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Analysis Of Homogeneous And Non-Homogeneous Plates

  • Holm AltenbachEmail author
Part of the Solid Mechanics And Its Applications book series (SMIA, volume 154)

Plate theory is an old branch of solid mechanics — the first development of a general plate theory was made by Kirchhoff more than 150 years ago. After that many improvements were suggested; at the same time some research was focussed on the establishment of a consistent plate theory. Plate-like structural elements are widely used in classical application fields like mechanical and civil engineering, but also in some new fields (electronics, medicine among others). This paper gives a brief overview of the main theoretical directions in the theory of elastic plates. Additional information is available in the literature.

Keywords

structural analysis plates homogeneous and non-homogeneous cross-sections 

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References

  1. Altenbach, H. (1985). Untersuchung von Querschwingungen einschichtiger und mehrschichtiger Platten auf der Grundlage der linearen Theorie einfacher Schalen (in Russ.).Dinamika i prochnost mashin41:47–51Google Scholar
  2. Altenbach, H. (1987). Definition of elastic moduli for plates made from thickness-uneven anisotropic material.Mech. Solids22(1):135–0141Google Scholar
  3. Altenbach, H. (1988). Eine direkt formulierte lineare Theorie für viskoelastische Platten und Schalen.Ingenieur-Archiv58:215–228CrossRefGoogle Scholar
  4. Altenbach, H. (1991). Modeling of viscoelastic behaviour of plates. In Źyczkowski, M., editor,Creep in Structurespages 531–537, Berlin. SpringerGoogle Scholar
  5. Altenbach, H. (2000a). An alternative determination of transverse shear stiffnesses for sandwich and laminated plates.Int. J. Solids Struct.37(25):3503–3520CrossRefGoogle Scholar
  6. Altenbach, H. (2000b). On the determination of transverse shear stiffnesses of orthotropic plates. ZAMP, 51:629–649CrossRefGoogle Scholar
  7. Altenbach, J. and Altenbach, H. (1994).Einführung in die Kontinuumsmechanik. B. G. Teubner, StuttgartGoogle Scholar
  8. Altenbach, H., Altenbach, J., and Kissing, W. (2004).Mechanics of Composite Structural Elements. Springer, BerlinGoogle Scholar
  9. Altenbach, H., Altenbach, J., and Naumenko, K. (1998).Ebene Flächentragwerke. Grundlagen der Modellierung und Berechnung von Scheiben und Platten. Springer, BerlinGoogle Scholar
  10. Altenbach, H., Altenbach, J., and Rikards, R. (1996).Einführung in die Mechanik der Laminat-und Sandwichtragwerke. Modellierung und Berechnung von Balken und Platten aus Verbundwerkstoffen. Deutscher Verlag für Grundstoffindustrie, StuttgartGoogle Scholar
  11. Altenbach, H. and Becker, W., editors (2003).Modern Trends in Composite Laminates Mechanicsvolume 448 ofCISM Courses and Lectures. Springer, WienGoogle Scholar
  12. Altenbach, H. and Fedorov, V.A. (1985). Untersuchung des Randeffektes einer in Dickenrich-tung inhomogenen Kreiszylinderschale (in Russ.).Dinamika i prochnost mashin42:20–24Google Scholar
  13. Altenbach, J., Kissing, W., and Altenbach, H. (1994). Dünnwandige Stab- und Stabschalentrag-werke. Vieweg, Braunschweig/WiesbadenGoogle Scholar
  14. Altenbach, H. and Matzdorf, V. (1988). Zu einigen Anwendungen direkt formulierter Platten-theorien.Wiss. Z. der TU Magdeburg32(4):95–99Google Scholar
  15. Altenbach, H., Naumenko, K., and Zhilin, P.A. (2005). A direct approach to the formulation of constitutive equations for roads and shells. In Pietraszkiewicz, W. and Szymczak, C., editors,Shell Structures — Theory and Application 2005pages 87–90, London. Taylor & Francis/BalkemaGoogle Scholar
  16. Altenbach, H. and Shilin, P.A. (1982). Eine nichtlineare Theorie dünner Dreischichtschalen und ihre Anwendung auf die Stabilitätsuntersuchung eines dreischichtigen Streifens.Technische Mechanik3(2):23–30Google Scholar
  17. Altenbach, H. and Zhilin, P.A. (1988). A general theory of elastic simple shells (in Russ.).Uspekhi Mekhaniki11(4):107–148MathSciNetGoogle Scholar
  18. Altenbach, H. and Zhilin, P.A. (2004). The theory of simple elastic shells. In Kienzler, R., Altenbach, H., and Ott, I., editors,Critical Review of the Theories of Plates and Shells and new ApplicationsLect. Notes Appl. Comp. Mech. 16, pages 1–12. Springer, BerlinGoogle Scholar
  19. Ambarcumyan, S.A. (1987).Theory of Anisotropic Plates (in Russ.). Nauka, MoscowGoogle Scholar
  20. Başar, Y. and Krätzig, W.B. (1985).Mechanik der Flächentragwerke. Vieweg, Braunschweig, WiesbadenGoogle Scholar
  21. Bollé, L. (1947a). Contribution au problème linéaire de flexin d'une plaque élastique.Bull. Techn. Suisse Romande73(21):281–285Google Scholar
  22. Bollé, L. (1947b). Contribution au problème linéaire de flexin d'une plaque élastique.Bull. Techn. Suisse Romande73(22):293–298. Burton, W.S. and Noor, A.K. (1995). Assessment of computational models for sandwich panels and shells.Comp. Methods Appl. Mech. Eng., 124(1–2):125–151Google Scholar
  23. Burton, W.S. and Noor, A.K. (1995). Assessment of computational models for sandwich panels and shells.Comp. Methods Appl. Mech. Eng., 124(1–2):125–151CrossRefGoogle Scholar
  24. Ciarlet, P.G. (1990).Plates and Junctions in Elastic Multi-Structures: An Asymptotic Analysisvolume 14 ofCollection Recherches en Mathématiques Appliquées. Masson, Paris; Springer, HeidelbergGoogle Scholar
  25. Cosserat, E. and Cosserat, F. (1896). Sur la théorie de l'élasticité.Ann. Toulouse10:1–116Google Scholar
  26. Cosserat, E. and Cosserat, F. (1909).Théorie des corps déformables. Herman, ParisGoogle Scholar
  27. Courant, R. and Hilbert, D. (1989).Methods of Mathematical Physics, Vol. 2. Partial Differential Equations. Wiley Interscience Publication, New YorkGoogle Scholar
  28. Dyszlewicz, J. (2004).Micropolar Theory of Elasticityvolume 15 ofLecture Notes in Applied and Computational Mechanics. Springer, BerlinGoogle Scholar
  29. Girkmann, K. (1986). Flächentragwerke. Springer, Wien, 6. editionGoogle Scholar
  30. Gould, P.L. (1988).Analysis of Shells and Plates. Springer, New York et alGoogle Scholar
  31. Green, A.E., Naghdi, P.M., and Waniwright, W.L. (1965). A general theory of Cosserat surface.Arch. Rat. Mech. Anal., 20:287–308Google Scholar
  32. Grigolyuk, E.I. and Kogan, A.F. (1972). Present state of the theory of multilayered shells (in Russ.).Prikl. Mekh., 8(6):3–17Google Scholar
  33. Grigolyuk, E.I. and Seleznev, I.T. (1973). Nonclassical theories of vibration of beams, plates and shelles (in Russ.). InItogi nauki i tekhnikivolume 5 ofMekhanika tverdogo deformiruemogo tela. VINITI, MoskvaGoogle Scholar
  34. Günther, W. (1961). Analoge Systeme von Schalengleichungen.Ingenieur–Archiv, 30:160–188zbMATHCrossRefGoogle Scholar
  35. Hencky, H. (1947). Über die Berücksichtigung der Schubverzerrung in ebenen Platten. Ingenieur–Archiv, 16:72–76CrossRefGoogle Scholar
  36. Irschik, H. (1993). On vibrations of layered beams and plates.ZAMM73:T34–T45Google Scholar
  37. Jaiani, G., Podio-Guidugli, P., editor (2008). IUTAM Symposium on Relations of Shell, Plate, Beam and 3D Models, Springer, BerlinzbMATHGoogle Scholar
  38. Kączkowski, Z. (1980).Plyty obliczenia statyczne. Arkady, WarszawaGoogle Scholar
  39. Kienzler, R. (1982). Erweiterung der klassischen Schalentheorie; der Einfluβ von Dickenverz-errung und Querschnittsverwölbungen.Ingenieur-Archiv52:311–322CrossRefGoogle Scholar
  40. Kienzler, R. (2002). On the consistent plate theories.Arch. Appl. Mech., 72:229–247Google Scholar
  41. Kienzler, R., Altenbach, H., and Ott, I., editors (2004).Critical Review of the Theories of Plates and Shells, New Applicationsvolume 16 ofLect. Notes Appl. Comp. Mech.Springer, BerlinGoogle Scholar
  42. Kirchhoff, G.R. (1850). Über das Gleichgewicht und die Bewegung einer elastischen Scheibe.Crelles Journal für die reine und angewandte Mathematik40:51–88CrossRefGoogle Scholar
  43. Kromm, A. (1953). Verallgemeinerte Theorie der Plattenstatik.Ingenieur-Archiv21:266–286CrossRefGoogle Scholar
  44. Kröner, E., editor (1967).Mechanics of Generalized ContinuaBerlin. Proc. of the IUTAM-Symposium on Generalized Cosserat Continuum and the Continuum Theory of Dislocations with Applications, Springer, Freudenstatt-StuttgartGoogle Scholar
  45. Levinson, M. (1980). An accurate, simple theory of the statics and dynamics of elastic plates.Mech. Res. Commun.7(6):343–350CrossRefGoogle Scholar
  46. Lo, K. H., Christensen, R. M., and Wu, E. M. (1977a). A high-order theory of plate deformation. Part I: Homogeneous plates.Trans. ASME. J. Appl. Mech.44(4):663–668zbMATHGoogle Scholar
  47. Lo, K. H., Christensen, R. M., and Wu, E. M. (1977b). A high-order theory of plate deformation. Part II: Laminated plates.Trans. ASME. J. Appl. Mech.44(4):669–676zbMATHGoogle Scholar
  48. Lurie, A.I. (2005).Theory of Elasticity. Foundations of Engineering Mechanics. Springer, BerlinGoogle Scholar
  49. Meenen, J. and Altenbach, H. (2001). A consistent deduction of von Kármán-type plate theories from threedimensional non-linear continuum mechanics.Acta Mech.147:1–17CrossRefGoogle Scholar
  50. Mindlin, R.D. (1951). Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates.Trans. ASME. J. Appl. Mech.18:31–38Google Scholar
  51. Naghdi, P. (1972). The theory of plates and shells. In Flügge, S., editor,Handbuch der Physikvolume VIa/2, pages 425–640. Springer, Berlin, Heidelberg, New YorkGoogle Scholar
  52. Naumenko, K. and Altenbach, H. (2007).Modeling of Creep for Structural Analysis. Foundations of Engineering Mechanics. Springer, BerlinGoogle Scholar
  53. Neff, P. (2006). The cosserat couple modulus for continuous solids is zero viz the linearized cauchy-stress tensor is symmetric.ZAMM86(11):892–912CrossRefMathSciNetGoogle Scholar
  54. Noor, A.K. and Burton, W.S. (1989a). Assessment of shear deformation theories for multilay-ered composite plates.Appl. Mech. Rev.42(1):1–13Google Scholar
  55. Noor, A.K. and Burton, W.S. (1989b). Stress and free vibration analysis of multilayered composite plates.Comp. Struct.11(3):183–204CrossRefGoogle Scholar
  56. Noor, A.K. and Burton, W.S. (1990a). Assessment of computational models for multilayered anisotropic plates.Comp. Struct.14(3):233–265CrossRefGoogle Scholar
  57. Noor, A.K. and Burton, W.S. (1990b). Assessment of computational models for multilayered composite shells.Appl. Mech. Rev.43(4):67–96Google Scholar
  58. Noor, A.K., Burton, W.S., and Bert, C.W. (1996). Computational models for sandwich panels and shells.Appl. Mech. Rev.49(3):155–199Google Scholar
  59. Nowacki, W. (1986).Theory of Asymmetric Elasticity. Pergamon, OxfordGoogle Scholar
  60. Nye, J.F. (1992).Physical Properties of Crystals. Oxford Science Publications, OxfordGoogle Scholar
  61. Pal'mov, V.A. (1964). Fundamental equations of the theory of asymmetric elasticity (in Russ.).Prikladnaya Matematika i Mekhanika28(6):1117MathSciNetGoogle Scholar
  62. Palmow, W.A. and Altenbach, H. (1982). Über eine Cosseratsche Theorie für elastische Platten.Technische Mechanik3(3):5–9Google Scholar
  63. Panc, V. (1975).Theories of Elastic Plates. Noordhoff International Publishing, LeydenGoogle Scholar
  64. Pietraszkiewicz, W. and Szymczak, C., editors (2005).Shell Structures – Theory and ApplicationTaylor & Francis/Balkema, LondonGoogle Scholar
  65. Preuβer, G. (1984). Eine systematische Herleitung verbesserter Plattentheorien.Ingenieur-Archiv54:51–61CrossRefGoogle Scholar
  66. Reddy, J.N. (1984). A simple higher–order theory for laminated composite plates.Trans. ASME. J. Appl. Mech.51:745–752CrossRefGoogle Scholar
  67. Reddy, J.N. (1990). A general non-linear third order theory of plates with transverse deformations.J. Non-linear Mech.25(6):667–686Google Scholar
  68. Reddy, J.N. (1996).Mechanics of Laminated Composite Plates: Theory and Analysis. CRC Press, Boca RatonGoogle Scholar
  69. Reissner, E. (1944). On the theory of bending of elastic plates.J. Math. Phys.23:184–194Google Scholar
  70. Reissner, E. (1945). The effect of transverse shear deformation on the bending of elastic plates.J. Appl. Mech.12(11):A69–A77Google Scholar
  71. Reissner, E. (1947). On bending of elastic plates.Quart. Appl. Math.5:55–68MathSciNetGoogle Scholar
  72. Reissner, E. (1985). Reflection on the theory of elastic plates.Appl. Mech. Rev.38(11): 1453–1464CrossRefGoogle Scholar
  73. Rothert, H. (1973).Direkte Theorie von Linien- und Flächentragwerken bei viskoelastischen Werkstoffverhalten. Techn.-Wiss. Mitteilungen des Instituts für Konstruktiven Ingenieurbaus 73-2Ruhr-Universität, BochumGoogle Scholar
  74. Rothert, H. and Zastrau, B. (1981). Herleitung einer Direktortheorie für Kontinua mit lokalen Krümmungseigenschaften.ZAMM, 61:567–581CrossRefGoogle Scholar
  75. Rubin, M.B. (2000).Cosserat Theories: Shells, Rods and Pointsvolume 79 ofSolid Mechanics and Its Applications. Springer, BerlinGoogle Scholar
  76. Schaefer, H. (1967). Das Cosserat-Kontinuum.ZAMM, 47(8):485–498CrossRefGoogle Scholar
  77. Timoshenko, S. and Woinowsky Krieger, S. (1985).Theory of Plates and Shells. McGraw Hill, New YorkGoogle Scholar
  78. Todhunter, I. and Pearson, K. (1893).A History of the Theory of Elasticity and of the Strength of Materialsvolume II. Saint-Venant to Lord Kelvin, Part II. University Press, CambridgeGoogle Scholar
  79. Touratier, M. (1991). An efficient standard plate theory.Int. J. Eng. Sci.29(8):901–916CrossRefGoogle Scholar
  80. Truesdell, C. (1964). Die Entwicklung des Drallsatzes.ZAMM, 44(4/5):149–158Google Scholar
  81. Vlasov, V.Z. (1958).Thinwalled Spatial Systems. Gosstroiizdat, Moskwa. (in Russ.)Google Scholar
  82. Woźniak, C. (2001).Mechanik sprȩźystych plyt i powlokvolume VIII ofMechanika techniczna. Wydawnictwo Naukowe PWN, WarszawaGoogle Scholar
  83. Wunderlich, W (1973). Vergleich verschiedener Approximationen der Theorie dünner Schalen (mit numerischen Beispielen). Techn.-Wiss. Mitteilungen des Instituts für Konstruktiven In-genieurbaus 73-1, Ruhr-Universität, BochumGoogle Scholar
  84. Zhilin, P.A. (1976). Mechanics of deformable directed surfaces.Int. J. Solids Struct.12:635– 648CrossRefGoogle Scholar
  85. Zhilin, P.A. (1982). Basic equations of non-classical theory of shells (in Russ.). InTrudy LPI (Trans. Leningrad Polytechnical Institute) — Dinamika i prochnost mashin (Dynamics and strength of machines)number Nr. 386, pages 29–46. Leningrad Polytechnical Institute, LeningradGoogle Scholar
  86. Zhilin, P.A. (1992). The view on Poisson's and Kirchhoff's theories of plates in terms of modern theory of plates (in Russ.).Izvestiya RAN. Mekhanika tverdogo tela (Trans. Russ. Acad. Sci. Mech. Solids)Nr. 3:48–64Google Scholar
  87. Zhilin, P.A. (1995). On the classical theory of plates and the Kelvin-Teit transformation (in Russ.). Izvestiya RAN.Mekhanika tverdogo tela (Trans. Russ. Acad. Sci. Mech. Solids)Nr. 4:133–140Google Scholar
  88. Zhilin, P.A. (2006).Applied Mechanics. Theory of Thin Elastic Rods (in Russ.). St. Petersburg State Polytechnical University, St. PetersburgGoogle Scholar
  89. Zhilin, P.A. (2007).Applied Mechanics. Foundations of the Theory of Shells (in Russ.). St. Petersburg State Polytechnical University, St. PetersburgGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Lehrstuhl für Technische Mechanik Zentrum für IngenieurwissenschaftenMartin-Luther-Universität Halle-WittenbergHalle (Saale)Germany

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