Skip to main content
Log in

Post-buckling behavior of a non-linearly hyperelastic thin rod with cross-section invariant under the dihedral group Dn

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  • J. C. Alexander & S. S. Antman (1982), The ambiguous twist of Love. Quart. Appl. Math. 40, 83–92.

    Google Scholar 

  • S. S. Antman (1972), The Theory of Rods, Handbuch der Physik, Vol. VIa/2, Springer-Verlag, Berlin-Heidelberg-New York.

    Google Scholar 

  • S. S. Antman (1976), Ordinary differential equations of nonlinear elasticity I: Foundations of the theories of nonlinearly elastic rods and shells. Arch. Rational Mech. Anal. 61, 307–351.

    Google Scholar 

  • S. S. Antman & K. B. Jordan (1975), Qualitative aspects of the spatial deformation of nonlinearly elastic rods. Proc. Roy. Soc. Edinburgh 73A, 85–105.

    Google Scholar 

  • S. S. Antman & C. S. Kenney (1981), Large buckled states of nonlinearly elastic rods under torsion, thrust, and gravity. Arch. Rational Mech. Anal. 76, 339–354.

    Google Scholar 

  • J. M. Ball & D. G. Schaeffer (1983), Bifurcation and stability of homogeneous equilibrium configurations of an elastic body under dead-load tractions. Math. Proc. Camb. Phil. Soc. 94, 315–339.

    Google Scholar 

  • E. Buzano (1984), Secondary bifurcations of a thin rod under axial compression. To appear in SIAM J. of Math. Anal.

  • E. Buzano & G. Geymonat (1983), Geometrical methods in some bifurcation problems of elasticity. Trends and Applications of Pure Mathematics to Mechanics, P. G. Ciarlet & M. Roseau (eds.), Springer Lecture Notes in Physics 195, 5–19.

  • P. G. Ciarlet (1980), A justification of the von Kármán equations. Arch. Rational Mech. Anal. 80, 295–331.

    Google Scholar 

  • G. Geymonat & G. Raugel (1984), Finite dimensional approximation of some bifurcation problems in presence of symmetries. Numerical Methods for Bifurcation Problems, T. Küpper, H. Mittelmann & H. Weber (eds.), Birkhäuser-Verlag, Basel, 359–384.

    Google Scholar 

  • M. Golubitsky & J. Marsden (1983), The Morse Lemma in infinite dimensions via singularity theory. SIAM J. of Math. Anal. 14, 1037–1044.

    Google Scholar 

  • M. Golubitsky & D. Schaeffer (1979), A theory for imperfect bifurcation via singularity theory. Comm. Pure Appl. Math. 32, 21–98.

    Google Scholar 

  • M. Golubitsky & D. Schaeffer (1979), Imperfect bifurcation in the presence of symmetry. Comm. Math. Phys. 67, 205–232.

    Google Scholar 

  • M. Golubitsky & D. Schaeffer (1982), Bifurcations with O(3)-symmetry including applications to the Bènard problem. Comm. Pure Appl. Math. 35, 81–111.

    Google Scholar 

  • M. Golubitsky & D. Schaeffer (1983), A discussion of symmetry and symmetry breaking. Proc. Symposia in Pure Math. of AMS, 40, Part 1, 499–515.

    Google Scholar 

  • A. E. Green & J. E. Adkins (1960), Large Elastic Deformations and Non-Linear Continuum Mechanics. Oxford, Clarendon Press.

    Google Scholar 

  • D. Gromoll & W. Meyer (1969), On differentiable functions with isolated critical points. Topology 8, 361–369.

    Google Scholar 

  • J. B. Keller (1960), The shape of the strongest column. Arch. Rational Mech. Anal. 5, 275–285.

    Google Scholar 

  • G. Kirchhoff (1859), Über das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes. J. reine angew. Math. (Crelle) 56, 285–313.

    Google Scholar 

  • A. E. Love (1927), A Treatise on the Mathematical Theory of Elasticity. Fourth Edition, Cambridge University Press, (Dover reprint, 1944).

  • R. Magnus (1979), Universal unfoldings in Banach spaces: reduction and stability. Math. Proc. Camb. Phil. Soc. 86, 41–55.

    Google Scholar 

  • R. Magnus (1980), A splitting lemma for non-reflexive Banach spaces. Math. Scan. 46, 118–128.

    Google Scholar 

  • N. Olhoff & S. H. Rasmussen (1977), On single and bimodal optimum buckling loads of clamped columns. Inter. J. Solids Structures 13, 605–614.

    Google Scholar 

  • V. Poenaru (1976), Singularités C en présence de symétrie. Lecture Notes in Mathematics No. 105, Springer-Verlag, Berlin-Heidelberg-New York.

    Google Scholar 

  • T. Poston & I. Stewart (1978), Catastrophe Theory and its Applications. Pitman, London.

    Google Scholar 

  • G. Raugel (1984), Approximation numérique à problèmes nonlinéaires. Thèse d'Etat, Université de Rennes-Beaulieu, Rennes, France.

    Google Scholar 

  • D. H. Sattinger (1983), Branching in the presence of symmetry. CBMS-NSF Regional Conference Series in Applied Mathematics # 40, SIAM, Philadelphia.

    Google Scholar 

  • G. Schwarz (1975), Smooth functions invariant under the action of a compact Lie group. Topology 14, 63–68.

    Google Scholar 

  • G. F. Smith & R. S. Rivlin (1958), The strain-energy function for anisotropic elastic materials. Trans. Amer. Math. Soc. 88, 175–193.

    Google Scholar 

  • F. Treves (1967), Topological Vector Spaces, Distributions and Kernels. Academic Press, New York.

    Google Scholar 

  • C. Truesdell & R. Toupin (1960), The Classical Field Theories. Handbuch der Physik, vol. III/1, 225–739. Springer-Verlag, Berlin-Heidelberg-New York.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by S. Antman

Rights and permissions

Reprints and permissions

About this article

Cite this article

Buzano, E., Geymonat, G. & Poston, T. Post-buckling behavior of a non-linearly hyperelastic thin rod with cross-section invariant under the dihedral group Dn . Arch. Rational Mech. Anal. 89, 307–388 (1985). https://doi.org/10.1007/BF00250729

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00250729

Keywords

Navigation