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Exact and numerical elastodynamic solutions for spherically symmetric problems of functionally graded thick-walled spheres subjected to pressure shocks

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Abstract

In the present paper, analytical and numerical elastodynamic solutions are developed for spherically symmetric problems of functionally graded thick-walled spheres subjected to arbitrary dynamic and shock loads. Both transient dynamic response and elastic wave propagation characteristics are studied in the mentioned nonhomogeneous structures. Variations of the material properties across the thickness are described according to both polynomial and power law functions. The numerical consistent transfinite element formulation is presented for both functions whereas the exact solution is presented for the power law function. The functionally graded material sphere is not divided into isotropic sub-spheres. An approach associated with dividing the dynamic radial displacement expression into quasi-static and dynamic parts and expansion of the transient wave functions in terms of a series of eigenfunctions is employed to propose the exact solution. Results are obtained for various exponents of the functions of the material properties distributions, various radius ratios, and variety of dynamic and shock loads.

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References

  • Ait Atmane, H.A., Tounsi, A., Mechab, I., Adda Bedia, E.A.: Free vibration analysis of functionally graded plates resting on Winkler–Pasternak elastic foundations using a new shear deformation theory. Int. J. Mech. Mater. Des. 6, 113–121 (2010)

    Article  Google Scholar 

  • Allahverdizadeh, A., Naei, M.H., Rastgo, A.: The effects of large vibration amplitudes on the stresses of thin circular functionally graded plates. Int. J. Mech. Mater. Des. 3, 161–174 (2006)

    Article  Google Scholar 

  • Bagri, A., Eslami, M.R.: Analysis of thermoelastic waves in functionally graded hollow spheres based on the Green–Lindsay theory. J. Therm. Stress. 30(12), 1175–1193 (2008)

    Article  Google Scholar 

  • Bathe, K.J.: Finite Element Procedures. Prentice Hall, Englewood Cliffs (2007)

    Google Scholar 

  • Bayat, M., Sahari, B.B., Saleem, M., Hamouda, A.M.S., Reddy, J.N.: Thermo elastic analysis of functionally graded rotating disks with temperature-dependent material properties: uniform and variable thickness. Int. J. Mech. Mater. Des. 5, 263–279 (2009)

    Article  Google Scholar 

  • Bruck, H.A.: A one-dimensional model for designing functionally graded materials to manage stress waves. Int. J. Solids Struct. 37, 6383–6395 (2000)

    Article  MATH  Google Scholar 

  • Chen, W.Q.: Stress distribution in a rotating elastic functionally graded material hollow sphere with spherical isotropy. J. Strain Anal. 35(1), 13–20 (2000)

    Article  Google Scholar 

  • Chen, W.Q., Ding, H.J.: Free vibration of multilayered spherically isotropic hollow spheres. Int. J. Mech. Sci. 43, 667–680 (2001)

    Article  MATH  Google Scholar 

  • Chen, Y.Z., Lin, X.Y.: An alternative numerical solution of thick-walled cylinders and spheres made of functionally graded materials. Comput. Mater. Sci. 48(3), 640–647 (2010)

    Article  Google Scholar 

  • Dai, H.L., Xiao, X., Fu, Y.M.: Analytical solutions of stresses in functionally graded piezoelectric hollow structures. J. Solid State Commun. 150, 763–767 (2010)

    Article  Google Scholar 

  • Ding, H.J., Wang, H.M., Chen, W.Q.: Transient responses in a piezoelectric spherically isotropic hollow sphere for symmetric problem. J. Appl. Mech. 70, 436–445 (2003)

    Article  MATH  Google Scholar 

  • Ding, H.J., Wang, H.M., Chen, W.Q.: Elastodynamic solution for spherically symmetric problems of a multilayered hollow sphere. Arch. Appl. Mech. 73, 753–768 (2004)

    Article  MATH  Google Scholar 

  • El-Raheb, M.: Transient waves in an inhomogeneous hollow infinite cylinder. Int. J. Solids Struct. 42, 5356–5376 (2005)

    Article  MATH  Google Scholar 

  • Eringen, A.C., Suhubi, E.S.: Elastodynamics, Vol. II, Linear Theory. Academic press, New York (1975)

    MATH  Google Scholar 

  • Eslami, M.R., Babaei, M.H., Poulatngari, R.: Thermal and mechanical stresses in a functionally graded thick sphere. Int. J. Press. Vessel. Pip. 82, 522–527 (2005)

    Article  Google Scholar 

  • Fung, Y.C., Tong, P.: Classical and Computational Solid Mechanics. World Scientific Publication Co Inc, Singapore (2002)

    Google Scholar 

  • Ghosh, M.K., Kanoria, M.: Generalized thermoelastic functionally graded spherically isotropic solid containing a spherical cavity under thermal shock. J. Appl. Math. Mech. 29(10), 1263–1278 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  • Ghosh, M.K., Kanoria, M.: Analysis of thermoelastic response in a functionally graded spherically isotropic hollow sphere based on Green–Lindsay theory. Acta Mech. 207, 51–67 (2009)

    Article  MATH  Google Scholar 

  • Guven, U., Baykara, C.: On stress distributions in functionally graded isotropic spheres subjected to internal pressures. J. Mech. Commun. Research. 28(3), 277–281 (2001)

    Article  Google Scholar 

  • Han, H., Liu, G.R., Xi, Z.C., Lam, K.Y.: Transient waves in a functionally graded cylinder. Int. J. Solids Struct. 38, 3021–3037 (2001)

    Article  MATH  Google Scholar 

  • Heyliger, P., Jilania, A.: The free vibration of inhomogeneous elastic cylinders and spheres. Int. J. Solids Struct. 29, 2689–2708 (1992)

    Article  MATH  Google Scholar 

  • Honig, G., Hirdes, U.: A method for the numerical inversion of Laplace transforms. J. Comput. Appl. Math. 10, 113–132 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  • Hosseini, S.M., Akhlaghi, M., Shakeri, M.: Dynamic response and radial wave propagation velocity in thick hollow cylinder made of functionally graded materials. Eng. Comput. Int. J. Comput. Aided Eng. Softw. 24(3), 288–303 (2007)

    Article  Google Scholar 

  • Jiang, H., Young, P.G., Dickinson, S.M.: Natural frequency of vibration of layered hollow spheres using exact three-dimensional elasticity equations. J. Sound Vib. 196, 155–162 (1996)

    Article  Google Scholar 

  • Lutz, M.P., Zimmerman, R.W.: Thermal stresses and effective thermal expansion coefficient of a functionally graded sphere. J. Therm. Stress. 19, 39–54 (1996)

    Article  MathSciNet  Google Scholar 

  • Obata, Y., Noda, N.: Steady thermal stress in a hollow circular cylinder and a hollow sphere of functionally graded materials. J. Therm. Stress. 14, 471–487 (1994)

    Article  Google Scholar 

  • Ponnusamy, P.: Wave propagation in a generalized thermoelastic solid cylinder of arbitrary cross-section. Int. J. Solids Struct. 44, 5336–5348 (2007)

    Article  MATH  Google Scholar 

  • Poulatngari, R., Jabbari, M., Eslami, M.R.: Functionally graded hollow spheres under non-axisymmetric thermo-mechanical loads. Int. J. Press. Vessel. Pip. 85, 295–305 (2008)

    Article  Google Scholar 

  • Rade, L., Westergren, B.: Mathematics Handbook for Science and Engineering, 5th edn. Springer, Berlin (2004)

    MATH  Google Scholar 

  • Rao, S.S.: Vibration of Continuous Systems. Wiley, New York (2007)

    Google Scholar 

  • Reddy, J.N.: Analysis of functionally graded plates. Int. J. Numer. Mech. Eng. 47, 663–684 (2000)

    Article  MATH  Google Scholar 

  • Reddy, J.N.: An Introduction to the Finite Element Method, 3rd edn. McGraw-Hill, New York (2005)

    Google Scholar 

  • Rose, J.L., Chou, S.C., Chou, P.C.: Vibration analysis of thick-walled spheres and cylinders. J. Acoust. Soc. Am. 53, 771–776 (1973)

    Article  Google Scholar 

  • Samadhiya, R., Mukherjee, A., Schmauder, S.: Characterization of discretely graded materials using acoustic wave propagation. J. Comput. Mater. Sci. 37, 20–28 (2006)

    Article  Google Scholar 

  • Shakeri, M., Akhlaghi, M., Hoseini, S.M.: Vibration and radial wave propagation velocity in functionally graded thick hollow cylinder. Compos. Struct. 76, 174–181 (2006)

    Article  Google Scholar 

  • Shariyat, M., Nikkhah, M.: An exact elastodynamic solution for functionally graded thick-walled cylinders subjected to dynamic pressures. ISME 10(1), 55–76 (2009)

    Google Scholar 

  • Suresh, S., Mortensen, A.: Fundamentals of Functionally Graded Material. Processing and Thermomechanical Behavior of Graded Metals and Metal-Ceramic Composites. IOM Communications LTD, London (1998)

    Google Scholar 

  • Teymur, M., Chitkara, N.R., Yohngjo, K., Aboudi, J., Pindera, M.J., Arnold, S.M.: Thermoelastic theory for the response of materials functionally graded in two directions. Int. J. Solids Struct. 33(7), 931–966 (1996)

    Article  Google Scholar 

  • Tornabene, F., Viola, E.: Free vibration of functionally graded panels and shells of revolution. Meccanica 44, 255–281 (2009)

    Article  Google Scholar 

  • Tutuncu, N., Temel, B.: A novel approach to stress analysis of pressurized FGM cylinders, disks and spheres. J. Compos. Struct. 91, 385–390 (2009)

    Article  Google Scholar 

  • Wang, X.: An elastodynamic solution for an anisotropic hollow sphere. Int. J. Solids Struct. 31, 903–911 (1994)

    Article  MATH  Google Scholar 

  • Wang, H.M., Xu, Z.X.: Effect of material inhomogeneity on electromechanical behaviors of functionally graded piezoelectric spherical structures. Comput. Mater. Sci. 48, 440–445 (2010)

    Article  Google Scholar 

  • Wang, X., Lu, G., Guillow, S.R.: Stress wave propagation in orthotropic laminated thick-walled spherical shells. Int. J. Solids Struct. 39, 4027–4037 (2002)

    Article  MATH  Google Scholar 

  • Wang, H.M., Ding, H.J., Chen, Y.M.: Transient responses of a multilayered spherically isotropic piezoelectric hollow sphere. Arch. Appl. Mech. 74, 581–599 (2005)

    Article  MATH  Google Scholar 

  • Yu, J., Wu, B., He, C.: Characteristics of guided waves in graded spherical curved plates. Int. J. Solids Struct. 44, 3627–3637 (2007)

    Article  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. R&D Department, Iran Khodro Company, Tehran, Iran

    M. Nikkhah

  2. Department of Mechanical and Industrial Engineering, Concordia University, Montreal, QC, Canada

    A. Hadadian

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  1. M. Nikkhah
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  2. A. Hadadian
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Correspondence to M. Nikkhah.

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Nikkhah, M., Hadadian, A. Exact and numerical elastodynamic solutions for spherically symmetric problems of functionally graded thick-walled spheres subjected to pressure shocks. Int J Mech Mater Des 6, 239–255 (2010). https://doi.org/10.1007/s10999-010-9133-3

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  • DOI: https://doi.org/10.1007/s10999-010-9133-3

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