Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Bifurcation of hemitropic elastic rods under axial thrust

Authors: Timothy J. Healey and Christopher M. Papadopoulos
Journal: Quart. Appl. Math. 71 (2013), 729-753
MSC (2010): Primary 74K10, 74G60; Secondary 74B20, 37G40
DOI: https://doi.org/10.1090/S0033-569X-2013-01308-7
Published electronically: August 29, 2013
MathSciNet review: 3136993
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Abstract: In this work we consider the analysis of unshearable, hemitropic hyperelastic rods under end thrust alone. Roughly speaking, a nominally straight hemitropic rod is rotationally invariant about its centerline but lacks the reflection symmetries characterizing isotropic rods. Consequently a constitutive coupling between extension and twist is natural. We provide a rigorous bifurcation analysis for such structures under ``hard'' axial loading. First, we show that the initial post-buckling behavior depends crucially upon the boundary conditions: if both ends are clamped against rotation, the initial buckled shape is spatial (nonplanar); if at least one end is unrestrained against rotation, the buckled rod is twisted but the centerline is planar. Second, we show that as with isotropic rods, nontrivial equilibria of hemitropic rods occur in discrete modes, but unlike the isotropic case, such equilibria need not be compressive but could also be tensile. Finally, we prove an exchange of stability between the trivial line of solutions and ``mode 1'' bifurcating branches in accordance with the usual theory.

References [Enhancements On Off] (What's this?)

  • 1. S.S. Antman and C.S Kenney, Large buckled states of nonlinearly elastic rods under torsion, thrust and gravity, Arch. Rat. Mech Anal. 76, 289-338 (1981). MR 628172 (82k:73043)
  • 2. S.S. Antman, The Nonlinear Problems of Elasticity, 2nd Ed., Springer-Verlag, New York, 2005. MR 2132247 (2006e:74001)
  • 3. M.G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8, 321-340 (1971). MR 0288640 (44:5836)
  • 4. G. Costello, The Theory of Wire Rope, Springer-Verlag, New York, 2007. MR 1101811 (92c:73104)
  • 5. L. Euler, Additamentum I de curvis elasticis, methodus invieniendi lineas curvas maximi minimivi proprietate gaudentes, Bousquent, Lausanne, in Opera Omnia I, Vol. 24, 234-297 (1744).
  • 6. T.J. Healey, Global bifurcation and continuation in the presence of symmetry with an application to solid mechanics, SIAM J. Math. Anal., 19, 824-839 (1988). MR 946645 (89k:58058)
  • 7. T.J. Healey, Material symmetry and chirality in nonlinearly elastic rods, Math. Mech. Solids 7, 405-420 (2002). MR 1921465 (2003g:74053)
  • 8. T.J. Healey, A rigorous derivation of hemitropy in nonlinearly elastic rods, DCDS-B 16, 265-282 (2011). MR 2799551
  • 9. J.B. Keller, Bifurcation theory for ordinary differential equations, in Bifurcation Theory and Nonlinear Eigenvalue Problems, Eds. J.B. Keller & S.S. Antman, W.A. Benjamin, Inc., New York, 1969.
  • 10. H. Kielheofer, Bifurcation Theory: An Introduction with Applications to PDE's, Springer-Verlag, New York, 2004. MR 2004250 (2004i:47133)
  • 11. A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th Ed, Cambridge University Press, Cambridge, 1927.
  • 12. R.S. Manning, K.A. Rogers, and J.H. Maddocks, Isoperimetric conjugate points with application to the stability of DNA minicircles, Proc. Roy. Soc. Lond. A 454, 3047-3074 (1998). MR 1664277 (99j:49032)
  • 13. J.F. Marco, Stretching must twist DNA, Europhys. Lett. 38, 183-188 (1997). MR 1446825 (98b:92004)
  • 14. C.M. Papadopoulos, Buckled States of Compressed Hemitropic Rods, Ph.D. Dissertation, Cornell University, Ithaca, NY, 1999.
  • 15. S.P. Timoshenko and J.M. Gere, Theory of Elastic Stability, 2nd Ed., McGraw-Hill, New York, 1951. MR 0134026 (24:B80)
  • 16. J.L. Synge and B.A. Griffith, Principles of Mechanics, 3rd Ed., McGraw-Hill, New York, 1959. MR 0106562 (21:5293)

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Additional Information

Timothy J. Healey
Affiliation: Department of Mathematics and Department of Mechanical & Aerospace Engineering, Cornell University, Ithaca, New York
Email: tjh10@cornell.edu

Christopher M. Papadopoulos
Affiliation: Department of Engineering Science and Materials, University of Puerto Rico, Mayagüez, Puerto Rico
Email: christopher.papadopoulos@upr.edu

DOI: https://doi.org/10.1090/S0033-569X-2013-01308-7
Keywords: Hemitropy, Cosserat rod, symmetry, bifurcation
Received by editor(s): December 14, 2011
Published electronically: August 29, 2013
Article copyright: © Copyright 2013 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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