Skip to main content
SpringerLink
Log in

Hamiltonian partial mixed finite element-state space symplectic semi-analytical approach for the piezoelectric smart composites and FGM analysis

  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

A partial mixed finite element (FE)–state space method (SSM) semi-analytical approach is presented for the static analysis of piezoelectric smart laminate composite and functionally graded material (FGM) plates. Hence, using the Hamiltonian formalism, the three-dimensional piezoelectricity equations are first worked so that a partial mixed variational formulation, which retains the translational displacements, electric potential, transverse stresses, and transverse electric displacement as primary variables, is obtained; this allows, in particular, straightforward fulfillment of the electromechanical continuity constraints at the laminate interfaces. After an in-plane FE discretization only, the problem is first reduced, for a single layer, to a Hamiltonian eigenvalue problem that is solved using the symplectic approach; then, the multilayer solution is reached via the SSM propagator matrix. The proposed methodology is finally applied to the static analysis of piezoelectric-cross-ply hybrid laminated composite and FGM plates. In a comparison with open literature, available tabulated results show good agreements, thus validating the proposed approach.

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. Kreja I.: A literature review on computational models for laminated composite and sandwich panels. Cent. Eur. J. Eng. 1, 59–80 (2011)

    Article  Google Scholar 

  2. Kapuria S., Kumari P., Nath J.K.: Efficient modeling of smart piezoelectric composite laminates: a review. Acta Mech. 214, 31–48 (2010)

    Article  MATH  Google Scholar 

  3. Birman V., Byrd L.W.: Modeling and analysis of functionally graded materials and structures. Appl. Mech. Rev. 60, 195–216 (2007)

    Article  Google Scholar 

  4. Benjeddou A.: Advances in piezoelectric finite element modeling of adaptive structural elements: a survey. Comput. Struct. 76, 347–363 (2000)

    Article  Google Scholar 

  5. Sze K.Y, Pan Y.S: Hybrid finite element models for piezoelectric materials. J. Sound Vib. 226, 519–547 (1999)

    Article  Google Scholar 

  6. Benjeddou A., Deü J.F.: Piezoelectric transverse shear actuation and sensing of plates-part 1: a three-dimensional mixed state space formulation. J. Intell. Mater. Syst. Struct. 12, 435–449 (2001)

    Google Scholar 

  7. Steele C.R., Kim Y.Y.: Modified mixed variational principle and the state-vector equation for elastic bodies and shells of revolution. J. Appl. Mech. 59, 587–595 (1992)

    Article  MATH  Google Scholar 

  8. D’Ottavio M., Kröplin B.: An extension of Reissner mixed variational theorem to piezoelectric laminates. Mech. Adv. Mater. Struct. 13, 139–150 (2006)

    Article  Google Scholar 

  9. Benjeddou A., Andrianarison O.: A piezoelectric mixed variational theorem for smart multilayered composites. Mech. Adv. Mater. Struct. 12, 1–11 (2005)

    Article  Google Scholar 

  10. Goldstein H., Poole C., Safko J.: Classical Mechanics. Addison Wesley, Reading (2002)

    Google Scholar 

  11. Zou G., Tang L.: A semi-analytical solution for laminated composite plates in Hamiltonian system. Comput. Methods. Appl. Mech. Eng. 128, 395–404 (1995)

    Article  MATH  Google Scholar 

  12. Sheng H.Y., Ye J.Q.: A state space finite element composite plates. Comput. Methods. Appl. Mech. Eng. 191, 4259–4276 (2002)

    Article  MATH  Google Scholar 

  13. Cheung Y.K., Jiang C.P.: Finite layer method in analyses of piezoelectric composite laminates. Comput. Methods. Appl. Mech. Eng. 191, 879–901 (2001)

    Article  MATH  Google Scholar 

  14. Qing G., Qiu J., Liu Y.: A semi-analytical solution for static and dynamic analysis of plates with piezoelectric patches. Int. J. Solids Struct. 43, 1388–1403 (2006)

    Article  MATH  Google Scholar 

  15. Zhou Y.Y., Chen W.Q., Lü C.F.: Semi-analytical solution for orthotropic piezoelectric laminates in cylindrical bending with interfacial imperfections. Compos. Struct. 92, 1009–1018 (2010)

    Article  Google Scholar 

  16. Zhao L., Chen W.Q.: Symplectic analysis of plane problems of functionally graded piezoelectric materials. Mech. Mater. 41, 1330–1339 (2009)

    Article  Google Scholar 

  17. Wu C.P., Li H.Y.: The RMVT- and PVD-based finite layer methods for the three-dimensional analysis of multilayered composite and FGM plates. Compos. Struct. 92, 2476–2496 (2010)

    Article  Google Scholar 

  18. Li R., Zhong Y., Tian B.: On new symplectic superposition method for exact bending solutions of rectangular cantilever thin plates. Mech. Res. Commun. 38, 111–116 (2011)

    Article  Google Scholar 

  19. Li R., Zhong Y.: On a new symplectic geometry method for exact bending solutions of orthotropic rectangular plates with two opposite sides clamped. Acta Mech. 216, 333–343 (2011)

    Article  MATH  Google Scholar 

  20. Van Loan C.: A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix. Linear Algebra Appl. 61, 233–251 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  21. Benner P., Mehrmann V., Xu H.: A numerically stable, structure preserving method for computing the eigenvalues of real Hamiltonian or symplectic pencils. Numerische Mathematik 78, 3229–3258 (1998)

    Article  MathSciNet  Google Scholar 

  22. Andrianarison, O., Benjeddou, A.: A semi-analytical mixed solution for the static analysis of multilayered composite and FGM plates. In: Proceedings of 4th International Congress Des. Model. Mech. Syst., Sousse (Tunisia), May 30th–June 1st (2011)

  23. Kapuria S., Kulkarni S.D.: Static electromechanical response of smart composite/sandwich plates using an efficient finite element with physical and electric nodes. Int. J. Mech. Sci. 51, 1–20 (2009)

    Article  Google Scholar 

  24. Cheng Z.Q., Batra R.C.: Three-dimensional asymptotic analysis of multiple-electroded piezoelectric laminates. AIAA J. 38, 317–324 (2000)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut Supérieur de l’Automobile et des Transports de Nevers, 49 rue Mademoiselle Bourgeois, BP 31, 58027, Nevers Cedex, France

    Orlando Andrianarison

  2. Institut Supérieur de Mécanique de Paris, 3 rue Fernand Hainaut, 93407, Saint Ouen Cedex, France

    Ayech Benjeddou

Authors
  1. Orlando Andrianarison
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ayech Benjeddou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ayech Benjeddou.

Additional information

Dedicated to Professor Hans Irschik on the occasion of his 60th birthday.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Andrianarison, O., Benjeddou, A. Hamiltonian partial mixed finite element-state space symplectic semi-analytical approach for the piezoelectric smart composites and FGM analysis. Acta Mech 223, 1597–1610 (2012). https://doi.org/10.1007/s00707-012-0646-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-012-0646-8

Keywords

Search

Navigation