Skip to main content
Springer Nature Link
Log in

Using high-order shear deformation theory in the analysis of Lamb’s waves propagation in materials reinforced with two families of fibers

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

The paper analyses the problem of elastic waves propagation through the material reinforced by two families of extensible fibers using high-order shear deformation theories. The material reinforced with two families of fibers can be seen as a material made of an arbitrary number of plies reinforced with one family of fibers, which are at angles \(\phi \) and \(-\phi \), in relation to the lines of symmetry, so that it can be considered to be a composite laminate. Elasticity constants are derived using strain energy and the theory of invariants. The second-order shear deformation theory was used for creating dynamic equations of motion. Using analytical methods, dispersion curves and diagrams of phase velocities in the polar coordinate system were obtained. The emergence of quasi-modes, in the forms of quasi- bending qA, quasi-extensional qS and quasi-horizontally polarized modes qSH, was shown. The numerical part was done in MATLAB software using combinations of symbolic and numeric values.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
€32.70 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Austria)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Kirchhoff, V.G.: Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. J. die Reine Angew Math. 40, 51–88 (1850)

    Article  MathSciNet  Google Scholar 

  2. Mindlin, R.D.: Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J. Appl. Mech. 18, 31–38 (1951)

    MATH  Google Scholar 

  3. Reissner, E.: On bending of elastic plates. Q. Appl. Math. 5, 55–68 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  4. Reissner, E.: The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech. 12, 69–72 (1945)

    MathSciNet  MATH  Google Scholar 

  5. Della Croce, L., Venini, P.: Finite elements for functionally graded Reissner–Mindlin plates. Comput. Methods Appl. Mech. Eng. 193, 705–725 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kim, K.D., Lomboy, G.R., Han, S.C.: Geometrically non-linear analysis of functionally graded material (FGM) plates and shells using a four-node quasi-conforming shell element. J. Compos. Mater. 42, 485–511 (2008)

    Article  Google Scholar 

  7. Alijani, F., Bakhtiari-Nejad, F., Amabili, M.: Nonlinear vibrations of FGM rectangular plates in thermal environments. Nonlinear Dyn. 66, 251–270 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fallah, A., Aghdam, M.M., Kargarnovin, M.H.: Free vibration analysis of moderately thick functionally graded plates on elastic foundation using the extended Kantorovich method. Arch. Appl. Mech. 83, 177–191 (2013)

    Article  MATH  Google Scholar 

  9. Lanhe, W.: Thermal buckling of a simply supported moderately thick rectangular FGM plate. Compos. Struct. 64, 211–218 (2004)

    Article  Google Scholar 

  10. Bouazza, M., Tounsi, A., Adda-Bedia, E.A., Megueni, A.: Thermoelastic stability analysis of functionally graded plates: an analytical approach. Comput. Mater. Sci. 49, 865–870 (2010)

    Article  Google Scholar 

  11. Reddy, J.N.: A simple higher-order theory for laminated composite plates. J. Appl. Mech. 51, 745–752 (1984)

    Article  MATH  Google Scholar 

  12. Khdeir, A.A., Reddy, J.N.: On the forced motions of antisymmetric cross-ply laminated plates. Int. J. Mech. Sci. 31, 499–510 (1989)

    Article  MATH  Google Scholar 

  13. Ghugal, Y.M., Shimpi, R.P.: A review of refined shear deformation theories of isotropic and anisotropic laminated plates. J. Reinf. Plast. Compos. 21, 775–813 (2002)

    Article  Google Scholar 

  14. Khandan, R., Noroozi, S., Sewell, P., Vinney, J.: The development of laminated composite plate theories: a review. J. Mater. Sci. 47, 5901–5910 (2012)

    Article  Google Scholar 

  15. Swaminathan, K., Naveenkumar, D.T., Zenkour, A.M., Carrera, E.: Stress, vibration and buckling analyses of FGM plates A state-of-the-art review. Compos. Struct. 120, 10–31 (2015)

    Article  Google Scholar 

  16. Hutchinson, J.R.: A comparison of Mindlin and Levinson plate theories. Mech. Res. Comm. 14, 165–170 (1987)

    Article  MATH  Google Scholar 

  17. Levinson, M.: An accurate, simple theory of the statics and dynamics of elastic plates. Mech. Res. Commun. 7, 343–350 (1980)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  19. Pandya, B.N., Kant, T.: Higher-order shear deformable theories for flexure of sandwich plates-finite element evaluations. Int. J. Solids Struct. 24, 1267–1286 (1988)

    Article  MATH  Google Scholar 

  20. Matsunaga, H.: Vibration and stability of cross-ply laminated composite plates according to a global higher-order plate theory. Compos. Struct. 48, 231–244 (2000)

    Article  Google Scholar 

  21. Matsunaga, H.: Free vibration and stability of functionally graded plates according to a 2-D higher-order deformation theory. Compos. Struct. 82, 499–512 (2008)

    Article  Google Scholar 

  22. Levy, M.: Memoire sur la theorie des plaques elastiques planes. J. Math. Pure Appl. 3, 219–306 (1877)

  23. Stein, M.: Nonlinear theory for plates and shells including the effects of transverse shearing. AIAA J. 24, 1537–1544 (1986)

    Article  MATH  Google Scholar 

  24. Touratier, M.: An efficient standard plate theory. Int. J. Eng. Sci. 29, 901–916 (1991)

    Article  MATH  Google Scholar 

  25. Soldatos, K.: A transverse shear deformation theory for homogeneous monoclinic plates. Acta Mech. 94, 195–220 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  26. Messina, A., Soldatos, K.P.: Vibration of completely free composite plates and cylindrical shell panels by a higher-order theory. Int. J. Mech. Sci. 41, 891–918 (1999)

    Article  MATH  Google Scholar 

  27. Soldatos, K.P., Watson, P.: A method for improving the stress analysis performance of one- and two-dimensional theories for laminated composites. Acta Mech. 123, 163–186 (1997)

    Article  MATH  Google Scholar 

  28. Soldatos, K.P., Watson, P.: A general theory for the accurate stress analysis of homogeneous and laminated composite beams. Int. J Solids Struct. 34, 2857–2885 (1997)

    Article  MATH  Google Scholar 

  29. Soldatos, K.P., Watson, P.: Accurate stress-analysis of laminated plates combining a 2-dimensional theory with the exact 3-dimensional solution for simply supported edge. Math. Mech. Solids 2, 459–489 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  30. Liu, S., Soldatos, K.P.: On the prediction improvement of transverse stress distributions in cross-ply laminated beams: advanced versus conventional beam modelling. Int. J. Mech. Sci. 44, 287–304 (2002)

    Article  MATH  Google Scholar 

  31. Shu, X., Soldatos, K.P.: An accurate de-lamination model for weakly bonded laminates subjected to different sets of edge boundary conditions. Int. J. Mech. Sci. 43, 935–959 (2001)

    Article  MATH  Google Scholar 

  32. Soldatos, K.P., Shu, X.: Improving the efficiency of finite element formulations in laminated composites. Commun. Numer. Meth. Eng. 18, 605–613 (2002)

    Article  MATH  Google Scholar 

  33. Karama, M., Afaq, K.S., Mistou, S.: Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity. Int. J. Solids Struct. 40, 1525–1546 (2003)

    Article  MATH  Google Scholar 

  34. Aydogdu, M.: A new shear deformation theory for laminated composite plates. Compos. Struct. 89, 94–101 (2009)

    Article  Google Scholar 

  35. Mantari, J.L., Oktem, A.S., Guedes Soares, C.: A new trigonometric shear deformation theory for isotropic, laminated composite and sandwich plates. Int. J. Solids Struct. 49, 43–53 (2012)

    Article  MATH  Google Scholar 

  36. Mantari, J.L., Guedes Soares, C.: Bending analysis of thick exponentially graded plates using a new trigonometric higher order shear deformation theory. Compos. Struct. 94, 1991–2000 (2012)

    Article  Google Scholar 

  37. Mantari, J.L., Oktem, A.S., Guedes Soares, C.: A new higher order shear deformation theory for sandwich and composite laminated plates. Compos. B Eng. 43, 1489–1499 (2012)

    Article  MATH  Google Scholar 

  38. Mantari, J.L., Oktem, A.S., Guedes Soares, C.: Bending response of functionally graded plates by using a new higher order shear deformation theory. Compos. Struct. 94, 714–723 (2012)

    Article  MATH  Google Scholar 

  39. Mantari, J.L., Oktem, A.S., Guedes Soares, C.: Bending and free vibration analysis of isotropic and multilayered plates and shells by using a new accurate higher order shear deformation theory. Compos. B Eng. 43, 3348–3360 (2012)

    Article  MATH  Google Scholar 

  40. Mantari, J.L., Guedes Soares, C.: Analysis of isotropic and multilayered plates and shells by using a generalized higher-order shear deformation theory. Compos. Struct. 94, 2640–2656 (2012)

    Article  MATH  Google Scholar 

  41. Mantari, J.L.: Finite element formulation of a generalized higher order shear deformation theory for advanced composite plates. Compos. Struct. 96, 545–553 (2013)

    Article  Google Scholar 

  42. Mantari, J.L., Guedes Soares, C.: Static response of advanced composite plates by a new non-polynomial higher-order shear deformation theory. Int. J. Mech. Sci. 78, 60–71 (2014)

    Article  Google Scholar 

  43. Viola, E., Tornabene, F., Fantuzzi, N.: General higher-order shear deformation theories for the free vibration analysis of completely doubly-curved laminated shells and panels. Compos. Struct. 95, 639–666 (2013)

    Article  Google Scholar 

  44. Meiche, N.E., et al.: A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate. Int. J. Mech. Sci. 53, 237–247 (2011)

    Article  Google Scholar 

  45. Akavci, S.S.: Two new hyperbolic shear displacement models for orthotropic laminated composite plates. Mech. Compos. Mater. 46, 215–226 (2010)

    Article  Google Scholar 

  46. Belaid, M., Ismail, M.: Analysis of thick orthotropic laminated composite plates based on higher order shear deformation theory by the new function under thermo-mechanical loading, Composites: Part B. Benaissa Samir 43, 1453–1458 (2012)

    Google Scholar 

  47. Grover, N., Maiti, D.K., Singh, B.N.: Flexural behavior of general laminated composite and sandwich plates using a secant function based shear deformation theory. Latin Am. J. Solids Struct. 11, 1275–1297 (2014)

    Article  Google Scholar 

  48. Neeraj, G., Singh, B.N., Maiti, D.K.: Analytical and finite element modeling of laminated composite and sandwich plates: an assessment of a new shear deformation theory for free vibration response. Int. J. Mech. Sci. 67, 89–99 (2013)

    Article  Google Scholar 

  49. Kant, T., Jha, D.K., Singh, R.K.: A higher-order shear and normal deformation functionally graded plate model: some recent results. Acta Mech. 225, 2865–2876 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  50. Jha, D.K., Kant, T., Singh, R.K.: Stress analysis of transversely loaded functionally graded plates with a higher order shear and normal deformation theory. J. Eng. Mech. 139, 1663–1680 (2013)

    Article  Google Scholar 

  51. Jha, D.K., Kant, T., Singh, R.K.: Free vibration response of functionally graded thick plates with shear and normal deformations effects. Compos. Struct. 96, 799–823 (2013)

    Article  Google Scholar 

  52. Jha, D.K., Kant, T., Singh, R.K.: Higher order shear and normal deformation theory for natural frequency of functionally graded rectangular plates. Nucl. Eng. Des. 250, 8–13 (2012)

    Article  Google Scholar 

  53. Jha, D.K., Kant, T., Srinivas, K., Singh, R.K.: An accurate higher order displacement model with shear and normal deformations effects for functionally graded plates. Fusion Eng. Des. 88, 3199–3204 (2013)

    Article  Google Scholar 

  54. Kant, T., Manjunatha, B.S.: An unsymmetric FRC laminate finite element model with 12 degrees of freedom per node. Eng. Comput. 5, 300–308 (1988)

    Article  Google Scholar 

  55. Swaminathan, K., Naveenkumar, D.T.: Higher order refined computational models for the stability analysis of FGM plates—analytical solutions. Eur. J. Mech. A Solids 47, 349–361 (2014)

    Article  MathSciNet  Google Scholar 

  56. Wang, L.: Elastic Waves Propagation in Composites. North Carolina State University, Raleigh (2004)

    Google Scholar 

  57. Green, W.A.: Bending waves in strongly anisotropic elastic plates. Quart. J. Mech. Appl. Math. 35, 485–507 (1982)

    Article  MATH  Google Scholar 

  58. Green, W.A., Milosavljevic, D.: Extensional waves in strongly anisotropic elastic plates. Int. J. Solids Struct. 21, 343–353 (1985)

    Article  MATH  Google Scholar 

  59. Green, W.A., Rogerson, G.A., Milosavljević, D.I.: Transient Waves in Six-Ply and Eight-Ply Fibre Composite Plates. Composites Science and Technology, vol. 44, pp. 151–158. Elsevier Science Publishers Ltd, New York (1992)

  60. Milosavljević, D.I.: Rayleigh waves in materials reinforced by two families of fibres. In: Parker, D.F., Maugin, G.A. (eds.) Recent Development in Surface Acoustic Waves, Springers Series on Wave Phenomena, pp. 251–259. Springer, Berlin (1988)

  61. Milosavljević, D., Bogdanović, G., Lazić, V., Aleksandrović, S., Lazić, M.: Bulk waves and dynamical behaviour in elastic solids reinforced by two families of strong fibres. J. Eng. Math. 2014. doi:10.1007/s10665-014-9747-9

  62. Milosavljević, D., Bogdanović, G., Veljović, L., Radaković, A., Lazić, M.: Wave Propagation in Layer with Two Preferred Directions, International Journal of Non-Linear Mechanics. Elsevier Ltd., Amsterdam (2014). doi:10.1016/j.ijnonlinmec.2014.11.014

  63. Nayfeh, A., Chimenti, D.: Free wave propagation in plates of general anisotropic media. J. Appl. Mech. 56, 881–886 (1989)

    Article  MATH  Google Scholar 

  64. Spencer, A.J.M.: Constitutive theory for strongly anisotropic solids. In: Spencer, A.J.M. (ed.) Continuum Theory of the Mechanics of Fibre Reinforced Composites, pp. 1–32. Springer, Wien (1984)

  65. Spencer, A.J.M.: Deformations of Fibre Reinforced Materials. Clarendon Press, Oxford (1972)

    MATH  Google Scholar 

  66. Whitney, J.M., Sun, C.T.: Higher-order theory for extensional motion of laminated composites. J. Sound Vib. 30, 85–97 (1973)

    Article  MATH  Google Scholar 

  67. Wang, L., Yuan, F.G.: Lamb wave propagation in composite laminates using a higher-order plate theory. In: Proceedings of SPIE 6531, Nondestructive Characterization for Composite Materials, Aerospace Engineering, Civil Infrastructure, and Homeland Security (2007)

  68. Wang, L., Yuan, F.G.: Group velocity and characteristic wave curves of Lamb waves in composites: modeling and experiments. Compos. Sci. Technol. 67, 1370–1384 (2007)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. State University of Novi Pazar, Novi Pazar, 36300, Serbia

    Aleksandar Radaković

  2. Faculty of Engineering, University of Kragujevac, Kragujevac, 34000, Serbia

    Gordana Bogdanović, Dragan Milosavljević & Ljiljana Veljović

  3. Faculty of Technical Sciences, University of Priština, Kosovska Mitrovica, 38220, Serbia

    Dragan Čukanović

Authors
  1. Aleksandar Radaković
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gordana Bogdanović
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dragan Milosavljević
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ljiljana Veljović
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Dragan Čukanović
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gordana Bogdanović.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Radaković, A., Bogdanović, G., Milosavljević, D. et al. Using high-order shear deformation theory in the analysis of Lamb’s waves propagation in materials reinforced with two families of fibers. Acta Mech 228, 187–200 (2017). https://doi.org/10.1007/s00707-016-1707-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-016-1707-1