Skip to main content
SpringerLink
Log in

An eigenvalue problem for tensors used in mechanics and the number of independent Saint-Venant strain compatibility conditions

  • Published:
Moscow University Mechanics Bulletin Aims and scope

Abstract

A number of questions concerning the eigenvalue problem for a tensor \(\mathop A\limits_ \approx\) ∈ ℝ4(Ω) with special symmetries are considered; here Ω is a domain of a four-dimensional (three-dimensional) Riemannian space. It is proved that a nonsingular fourth-rank tensor has no more than six (three) independent components in the case of a four-dimensional (three-dimensional) Riemannian space. It is shown that the number of independent Saint-Venant strain compatibility conditions is less than six.

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

Similar content being viewed by others

References

  1. I. N. Vekua, Foundations of Tensor Analysis and the Theory of Covariants (Nauka, Moscow, 1978) [in Russian].

    MATH  Google Scholar 

  2. M. U. Nikabadze, “Construction of Eigentensor Columns in the Linear Micropolar Theory of Elasticity,” Vestn. Mosk. Univ. Ser. 1: Mat. Mekh., No. 1, 30–39 (2014) [Moscow Univ. Mech. Bull. 69 (1), 1–9 (2014)].

    MathSciNet  MATH  Google Scholar 

  3. M. U. Nikabadze, “On Some Questions of Tensor Calculus with Applications to Mechanics,” in Tensor Analysis (RUDN Univ., Moscow, 2015), Vol. 55, pp. 3–194.

    Google Scholar 

  4. A. I. Lurie, Nonlinear Theory of Elasticity (Nauka, Moscow, 1980; North-Holland, Amsterdam, 1990).

    MATH  Google Scholar 

  5. B. E. Pobedrya, Lectures on Tensor Analysis (Mosk. Gos. Univ., Moscow, 1986) [in Russian].

    MATH  Google Scholar 

  6. Ya. Rykhlevskii, Mathematical Structure of Elastic Bodies, Preprint No. 217 (Institute for Problems in Mechanics, Moscow, 1983).

    Google Scholar 

  7. N. I. Ostrosablin, Anisotropy and General Solutions of Equations in the Linear Theory of Elasticity, Doctoral Dissertation in Mathematics and Physics (Hydromechanics Inst., Novosibirsk, 2000).

    Google Scholar 

  8. B. D. Annin and N. I. Ostrosablin, “Anisotropy of Elastic Properties of Materials,” Zh. Prikl. Mekh. Tekh. Fiz. 49 (6), 131–151 (2008) [J. Appl. Mech. Tech. Phys. 49 (6), 998–1014 (2008)].

    MathSciNet  Google Scholar 

  9. B. E. Pobedrya, Numerical Methods in the Theory of Elasticity and Plasticity, 2nd ed. (Mosk. Gos. Univ., Moscow, 1995) [in Russian].

    Google Scholar 

  10. B. E. Pobedrya, S. V. Sheshenin, and T. Kholmatov, Problems in Terms of Stresses (Fan, Tashkent, 1988) [in Russian].

    Google Scholar 

  11. W. Nowacki, Teoria Sprezystosci (PWN, Warsaw, 1970; Mir, Moscow, 1975); Engl. transl. of the title: Theory of Elasticity.

    MATH  Google Scholar 

  12. M. U. Nikabadze, “Compatibility Conditions in the Linear Micropolar Theory,” Vestn. Mosk. Univ. Ser. 1: Mat. Mekh., No. 5, 48–51 (2010) [Moscow Univ. Mech. Bull. 65 (5), 110–113 (2010)].

    MathSciNet  Google Scholar 

  13. M. U. Nikabadze, “Compatibility Conditions and Equations of Motion in the Linear Micropolar Theory of Elasticity,” Vestn. Mosk. Univ. Ser. 1: Mat. Mekh., No. 1, 63–66 (2012) [Moscow Univ. Mech. Bull. 67 (1), 18–22 (2012)].

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Moscow State University, Leninskie Gory, Moscow, 119899, Russia

    M. U. Nikabadze

Authors
  1. M. U. Nikabadze
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. U. Nikabadze.

Additional information

Original Russian Text © M.U. Nikabadze, 2017, published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2017, Vol. 72, No. 3, pp. 54–58.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nikabadze, M.U. An eigenvalue problem for tensors used in mechanics and the number of independent Saint-Venant strain compatibility conditions. Moscow Univ. Mech. Bull. 72, 66–69 (2017). https://doi.org/10.3103/S0027133017030037

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0027133017030037

Search

Navigation