Skip to main content
SpringerLink
Log in

A refined analysis for the transversely isotropic plate under normal loads with the 3D Green’s function

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

Abstract

Many composite plates behave transversely isotropic. An accurate and efficient 3D analysis for the transversely isotropic plate will contribute much to the accurate design and quality promotion of this kind of structure. In this paper, a refined analysis for the transversely isotropic plate under normal loads is addressed based on the 3D Green’s function. The 3D Green’s function for the transversely isotropic plate loaded by a normal point force is derived. By adopting the general solution for the transversely isotropic material, a new harmonic function which contains undetermined constants is constructed. All stress components can be obtained by substituting the harmonic function into the general solution. The constants can be determined by considering the equilibrium and boundary conditions of the plate. For the reason that this solution satisfies the 3D governing equations rigorously everywhere in the plate, it is much more refined than that of the traditional plate theory, which is restricted by the thickness of the plate. Numerical analysis is given to discuss the efficiency and accuracy of this method and study the stress distributions in the transversely isotropic plate loaded by a normal point force and a normal distributed load, respectively.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Long, N.M.A.N., Khaldjigitov, A.A., Adambaev, U.: On the constitutive relations for isotropic and transversely isotropic materials. Appl. Math. Model. 37, 7726–7740 (2013)

    Article  MathSciNet  Google Scholar 

  2. Zobeiry, N., Malek, S., Vaziri, R., Poursartip, A.: A differential approach to finite element modelling of isotropic and transversely isotropic viscoelastic materials. Mech. Mater. 97, 76–91 (2016)

    Article  Google Scholar 

  3. Wang, L.J., Ai, Z.Y.: Plane strain and three-dimensional analyses for thermo-mechanical behavior of multilayered transversely isotropic materials. Int. J. Mech. Sci. 103, 199–221 (2015)

    Article  Google Scholar 

  4. Mokhtari, M., Schipper, D.J., Vileugels, N., Noordermeer, J.W.M.: Transversely isotropic viscoelastic materials: contact mechanics and friction. Tribol. Int. 97, 116–123 (2016)

    Article  Google Scholar 

  5. Willis, J.R.: Hertzian contact of anisotropic bodies. J. Mech. Phys. Solids 14(3), 163–176 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Borodich, F.M.: Hertz contact problems for an anisotropic physically nonlinear elastic medium. Strength Mater. 21(12), 1668 (1989)

    Article  Google Scholar 

  7. Li, P., Guo, Y.B., Shim, V.P.W.: A constitutive model for transversely isotropic material with anisotropic hardening. Int. J. Solids Struct. 000, 1–10 (2017)

    Google Scholar 

  8. Argatov, I.I.: Depth-sensing indentation of a transversely isotropic elastic layer: second-order asymptotic models for canonical indenter. Int. J. Solids Struct. 48(25–26), 3444–3452 (2011)

    Article  Google Scholar 

  9. Green, A.E., Zerna, W.: Theoretical Elasticity. Dover, New York (1968)

    MATH  Google Scholar 

  10. Conway, H.D., Farnham, K.A., Ku, T.C.: The indentation of a transversely isotropic half space by a rigid sphere. J. Appl. Mech. 34(2), 491–492 (1967)

    Article  Google Scholar 

  11. Xu, B., Wang, M.: The quasi Eshelby property for rotational symmetrical inclusions of uniform eigencurvatures within an infinite plate. Proc. Math. Phys. Eng. 461, 2899–2910 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Beom, H.G., Kim, I.B.: Analysis of a multilayered plate containing a cuboidal inclusion with eigenstrains. Mech. Mater. 31, 729–741 (1991)

    Article  Google Scholar 

  13. Zou, W.N., He, Q.C., Huang, M.J., Zheng, Q.S.: Eshelby’s problem of nonelliptical inclusions. J. Mech. Phys. Solids 58, 346–372 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ai, Z.Y., Hu, Y.D.: A coupled BEM-ALEM approach for analysis of elastic thin plates on multilayered soil with anisotropic permeability. Eng. Anal. Bound. Elem. 53, 40–45 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Seremet, V., Seremet, D.: Solution in integrals of a 3D BVP of thermoelasticity: Green’s functions and integration formula for thermal stresses within a semi-bounded parallelepiped. Acta Mech. 12, 4471–4490 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, L., Zhang, X.L., Li, Y.H.: Analysis of coupled vibration characteristics of wind turbine blade based on Green’s functions. Acta Mech. 29, 620–630 (2016)

    Google Scholar 

  17. Moslemi, A., Neya, B.N., Amiri, J.V.: Benchmark solution for buckling of thick rectangular transversely isotropic plates under biaxial load. Int. J. Mech. Sci. 131, 356–367 (2017)

    Article  Google Scholar 

  18. Yang, B., Chen, W.Q., Ding, H.J.: 3D elastostatic solutions for transversely isotropic functionally graded material plates containing elastic inclusion. Appl. Math. Mech. Engl. 36(4), 417–426 (2015)

    Article  Google Scholar 

  19. Yang, B., Chen, W.Q., Ding, H.J.: Equilibrium of transversely isotropic FGM plates with an elliptical hole: 3D elasticity solutions. Arch. Appl. Mech. 86(8), 1391–1414 (2016)

    Article  Google Scholar 

  20. Lee, H.S., Kim, Y.Y.: Multipole expansion of Green’s function for guided waves in a transversely isotropic plate. J. Mech. Sci. Technol. 29(5), 1899–1906 (2015)

    Article  Google Scholar 

  21. Kotousov, A., Wang, C.H.: 3D solutions for transversally isotropic composite plates. Compos. Struct. 57, 445–452 (2002)

    Article  Google Scholar 

  22. Woodward, B., Kashtalyan, M.: 3D elasticity solution for bending of transversely isotropic functionally graded plates. Eur. J. Mech. A Solids 30(5), 705–718 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jiang, J.L., Huang, D.J., Yang, B., Chen, W.Q., Ding, H.J.: Elasticity solutions for a transversely isotropic functionally graded annular sector plate. Acta Mech. 228(7), 2603–2621 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhao, B.S., Wu, D., Wang, M.Z.: The refined theory and the decomposed theorem of a transversely isotropic elastic plate. Eur. J. Mech. A Solids 39, 243–250 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Huang, J., Huang, X., Han, W.: A new Co discontinuous Galerkin method for Kirchhoff plates. Comput. Methods Appl. Mech. Eng. 199, 1446–1454 (2010)

    Article  MATH  Google Scholar 

  26. Lim, C.W., Lü, C.F., Xiang, Y., Yao, W.: On new symplectic elasticity approach for exact free vibration solutions of rectangular Kirchhoff plates. Int. J. Eng. Sci. 47, 131–140 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ding, H.J., Chen, W.Q., Zhang, L.Z.: Elasticity of Transversely Isotropic Materials. Springer, Dordrecht (2006)

    MATH  Google Scholar 

  28. Sharma, J.N., Sharma, P.K.: Free vibration of homogeneous transversely isotropic cylindrical panel. J. Therm. Stresses 25, 169–182 (2002)

    Article  Google Scholar 

  29. Hanson, M.T.: The elastic field for spherical indentation including sliding friction for transversely isotropy. J. Tribol. 114, 606–611 (1992)

    Article  Google Scholar 

  30. Hanson, M.T.: The elastic field for conical indentation including sliding friction for transversely isotropy. ASME J. Appl. Mech. 59, 123–130 (1992)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, People’s Republic of China

    Peng-Fei Hou & Jia-Yun Chen

  2. Department of Engineering Mechanics, Hunan University, Changsha, 410082, People’s Republic of China

    Peng-Fei Hou & Jia-Yun Chen

Authors
  1. Peng-Fei Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jia-Yun Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jia-Yun Chen.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hou, PF., Chen, JY. A refined analysis for the transversely isotropic plate under normal loads with the 3D Green’s function. Acta Mech 229, 3767–3779 (2018). https://doi.org/10.1007/s00707-018-2185-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-018-2185-4

Search

Navigation