A rigorous derivation of hemitropy in nonlinearly elastic rods

Timothy J. Healey

doi:10.3934/dcdsb.2011.16.265

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2011, Volume 16, Issue 1: 265-282. Doi: 10.3934/dcdsb.2011.16.265

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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

Received: April 2010

Revised: September 2010

Published online: April 2011

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Abstract

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.

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Mathematics Subject Classification: Primary: 74K10, 74Q05; Secondary: 34L30, 49J05.

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