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On the Stability of Incompressible Elastic Cylinders in Uniaxial Extension

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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.

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References

  1. Ambrosio, L., Fonseca, I., Marcellini, P., Tartar, L.: On a volume-constrained variational problem. Arch. Ration. Mech. Anal. 149, 23–47 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403 (1977)

    Article  MATH  Google Scholar 

  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. Ciarlet, P.G., Nečas, J.: Injectivity and self-contact in non-linear elasticity. Arch. Ration. Mech. Anal. 97, 171–188 (1987)

    Article  Google Scholar 

  5. Ciarlet, P.G.: Mathematical Elasticity, vol. 1. Elsevier, Amsterdam (1988)

    MATH  Google Scholar 

  6. Del Piero, G.: Lower bounds for the critical loads of elastic bodies. J. Elast. 10, 135–143 (1980)

    Article  MATH  Google Scholar 

  7. Del Piero, G., Rizzoni, R.: Weak local minimizers in finite elasticity. J. Elast. 93, 203–244 (2008)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ericksen, J.L., Rivlin, R.S.: Large elastic deformations of homogeneous anisotropic materials. J. Ration. Mech. Anal. 3, 281–301 (1954)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  Google Scholar 

  11. Gelfand, I.M., Fomin, S.V.: Calculus of Variations, Dover, New York (1963)

    Google Scholar 

  12. Green, A.E., Adkins, J.E.: Large Elastic Deformations, 2nd edn. Clarendon Press, Oxford (1970)

    MATH  Google Scholar 

  13. Gurtin, M.E.: An Introduction to Continuum Mechanics. Academic Press, San Diego (1981)

    MATH  Google Scholar 

  14. Hill, R., Hutchinson, J.W.: Bifurcation phenomena in the plane tension test. J. Mech. Phys. Solids 23, 239–264 (1975)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Mielke, A.: Necessary and sufficient conditions for polyconvexity of isotropic functions. J. Convex Anal. 12, 291–314 (2005)

    MathSciNet  MATH  Google Scholar 

  16. Mizel, V.J.: On the ubiquity of fracture in nonlinear elasticity. J. Elast. 52, 257–266 (1998)

    Article  MathSciNet  Google Scholar 

  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. 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)

    MATH  Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  20. Ogden, R.W.: Non-Linear Elastic Deformations. Dover, New York (1984)

    Google Scholar 

  21. Rosakis, P.: Characterization of convex isotropic functions. J. Elast. 49, 257–267 (1997)

    Article  MathSciNet  Google Scholar 

  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. Sivaloganathan, J., Spector, S.J.: Energy minimising properties of the radial cavitation solution in incompressible nonlinear elasticity. J. Elast. 93, 177–187 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. Spector, S.J.: On the absence of bifurcation for elastic bars in uniaxial tension. Arch. Ration. Mech. Anal. 85, 171–199 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  28. Steigmann, D.J.: On isotropic, frame-invariant, polyconvex strain-energy functions. Q. J. Mech. Appl. Math. 56, 483–491 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  30. Wesołowski, Z.: The axially symmetric problem of stability loss of an elastic bar subject to tension. Arch. Mech. Stosow. 15, 383–394 (1963)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

    Jeyabal Sivaloganathan

  2. Department of Mathematics, Southern Illinois University, Carbondale, IL, 62901, USA

    Scott J. Spector

Authors
  1. Jeyabal Sivaloganathan
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  2. Scott J. Spector
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Correspondence to Scott J. Spector.

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Dedicated to the memory of Donald E. Carlson.

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Sivaloganathan, J., Spector, S.J. On the Stability of Incompressible Elastic Cylinders in Uniaxial Extension. J Elast 105, 313–330 (2011). https://doi.org/10.1007/s10659-011-9330-9

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  • DOI: https://doi.org/10.1007/s10659-011-9330-9

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