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July  2011, 16(1): 265-282. doi: 10.3934/dcdsb.2011.16.265

A rigorous derivation of hemitropy in nonlinearly elastic rods

Timothy J. Healey 1,

1. 

Department of Mathematics and Department of Mechanical & Aerospace Engineering, Cornell University, Ithaca, NY, United States

Received  April 2010 Revised  September 2010 Published  April 2011

We consider a class of nonlinearly hyperelastic rods with helical symmetry, cf. [7]. Such a rod is mechanically invariant under the symmetries of a circular-cylindrical helix. Examples include idealized DNA molecules, wire ropes and cables. We examine the limit as the pitch of the helix characterizing the symmetry approaches zero and show that the resulting model is a hemitropic rod. The former is mechanically invariant under all proper rotations about its centerline and generally possesses chirality or handedness in its mechanical response, cf. [7]. An isotropic rod is also rotationally invariant but, in addition, enjoys certain reflection symmetries, which rule out chirality. Isotropy implies hemitropy, but the converse is not generally true. We employ both averaging methods and methods of gamma convergence to obtain the effective or homogenized (hemitropic) problem, the latter not corresponding to a naïve average.
Keywords: Cosserat rods, effective elastic properties., nonlinearly elastic, Hemitropy.
Mathematics Subject Classification: Primary: 74K10, 74Q05; Secondary: 34L30, 49J0.
Citation: Timothy J. Healey. A rigorous derivation of hemitropy in nonlinearly elastic rods. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 265-282. doi: 10.3934/dcdsb.2011.16.265
References:
[1]

S. S. Antman, "Problems of Nonlinear Elasticity," Springer-Verlag, New York, 2cnd edition, 2005. Google Scholar

[2]

J. M. Ball, Remarques sur l'existence et la régularité des solutions d'elatostatique non linéar, Recent Contributions to Nonlinear Partial Differential Equations, eds., H Berestycki & H. Brezis, Pitman, London, (1981), 50-62.  Google Scholar

[3]

A. Braides, "Gamma Convergence for Beginners,'' Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.  Google Scholar

[4]

D. Cioranescu and P. Donato, "An Introduction to Homogenization,'' Oxford University Press, Oxford, 2002.  Google Scholar

[5]

B. Dacorogna, "Direct Methods in the Calculus of Variations,'' Springer-Verlag, New York, 1989.  Google Scholar

[6]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillation, Dynamical Systems, and Bifurcations of Vector Fields,'' Springer-Verlag, New York, 1983.  Google Scholar

[7]

T. J. Healey, Material symmetry and chirality in nonlinearly elastic rods, Math. Mech. Solids, 7 (2002), 405-420.  Google Scholar

[8]

T. J. Healey and P. Mehta, Straightforward computation of spatial equilibria of geometrically exact Cosserat rods, Int. J. Bifur.Chaos, 15 (2005), 949-965. doi: 10.1142/S0218127405012387.  Google Scholar

[9]

J. Jost and X. Li-Jost, "Calculus of Variations,'' Cambridge University Press, Cambridge, 2008.  Google Scholar

[10]

S. Kerhbaum and J. H. Maddocks, Effective properties of elastic rods with high intrinsic twist, in M. Deville and R. Owens, editors, "Proceedings of the 16th IMACS World Congress 2000,'' pp. 1-8, 2000. Google Scholar

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S. Mudaliar, M. Eng. Report, Cornell University, 2009. Google Scholar

[12]

R. T. Rockafellar, "Convex Analysis,'' Princeton University Press, Princeton, N.J., 1970.  Google Scholar

show all references

References:
[1]

S. S. Antman, "Problems of Nonlinear Elasticity," Springer-Verlag, New York, 2cnd edition, 2005. Google Scholar

[2]

J. M. Ball, Remarques sur l'existence et la régularité des solutions d'elatostatique non linéar, Recent Contributions to Nonlinear Partial Differential Equations, eds., H Berestycki & H. Brezis, Pitman, London, (1981), 50-62.  Google Scholar

[3]

A. Braides, "Gamma Convergence for Beginners,'' Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.  Google Scholar

[4]

D. Cioranescu and P. Donato, "An Introduction to Homogenization,'' Oxford University Press, Oxford, 2002.  Google Scholar

[5]

B. Dacorogna, "Direct Methods in the Calculus of Variations,'' Springer-Verlag, New York, 1989.  Google Scholar

[6]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillation, Dynamical Systems, and Bifurcations of Vector Fields,'' Springer-Verlag, New York, 1983.  Google Scholar

[7]

T. J. Healey, Material symmetry and chirality in nonlinearly elastic rods, Math. Mech. Solids, 7 (2002), 405-420.  Google Scholar

[8]

T. J. Healey and P. Mehta, Straightforward computation of spatial equilibria of geometrically exact Cosserat rods, Int. J. Bifur.Chaos, 15 (2005), 949-965. doi: 10.1142/S0218127405012387.  Google Scholar

[9]

J. Jost and X. Li-Jost, "Calculus of Variations,'' Cambridge University Press, Cambridge, 2008.  Google Scholar

[10]

S. Kerhbaum and J. H. Maddocks, Effective properties of elastic rods with high intrinsic twist, in M. Deville and R. Owens, editors, "Proceedings of the 16th IMACS World Congress 2000,'' pp. 1-8, 2000. Google Scholar

[11]

S. Mudaliar, M. Eng. Report, Cornell University, 2009. Google Scholar

[12]

R. T. Rockafellar, "Convex Analysis,'' Princeton University Press, Princeton, N.J., 1970.  Google Scholar

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