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Chiral effects in uniformly loaded rods. (English) Zbl 1208.74071
Summary: Examples of chiral materials include some auxetic materials, bones, some honeycomb structures, as well as composites with inclusions. The chiral effects cannot be described within classical elasticity. In the context of the linear theory of Cosserat elastic solids, we investigate the deformation of a chiral rod subjected to tractions on the lateral surface, to body loads, and to resultant forces and moments on the ends. The work is motivated by the recent interest in the using of the Cosserat elastic solid as model for auxetic composites, carbon nanotubes and bones. The three-dimensional problem is reduced to the study of some generalized plane strain problems. New chiral effects are presented. In the case of cylinders of arbitrary cross-section, the flexure produced by a transversal force, in contrast with the case of achiral materials, is accompanied by extension and bending by terminal couples. The body loads and the tractions on the lateral surface produce extension, flexure, torsion, bending by terminal couples and a plane strain. It is shown that a uniform pressure acting on the lateral surface of a chiral circular cylinder does not produce bending effects.

MSC:
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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[1] Chandraseker, K.; Mukherjee, S., Coupling of extension and twist in single-walled carbon nanotubes, J. appl. mech., 73, 315-326, (2006) · Zbl 1111.74348
[2] Chandraseker, K.; Mukherjee, S.; Paci, J.T.; Schatz, G.C., An atomistic-continuum Cosserat rod model of carbon nanotubes, J. mech. phys. solids, 57, 932-958, (2009)
[3] Donescu, S.; Chiroiu, V.; Munteanu, L., On the Young’s modulus of an auxetic composite structure, Mech. res. commun., 36, 294-301, (2009) · Zbl 1258.74177
[4] Dyszlewicz, J., Micropolar theory of elasticity, (2004), Springer Berlin, Heidelberg, New York · Zbl 0247.73002
[5] Eringen, A.C., Microcontinuum field theories, I: foundations and solids, (1999), Springer New York, Berlin, Heidelberg · Zbl 0953.74002
[6] Fichera, G., Existence theorems in elasticity, () · Zbl 0317.73008
[7] Guz, I.A.; Rodger, A.A.; Guz, A.N.; Rushchitsky, I.I., Developing the mechanical models for nanomaterials, Compos. part A appl. sci. manuf., 38, 1234-1250, (2007)
[8] Haijun, Z.; Zhong-can, O., Bending and twisting elasticity: a revised marko – sigga model of DNA chirality, Phys. rev. E, 58, 4816-4821, (1998)
[9] Healey, T.J., Material symmetry and chirality in nonlinearly elastic rods, Math. mech. solids, 7, 405-420, (2002) · Zbl 1090.74610
[10] Ieşan, D., On saint-Venant’s problem, Arch. ration. mech. anal., 91, 363-373, (1986) · Zbl 0602.73005
[11] Ieşan, D., Classical and generalized models of elastic rods, (2009), Chapman & Hall, CRC Press London, New York, Boca Raton · Zbl 1177.74001
[12] Jasiuk, I.; Ostoja-Starzewski, M., From lattices and composites to micropolar continua. analysis of materials with complex microstructure, () · Zbl 1090.74556
[13] Khatiashvili, G.M., 1983. Almansi-Michell problem for homogeneous and composed bodies, Izd. Metzniereba, Tbilisi. (in Russian). · Zbl 0545.73012
[14] Khurana, A.; Tomar, S.K., Longitudinal wave response of a chiral slab interposed between micropolar solid half-spaces, Int. J. solids struct., 46, 135-150, (2009) · Zbl 1168.74376
[15] Krishna Reddy, G.V.; Venkatasubramanian, N.K., On the flexural rigidity of a micropolar elastic circular cylinder, J. appl. mech., 45, 429-431, (1978)
[16] Kunin, I.A., Elastic media with microstructure II. Springer series in solid state sciences, vol. 44, (1983), Springer-Verlag Berlin · Zbl 0167.54401
[17] Lakes, R.S.; Benedict, R.L., Noncentrosymmetry in micropolar elasticity, Int. J. eng. sci., 29, 1161-1167, (1982) · Zbl 0491.73004
[18] Lakes, R.S.; Yoon, H.S.; Katz, J.L., Slow compressional wave propagation in wet human and bovine cortical bone, Science, 220, 513-515, (1983)
[19] Lakes, R.S., Elastic freedom in cellular solids and composite materials, (), 129-153 · Zbl 0937.74017
[20] Lakes, R., Elastic and viscoelastic behaviour of chiral materials, Int. J. mech. sci., 43, 1579-1589, (2001) · Zbl 1049.74012
[21] Natroshvili, D.; Giorgashvili, L.; Stratis, I.G., Representation formulae of general solutions in the theory of hemitropic elasticity, Q.J. mech. appl. math., 59, 451-474, (2006) · Zbl 1107.74005
[22] Prall, D.; Lakes, R.S., Properties of a chiral honeycomb with a Poisson’s ratio-1, Int. J. mech. sci., 39, 305-314, (1997) · Zbl 0894.73018
[23] Ro, R., Elastic activity of the chiral medium, J. appl. phys., 85, 2508-2513, (1999)
[24] Rubin, M.B., Cosserat theories: shells, rods and points, (2000), Kluwer Academic Publisher Dordrecht · Zbl 0984.74003
[25] Truesdell, C.; Toupin, R., The classical field theories, ()
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