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Journal of Elasticity

, Volume 105, Issue 1–2, pp 313–330 | Cite as

On the Stability of Incompressible Elastic Cylinders in Uniaxial Extension

  • Jeyabal Sivaloganathan
  • Scott J. Spector
Article

Abstract

Consider a cylinder (not necessarily of circular cross-section) that is composed of a hyperelastic material and which is stretched parallel to its axis of symmetry. Suppose that the elastic material that constitutes the cylinder is homogeneous, transversely isotropic, and incompressible and that the deformed length of the cylinder is prescribed, the ends of the cylinder are free of shear, and the sides are left completely free. In this paper it is shown that mild additional constitutive hypotheses on the stored-energy function imply that the unique absolute minimizer of the elastic energy for this problem is a homogeneous, isoaxial deformation. This extends recent results that show the same result is valid in 2-dimensions. Prior work on this problem had been restricted to a local analysis: in particular, it was previously known that homogeneous deformations are strict (weak) relative minimizers of the elastic energy as long as the underlying linearized equations are strongly elliptic and provided that the load/displacement curve in this class of deformations does not possess a maximum.

Keywords

Uniaxial tension Incompressible Homogeneous absolute minimizer 

Mathematics Subject Classification (2000)

74B20 35J50 49K20 74G65 

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References

  1. 1.
    Ambrosio, L., Fonseca, I., Marcellini, P., Tartar, L.: On a volume-constrained variational problem. Arch. Ration. Mech. Anal. 149, 23–47 (1999) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403 (1977) MATHCrossRefGoogle Scholar
  3. 3.
    Ball, J.M.: Constitutive inequalities and existence theorems in nonlinear elastostatics. In: Knops, R.J. (ed.) Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, vol. 1, pp. 187–241. Pitman, London (1977) Google Scholar
  4. 4.
    Ciarlet, P.G., Nečas, J.: Injectivity and self-contact in non-linear elasticity. Arch. Ration. Mech. Anal. 97, 171–188 (1987) CrossRefGoogle Scholar
  5. 5.
    Ciarlet, P.G.: Mathematical Elasticity, vol. 1. Elsevier, Amsterdam (1988) MATHGoogle Scholar
  6. 6.
    Del Piero, G.: Lower bounds for the critical loads of elastic bodies. J. Elast. 10, 135–143 (1980) MATHCrossRefGoogle Scholar
  7. 7.
    Del Piero, G., Rizzoni, R.: Weak local minimizers in finite elasticity. J. Elast. 93, 203–244 (2008) MATHCrossRefGoogle Scholar
  8. 8.
    Duda, F.P., Martins, L.C., Podio-Guidugli, P.: On the representation of the stored energy for elastic materials with full transverse response-symmetry. J. Elast. 51, 167–176 (1998) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Ericksen, J.L., Rivlin, R.S.: Large elastic deformations of homogeneous anisotropic materials. J. Ration. Mech. Anal. 3, 281–301 (1954) MathSciNetMATHGoogle Scholar
  10. 10.
    Fosdick, R., Foti, P., Fraddosio, A., Piccioni, M.D.: A lower bound estimate of the critical load for compressible elastic solids. Contin. Mech. Thermodyn. 22, 77–97 (2010) MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    Gelfand, I.M., Fomin, S.V.: Calculus of Variations, Dover, New York (1963) Google Scholar
  12. 12.
    Green, A.E., Adkins, J.E.: Large Elastic Deformations, 2nd edn. Clarendon Press, Oxford (1970) MATHGoogle Scholar
  13. 13.
    Gurtin, M.E.: An Introduction to Continuum Mechanics. Academic Press, San Diego (1981) MATHGoogle Scholar
  14. 14.
    Hill, R., Hutchinson, J.W.: Bifurcation phenomena in the plane tension test. J. Mech. Phys. Solids 23, 239–264 (1975) MathSciNetADSMATHCrossRefGoogle Scholar
  15. 15.
    Mielke, A.: Necessary and sufficient conditions for polyconvexity of isotropic functions. J. Convex Anal. 12, 291–314 (2005) MathSciNetMATHGoogle Scholar
  16. 16.
    Mizel, V.J.: On the ubiquity of fracture in nonlinear elasticity. J. Elast. 52, 257–266 (1998) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Mora-Coral, C.: Explicit energy-minimizers of incompressible elastic brittle bars under uniaxial extension. C. R. Math. 348, 1045–1048 (2010) Google Scholar
  18. 18.
    Müller, S., Tang, Q., Yan, B.S.: On a new class of elastic deformation not allowing for cavitation. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 11, 217–243 (1994) MATHGoogle Scholar
  19. 19.
    Ogden, R.W.: Large deformation isotropic elasticity—on the correlation of theory and experiment for incompressible rubberlike solids. Proc. R. Soc. Lond. A 326, 565–584 (1972) ADSMATHCrossRefGoogle Scholar
  20. 20.
    Ogden, R.W.: Non-Linear Elastic Deformations. Dover, New York (1984) Google Scholar
  21. 21.
    Rosakis, P.: Characterization of convex isotropic functions. J. Elast. 49, 257–267 (1997) MathSciNetCrossRefGoogle Scholar
  22. 22.
    Šilhavý, M.: Convexity conditions for rotationally invariant functions in two dimensions. In: Sequeira, A., et al. (ed.) Applied Nonlinear Analysis, pp. 513–530. Plenum, New York (1999) Google Scholar
  23. 23.
    Sivaloganathan, J., Spector, S.J.: Energy minimising properties of the radial cavitation solution in incompressible nonlinear elasticity. J. Elast. 93, 177–187 (2008) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Sivaloganathan, J., Spector, S.J.: On the symmetry of energy-minimising deformations in nonlinear elasticity I: Incompressible materials. Arch. Ration. Mech. Anal. 196, 363–394 (2010) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Sivaloganathan, J., Spector, S.J.: On the symmetry of energy-minimising deformations in nonlinear elasticity II: Compressible materials. Arch. Ration. Mech. Anal. 196, 395–431 (2010) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Sivaloganathan, J., Spector, S.J.: On the global stability of two-dimensional incompressible, elastic bars in uniaxial extension. Proc. R. Soc. Lond. A 466, 1167–1176 (2010) MathSciNetADSMATHCrossRefGoogle Scholar
  27. 27.
    Spector, S.J.: On the absence of bifurcation for elastic bars in uniaxial tension. Arch. Ration. Mech. Anal. 85, 171–199 (1984) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Steigmann, D.J.: On isotropic, frame-invariant, polyconvex strain-energy functions. Q. J. Mech. Appl. Math. 56, 483–491 (2003) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Wesołowski, Z.: Stability in some cases of tension in the light of the theory of finite strain. Arch. Mech. Stosow. 14, 875–900 (1962) MATHGoogle Scholar
  30. 30.
    Wesołowski, Z.: The axially symmetric problem of stability loss of an elastic bar subject to tension. Arch. Mech. Stosow. 15, 383–394 (1963) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of BathBathUK
  2. 2.Department of MathematicsSouthern Illinois UniversityCarbondaleUSA

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