Applied Mathematics and Mechanics

, Volume 9, Issue 7, pp 649–657 | Cite as

Variational principle of hybrid energy and the fundamentals of 3-D laminate theory—A new approach for the analysis of interlaminar stresses in composite laminates

  • Huang Qian


This paper discusses the discontinuity of stresses and strains at interlaminar surfaces of the composue laminate and presents a 3-D laminate theory for composite materials. This paper also presents a new type of elastic energy based on the globally continuous variables in laminates, different from the traditional potential energy and complementary energy. Then, a variational principle corresponding to the 3-D laminate theory is developed. The theory and the principle could be a basis of verifying the 2-D laminate theory and determining the interlaminar stresses near the free edges.


Variational Principle Forced Constraint Conjunction Condition Stress Formulation Hybrid Energy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Lekhnitskii, S.G.,Theory of Elasticity of an Anisotropic Body, Gostekhizdat, Moscow (1950). (in Russian). Transl., Holden-Day, San Francisco (1963). Second Edition, Nauka, Moscow, (1977), (in Russian). Transl., Mir Publishers, Moscow (1981).zbMATHGoogle Scholar
  2. [2]
    Pipes, R.B.,J. Compos. Mater.,4, Oct (1970), 538–548.Google Scholar
  3. [3]
    Altus, E., A. Rotem, and M. Shmueli,J. Compos. Mater.,14 (1980), 21–30.Google Scholar
  4. [4]
    Engblom, J.J. and O.O. Ochoa,Int. J. Numer. Methods Eng.,21, 10. Oct. (1985), 1759–1776.zbMATHCrossRefGoogle Scholar
  5. [5]
    Heppler, G.R. and J.S. Hansen,Adv. in Compos. Mater.,1 Aug. (1980), 666–692.Google Scholar
  6. [6]
    Yeh, J.R. and I.G. Tabjbakhsh,J. Compos. Mater.,20, 4. Jul. (1986), 347–364.Google Scholar
  7. [7]
    Natarajan, R., S.V. Hoa, and T.S. Sankar,Int. Numer. Methods Eng.,23, 4. Apr. (1986). 623–633.zbMATHCrossRefGoogle Scholar
  8. [8]
    Lucking, W.M., S.V. Hoa and T.S. Sankar,J. Compos. Mater.,18, 2. Mar. (1984), 188–198.Google Scholar
  9. [9]
    Rybicki, E.F.,J. Compos. Mater.,5 (1971), 354–360.Google Scholar
  10. [10]
    Khalil, S.A., C.T. Sun and W.C. Hwang,Int. J. Fract.,31, 1. May (1986), 37–51.CrossRefGoogle Scholar
  11. [11]
    Moriya, K.,Nippon Kikai Gakkai Ronbunshu a Hen,52, 478, Jun. (1986), 1600–1607.Google Scholar
  12. [12]
    Spilker, R.L. and D.M. Jakobs,Int. Numer. Methods Eng.,23, 4. Apr. (1986), 555–578.zbMATHCrossRefGoogle Scholar
  13. [13]
    Wang, S.S. and F.G. Yuan,J. Appl. Mech.,50 (1983), 835–844.zbMATHCrossRefGoogle Scholar
  14. [14]
    Hsu, P.W. and C.T. Herakovich,J. Compos. Mater.,11 (1977), 422–427.Google Scholar
  15. [15]
    Tang, S.,J. Compos. Mater.,10, 1, Jan. (1976), 69–78.Google Scholar
  16. [16]
    Bar-Yoseph, P. and T.H.H. Pian,J. Compos. Mater.,15, 3, May (1981), 225–239.Google Scholar
  17. [17]
    Bar-Yoseph, P. and T.H.H. Pian,Comput. Methods Appl. Mech. Eng.,36, 3, Mar (1983), 309–329.zbMATHCrossRefGoogle Scholar
  18. [18]
    Pagano, N.J.,Int. J. Solids Structures,14 (1978), 385–400.zbMATHCrossRefGoogle Scholar
  19. [19]
    Wang, J.T.S. and J.N. Dickson,J. Compos. Mater.,12 (1978), 390–401.Google Scholar
  20. [20]
    Oery, H., H. Reimerdes and S. Dieker,Z. Flugwiss Weltraumforsch,8, 6, Nov.–Dec. (1984), 392–404.Google Scholar
  21. [21]
    Whitney, J.M. and C.E. Browning,J. Compos. Mater.,6 (1973), 300–303.Google Scholar
  22. [22]
    Berghaus, D.G. and R.W. Aderholdt,Exp. Mech.,15, 4, Jul. (1975), 173–176.CrossRefGoogle Scholar
  23. [23]
    Spilker, R.L. and S.C. Chou,J. Compos. Mater.,14 (1980), 2–20.Google Scholar
  24. [24]
    Chien, W.Z.,On the Generalized Variational Principles, Knowledge Press, Shanghai (1985). (in Chinese)Google Scholar
  25. [25]
    Tsai, S.W. and H.T. Hahn,Introduction to Composite Materials. Technomic Publishing Co. Inc. (1980).Google Scholar
  26. [26]
    Christensen, R.M.,Mechanics of Composite Materials, John Wiley and Sons Inc. (1979).Google Scholar
  27. [27]
    Hull, D.,An Introduction to Composite Materials, Cambridge University Press (1981).Google Scholar

Copyright information

© SUT 1988

Authors and Affiliations

  • Huang Qian
    • 1
    • 2
  1. 1.Concordia UniversityMontrealCanada
  2. 2.Shanghai University of TechnologyChina

Personalised recommendations