the vielbein satisfying the Einstein equations must also satisfy the gauge condition, which in the linear approximation leads to the elasticity equations for the displacement vector field. The gauge condition depends on the Poisson ratio, which can be experimentally measured. This indicates the existence of a privileged reference frame, which denies the relativity principle.

Keywords: dislocation, Riemann–Cartan geometry

DOI: https://doi.org/10.4213/tmf183

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English version:
Theoretical and Mathematical Physics, 2003, 135:2, 733–744

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Document Type: Article
Revised: 02.09.2002

Citation: M. O. Katanaev, “Wedge Dislocation in the Geometric Theory of Defects”, TMF, 135:2 (2003), 338–352; Theoret. and Math. Phys., 135:2 (2003), 733–744

Citation in format AMSBIB
\Bibitem{Kat03} \by M.~O.~Katanaev \paper Wedge Dislocation in the Geometric Theory of Defects \jour TMF \yr 2003 \vol 135 \issue 2 \pages 338--352 \mathnet{http://mi.mathnet.ru/tmf183} \crossref{https://doi.org/10.4213/tmf183} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2008768} \zmath{https://zbmath.org/?q=an:1178.74021} \transl \jour Theoret. and Math. Phys. \yr 2003 \vol 135 \issue 2 \pages 733--744 \crossref{https://doi.org/10.1023/A:1023687003017} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000183468400014} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. M. O. Katanaev, “One-Dimensional Topologically Nontrivial Solutions in the Skyrme Model”, Theoret. and Math. Phys., 138:2 (2004), 163–176
2. Carvalho AMD, Moraes F, Furtado C, “The self-energy of a charged particle in the presence of a topological defect distribution”, International Journal of Modern Physics A, 19:13 (2004), 2113–2122
3. M. O. Katanaev, “Geometric theory of defects”, Phys. Usp., 48:7 (2005), 675–701
4. Marques, GD, “Quantum effects due to a magnetic flux associated to a topological defect”, International Journal of Modern Physics A, 20:26 (2005), 6051
5. Furtado, C, “Aharonov-Bohm effect and disclinations in an elastic medium”, Modern Physics Letters A, 21:17 (2006), 1393
6. de Berredo-Peixoto, G, “Inside the BTZ black hole”, Physical Review D, 75:2 (2007), 024004
7. Carvalho, J, “Self-interaction in the von Karman cosmic string street configuration”, European Physical Journal C, 58:2 (2008), 331
8. Netto, ALS, “Elastic Landau levels”, Journal of Physics-Condensed Matter, 20:12 (2008), 125209
9. de Berredo-Peixoto, G, “Tube dislocations in gravity”, Journal of Mathematical Physics, 50:4 (2009), 042501
10. Carvalho, AMD, “TOPOLOGICAL DEFECT DISTRIBUTIONS AND THE SELF-ENERGY OF A CHARGED PARTICLE”, International Journal of Modern Physics D, 18:2 (2009), 237
11. de Berredo-Peixoto G., Katanaev M.O., Konstantinova E., Shapiro I.L., “Schrodinger equation in the space with cylindrical geometric defect and possible application to multi-wall nanotubes”, Nuovo Cimento Della Societa Italiana Di Fisica B-Basic Topics in Physics, 125:8 (2010), 915–931
12. Mesaros A., Sadri D., Zaanen J., “Parallel transport of electrons in graphene parallels gravity”, Physical Review B, 82:7 (2010), 073405
13. Katanaev M.O., Mannanov I.G., “Wedge Dislocations, Three-Dimensional Gravity, and the Riemann–Hilbert Problem”, Phys. Part. Nuclei, 43:5 (2012), 639–643
14. Bohmer C.G., Obukhov Yu.N., “A Gauge-Theoretical Approach to Elasticity with Microrotations”, Proc. R. Soc. A-Math. Phys. Eng. Sci., 468:2141 (2012), 1391–1407
15. Boehmer C.G., Tamanini N., “Rotational Elasticity and Couplings To Linear Elasticity”, Math. Mech. Solids, 20:8 (2015), 959–974
16. Katanaev M.O., “Rotational Elastic Waves in Double Wall Tube”, Phys. Lett. A, 379:24-25 (2015), 1544–1548
17. Katanaev M.O., “Rotational Elastic Waves in a Cylindrical Waveguide With Wedge Dislocation”, J. Phys. A-Math. Theor., 49:8 (2016), 085202
18. Yajima T., Nagahama H., “Finsler geometry of topological singularities for multi-valued fields: Applications to continuum theory of defects”, Ann. Phys.-Berlin, 528:11-12 (2016), 845–851
19. Katanaev M.O., “Chern–Simons Term in the Geometric Theory of Defects”, Phys. Rev. D, 96:8 (2017), 084054
20. M. O. Katanaev, “Chern–Simons action and disclinations”, Proc. Steklov Inst. Math., 301 (2018), 114–133
21. Yajima T., Yamasaki K., Nagahama H., “Non-Holonomic Geometric Structures of Rigid Body System in Riemann-Cartan Space”, J. Phys. Commun., 2:8 (2018), UNSP 085008
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