Advertisement

Applied Mathematics and Mechanics

, Volume 26, Issue 4, pp 486–494 | Cite as

Equivalence of refined theory and decomposed theorem of an elastic plate

Article

Abstract

A connection between Cheng's refined theory and Gregory's decomposed theorem is analyzed. The equivalence of the refined theory and the decomposed theorem is given. Using operator matrix determinant of partial differential equation, Cheng gained one equation, and he substituted the sum of the general integrals of three differential equations for the solution of the equation. But he did not prove the rationality of substitute. There, a whole proof for the refined theory from Papkovich-Neuber solution was given. At first expressions were obtained for all the displacements and stress components in term of the midplane displacement and its derivatives. Using Lur'e method and the theorem of appendix, the refined theory was given. At last, using basic mathematic method, the equivalence between Cheng's refined theory and Gregory's decomposed theorem was proved,i. e., Cheng's bi-harmonic equation, shear equation and transcendental equation are equivalent to Gregory's interior state, shear state and Papkovich-Fadle state, respectively.

Key words

elastic plate isotropic plate refined theory decomposed theorem Papkovich-Neuber general solution 

Chinese Library Classification

O343 

2000 Mathematics Subject Classification

74B20 74K20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Cheng Shun. Elasticity theory of plates and a refined theory[J].Journal of Application Mechanics, 1979,46(2): 644–650.MATHGoogle Scholar
  2. [2]
    Lur'e A I.Three-Dimensional Problems in the Theory of Elasticity[M]. Interscience New York, 1964, 148–166.Google Scholar
  3. [3]
    Wang Feiyue. A refined elasticity theory for transversely isotropic plates[J].Shanghai Mechanics, 1985,6(2): 10–21 (in Chinese).Google Scholar
  4. [4]
    Gregory R D. The general form of the three-dimensional elastic field inside an isotropic plate with free faces[J].Journal of Elasticiy, 1992,28(1): 1–28.MATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    Gregory R D. The semi-infinite stripx≥0, −1≤y≤1; completeness of the Papkovich-Fadle eigenfunctions whenφ xx(0,y),φyy(0,y) are prescribed[J].Journal of Elasticity, 1980,10(1): 57–80.MATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    Gregory R D. The traction boundary value problems for the elastostatic semi-infinite strip; existence of solution, and completeness of the Papkovich-Fadle eigenfunctions[J].Journal of Elasticity, 1980,10(3): 295–327.MATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    Wang Minzhong, Zhao Baosheng. The decomposed form of the three-dimensional elastic plate[J].Acta Mechanica, 2003,166(3): 207–216.Google Scholar
  8. [8]
    Zhao Baosheng, Wang Minzhong. The decomposed theorem of the transversely isotropic elastic plate[J].Acta Mechanica Sinica, 2004,36(1): 57–63 (in Chinese).Google Scholar
  9. [9]
    Wang Minzhong, Wang Wei. Completeness and nonuniqueness of general solutions of transversely isotropic elasticity[J].International Journal of Solids and Structures, 1995,32(3/4): 501–513.MathSciNetGoogle Scholar
  10. [10]
    Wang Wei, Wang Minzhong. Constructivity and completeness of the general solutions in elastodynamics[J].Acta Mechanica, 1992,91(1): 209–214.MathSciNetGoogle Scholar
  11. [11]
    Wang Wei, Shi Mingxing. Thick plate theory based on general solutions of elasticity[J].Acta Mechanica, 1997,123(1): 27–36.MathSciNetCrossRefGoogle Scholar

Copyright information

© Editorial Committee of Appl. Math. Mech 2005

Authors and Affiliations

  1. 1.School of Mechanical Engineering and AutomationAnshan University of Science and TechnologyAnshan Liaoning ProvinceP. R. China
  2. 2.State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering SciencePeking UniversityBeijingP. R. China

Personalised recommendations