A fundamental solution for the harmonic vibration of laminated composite plates with coupled dynamic bending and quasistatic extension

Abstract

We obtain here a new fundamental solution for the harmonic vibration of asymmetric, laminated, anisotropic plates. The fundamental solution is derived via the Fourier transform and its final form is given in terms of definite integrals, which are evaluated numerically. Moreover, we present some of the higher order derivatives of the solution and their explicit spatial singularities, which are necessary for a boundary element method implementation.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

References

  1. 1.

    Becker, W.: A complex potential method for plate problems with bending extension coupling. Arch. Appl. Mech. 61, 318–326 (1991)

    Article  Google Scholar 

  2. 2.

    Becker, W.: Concentrated forces and moments on laminates with bending extension coupling. Compos. Struct. 30, 1–11 (1995)

    Article  Google Scholar 

  3. 3.

    Bui, T.Q., Nguyen, M.N., Zhang, C.: An efficient meshfree method for vibration analysis of laminated composite plates. Comput. Mech. 48(2), 175–193 (2011)

    Article  Google Scholar 

  4. 4.

    Daros, C.H.: The dynamic fundamental solution and BEM formulation for laminated anisotropic kirchhoff plates. Eng. Anal. Bound. Elem. 54, 19–27 (2015)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Daros, C.H.: A new Fourier transform-based fundamental solution for laminated composite plates with coupled bending and extension. Compos. Struct. 222, 110918 (2019)

    Article  Google Scholar 

  6. 6.

    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals. Series and Products, 7th edn. Academic Press, New York (2007)

    Google Scholar 

  7. 7.

    Hwu, C.: Green’s function for the composite laminates with bending extension coupling. Compos. Struct. 63, 283–292 (2004)

    Article  Google Scholar 

  8. 8.

    Hwu, C.: Boundary integral equations for general laminated plates with coupled stretching-bending deformation. Proc. R. Soc. A 466, 1027–1054 (2010)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Hwu, C.: Boundary element formulation for the coupled stretching-bending analysis of thin laminated plates. Eng. Anal. Bound. Elem. 36, 1027–1039 (2012)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Hwu, C., Chang, H.W.: Coupled stretching-bending analysis of laminated plates with corners via boundary elements. Compos. Struct. 120, 300–314 (2015)

    Article  Google Scholar 

  11. 11.

    Payton, R.G.: Elastic Wave Propagation in Transversely Isotropic Media. Mechanics of Elastic and Inelastic Solids 4. Martinus Nijhoff Publishers, The Hague (1983)

    Google Scholar 

  12. 12.

    Rangelov, T.V., Manolis, G., Dineva, P.S.: Elastodynamic fundamental solutions for certain families of 2D inhomogeneous anisotropic domains: basic derivations. Eur. J. Mech. A/Solids 24, 820–836 (2005)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Reddy, J.N.: Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2nd edn. CRC Press, Boca Raton (2004)

    Google Scholar 

  14. 14.

    Shojaee, S., Valizadeh, N., Izadpanah, E., Bui, T., Vu, T.V.: Free vibration and buckling analysis of laminated composite plates using the NURBS-based isogeometric finite element method. Compos. Struct. 94(5), 1677–1693 (2012)

    Article  Google Scholar 

  15. 15.

    Yin, S., Yu, T., Bui, T.Q., Xia, S., Hirose, S.: A cutout isogeometric analysis for thin laminated composite plates using level sets. Compos. Struct. 127, 152–164 (2015)

    Article  Google Scholar 

  16. 16.

    Yu, T., Yin, S., Bui, T.Q., Xia, S., Tanaka, S., Hirose, S.: NURBS-based isogeometric analysis of buckling and free vibration problems for laminated composites plates with complicated cutouts using a new simple FSDT theory and level set method. Thin-Walled Struct. 101, 141–156 (2016)

    Article  Google Scholar 

  17. 17.

    Zakharov, D.D.: Asymptotic analysis of three-dimensional dynamic elastic equations for a thin multilayer anisotropic plate of arbitrary structure. J. Appl. Math. Mech. 56(5), 637–644 (1992)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Zakharov, D.D.: Asymptotical integration of 3-D dynamic equations for thin multilayered anisotropic plates. Comptes Rendus Acad. Sci. Paris Ser. 2 315, 915–920 (1992)

    MATH  Google Scholar 

  19. 19.

    Zakharov, D.D.: Green’s tensor and the boundary integral equations for thin elastic multilayer asymmetric anisotropic plates. J. Appl. Math. Mech. 61(3), 483–492 (1997)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Zakharov, D.D., Becker, W.: 2D problems of thin asymmetric laminates. Z. Angew. Math. Phys. 51, 555–572 (2000)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Zakharov, D.D., Becker, W.: Singular potentials and double-force solutions for anisotropic laminates with coupled bending and stretching. Arch. Appl. Mech. 70, 659–669 (2000)

    Article  Google Scholar 

Download references

Acknowledgements

The corresponding author states that there is no conflict of interest.

Author information

Affiliations

Authors

Corresponding author

Correspondence to C. H. Daros.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Daros, C.H. A fundamental solution for the harmonic vibration of laminated composite plates with coupled dynamic bending and quasistatic extension. Arch Appl Mech (2020). https://doi.org/10.1007/s00419-020-01717-z

Download citation

Keywords

  • Fundamental solution
  • Fourier transform
  • Composite plates
  • Harmonic vibration
  • Bending extension coupling
  • Anisotropic plates