Structural Engineering and Mechanics

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Torsional wave dispersion in a bi-layered hollow cylinder with inhomogeneous initial stresses caused by internal and external radial pressures

  • Akbarov, Surkay D. (Department of Mechanical Engineering, Faculty of Mechanical Engineering, Yildiz Technical University) ;
  • Bagirov, Emin T. (Institute of Mathematics and Mechanics of Azerbaijan National Academy of Sciences)
  • Received : 2019.08.30
  • Accepted : 2020.12.01
  • Published : 2021.03.10
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Abstract

The present paper studies the influence of the inhomogeneous initial stresses in the bi-layered hollow cylinder and it is assumed that these stresses are caused by the hydrostatic pressures acting on the interior and outer free surfaces of the cylinder. The study is made by utilizing the version of the three-dimensional linearized theory of elastic waves in bodies with initial stresses for which the initial stress-strain state in bodies is determined within the scope of the classical linear theory of elasticity. For the solution to the corresponding eigenvalue problem, the discrete-analytical method is employed. Numerical results are presented and analyzed for concrete selected pairs of materials. According to these results and their analyses, it is established that, unlike homogeneous initial stresses, the influence of the inhomogeneous initial stresses on the torsional wave dispersion has not only quantitative but also qualitative character. For instance, in particular, it is established that as a result of the initial stresses caused by the hydrostatic pressure acting in the interior free surface of the cylinder, the cut-off frequency appears for the fundamental dispersive mode and the values of this frequency increase with the intensity of this pressure.

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References

  1. Akbarov, S.D. (2006), "Frequency response of the axisymmetrically finite pre-stretched slab from incompressible functionally graded material on a rigid foundation", Int. J. Eng. Sci., 44, 93-112. https://doi.org/10.1016/j.ijengsci.2006.04.004.
  2. Akbarov, S.D. (2012), "The influence of the third order elastic constants on axisymmetric wave propagation velocity in the two-layered pre-stressed hollow cylinder", CMC: Comput. Mater. Continua, 32(1), 29-60. https://doi/10.3970/cmc.2012.032.029.html.
  3. Akbarov, S.D. (2014), "Flexural wave dispersion in finitely prestrained solid and hollow circular cylinders made of compressible materials", CMES: Comput. Model. Eng. Sci., 92(4), 387-427. https://doi/10.3970/cmes.2014.097.407.
  4. Akbarov, S.D. (2015), Dynamics of Pre-Strained Bi-Material Elastic Systems: Linearized Three- Dimensional Approach, Springer, Cham.
  5. Akbarov, S.D. and Bagirov, E.T. (2019), "Axisymmetric longitudinal wave dispersion in a bi-layered circular cylinder with inhomogeneous initial stresses", J. Sound Vib., 450, 1-27. https://doi.org/10.1016/j.jsv.2019.03.003.
  6. Akbarov, S.D. and Guliev, M.S. (2009), "Axisymmetric longitudinal wave propagation in a finite prestrained compound circular cylinder made from compressible materials", CMES: Comput. Model. Eng. Sci., 39(2), 155-177. https://doi/10.3970/cmes.2009.039.155.
  7. Akbarov, S.D., Guliev, M.S. and Kepceler, T. (2011a), "Dispersion relations of axisymmetric wave propagation in initially twisted bi-material compounded cylinders", J. Sound Vib., 330, 1644-1664. https://doi.org/10.1016/j.jsv.2010.10.020.
  8. Akbarov, S.D., Guliyev, H.H. and Yahnioglu, N. (2017), "Three-dimensional analysis of the natural vibration of the three-layered hollow sphere with middle layer made of FGM", Struct. Eng. Mech., 61(5), 563-576. https://doi.org/10.12989/sem.2017.61.5.563.
  9. Akbarov, S.D., Guliyev, H.H., Sevdimaliyev, Y.M. and Yahnioglu, N. (2018), "The discrete-analytical solution method for investigation dynamics of the sphere with inhomogeneous initial stresses", CMC: Comput. Mater. Continua, 55(2), 359-380. https://doi:10.3970/cmc.2018.00173.
  10. Akbarov, S.D., Kepceler, T. and Egilmez, M.M. (2011b), "Torsional wave dispersion in a finitely pre-strained hollow sandwich circular cylinder", J. Sound Vib., 330, 4519-4537. https://doi.org/10.1016/j.jsv.2011.04.009.
  11. Akbarov, S.D., Kocal, T. and Kepceler, T. (2016), "On the dispersion of the axisymmetric longitudinal wave propagating in a bi-layered hollow cylinder made of viscoelastic materials", Int. J. Solid. Struct., 100-101(1), 195-210. https://doi.org/10.1016/j.ijsolstr.2016.08.016.
  12. Armenakas, A.E. (1965), "Torsional waves in composite rods", J. Acoust. Soc. Am., 39, 439-446. https://doi.org/10.1121/1.1909709
  13. Armenakas, A.E. (1970), "Propagation of harmonic waves in composite circular cylindrical rods", J. Acoust. Soc. Am., 47, 822-837. https://doi.org/10.1121/1.1911965
  14. Armenakas, A.E. (1971), "Propagation of harmonic waves in composite circular cylindrical shells. Pt. II. Numerical investigation", AIAA J., 9(4), 5999.
  15. Batra, R.C. and Bahrami, A. (2009), "Inflation and eversion of functionally graded non-linear elastic incompressible circular cylinders", Int. J. Nonlin. Mech., 44, 311-323. https://doi:10.1016/j.ijnonlinmec.2008.12.005.
  16. Chen, W.Q. and Ding, H.J. (2012), "The state-space method and its application in analyses of FGM structures", Eds. Zhong, Z., Wu, L.Z. and Chen, W.Q., Mechanics of Functionally Graded Materials and Structures, Nova Science Publishers, New York.
  17. Chen, W.Q., Liu, D.Y., Kitipornchai, S. and Yang, J. (2017), "Bifurcation of pressurized functionally graded elastomeric hollow cylinders", Compos. Part B Eng., 109, 259-276. https://doi:10.1016/j.compositesb.2016.10.063.
  18. Chen, W.T. and Wright, T.W. (1966), "Frequency equations for wave propagation in an initially stressed circular cylinder", J. Acoust. Soc. Am., 39, 847-848. https://doi.org/10.1121/1.1909962
  19. Demiray, H. and Suhubi, E.S. (1970), "Small torsional oscillation in initially twisted circular rubber cylinder", Int. J. Eng. Sci., 8, 19-30. https://doi.org/10.1016/0020-7225(70)90011-X
  20. Destrade, M. and Saccomandi, G. (2007), Waves in Nonlinear Prestressed Materials, Springer Wien, New York.
  21. Ding, H.J. and Chen, W.Q. (2001), Three Dimensional Problems of Piezoelasticity, Nova Science Publishers, New York.
  22. Ebrahimi, F., Seyfi Dabbagh, A. and Tornabene, F. (2019): "Wave dispersion characteristics of porous graphene platelet-reinforced composite shells", Struct. Eng. Mech., 71(1), 99-107. https://doi.org/10.12989/sem.2019.71.1.099.
  23. Engin, H. and Suhubi, E.S. (1978), "Torsional oscillations of an infinite cylindrical elastic tube under large internal and external pressure", Int. J. Eng. Sci., 16, 387-396. https://doi.org/10.1016/0020-7225(78)90028-9
  24. Eringen, A.C. and Suhubi, E. (1975), Elastodynamics, I: Finite Motions, II: Linear theory, Academic Press, New York.
  25. Green, A.E. (1961), "Torsional vibration of an initially stressed circular cylinder", Problems of Continuum Mechanics, (Muskhelishvili Anniversary Vol.) Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania.
  26. Guz, A.N. (1986a), Elastic Waves in Bodies with Initial Stresses: 2 volumes, Vol. 1. General Questions, Naukova Dumka, Kiev. (in Russian)
  27. Guz, A.N. (1986b), Elastic Waves in Bodies with Initial Stresses: 2 Volumes, Vol. 2. Propagation Laws, Naukova Dumka, Kiev. (in Russian)
  28. Guz, A.N. (2004), Elastic Waves in Bodies with Initial (Residual) Stresses, A.C.K. Kiev. (in Russian)
  29. Karami, B., Janghorban, M. and Tounsi, A. (2019), "Wave propagation of functionally graded anisotropic nanoplates resting on Winkler-Pasternak foundation", Struct. Eng. Mech., 70(1), 55-66. https://doi.org/10.12989/sem.2019.70.1.055.
  30. Kocal, T. and Akbarov, S.D. (2017), "On the attenuation of the axisymmetric longitudinal waves propagating in the bi-layered hollow cylinder made of viscoelastic materials", Struct. Eng. Mech., 61(2), 145-165. https://doi.org/10.12989/sem.2017.61.2.145.
  31. Kushnir, V.P. (1979), "Longitudinal waves in the field of a transversally isotropic cylinder with initial stresses", Int. Appl. Mech., 15(9), 884-886.
  32. Kushnir, V.P. (1983), "Numerical study of the laws governing the propagation of bending waves in a hollow compressible cylinder with initial stresses", Prikladnaya Mekhanica, 19(12), 113-115.
  33. Lashhab, M.I., Rogerson, G.A. and Sandiford, K.J. (2015), "Dispersion phenomena in symmetric pre-stressed layered elastic structures", Int. J. Solid. Struct., 58, 220-232. https://doi.org/10.1016/j.ijsolstr.2015.01.006.
  34. Li, G.Y., He, Q., Mangan, R., Xu, G.Q., Mo, C., Luo, C.J., Destrade, M. and Cao, Y.P. (2017), "Guided waves in prestressed hyperelastic plates and tubes: application to the ultrasound elastography of thin-walled soft materials", J. Mech. Phys. Solid., 102, 67-79. https://doi:10.1016/j.jmps.2017.02.008.
  35. Nam, N.T., Merodio, J., Ogden, R.W. and Vinh, P.C. (2016), "The effect of initial stress on the propagation of surface waves in a layered half-space", Int. J. Solid. Struct., 88/89(15), 88-100. https://doi.org/10.1016/j.ijsolstr.2016.03.019.
  36. Nayfeh, A.N. (1995), Wave Propagation in Layered Anisotropic Media: with Applications to Composites, Elsevier Science, Amsterdam.
  37. Ogden, R.W. (1997), Non-Linear Elastic Deformation, Dover, New York.
  38. Ozturk, A. and Akbarov, S.D. (2008), "Propagation of torsional waves in a pre-stretched compound hollow circular cylinder", Mech. Compos. Mater., 44(1), 77-86. https://doi.org/10.1007/s11029-008.
  39. Ozturk, A. and Akbarov, S.D. (2009a), "Torsional wave dispersion relations in a pre-stressed bi-material compounded cylinder", Z. Angew. Math. Mech. (ZAMM), 89(9), 754-766. https://doi.org/10.1002/zamm.200800201.
  40. Ozturk, A. and Akbarov, S.D. (2009b), "Torsional wave propagation in a pre-stressed circular cylinder embedded in a pre-stressed elastic medium", Appl. Math. Model., 33, 3636- 3649. https://doi.org/10.1016/j.apm.2008.12.003.
  41. Shearer, T., Abrahams, I.D., Parnell, W.J. and Daros, C.H. (2013), "Torsional wave propagation in a pre-stressed hyperelastic annular circular cylinder", Q. J. Mech. Appl. Math., 66, 465-487. https://doi:10.1093/qjmam/hbt014.
  42. Suhubi, E.S. (1965), "Small longitudinal vibration of an initially stretched circular cylinder", Int. J. Eng. Sci., 2, 509-517. https://doi.org/10.1016/0020-7225(65)90006-6
  43. Thurston, R.N. (1976), "Torsional acoustic modes in clad rods", IEEE Tran. Sonic. Ultrason., 23(3), 154-161. https://doi.org/10.1109/T-SU.1976.30855
  44. Timoshenko S. and Goodier, J.N. (1951), Theory of elasticity. McGraw-Hill.
  45. Wu, B., Su, Y., Liu, D., Chen, W. and Zhang, C. (2018), "On propagation of axisymmetric waves in pressurized functionally graded elastomeric hollow cylinders", J. Sound Vib., 421, 17-47. https://doi:10.1016/j.jsv.2018.01.055.
  46. Wu, B., Su, Y.P., Chen, W.Q. and Zhang, C.Z. (2017), "On guided circumferential waves in soft electroactive tubes under radially inhomogeneous biasing fields", J. Mech. Phys. Solid., 99, 116-145. https://doi.org/10.1016/j.jmps.2016.11.004.