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Finite deformations of a hyperelastic, compressible and fibre reinforced tube. (English) Zbl 1027.74004
Summary: Finite torsion and axial stretch of a long, hyperelastic, compressible and circular tube is studied for the design of a prototype of small diameter vascular prosthesis. The analysis is carried out in the context of the finite elasticity theory by using a class of Ogden strain energy function augmented with unidirectional reinforcing that is characterized by a single additional constitutive parameter for strength of reinforcement. The highly nonlinear differential equations with variable coefficients governing the problem are solved numerically using a Runge–Kutta method. For different prestresses supported by the tube, the effects of the combined deformation on the stress distributions are presented.
Reviewer: Reviewer (Berlin)

74B20 Nonlinear elasticity
74L15 Biomechanical solid mechanics
92C10 Biomechanics
Full Text: DOI
[1] Beatty, M.F., Topics in finite elasticity: hyperelasticity of rubber, elastomers and biological tissues-with examples, Appl. mech. rev., 40, 1699-1734, (1987)
[2] Carroll, M.M., Finite strain solutions in compressible isotropic elasticity, J. elasticity, 20, 65-92, (1988) · Zbl 0654.73030
[3] Carroll, M.M.; Horgan, C.O., Finite strain solutions for a compressible elastic solid, Quart. appl. math., 48, 4, 767-780, (1990) · Zbl 0716.73037
[4] Cheref M., 1998. Approche mécanique à la conception d’une prothèse vasculaire de petit diamètre. PhD thesis, University Paris Val de Marne
[5] Ericksen, J.L., Deformations possible in every compressible isotropic, perfectly elastic material, J. math. phys., 34, 126-128, (1955) · Zbl 0064.42105
[6] Fung, Y.C., Biomechanics. mechanical properties of living tissues, (1993), Springer-Verlag New York
[7] Hayashi, K., Experimental approaches on measuring the mechanical properties ans constitutive laws of arterials walls, ASME J. biomech. engrg., 115, 481-488, (1993)
[8] How, T.V.; Clarke, R.M., The elastic properties of a polyurethane arterial prosthesis, J. biomech., 17, 8, 597-608, (1984)
[9] How, T.V.; Guidoin, R.; Young, S.K., Engineering design of vascular prostheses, J. engrg. med. part H, 61-71, (1992)
[10] Jiang, X.; Ogden, R.W., Some new solutions for the axial shear of a circular cylindrical tube of compressible elastic material, Internat. J. non-linear mech., 35, 361-369, (2000) · Zbl 1006.74013
[11] Kurashige, M., Instability of a transversely isotropic elastic slab subjected to axial loads, ASME J. appl. mech., 48, 351-356, (1981) · Zbl 0471.73050
[12] Le Tallec, P.; Vidrascu, M., A numerical method for solving equilibrium problems of compressible hyperelastic bodies in large deformations, Numer. math., 43, 199-224, (1984)
[13] Ogden, R.W., Large deformation isotropic elasticity: on the correlation of theory and experiment for compressible rubber-like solids, Proc. roy. soc. London ser. A, 328, 567-583, (1972) · Zbl 0245.73032
[14] Qiu, G.Y.; Pence, T.J., Remarks on the behavior of simple directionally reinforced incompressible nonlinearly elastic solids, J. elasticity, 49, 1-30, (1997) · Zbl 0964.74008
[15] Salacinski, S.; Goldner, G.S.; Guidiceandrea, A.; Hamilton, G.; Seifalian, A.M.; Edwards, A.; Carson, R.J., The mechanical behaviour of vascular grafts: a review, J. biomech. appl., 15, 3, 241-278, (2001)
[16] Sensenig, C.B., Non linear theory for the deformation of pre-stressed circular plates and ring, Comm. pure appl. math., 18, 147-161, (1965) · Zbl 0143.46101
[17] Spencer, A.J.M., Continuum theory of the mechanics of fibre-reinforced composites, (1984), Springer-Verlag New York
[18] Triantafyllidis, N.; Abeyaratne, R., Instability of finitely deformed fiber-reinforced elastic material, ASME J. appl. mech., 50, 149-156, (1983) · Zbl 0511.73036
[19] Vorp, D.A.; Rajagopal, K.R.; Smolinski, P.J.; Borovetz, H.S., Identification of elastic properties of homogeneous, orthotropic vascular segments in distension, J. biomech., 28, 5, 501-512, (1995)
[20] Wineman, A.S.; Waldon, W.K., Normal stress effects induced during circular shear of a compressible non-linear elastic cylinder, Internat. J. non-linear mech., 30, 3, 323-339, (1995) · Zbl 0853.73014
[21] Zidi, M.; Cheref, M.; Oddou, C., Finite elasticity modeling of vascular prostheses mechanics, Eur. phys. J. appl. phys., 7, 3, 271-275, (1999)
[22] Zidi, M., Combined torsion, circular and axial shearing of a compressible hyperelastic and prestressed tube, ASME J. appl. mech., 67, 33-40, (2000) · Zbl 1110.74808
[23] Zidi, M., Effects of a prestress on reinforced, nonlinearly elastic and compressible tube subjected to combined deformations, Internat. J. solids structures, 38, 26-27, 4657-4669, (2001) · Zbl 0997.74007
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