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Unusual extension-torsion-inflation couplings in pressurized thin circular tubes with helical anisotropy. (English) Zbl 07273334
Summary: We present a thin tube formulation for coupled extension-torsion-inflation deformation in helically reinforced pressurized circular tubes. Both compressible and incompressible tubes are considered. On applying the thin tube limit, the nonlinear ordinary differential equation to obtain the in-plane radial displacement is converted into a set of two simple algebraic equations for the compressible case and one equation for the incompressible case. This allows us to obtain analytical expressions, in terms of the tube’s intrinsic twist, material constants, and the applied pressure, which can predict whether such tubes would overwind/unwind on being infinitesimally stretched or exhibit positive/negative Poisson’s effect. We further show numerically that such tubes can be tuned to generate initial overwinding followed by rapid unwinding as observed during finite stretching of a torsionally relaxed DNA. Finally, we demonstrate that such tubes can also exhibit usual deflation initially followed by unusual inflation as the tube is finitely stretched.
74 Mechanics of deformable solids
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[1] Holzapfel, GA, Ogden, RW. Constitutive modelling of arteries. Proc R Soc A 2010; 466: 1551-1597. · Zbl 1197.74075
[2] Ghosh, A, Fischer, P. Controlled propulsion of artificial magnetic nanostructured propellers. Nano Lett 2009; 9: 2243-2245
[3] Connolly, F, Polygerinos, P, Walsh, CJ, et al. Mechanical programming of soft actuators by varying fiber angle. Soft Robot 2015; 2(1): 26-32.
[4] Kumar, A, Mukherjee, S, Paci, JT, et al. A rod model for three dimensional deformations of single-walled carbon nanotubes. Int J Sol Structs 2011; 48(20): 2849-2858.
[5] Kumar, A, Kumar, S, Gupta, P. A helical Cauchy-Born rule for special Cosserat rod modeling of nano and continuum rods. J Elast 2016; 124(1): 81-106. · Zbl 1338.74012
[6] Lionnet, T, Joubaud, S, Lavery, R, et al. Wringing out DNA. Phys Rev Lett 2006; 96(17): 178102.
[7] Bozec, L, van der Heijden, G, Horton, M. Collagen fibrils: nanoscale ropes. Biophys J 2007 92(1): 70-75.
[8] Healey, TJ. Material symmetry and chirality in nonlinearly elastic rods. Math Mech Solids 2002; 7(4): 405-420. · Zbl 1090.74610
[9] Chandraseker, K, Mukherjee, S. Coupling of extension and twist in single-walled carbon nanotubes. J App Mech 2006; 73(2): 315-326. · Zbl 1111.74348
[10] Upmanyu, M, Wang, HL, Liang, HY, et al. Strain-dependent twist-stretch elasticity in chiral filaments. J R Soc Interface 2008; 5(20): 303-310.
[11] Kumar, A, Mukherjee, S. A geometrically exact rod model including in-plane cross-sectional deformation. J App Mech 2011; 78(1): 011010.
[12] Durickovic, B, Goriely, A, Maddocks, JH. Twist and stretch of helices explained via the Kirchhoff-Love rod model of elastic filaments. Phys Rev Lett 2013; 111(10): 108103.
[13] Singh, R, Kumar, S, Kumar, A. Effect of intrinsic twist and orthotropy on extension-twist-inflation coupling in compressible circular tubes. J Elast 2017; 128(2): 175-201. · Zbl 1374.74017
[14] Gore, J, Bryant, Z, Nöllmann, M, et al. DNA overwinds when stretched. Nature 2006; 442(7104): 836-839.
[15] Gross, P, Laurens, N, Oddershede, LB, et al. Quantifying how DNA stretches, melts and changes twist under tension. Nat Phys 2011; 7(9): 731-736.
[16] Fozdar, DY, Soman, P, Lee, JW, et al. Three-dimensional polymer constructs exhibiting a tunable negative Poisson’s ratio. Adv Funct Mater 2011; 21(14): 2712-2720.
[17] Lee, JW, Soman, P, Park, JH, et al. A tubular biomaterial construct exhibiting a negative Poisson’s ratio. PloS One 2016; 11(5): 0155681.
[18] Raamachandran, J, Jayavenkateshwaran, K. Modeling of stents exhibiting negative Poisson’s ratio effect. Methods Biomech Biomed Eng 2007; 10(4): 245-255.
[19] Gent, AN, Rivlin, RS. Experiments on the mechanics of rubber II: the torsion, inflation and extension of a tube. Proc Phys Soc B 1952; 65(7): 487.
[20] Green, AE. Finite elastic deformation of compressible isotropic bodies. Proc Roy Soc A 1955; 227: 271-278. · Zbl 0064.19403
[21] Ogden, RW, Chadwick, P. On the deformation of solid and tubular cylinders of incompressible isotropic elastic material. J Mech Phys Solids 1972; 20(2), 77-90. · Zbl 0232.73074
[22] Kirkinis, E, Ogden, RW. On extension and torsion of a compressible elastic circular cylinder. Math Mech Solids 2002; 7(4): 373-392. · Zbl 1045.74013
[23] Goriely, A, Tabor, M. Rotation, inversion and perversion in anisotropic elastic cylindrical tubes and membranes. Proc R Soc A 2013; 256(2153). DOI: 10.1098/rspa.2013.0011. · Zbl 1371.74210
[24] Horgan, CO, Murphy, JG. Extension or compression induced twisting in fiber-reinforced nonlinearly elastic circular cylinders. J Elast 2016; 125(1): 73-85. · Zbl 1348.74050
[25] Merodio, J, Ogden, RW. Extension, inflation and torsion of a residually stressed circular cylindrical tube. Continuum Mech Therm 2016; 28(1-2): 157. · Zbl 1348.74052
[26] Polignone, DA, Horgan, CO. Pure torsion of compressible non-linearly elastic circular cylinders. Quart Appl Math 1991; 49(3): 591-607. · Zbl 0751.73014
[27] Zidi, M. Finite torsion and anti-plane shear of a compressible hyperelastic and transversely isotropic tube. Int J Eng Sci 2000; 38(13): 1487-1496.
[28] Zubov, LM. The non-linear Saint-Venant problem of the torsion, stretching and bending of a naturally twisted rod. J Appl Math Mech 2006; 70(2): 300-310.
[29] Iesan, D. Chiral effects in uniformly loaded rods. J Mech Phys Solids 2010; 58(9): 1272-1285. · Zbl 1208.74071
[30] Haughton, DM, Ogden, RW. Bifurcation of inflated circular cylinders of elastic material under axial loading - I: membrane theory for thin-walled tubes. J Mech Phys Solids 1979; 27(3): 179-212. · Zbl 0412.73065
[31] Itskov, M, Aksel, N. Elastic constants and their admissible values for incompressible and slightly compressible anisotropic materials. Acta Mech 2002; 157(1): 81-96. · Zbl 1027.74003
[32] 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 2004; 41(14): 3833-3848. · Zbl 1079.74516
[33] Blatz, PJ, Ko, WL. Application of finite elastic theory to the deformation of rubbery materials. Trans Soc Rheology 1962; 6(1): 223-252.
[34] Horgan, CO. Remarks on ellipticity for the generalized Blatz-Ko constitutive model for a compressible nonlinearly elastic solid. J Elast 1996; 42(2): 165-176. · Zbl 0852.73019
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