Skip to main content
SpringerLink
Log in

CYLINDRICAL DISLOCATION IN A NONLINEAR ELASTIC INCOMPRESSIBLE MATERIAL

  • Published:
Journal of Applied Mechanics and Technical Physics Aims and scope

Abstract

A cylindrical dislocation in an incompressible Mooney–Rivlin solid is studied using the nonlinear equations of the theory of elasticity and the geometric theory of defects. A cylindrical dislocation consists of two hollow concentric cylinders, one of which is inserted into the other and adhered after a corresponding symmetrical deformation. The approaches of the classical theory of elasticity and the geometric theory of defects are compared, which makes it possible to ensure a physical interpretation of the tensor density of momentum energy in Einstein equations for a cylindrical dislocation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

REFERENCES

  1. J. P. Hirth and J. Lothe, Theory of Dislocations (McGraw-Hill, 1972).

    MATH  Google Scholar 

  2. K. Kondo, “On the Geometrical and Physical Foundations of the Theory of Yielding," Proc. of the 2nd Japan Nat. Congress for Applied Mechanics, Tokio (Japan), 1952.

  3. B. A. Bilby, R. Bollough, and E. Smith, “Continuous Distributions of Dislocations: A New Application of the Methods of Non-Riemannian Geometry," Proc. Roy. Soc. London. Ser. A 231 (1185), 263–273 (1955).

    Article  ADS  MathSciNet  Google Scholar 

  4. A. Kadic and D. G. B. Edelen, A Gauge Theory of Dislocation and Disclinations: Lecture Notes in Physics (Springer-Verlag, 1983).

    Book  Google Scholar 

  5. S. P. Kiselev, “Internal Stresses in a Solid with Dislocations," Prikl. Mat. Mekh. 45 (4), 131–136 (2004) [J. Appl. Math. Mech. 45 (4), 567–571 (2004)].

    ADS  MATH  Google Scholar 

  6. G. de Berredo-Peixoto and M. O. Katanaev, “Tube Dislocations in Gravity," J. Math. Phys. 50, 042501 (2009).

    Article  ADS  MathSciNet  Google Scholar 

  7. A. F. Andrade and G. de Berredo-Peixoto, “Geodesics in a Space with a Spherically Symmetric Dislocation," Gravitat. Cosmology. 19 (1), 29–34 (2013).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern Geometry - Methods and Applications Vol. 1: Part I. The Geometry of Surfaces, Transformation Groups, and Fields (Springer New York, NY, 1984).

    Book  Google Scholar 

  9. A. I. Lurie, Nonlinear Theory of Elasticity (North-Holland, 1990).

    MATH  Google Scholar 

  10. S. A. Lychev and A. V. Mark, “Axisymmetric Growth of a Hollow Hyperelastic Cylinder," Izv. Saratov. Univ. 14 (2), 209–227 (2014).

    MATH  Google Scholar 

  11. M. O. Katanaev, Geometric Methods in Mathematical Physics (Center of Research and Education of the St. Petersburg Department of Steklov Mathematical Institute of the Russian Academy of Sciences, 2016) [in Russian].

  12. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics. Vol. 2. The Classical Theory of Fields (Addison-Wesley, 1951).

    MATH  Google Scholar 

  13. M. O. Katanaev, “Wedge Dislocation in the Geometric Theory of Defects," Teoret. Mat. Fiz. 135 (2), 338–352 (2003) [Theoretical and Mathematical Physics 135 (2), 733–744 (2003)].

    Article  MathSciNet  Google Scholar 

  14. M. O. Katanaev, “Geometric Theory of Defects," Uspekhi Fiz. Nauk 175, 705–733 (2005).

    Article  Google Scholar 

  15. V. S. Vladimirov, Equations of Mathematical Physics (Nauka, Moscow, 1981) [in Russian].

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Academy of State Fire Service of the EMERCOM of Russia, 129366, Moscow, Russia

    A. V. Mark

Authors
  1. A. V. Mark
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Mark.

Additional information

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, 2021, Vol. 63, No. 4, pp. 172-182. https://doi.org/10.15372/PMTF20220418.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mark, A.V. CYLINDRICAL DISLOCATION IN A NONLINEAR ELASTIC INCOMPRESSIBLE MATERIAL. J Appl Mech Tech Phy 63, 702–710 (2022). https://doi.org/10.1134/S0021894422040186

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0021894422040186

Keywords

Search

Navigation