Journal of Engineering Mathematics

, Volume 62, Issue 3, pp 203–221 | Cite as

Nonlocally related PDE systems for one-dimensional nonlinear elastodynamics



Complete dynamical PDE systems of one-dimensional nonlinear elasticity satisfying the principle of material frame indifference are derived in Eulerian and Lagrangian formulations. These systems are considered within the framework of equivalent nonlocally related PDE systems. Consequently, a direct relation between the Euler and Lagrange systems is obtained. Moreover, other equivalent PDE systems nonlocally related to both of these familiar systems are obtained. Point symmetries of three of these nonlocally related PDE systems of nonlinear elasticity are classified with respect to constitutive and loading functions. Consequently, new symmetries are computed that are: nonlocal for the Euler system and local for the Lagrange system; local for the Euler system and nonlocal for the Lagrange system; nonlocal for both the Euler and Lagrange systems. For realistic constitutive functions and boundary conditions, new dynamical solutions are constructed for the Euler system that only arise as symmetry reductions from invariance under nonlocal symmetries.


Nonlinear elasticity Nonlocal symmetries Nonlocally related systems Group invariant solutions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bluman GW, Anco SC (2002) Symmetry and integration methods for differential equations. Springer, New YorkMATHGoogle Scholar
  2. 2.
    Olver PJ (1993) Application of lie groups to differential equations. Springer, New YorkGoogle Scholar
  3. 3.
    Ovsiannikov LV (1982) Group analysis of differential equations. Academic Press, New YorkMATHGoogle Scholar
  4. 4.
    Bluman GW, Kumei S (1989) Symmetries and differential equations. Springer, New YorkMATHGoogle Scholar
  5. 5.
    Dorodnitsyn V, Winternitz P (2000) Lie point symmetry preserving discretizations for variable coefficient Korteweg-de Vries equations. Modern group analysis. Nonlin Dynam 22: 49–59MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Vassilev VM, Djondjorov PA (2003) Application of Lie transformation group methods to classical linear theories of rods and plates. Int J Sol Struct 40: 1585–1614MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Ozer T (2003) Symmetry group classification of one-dimensional elastodynamics problems in nonlocal elasticity. Mech Res Comm 30: 539–546CrossRefMathSciNetGoogle Scholar
  8. 8.
    Ozer T (2003) Symmetry group classification for two-dimensional elastodynamics problems in nonlocal elasticity. Int J Eng Sci 41: 2193–2211CrossRefMathSciNetGoogle Scholar
  9. 9.
    Suhubi ES (2000) Explicit determination of isovector fields of equivalence groups for second order balance equations. Int J Eng Sci 38: 715–736CrossRefMathSciNetGoogle Scholar
  10. 10.
    Ozer T (2003) The solution of Navier equations of classical elasticity using Lie symmetry groups. Mech Res Comm 30: 193–201CrossRefMathSciNetGoogle Scholar
  11. 11.
    Suhubi ES, Bakkaloglu A (1997) Symmetry groups for arbitrary motions of hyperelastic solids. Int J Eng Sci 35: 637–657MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Bland DR (1969) Nonlinear dynamic elasticity. Ginn, BostonMATHGoogle Scholar
  13. 13.
    Horgan CO, Murphy JH (2005) Lie group analysis and plane strain bending of cylindrical sectors for compressible nonlinearly elastic materials. IMA J Appl Math 70: 80–91MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Horgan CO, Murphy JH (2005) A Lie group analysis of the axisymmetric equations of finite elasttostatics for compressible materials. Math Mech Sol 10: 311–333MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Budiansky B, Rice JR (1968) Conservation laws and energy release rates. J Appl Mech 40: 201–203Google Scholar
  16. 16.
    Hatfield GA, Olver PJ (1998) Canonical forms and conservation laws in linear elastostatics. Arch Mech 50: 389–404MATHGoogle Scholar
  17. 17.
    Yavari A, Marsden JE, Ortiz M (2006) On spatial and material covariant balance laws in elasticity. J Math Phys 47:042903 CrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Anco S, Bluman G (2002) Direct construction method for conservation laws of partial differential equations. Part I: examples of conservation law classfications. Eur J Appl Math 13: 545–566MATHMathSciNetGoogle Scholar
  19. 19.
    Anco S, Bluman G (2002) Direct construction method for conservation laws of partial differential equations. Part II: general treatment. Eur J Appl Math 13: 567–585MATHMathSciNetGoogle Scholar
  20. 20.
    Bluman G, Cheviakov AF (2005) Framework for potential systems and nonlocal symmetries: algorithmic approach. J Math Phys 46: 123506CrossRefADSMathSciNetGoogle Scholar
  21. 21.
    Bluman G, Cheviakov AF, Ivanova NM (2006) Framework for nonlocally related PDE systems and nonlocal symmetries: extension, simplification, and examples. J Math Phys 47: 113505CrossRefADSMathSciNetGoogle Scholar
  22. 22.
    Ciarlet PG (1988) Mathematical elasticity. Volume I: three-dimensional Elasticity. Collection studies in mathematics and applications vol 20. North-HollandGoogle Scholar
  23. 23.
    Ogden R (1997) Nonlinear elastic deformations. DoverGoogle Scholar
  24. 24.
    Truesdell C, Noll W (1965) The non-linear field theories of mechanics. In Handbuch der Physik, vol III/3. Springer, BerlinGoogle Scholar
  25. 25.
    Marsden JE, Hughes TJR (1983) Mathematical foundations of elasticity. DoverGoogle Scholar
  26. 26.
    Bluman G, Cheviakov AF (2007) Nonlocally related systems, linearization and nonlocal symmetries for the nonlinear wave equation. J Math An App 333: 93–111MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Bluman G, Cheviakov AF, Senthilvelan M (2008) Solution and asymptotic/blow-up behaviour of a class of nonlinear dissipative systems. J Math An App 339: 1199–1209MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Cheviakov AF (2007) Comp Phys Comm 176(1):48–61 (The GeM package and documentation is available at
  29. 29.
    Wolf T (2002) Crack, LiePDE, ApplySym and ConLaw. In: Grabmeier J, Kaltofen E, Weispfenning V (eds) Computer algebra handbook. Springer, pp 465–468Google Scholar
  30. 30.
    Horgan CO (2001) Equilibrium solutions for compressible nonlinearly elastic materials. In: Fu YB, Ogden RW (eds) Nonlinear elasticity: theory and applications. Cambridge University Press, pp 135–159Google Scholar
  31. 31.
    Varley E (1965) Simple waves in general elastic materials. Arch Rat Mech Anal 20: 309–328CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  2. 2.LEMTA – ENSEMVandoeuvre les Nancy, CedexFrance

Personalised recommendations