Skip to main content
SpringerLink
Log in

Diagonalization of the system of static Lamé equations of isotropic linear elasticity

  • Published:
Journal of Applied and Industrial Mathematics Aims and scope Submit manuscript

Abstract

We find a simplest representation for the general solution to the system of the static Lamé equations of isotropic linear elasticity in the form of a linear combination of the first derivatives of three functions that satisfy three independent harmonic equations. The representation depends on 12 free parameters choosing which it is possible to obtain various representations of the general solution and simplify the boundary value conditions for the solution of boundary value problems as well as the representation of the general solution for dynamic Lamé equations. The system of Lamé equations diagonalizes; i.e., it is reduced to the solution of three independent harmonic equations. The representation implies three conservation laws and some formula for producing new solutions which makes it possible, given a solution, to find new solutions to the static Lamé equations by derivations. In the two-dimensional case of a plane deformation, the so-found solution immediately implies the Kolosov-Muskhelishvili representation for shifts by means of two analytic functions of complex variable. Two examples are given of applications of the proposed method of diagonalization of the two-dimensional elliptic systems.

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.

Similar content being viewed by others

References

  1. N. I. Ostrosablin, “The General Solution and Reduction to Diagonal Form of a System of Equations of Isotropic Linear Elasticity,” Sibirsk. Zh. Indust. Mat. 12(2), 79–83 (2009) [J. Appl. Indust. Math. 4 (3), 354–358 (2010)].

    MathSciNet  MATH  Google Scholar 

  2. N. I. Ostrosablin, “Canonical Moduli and General Solution of Equations of a Two-Dimensional Static Problem of Anisotropic Elasticity,” Prikl. Mekh. Tekhn. Fiz. 51(3), 94–106 (2010) [J. Appl. Mech. Tech. Phys. 51 (3), 377–388 (2010)].

    MathSciNet  Google Scholar 

  3. A. I. Lur’e, Theory of Elasticity (Nauka, Moscow, 1970; Springer, Berlin, 2005).

    Google Scholar 

  4. W. Nowacki, Teoria Spr eżistości (Elasticity Theory) (Państwowe Wydawnictwo Naukowe, Warszawa, 1970; Mir, Moscow, 1975).

    Google Scholar 

  5. N. I. Ostrosablin, “Symmetry Operators and General Solutions of the Equations of the Linear Theory of Elasticity,” Prikl. Mekh. Tekhn. Fiz. 36(5), 98–104 (1995) [J. Appl. Mech. Tech. Phys. 36 (5), 724–729 (1995)].

    MathSciNet  MATH  Google Scholar 

  6. S. Chiriţă, A. Danescu, and M. Ciarletta, “On the Strong Ellipticity of the Anisotropic Linearly Elastic Materials,” J. Elast. 87(1), 1–27 (2007).

    Article  MATH  Google Scholar 

  7. P. Olver, Applications of Lie Groups to Differential Equations (Springer, New York, 1986; Mir, Moscow, 1989).

    Book  MATH  Google Scholar 

  8. P. P. Kiryakov,S. I. Senashov, and A. N. Yakhno, Application of Symmetries and Conservation Laws for Solving Differential Equations (Izd. Siberian Division of Russ. Acad. Sci. Novosibirsk, 2001) [in Russian].

  9. N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity (Nauka, Moscow, 1966; Noordhoff, Leyden, 1975).

    Google Scholar 

  10. A. V. Bitsadze, Boundary Value Problems for Second Order Elliptic Equations (Nauka, Moscow, 1966; North-Holland, Amsterdam, 1968).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lavrent’ev Institute of Hydrodynamics, pr. Lavrent’eva 15, Novosibirsk, 630090, Russia

    N. I. Ostrosablin

Authors
  1. N. I. Ostrosablin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. I. Ostrosablin.

Additional information

Original Russian Text © N.I. Ostrosablin, 2012, published in Sibirskii Zhurnal Industrial’noi Matematiki, 2012, Vol. XV, No. 3, pp. 87–98.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ostrosablin, N.I. Diagonalization of the system of static Lamé equations of isotropic linear elasticity. J. Appl. Ind. Math. 7, 89–99 (2013). https://doi.org/10.1134/S1990478913010092

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1990478913010092

Keywords

Search

Navigation