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Journal of Elasticity

, Volume 107, Issue 2, pp 151–178 | Cite as

Violation of the Complementing Condition and Local Bifurcation in Nonlinear Elasticity

  • Pablo V. Negrón-Marrero
  • Errol Montes-Pizarro
Article

Abstract

The complementing condition (CC) is an algebraic compatibility requirement between the principal part of a linear elliptic differential operator and the principal part of the corresponding boundary operators. We study the implications of failure of the CC in the context of nonlinear elasticity. In particular we show that for axisymmetric deformations of cylinders and for any homogeneous isotropic material, failure of the CC is equivalent to the existence of sequences of possible bifurcation points accumulating at the point where the CC fails. For non axisymmetric deformations and for Hadamard–Green type materials, we show for axial compressions of the cylinder that the CC fails on a full interval of values of the loading parameter, and for the lateral compression problem it fails at least once.

Keywords

Nonlinear elasticity Complementing condition Global bifurcation Wrinkling 

Mathematics Subject Classification (2000)

74B20 74G60 35J57 35Q74 

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References

  1. 1.
    Agmon, S.: On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems. Commun. Pure Appl. Math. 15, 119–147 (1962) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I. Commun. Pure Appl. Math. 12, 623–727 (1959) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Commun. Pure Appl. Math. 17, 35–92 (1964) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Antman, S.S.: Nonlinear Problems of Elasticity. Springer, New York (1995) MATHGoogle Scholar
  5. 5.
    Guo, Z.H.: Vibration and stability of a cylinder subject to finite deformation. Arch. Mech. Stosow. 5(14), 757–768 (1962) Google Scholar
  6. 6.
    Gurtin, M.: An Introduction to Continuum Mechanics. Academic Press, New York (1981) MATHGoogle Scholar
  7. 7.
    Healey, T., Montes-Pizarro, E.: Global bifurcation in nonlinear elasticity with an application to barrelling states of cylindrical columns. J. Elast. 71, 33–58 (2003) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Healey, T.J., Rosakis, P.: Unbounded branches of classical injective solutions to the forced displacement problem in nonlinear elastostatics. J. Elast. 49, 65–78 (1997) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Healey, T.J., Simpson, H.C.: Global continuation in nonlinear elasticity. Arch. Ration. Mech. Anal. 143, 1–28 (1998) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Lopatinskii, Ya.B.: On a method of reducing boundary value problems for a system of differential equations of elliptic type to regular equations. Ukr. Mat. Zh. 5, 123–151 (1953) MathSciNetGoogle Scholar
  11. 11.
    Montes-Pizarro, E., Negrón-Marrero, P.V.: Local bifurcation analysis of a second gradient model for deformations of a rectangular slab. J. Elast. 86(2), 173–204 (2006) CrossRefGoogle Scholar
  12. 12.
    Morrey, C.B.: Multiple Integrals in the Calculus of Variations. Springer, New York (1966) MATHGoogle Scholar
  13. 13.
    Negrón-Marrero, P.V.: An analysis of the linearized equations for axisymmetric deformations of hyperelastic cylinders. Math. Mech. Solids 4, 109–133 (1999) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Negrón-Marrero, P.V., Montes-Pizarro, E.: Axisymmetric deformations of buckling and barrelling type for cylinders under lateral compression—the linear problem. J. Elast. 65, 61–86 (2001) MATHCrossRefGoogle Scholar
  15. 15.
    Rabier, P.J., Oden, J.T.: Bifurcation in Rotating Bodies. Springer, New York (1989) MATHGoogle Scholar
  16. 16.
    Rabinowitz, P.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7, 487–513 (1971) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Sensenig, C.B.: Instability of thick elastic shells. Commun. Pure Appl. Math. 17, 451–491 (1964) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Shapiro, Z.Ja.: On general boundary value problems for elliptic equations. Izv. Akad. Nauk SSSR 17, 539–562 (1953) MATHGoogle Scholar
  19. 19.
    Simpson, H.C., Spector, S.J.: On bifurcation in finite elasticity: buckling of a rectangular rod. J. Elast. 92, 277–326 (2008) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Simpson, H.C., Spector, S.J.: On barrelling instabilities in finite elasticity. J. Elast. 14, 103–125 (1984) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Simpson, H.C., Spector, S.J.: On barrelling for a special material in finite elasticity. Q. Appl. Math. 14, 99–111 (1984) Google Scholar
  22. 22.
    Simpson, H.C., Spector, S.J.: On the positivity of the second variation in finite elasticity. Arch. Ration. Mech. Anal. 98, 1–30 (1987) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Thompson, J.L.: Some existence theorems for the traction boundary value problem of linearized elastostatics. Arch. Ration. Mech. Anal. 32, 369–399 (1969) MATHCrossRefGoogle Scholar
  24. 24.
    Valent, T.: Boundary Value Problems of Finite Elasticity. Local Theorems on Existence, Uniqueness, and Analytic Dependence on Data. Springer Tracts in Natural Philosophy, vol. 31. Springer, New York (1988) MATHGoogle Scholar
  25. 25.
    Wloka, J.: Partial Differential Equations. Cambridge University Press, New York (1987) MATHGoogle Scholar
  26. 26.
    Wloka, J., Rowley, B., Lawruk, B.: Boundary Value Problems for Elliptic Systems. Cambridge University Press, New York (1995) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Pablo V. Negrón-Marrero
    • 1
  • Errol Montes-Pizarro
    • 2
  1. 1.Department of MathematicsUniversity of Puerto RicoHumacaoPuerto Rico
  2. 2.Department of Mathematics and PhysicsUniversity of Puerto RicoCayeyPuerto Rico

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