DOI QR코드

DOI QR Code

Fractional wave propagation in radially vibrating non-classical cylinder

  • Fadodun, Odunayo O. (Department of Mathematics, Obafemi Awolowo University) ;
  • Layeni, Olawanle P. (Department of Mathematics, Obafemi Awolowo University) ;
  • Akinola, Adegbola P. (Department of Mathematics, Obafemi Awolowo University)
  • 투고 : 2017.01.07
  • 심사 : 2017.12.11
  • 발행 : 2017.11.25

초록

This work derives a generalized time fractional differential equation governing wave propagation in a radially vibrating non-classical cylindrical medium. The cylinder is made of a transversely isotropic hyperelastic John's material which obeys frequency-dependent power law attenuation. Employing the definition of the conformable fractional derivative, the solution of the obtained generalized time fractional wave equation is expressed in terms of product of Bessel functions in spatial and temporal variables; and the resulting wave is characterized by the presence of peakons, the appearance of which fade in density as the order of fractional derivative approaches 2. It is obtained that the transversely isotropic structure of the material of the cylinder increases the wave speed and introduces an additional term in the wave equation. Further, it is observed that the law relating the non-zero components of the Cauchy stress tensor in the cylinder under consideration generalizes the hypothesis of plane strain in classical elasticity theory. This study reinforces the view that fractional derivative is suitable for modeling anomalous wave propagation in media.

키워드

참고문헌

  1. Abd-alla, A.N., Alshaikah, F., Giorgio, I. and Della Corte, A. (2015), "A mathematical model for longitudinal wave propagation in a magnetoelastic hollow cylinder of anisotropic material under influence of initial hydrostatics stress", Math. Mech. Solid., 21(1), 104-118. https://doi.org/10.1177/1081286515582883
  2. Akinola, A. (1999), "On interating longitudinal-shear waves in large elastic deformation of a composite cylindrical laminate", Int. J. Nonlin. Mech., 34, 405-414. https://doi.org/10.1016/S0020-7462(98)00019-5
  3. Akinola, A. (2001), "An application of nonlinear fundamental problems of a transversely isotropic layer in finite deformation", Int. J. Nonlin. Mech., 91(3), 307-321.
  4. Akinola, A.P., Layeni, O.P. and Olagunju, M.A. (2004), "A solution by anisotropic expansion for a composite parallelepiped deformed into cylinder", Appl. Math. Comput., 149, 599-611.
  5. Amenzade, Y.A. (1979), Theory of Elasticity, MIR Publishers, Moscow.
  6. Bennoun, M., Houari, M.S.A. and Tounsi, A. (2016), "A novel five variable refined plate theory for vibration analysis of functionally graded sandwich plates", Mech. Adv. Mater. Struct., 23(4) 423-431.
  7. Berferhat, R., Daouadji, T.H., Mansour, M.S. and Hadji, L. (2016), "Effect of porosity on the bending and free vibration response of functionally graded plates resting on Winkler-Pasternak foundations", Earthq. Struct., 10(6), 1429-1449. https://doi.org/10.12989/eas.2016.10.6.1429
  8. Bourada, F., Amara, K. and Tounsi, A. (2016), "Buckling analysis of isotropic and orthotropic plates using a novel four variable refined plate theory", Steel Compos. Struct., 21(6), 1287-1306. https://doi.org/10.12989/scs.2016.21.6.1287
  9. Calim, F.F. (2016), "Dynamic response of curved Timoshenko beams resting on viscoelastic foundation", Struct. Eng. Mech., 59(4), 761-774. https://doi.org/10.12989/sem.2016.59.4.761
  10. Chen, W., Ye, L. and Sun, H. (2010), "Fractional diffusion equation by the Kansa method", Comput. Math. Appl., 59, 1614-1620. https://doi.org/10.1016/j.camwa.2009.08.004
  11. Du, R., Cao, W.R. and Sun, Z.Z. (2010), "A compact difference scheme for the fractional diffusion-wave equation", Appl. Math. Model., 34, 2998-3007. https://doi.org/10.1016/j.apm.2010.01.008
  12. Fadodun, O.O. (2014), "Two-dimensional theory for a transversely isotropic thin plate in nonlinear elasticity", Ph.D. dissertation, Obafemi Awolowo University, Ile-Ife, Nigeria.
  13. Fadodun, O.O. and Akinola, A.P. (2017), "Bending of an isotropic non-classical thin rectangular plate", Struct. Eng. Mech., 61(4), 437-440. https://doi.org/10.12989/sem.2017.61.4.437
  14. Fu, Z.J., Chen, W. and Yang, H.T. (2013), "Boundary particle method for Laplace transformed time fractional diffusion equations", J. Comput. Phys., 235, 52-66. https://doi.org/10.1016/j.jcp.2012.10.018
  15. Hadji, L., Khelifa, Z. and Adda Bedia, E.A. (2016), "A new higher-order shear deformation model for functionally graded beams", KSCE J. Civil Eng., 20(5), 1835-1841. https://doi.org/10.1007/s12205-015-0252-0
  16. Hadji, L., Zouatnia, N. and Kassoul, A. (2017), "Wave propagation in functionally graded beams using various higher-order shear deformation beams theories", Struct. Eng. Mech., 62(2), 143-149. https://doi.org/10.12989/sem.2017.62.2.143
  17. Holm, B. and Sinkus, R. (2003), "Fractional laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power law dependency", J. Acoust. Soc. Am., 115(4), 1424-1430. https://doi.org/10.1121/1.1646399
  18. Honarvar, F., Enjilela, E. and Sinclair, A.N. (2007), "Wave propagation in transversely isotropic cylinders", Int. J. Solid. Struct., 44, 5236-5246. https://doi.org/10.1016/j.ijsolstr.2006.12.029
  19. Khalil, R., Al Horani, M., Yousef, A. and Sababheh, M. (2014), "A new definition of fractional derivative", J. Comput. Appl. Mech., 264(1), 65-70. https://doi.org/10.1016/j.cam.2014.01.002
  20. Korabathina, R. and Koppanati, M.S. (2016), "Linear free vibration analysis of tapered Timoshenko beams using coupled displacement field method", J. Vibroeng. (JVE International Ltd.), 2(1), 27-34.
  21. Li, X.C. (2014), "Analytical solution to a fractional generalized two phase Lame-ClapeyronStefan problem", Int. J. Numer. Meth. Heat. Fluid Flow, 24(6), 1251-1259. https://doi.org/10.1108/HFF-03-2013-0102
  22. Minardi, F. (1995), "The time fractional diffusion-wave equation", Radiophys. Quant. Elect., 38(1), 13-24. https://doi.org/10.1007/BF01051854
  23. Ponnusamy, P. and Rajagopal, M. (2010), "Wave propagation in a transversely isotropic solid cylinder of arbitrary cross-sections immersed in fluid", Eur. J. Mech. A Solid., 29, 158-165. https://doi.org/10.1016/j.euromechsol.2009.09.002
  24. Rao, S.S. (2007), Vibration of Continuous Systems, John Wiley and Son, Inc., Hobokean, New Jersey, U.S.A.
  25. Torres, F.J. (2013), "Existence of positive solutions for a boundary value problem of a nonlinear fractional differential equation", Bull. Iran. Math. Soc., 37(2), 307-325.
  26. Treeby, B.E. and Cox, B.T. (2010), "Modeling power law absorption and dispersion for acoustic propagtion using the fractional Laplacian", J. Acoust. Soc. Am., 195(5), 2741-2748.
  27. Yuste, S.B. (2006), "Weighted average finite difference method for fractional diffusion equation", J. Comput. Phys., 216(1), 264-274. https://doi.org/10.1016/j.jcp.2005.12.006

피인용 문헌

  1. Wave dispersion properties in imperfect sigmoid plates using various HSDTs vol.33, pp.5, 2017, https://doi.org/10.12989/scs.2019.33.5.699