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A nonlinear composite plate theory

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Abstract

A general nonlinear theory for the dynamics of elastic anisotropic plates undergoing moderate-rotation vibrations is presented. The theory fully accounts for geometric nonlinearities (moderate rotations and displacements) by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvatures and strain-displacement expressions that contain the von Karman strains as a special case. The theory accounts for transverse shear deformations by using a third-order theory and for extensionality and changes in the configuration due to in-plane and transverse deformations. Five third-order nonlinear partial-differential equations of motion describing the extension-extension-bending-shear-shear vibrations of plates are obtained by an asymptotic analysis, which reveals that laminated plates display linear elastic and nonlinear geometric couplings among all motions.

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References

  1. Seide, P., Small Elastic Deformations of Thin Shells, Noordhoff, The Netherlands, 1975.

    Google Scholar 

  2. Srinivas, S. and Rao, A.K., ‘Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates’, International Journal of Solids and Structures, 6, 1970, 1463–1481.

    Google Scholar 

  3. Srinivas, S., Joga Rao, C. V., and Rao, A. K., ‘An exact analysis for vibration of simply-supported and laminate thick rectangular plates’, Journal of Sound and Vibration 12, 1970, 187–199.

    Google Scholar 

  4. Pagano, N. J., ‘Exact solutions for rectangular bidirectional composites and sandwich plates’, Journal of Composite Materials 4, 1970, 20–34.

    Google Scholar 

  5. Pagano, N. J. and Hatfield, S. J., ‘Elastic behavior of multilayer bidirectional composites’, AIAA Journal 10, 1972, 931–933.

    Google Scholar 

  6. Yang, P. C., Norris, C. H., and Stavsky, Y., ‘Plastic wave propagation in heterogeneous plates’, International Journal of Solids and Structures 2, 1966, 648–665.

    Google Scholar 

  7. Adams, R. D. and Bacon, D. G. C., ‘Measurement of the flexural damping capacity and dynamic Young's modulus of metals and reinforced plastics’, Journal of Physics D: Applied Physics 6, 1973, 27–41.

    Google Scholar 

  8. Reissner, E., ‘On the theory of bending of elastic plates’, Journal of Mathematical Physics 23, 1994, 184–191.

    Google Scholar 

  9. Reissner, E., ‘The effect of transverse shear deformation on the bending of elastic plates’, Journal of Applied Mechanics 12(1), 1945, A-69-A-77.

    Google Scholar 

  10. Reissner, E., ‘On bending of elastic plates’, Quarterly of Applied Mathematics 5, 1947, 55–68.

    Google Scholar 

  11. Basset, A. B., ‘On the extension and flexure of cylindrical and spherical thin elastic shells’, Philosophical Transactions of the Royal Society of London Ser. A, 181(6), 1890, 433–480.

    Google Scholar 

  12. Hildebrand, F. B., Reissner, E., and Thomas, G. B., ‘Note on the foundations of the theory of small displacements of orthotropic shells’, NACA Technical Note No. 1833, March 1949.

  13. Hencky, H., ‘Uber die Berucksichtigung der Schubverzerrung in ebenen Platten’, Ingenieur Archiv 16, 1947, 72–76.

    Google Scholar 

  14. Mindlin, R. D., ‘Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates’, Journal of Applied Mechanics 18, 1951, 31–38.

    Google Scholar 

  15. Librescu, L., ‘The thermoelastic problem of shells and plates approached by eliminating the Love-Kirchhoff hypothesis’, Studii si Cercetari de Mecanica Aplicata 21, 1966, 351–365.

    Google Scholar 

  16. Labrescu, L., ‘The elasto-kinetic problems in the theory of anisotropic shells and plates—II. Theory of plates’, Revue Roumaine des Sciences Techniques, Serie de Mechanique Appliquee 14(3), 1969, 637–654.

    Google Scholar 

  17. Nelson, R. B. and Lorch, D. R., ‘A refined theory for laminated orthotropic plates’, Journal of Applied Mechanics 41, 1974, 177–183.

    Google Scholar 

  18. Lo, K. H., Christensen, R. M. and Wu, E. M., ‘A higher-order theory of plate deformation-part I and II’, Journal of Applied Mechanics 44(4), 1977, 663–676.

    Google Scholar 

  19. Levinson, M., ‘An accurate simple theory of the statics and dynamics of elastic plates’, Mechanics Research Communications 7, 1980, 343–350.

    Google Scholar 

  20. Murthy, M. V. V., ‘An improved transverse shear deformation theory for laminated anisotropic plates’, NASA Technical Paper No. 1903, November 1981.

  21. Schmidt, R., ‘A refined nonlinear theory of plates with transverse shear deformation’, Journal of Industrial Mathematics Society 27, 1977, 23–38.

    Google Scholar 

  22. Whitney, J. M. and Sun, C. T., ‘A higher order theory for extensional motion of laminated composites’, Journal of Sound and Vibration 30(1), 1973, 85–97.

    Google Scholar 

  23. Reddy, J. N., ‘A simple higher-order theory for laminated composite plates’, Journal of Applied Mechanics 45, 1984, 745–752.

    Google Scholar 

  24. Bhimaraddi, A. and Stevens, L. K., ‘A higher-order theory for free vibration of orthotropic, homogeneous, and laminated rectangular plates’, Journal of Applied Mechanics 51, 1984, 195–198.

    Google Scholar 

  25. Librescu, L. and Reddy, J. N., ‘A few remarks concerning several refined theories of anisotropic composite laminated plates’, International Journal of Engineering Science 27, 1989, 515–527.

    Google Scholar 

  26. Reddy, J. N., ‘A review of the literature on finite-element modeling of laminated composite plates’, Shock and Vibration Digest 17, 1985, 3–8.

    Google Scholar 

  27. Librescu, L., Khdeir, A. A., and Reddy, J. N., ‘A comprehensive analysis of the state of stress of elastic anisotropic flat plates using refined theories’, Acta Mechanica 70, 1987, 57–81.

    Google Scholar 

  28. Librescu, L. and Khdeir, A. A. ‘Analysis of symmetric cross-ply laminated elastic plates using a higher-order theory: part I-stress and displacement’, Composite Structures, 9, 1988, 189–213.

    Google Scholar 

  29. Khdeir, A. A. and Librescu, L., ‘Analysis of symmetric cross-ply laminated elastic plates using a higher-order theory: part II-puckling and free vibration’, Composite Structures, 9, 1988, 259–277.

    Google Scholar 

  30. Reissner, E., ‘Finite deflections of sandwich plates’, Journal of the Aeronautical Sciences 15, 1948, 435–440.

    Google Scholar 

  31. Reissner, E., ‘On variational theory for finite elastic deformation’, Journal of Mathematical Physics 32, 1953, 129–135.

    Google Scholar 

  32. Reddy, J. N., ‘A refined nonlinear theory of plates with transverse shear deformation’ International Journal of Solids and Structures, 20, 1984, 881–896.

    Google Scholar 

  33. Bhimaraddi, A., ‘Nonlinear flexural vibrations of rectangular plates subjected to in-plane forces using a new shear deformation theory’, Thin-Walled Structures 5, 1987, 309–327.

    Google Scholar 

  34. Hodges, D. H. and Dowell, E. H., ‘Nonlinear Equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades’,NASA TN D-7818, 1974.

  35. Dowell, E. H., Traybar, J., and Hodges, D. H., ‘An experimental-theoretical correlation study of nonlinear bending and torsion deformations of a cantilever beam’, Journal of Sound and Vibration 50, 1977, 533–544.

    Google Scholar 

  36. Crespo de Silva, M. R. M. and Glynn, C. C., ‘Nonlinear flexural-flexural-torsional dynamics of inextensional beams-I, equations of motion’, Journal of Structural Mechanics 6, 1978, 437–448.

    Google Scholar 

  37. Alkire, K., ‘An analysis of rotor blade twist variables associated with different Euler sequences and pretwist treatments’, NASA TM 84394, 1984.

  38. Hodges, D. H., Crespo de Silva, M. R. M. and Peters, D. A., ‘Nonlinear effects in the static and dynamic behavior of beams and rotor blades’, Vertica 12, 1988, 243–256.

    Google Scholar 

  39. Pai, P. F. and Nayfeh, A. H., ‘Thee-dimensional nonlinear vibrations of composite beams—I. Equations of motion’, Nonlinear Dynamics 1, 1990, 477–502.

    Google Scholar 

  40. Whitney, J. M., Structural Analysis of Laminated Anisotropic Plates, Technomic Publishing Company, Inc., Pennsylvania. 1987.

    Google Scholar 

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Authors and Affiliations

  1. Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, 24061, Blacksburg, Virginia, U.S.A.

    Perngjin F. Pai & Ali H. Nayfeh

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  1. Perngjin F. Pai
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  2. Ali H. Nayfeh
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Pai, P.F., Nayfeh, A.H. A nonlinear composite plate theory. Nonlinear Dyn 2, 445–477 (1991). https://doi.org/10.1007/BF00045438

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  • DOI: https://doi.org/10.1007/BF00045438

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