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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**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)**

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC (2010):
74K10,
74G60,
74B20,
37G40

Retrieve articles in all journals with MSC (2010): 74K10, 74G60, 74B20, 37G40

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.