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Helical buckling of thick-walled, pre-stressed, cylindrical tubes under a finite torsion. (English) Zbl 1327.74063
Summary: We study the occurrence of torsional instabilities in soft, incompressible, thick-walled tubes with both circumferential and axial pre-stretches. Assuming a neo-Hookean strain energy function, we investigate the helical buckling under a finite torsion in three different classes of boundary conditions: (a) no applied loads at the internal and external surfaces of the cylindrical tube, (b) a pressure load \(P\) acting on the external surface or (c) on the internal surface. We perform a linear stability analysis on the axisymmetric solutions using the method of small deformations superposed on finite strains. Applying a helical perturbation, we derive the Stroh formulation of the incremental boundary value problems and we solve it using a numerical procedure based on the surface impedance method. The threshold values of the torsion rate and the associated critical circumferential and longitudinal modes at the onset of the instability are examined in terms of the circumferential and axial pre-stretches, and of the initial geometry of the soft tube.
MSC:
74G60 Bifurcation and buckling
74B20 Nonlinear elasticity
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[1] Rivlin RS, Philos Trans R Soc S-A 241 pp 379– (1948) · Zbl 0031.42602 · doi:10.1098/rsta.1948.0024
[2] Rivlin RS, Math Proc Cambridge 45 (3) pp 485– (1949) · doi:10.1017/S0305004100025135
[3] Ericksen LA, Z Angew Math Phys 5 pp 466– (1954) · Zbl 0059.17509 · doi:10.1007/BF01601214
[4] Truesdell C, Handbuch der Physik (3) (1965)
[5] Carroll MM, Int J Eng Sci 5 pp 515– (1967) · doi:10.1016/0020-7225(67)90038-9
[6] Fosdick RL, Arch Rational Mech Anal 29 pp 272– (1968) · Zbl 0164.27305 · doi:10.1007/BF00276728
[7] Rivlin RS, Philos Trans R Soc S-A 242 pp 173– (1949) · Zbl 0035.41503 · doi:10.1098/rsta.1949.0009
[8] Ogden RW, Q J Mech Appl Math 26 (1) pp 23– (1973) · Zbl 0278.73027 · doi:10.1093/qjmam/26.1.23
[9] Gent AN, Int J Nonlin Mech 39 pp 433– (2004)
[10] Yamaki N, Elastic stability of circular cylindrical shells (1984) · Zbl 0544.73062
[11] Green AE, J Math Phys 37 pp 316– (1959) · Zbl 0088.16503 · doi:10.1002/sapm1958371316
[12] Duka ED, Acta Mech 98 pp 107– (1993) · Zbl 0771.73030 · doi:10.1007/BF01174297
[13] Flügge W, Stresses in shells, 2. ed. (1973) · doi:10.1007/978-3-642-88291-3
[14] Ertepinar A, Int J Solids Struct 11 pp 329– (1975) · Zbl 0294.73044 · doi:10.1016/0020-7683(75)90072-4
[15] Holzapfel GA, J R Soc Interface 7 pp 787– (2010) · doi:10.1098/rsif.2009.0357
[16] Ciarletta P, J Mech Phys Solids 60 pp 525– (2012) · Zbl 1244.74085 · doi:10.1016/j.jmps.2011.11.004
[17] Ogden RW, Non-linear elastic deformations (1997)
[18] Hoger A, Arch Ration Mech Anal 88 pp 271– (1985) · Zbl 0571.73011 · doi:10.1007/BF00752113
[19] Destrade M, Int J Eng Sci 48 pp 1212– (2010) · Zbl 1231.74040 · doi:10.1016/j.ijengsci.2010.09.011
[20] DOI: 10.1016/0021-9290(87)90262-4 · doi:10.1016/0021-9290(87)90262-4
[21] Stroh AN, J Math Phys 41 pp 77– (1962) · Zbl 0112.16804 · doi:10.1002/sapm196241177
[22] Biryukov SV, Sov Phys Acoust 31 pp 350– (1985)
[23] Norris AN, Quart J Mech Appl Math 63 pp 401– (2010) · Zbl 1242.74038 · doi:10.1093/qjmam/hbq010
[24] Destrade M, Int J Solids Struct 46 pp 4322– (2009) · Zbl 1176.74068 · doi:10.1016/j.ijsolstr.2009.08.017
[25] De Pascalis R, J Elast 102 pp 191– (2011) · Zbl 1273.74104 · doi:10.1007/s10659-010-9265-6
[26] Fu YB, Proc Roy Soc Lond A 463 pp 3073– (2007) · Zbl 1153.74006 · doi:10.1098/rspa.2007.0093
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