zbMATH — the first resource for mathematics

Extension or compression induced twisting in fiber-reinforced nonlinearly elastic circular cylinders. (English) Zbl 1348.74050
Summary: The problem of pure extension or compression of a solid circular fiber-reinforced cylinder is considered in the context of the theory of nonlinear elasticity. Here we are interested in a particular feature of this problem which only occurs for certain classes of incompressible anisotropic hyperelastic materials and is completely absent in the isotropic case. We show that pure extension or compression of a transversely isotropic fiber-reinforced cylinder can induce twisting of the cylinder. The transverse isotropy is due to a single family of helically wound extensible fibers distributed throughout the cylinder. For general pitch angles, it is shown that in the absence of an applied torsional moment, the cylinder subjected to an applied tensile or compressive force exhibits an induced twisting due to mechanical anisotropy. Such an induced twist does not occur when the fibers are oriented longitudinally or azimuthally. It is also shown that at a special pitch angle ( a “magic” angle), the initial slope of the axial force and induced moment with respect to stretch is independent of the Young’s modulus of the fibers. The results on induced twist are described in detail for three specific strain-energy densities. These models are quadratic in the anisotropic invariants, linear in the isotropic strain invariants and are consistent with the linearized theory of elasticity for fiber-reinforced materials. The results are illustrated on using elastic moduli for muscles based on experimental findings of other authors.

74B20 Nonlinear elasticity
74G55 Qualitative behavior of solutions of equilibrium problems in solid mechanics
Full Text: DOI
[1] Adkins, J.E.; Rivlin, R.S., Large elastic deformations of isotropic materials. X: reinforcement by inextensible cords, Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci., 248, 201-223, (1955) · Zbl 0066.18802
[2] Beatty, M.F., Topics in finite elasticity: hyperelasticity of rubber, elastomers and biological tissue, Appl. Mech. Rev., 40, 1699-1734, (1989)
[3] Destrade, M.; Horgan, C.O.; Murphy, J.G., Dominant negative Poynting effect in simple shearing of soft tissues, J. Eng. Math., 95, 87-98, (2015) · Zbl 1360.74110
[4] Destrade, M.; Donald, B.; Murphy, J.G.; Saccomandi, G., At least three invariants are necessary to model the mechanical response of incompressible, transversely isotropic materials, Comput. Mech., 52, 959-969, (2013) · Zbl 1311.74117
[5] Demirkoparan, H.; Pence, T.J., Torsional swelling of a hyperelastic tube with helically wound reinforcement, J. Elast., 92, 61-90, (2008) · Zbl 1143.74017
[6] Demirkoparan, H.; Pence, T.J., Magic angles for fiber reinforcement in rubber-elastic tubes subject to pressure and swelling, Int. J. Non-Linear Mech., 68, 87-95, (2015)
[7] Fang, Y.; Pence, T.J.; Tan, X., Fiber-directed conjugated-polymer torsional actuator: nonlinear elasticity modeling and experimental validation, IEEE/ASME Trans. Mechatron., 16, 656-664, (2011)
[8] Feng, Y.; Okamoto, R.J.; Namani, R.; Genin, G.M.; Bayly, P.V., Measurements of mechanical anisotropy in brain tissue and implications for transversely isotropic material models of white matter, J. Mech. Behav. Biomed. Mater., 23, 117-132, (2013)
[9] Gennisson, J.-L.; Catheline, S.; Chaffa, S.; Fink, M., Transient elastography in anisotropic medium: application to the measurement of slow and fast shear wave speeds in muscles, J. Acoust. Soc. Am., 114, 536-541, (2003)
[10] Goriely, A.; Tabor, M., Rotation, inversion and perversion in anisotropic elastic cylindrical tubes and membranes, Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci., 469, 20130011, (2013) · Zbl 1371.74210
[11] Horgan, C.O.; Murphy, J.G., Finite extension and torsion of fiber-reinforced non-linearly elastic circular cylinders, Int. J. Non-Linear Mech., 47, 97-104, (2012)
[12] Horgan, C.O.; Murphy, J.G., On the modeling of extension-torsion experimental data for transversely isotropic biological soft tissues, J. Elast., 108, 179-191, (2012) · Zbl 1243.74016
[13] Horgan, C.O.; Murphy, J.G., Reverse Poynting effects in the torsion of soft biomaterials, J. Elast., 118, 127-140, (2015) · Zbl 1305.74021
[14] Horgan, C.O.; Smayda, M., The importance of the second strain invariant in the constitutive modeling of elastomers and soft biomaterials, Mech. Mater., 51, 43-52, (2012)
[15] Kim, D.; Segev, R., Various issues raised by the mechanics of an octopus’s arm, Math. Mech. Solids, (2016) · Zbl 1371.74211
[16] Morrow, D.A.; Haut Donahue, T.L.; Odegard, G.M.; Kaufman, K.R., Transversely isotropic tensile material properties of skeletal muscle tissue, J. Mech. Behav. Biomed. Mater., 3, 124-129, (2010)
[17] Murphy, J.G., Transversely isotropic biological, soft tissue must be modelled using both anisotropic invariants, Eur. J. Mech. A, Solids, 42, 90-96, (2013) · Zbl 1406.74501
[18] Papazoglou, S.; Rump, J.; Braun, J.; Sack, I., Shear wave group velocity inversion in MR elastography of human skeletal muscle, Magn. Reson. Med., 56, 489-497, (2006)
[19] Sinkus, R.; Tanter, M.; Catheline, S.; Lorenzen, J.; Kuhl, C.; Sondermann, E.; Fink, M., Imaging anisotropic and viscous properties of breast tissue by magnetic resonance-elastography, Magn. Reson. Med., 53, 372-387, (2005)
[20] Truesdell, C.; Noll, W.; Flugge, S. (ed.), The non-linear field theories of mechanics, No. III/3, (2004), Berlin
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.