Skip to main content
SpringerLink
Log in

Generalized theory of elasticity

  • Published:
Mechanics of Solids Aims and scope Submit manuscript

Abstract

We obtain elasticity equations of higher (in the general case, infinite) order than the equations of the classical theory. In contrast to the numerous known versions of the nonclassical theory (Cosserat, nonsymmetric, microstructure, micropolar, multipolar, and gradient), which also result in higher-order equations and contain elasticity relations for traditional and couple stresses with a large number of elastic constants, our theory, regardless of the order of the equations, contains only one additional constant, which can be expressed in terms of the microstructure parameter of the medium. The basic equations of the generalized theory are presented for one-, two-, and three-dimensional problems; these equations take into account the stress gradients and can be written in terms of generalized stresses, strains, and displacements. A boundary value problem that does not require the introduction of couple stresses is stated for the generalized theory of elasticity.

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. E. Reissner, “Note on the Theorem of the Symmetry of the Stress Tensor,” in Proc. of the 4th Midwestern Conf. on Solid Mechanics (The University of Texas, 1959), pp. 192–194.

    Google Scholar 

  2. V. V. Vasil’ev, “Stress Tensor Symmetry and Singular Solutions in the Theory of Elasticity,” Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 2, 62–72 (2010) [Mech. Solids (Engl. Transl.) 45 (2), 205–213 (2010)].

    Google Scholar 

  3. D. B. Body and L. Sterberg, “The Effect of Couple-Stresses on the Corner Singularity due to an Asymmetric Shear Loading,” Int. J. Solids Struct. 4, 159–174 (1968).

    Article  Google Scholar 

  4. A. C. Eringen, Theory of Micropolar Elasticity. Fracture, Vol. 2: Mathematical Foundations of the Fracture Theory, Ed. by A. Yu. Ishlinskii (Mir, Moscow, 1975) [in Russian].

  5. G. A. Brovko, “On One Structural Model of the Cosserat Medium,” Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 1, 75–91 (2002) [Mech. Solids (Engl. Transl.) 37 (1), 60–73 (2002)].

    Google Scholar 

  6. R. Lakes, “Cosserat Micromechanics,” in Proc. of the 3rd Tech. Conf. of American Society for Composites. Technomic (1988), pp. 505–516.

    Google Scholar 

  7. R. D. Mindlin, “Microstructure in Linear Elasticity,” Arch. Rat. Mech. Anal. 16 (1), 51–78 (1964).

    Article  MathSciNet  MATH  Google Scholar 

  8. R. D. Mindlin, “Second Gradient of Strain and Surface-Tension in Linear Elasticity,” Int. J. Solids Struct. 1, 417–438 (1965).

    Article  Google Scholar 

  9. S. A. Lurie, P. A. Belov, and N. P. Tuchkova, “Gradient Theory of Media with Conversed Dislocations: Applications to Microstructured Materials,” in Advances in Mechanics and Mathematics, Vol. 21: Book of Mechanics of Generalized Continua: One Hundred Years after the Cosserats, Ed. by G. A. Maugin and A. V. Metrikine (Springer, 2010), pp. 223–232.

  10. S. Lurie, D. Volkov-Bogorodsky, V. Zubov, and N. Tuchkova, “Advanced Theoretical and Numerical Multiscale Modeling of Cohesion/Adhesion Interactions in Continuum Mechanics and Its Applications for Filled Nanocomposites,” Comput. Mater. Sci. 45 (3), 709–714 (2009).

    Article  Google Scholar 

  11. S. Lurie, D. Volkov-Bogorodsky, A. Leontiev, and E. Aifantis, “Eshelby’s Inclusion Problem in the Gradient Theory of Elasticity. Applications to CompositeMaterials,” Int. J. Engng Sci. 49, 1517–1525 (2011).

    Article  MathSciNet  Google Scholar 

  12. J. L. Synge, Relativity: The General Theory (North-Holland, Amsterdam, 1960; Inostr. Lit., Moscow 1963).

    MATH  Google Scholar 

  13. A. V. Andreev, Engineering Methods for Determining Stress Concentration in Machine Elements (Mashinostroenie, Moscow, 1976) [in Russian].

    Google Scholar 

  14. V. V. Vasil’ev and S. A. Lurie, “On the Solution Singularity in the Plane Elasticity Problem for a Cantilever Strip,” Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 4, 40–49 (2013) [Mech. Solids (Engl. Transl.) 48 (4), 388–396 (2013)].

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow State Aviation Technological University, ul. Orshanskaya 3, Moscow, 121552, Russia

    V. V. Vasil’ev

  2. Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333, Russia

    S. A. Lurie

  3. A. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr. Vernadskogo 101, str. 1, Moscow, 119526, Russia

    S. A. Lurie

Authors
  1. V. V. Vasil’ev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. A. Lurie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. V. Vasil’ev.

Additional information

Original Russian Text © V.V. Vasil’ev, S.A. Lurie, 2015, published in Izvestiya Akademii Nauk. Mekhanika Tverdogo Tela, 2015, No. 4, pp. 16–27.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vasil’ev, V.V., Lurie, S.A. Generalized theory of elasticity. Mech. Solids 50, 379–388 (2015). https://doi.org/10.3103/S0025654415040032

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0025654415040032

Keywords

Search

Navigation