zbMATH — the first resource for mathematics

Effect of intrinsic twist and orthotropy on extension-twist-inflation coupling in compressible circular tubes. (English) Zbl 1374.74017
Summary: We present effects of intrinsic twist and material orthotropy on extension-twist-inflation coupling in circular tubes about their stress-free state. Simple analytical expressions for coupling stiffnesses corresponding to extension-twist, twist-inflation and extension-inflation couplings are obtained. We show that the sign of the extension-twist coupling stiffness, which governs initial overwinding/unwinding in tubes during their extension, is not just dependent on the tube’s intrinsic twist but also on two other parameters: ratio of the Young’s moduli in the lateral surface of orthotropic tube and the excess of the Poisson’s ratio from an isotropy condition. By tuning these two parameters, one can generate the counter-intuitive overwinding as reported earlier in the case of DNA. Similarly, we show that even with positive Poisson’s ratio, an intrinsically twisted tube could inflate on being stretched. We also present a scheme to obtain all the relevant stiffnesses of chiral single-walled carbon nanotubes from a “one-atom unit cell” calculation. These stiffnesses, when plotted vs. the nanotube’s chirality, exhibit interesting periodicity which has its origin in the 6-fold symmetry of graphene. This trend is captured in the continuum model for a nanotube when it is assumed to be comprised of three families of parallel fibers on its lateral surface.

74B20 Nonlinear elasticity
74A25 Molecular, statistical, and kinetic theories in solid mechanics
74Q15 Effective constitutive equations in solid mechanics
Full Text: DOI
[1] Aggeli, A.; Nyrkova, I.A.; Bell, M.; Harding, R.; Carrick, L.; McLeish, T.C.B.; Semenov, A.N.; Boden, N., Hierarchical self-assembly of chiral rod-like molecules as a model for peptide \(β \)-sheet tapes, ribbons, fibrils, and fibers, Proc. Natl. Acad. Sci. USA, 98, 11857-11862, (2001)
[2] Antman, S.S.; Carbone, E.R., Shear and necking instabilities in nonlinear elasticity, J. Elast., 7, 125-151, (1977) · Zbl 0356.73048
[3] Antman, S.S.: Nonlinear Problems of Elasticity. Springer, New York (1995) · Zbl 0820.73002
[4] Audoly, B.; Hutchinson, J.W., Analysis of necking based on a one-dimensional model, J. Mech. Phys. Solids, 97, 68-91, (2016)
[5] Bertoldi, K.; Reis, P.M.; Willshaw, S.; Mullin, T., Negative poisson’s ratio behavior induced by an elastic instability, Adv. Mater., 22, 361-366, (2010)
[6] Bozec, L.; Heijden, G.; Horton, M., Collagen fibrils: nanoscale ropes, Biophys. J., 92, 70-75, (2007)
[7] Brenner, D.W., Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films, Phys. Rev. B, 42, 9458, (1990)
[8] Cai, W.; Fong, W.; Elsen, E.; Weinberger, C.R., Torsion and bending periodic boundary conditions for modeling the intrinsic strength of nanowires, J. Mech. Phys. Solids, 56, 3242-3258, (2008) · Zbl 1176.74020
[9] Chandraseker, K.; Mukherjee, S., Coupling of extension and twist in single-walled carbon nanotubes, J. Appl. Mech., 73, 315-326, (2006) · Zbl 1111.74348
[10] Chang, T., A molecular based anisotropic shell model for single-walled carbon nanotubes, J. Mech. Phys. Solids, 58, 1422-1433, (2010)
[11] Coleman, B.D., Necking and drawing in polymeric fibers under tension, Arch. Ration. Mech. Anal., 83, 115, (1983) · Zbl 0535.73016
[12] Durickovic, B.; Goriely, A.; Maddocks, J.H., Twist and stretch of helices explained via the Kirchhoff-love rod model of elastic filaments, Phys. Rev. Lett., 111, (2013)
[13] Gent, A.N.; Rivlin, R.S., Experiments on the mechanics of rubber II: the torsion, inflation and extension of a tube, Proc. Phys. Soc. B, 65, 487-501, (1952)
[14] Gore, J.; Bryant, Z.; Nöllmann, M.; Le, M.U.; Cozzarelli, N.R.; Bustamante, C., DNA overwinds when stretched, Nature, 442, 836-839, (2006)
[15] Gupta, P.; Kumar, A., Effect of material nonlinearity on spatial buckling of nanorods and nanotubes, J. Elast., 126, 155-171, (2017) · Zbl 1354.74023
[16] Goriely, A.; Tabor, M., Rotation, inversion and perversion in anisotropic elastic cylinderical tubes and membranes, Proc. R. Soc. A, Math. Phys. Eng. Sci., 469, (2013) · Zbl 1371.74210
[17] Green, A.E.; Laws, N., A general theory of rods, Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci., 293, 145-155, (1966) · Zbl 0181.52505
[18] Gross, P.; Laurens, N.; Oddershede, L.B.; Bockelmann, U.; Peterman, E.J.; Wuite, G.J., Quantifying how DNA stretches, melts and changes twist under tension, Nat. Phys., 7, 731-736, (2011)
[19] Healey, T.J., Material symmetry and chirality in nonlinearly elastic rods, Math. Mech. Solids, 7, 405-420, (2002) · Zbl 1090.74610
[20] Holzapfel, G.A.; Ogden, R.W., Constitutive modeling of arteries, Proc. R. Soc. A, 466, 1551-1597, (2010) · Zbl 1197.74075
[21] Horgan, C.O.; Murphy, J.G., Extension or compression induced twisting in fiber-reinforced nonlinearly elastic circular cylinders, J. Elast., 125, 73-85, (2016) · Zbl 1348.74050
[22] Iesan, D.; Quintanilla, R., On the deformation of inhomogeneous orthotropic elastic cylinders, Eur. J. Mech. A, Solids, 26, 999-1015, (2007) · Zbl 1122.74031
[23] Iesan, D., Chiral effects in uniformly loaded rods, J. Mech. Phys. Solids, 58, 1272-1285, (2010) · Zbl 1208.74071
[24] Itskov, M.; Aksel, N., A class of orthotropic and transversely isotropic hyperelastic constitutive models based on a polyconvex strain energy function, Int. J. Solids Struct., 41, 3833-3848, (2004) · Zbl 1079.74516
[25] James, R.D., Objective structures, J. Mech. Phys. Solids, 54, 2354-2390, (2006) · Zbl 1120.74312
[26] Kumar, A.; Healey, T.J., A generalized computational approach to stability of static equilibria of nonlinearly elastic rods in the presence of constraints, Comput. Methods Appl. Mech. Eng., 199, 1805-1815, (2010) · Zbl 1231.74484
[27] Kumar, A.; Mukherjee, S., A geometrically exact rod model including in-plane cross-sectional deformation, J. Appl. Mech., 78, (2011)
[28] Kumar, A.; Mukherjee, S.; Paci, J.T.; Chandraseker, K.; Schatz, G.C., A rod model for three dimensional deformations of single-walled carbon nanotubes, Int. J. Solids Struct., 48, 2849-2858, (2011)
[29] Kumar, A.; Kumar, S.; Gupta, P., A helical Cauchy-Born rule for special Cosserat rod modeling of nano and continuum rods, J. Elast., 124, 81-106, (2016) · Zbl 1338.74012
[30] Kurbatova, N.V.; Ustinov, Y.A., Saint-Venant problem for solids with helical anisotropy, Contin. Mech. Thermodyn., 28, 465-476, (2016) · Zbl 1348.74121
[31] Lakes, R.S.; Benedict, R.L., Noncentrosymmetry in micropolar elasticity, Int. J. Eng. Sci., 20, 1161-1167, (1982) · Zbl 0491.73004
[32] Lakes, R.S., Foam structures with negative poisson’s ratio, Science, 235, 1038-1040, (1987)
[33] Lionnet, T.; Joubaud, S.; Lavery, R.; Bensimon, D.; Croquette, V., Wringing out DNA, Phys. Rev. Lett., 96, (2006)
[34] Maultzsch, J.; Telg, H.; Reich, S.; Thomsen, C., Radial breathing mode of single-walled carbon nanotubes: optical transition energies and chiral-index assignment, Phys. Rev. B, 72, (2005)
[35] Merodio, J.; Ogden, R.W., Extension, inflation and torsion of a residually stressed circular cylinderical tube, Contin. Mech. Thermodyn., 28, 157-174, (2016) · Zbl 1348.74052
[36] Ogden, R.W.; Chadwick, P., On the deformation of solid and tubular cylinders of incompressible isotropic elastic materials, J. Mech. Phys. Solids, 20, 77-90, (1972) · Zbl 0232.73074
[37] Poynting, J.H., On pressure perpendicular to the shear planes in finite pure shears, and on the lengthening of loaded wires when twisted, Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci., 82, 546-559, (1909) · JFM 40.0875.02
[38] Raamachandran, J.; Jayavenkateshwaran, K., Modeling of stents exhibiting negative poisson’s ratio effect, Comput. Methods Biomech. Biomed. Eng., 10, 245-255, (2007)
[39] Ru, Q.C., Chirality-dependent mechanical behavior of carbon nanotubes based on an anisotropic elastic shell model, Math. Mech. Solids, 14, 88-101, (2009) · Zbl 1257.74119
[40] Upamanyu, M.; Wang, H.L.; Liang, H.Y.; Mahajan, R., Strain dependent twist stretch elasticity in chiral filaments, J. R. Soc. Interface, 20, 303-310, (2008)
[41] Wang, M.D.; Yin, H.; Landick, R.; Gelles, J.; Block, S.M., Stretching DNA with optical tweezers, Biophys. J., 72, 1335-1346, (1997)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.