Abstract
The classical plate theory can be applied to thin plates made of classical materials like steel. The first theory allowing the analysis of such plates was elaborated by Kirchhoff. But this approach was connected with various limitations (e.g., constant material properties in the thickness direction). In addition, some mathematical inconsistencies like the order of the governing equation and the number of boundary conditions exist. During the last century many suggestions for improvements of the classical plate theory were made. The engineering direction of improvements was ruled by applications (e.g., the use of laminates or sandwiches as the plate material), and so new hypotheses for the derivation of the governing equations were introduced. In addition, some mathematical approaches like power series expansions or asymptotic integration techniques were applied. A conceptional different direction is connected with the direct approach in the plate theory. This paper presents the extension of Zhilin’s direct approach to plates made of functionally graded materials.
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References
Abramowitz, M., Stegun, I.: (eds.) (1972). Handbook of Mathematical Functions. Dover, New York
Altenbach H. (1988). Eine direkt formulierte lineare Theorie für viskoelastische Platten und Schalen. Ingenieur Archiv 58: 215–228
Altenbach H. (2000). An alternative determination of transverse shear stiffnesses for sandwich and laminated plates. Int. J. Solids Struct. 37(25): 3503–3520
Altenbach H. (2000). On the determination of transverse shear stiffnesses of orthotropic plates. ZAMP 51: 629–649
Altenbach H. and Zhilin P. (1988). A general theory of elastic simple shells (in Russian). Uspekhi Mekhaniki 11(4): 107–148
Altenbach, H., Zhilin, P.A.: The theory of simple elastic shells. In: Kienzler, R., Altenbach, H., Ott, I. (eds.) Critical Review of the Theories of Plates and Shells and New Applications. Lect. Notes Appl. Comp. Mech, vol. 16, pp. 1–12. Springer, Berlin (2004)
Altenbach H., Altenbach J. and Naumenko K. (1998). Ebene Flächentragwerke. Grundlagen der Modellierung und Berechnung von Scheiben und Platten. Springer, Berlin
Altenbach H., Altenbach J. and Kissing W. (2004). Mechanics of Composite Structural Elements. Springer, Berlin
Ashby M.F., Evans A.G., Fleck N.A., Gibson L.J., Hutchinson J.W. and Wadley H.N.G. (2000). Metal Foams: A Design Guid. Butterworth-Heinemann, Boston
Banhart J. (2000). Manufacturing routes for metallic foams. J. Miner. 52(12): 22–27
Chróścielewski J., Makowski J. and Pietraszkiewicz W. (2004). Statics and dynamics of multifold shells. Non-linear theory and finite element method (in Polish). Wydawnictwo IPPT PAN, Warszawa
Gibson L.J. and Ashby M.F. (1997). Cellular Solids: Structure and Properties, 2nd edn. Cambridge Solid State Science Series. Cambridge University Press, Cambridge
Grigolyuk, E.I., Seleznev, I.T.: Nonclassical theories of vibration of beams, plates and shelles (in Russian). In: Itogi nauki i tekhniki, Mekhanika tverdogo deformiruemogo tela, vol. 5, VINITI, Moskva (1973)
Gupta N. and Ricci W. (2006). Comparison of compressive properties of layered syntactic foams having gradient in microballoon volume fraction and wall thickness. Mater. Sci. Eng. A 427: 331–342
El Hadek M.A. and Tippur H.V. (2003). Dynamic fracture parameters and constraint effects in functionally graded syntactic epoxy foams. Int. J. Solids struc. 40: 1885–1906
Kienzler R. (2002). On the consistent plate theories. Arch. Appl. Mech. 72: 229–247
Kienzler, R., Altenbach, H., Ott, I.: (eds.) Critical review of the theories of plates and shells, new applications. Lect. Notes Appl. Comp. Mech. vol. 16, Springer, Berlin (2004)
Kirchhoff G.R. (1850). Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. Crelles Journal für die reine und angewandte Mathematik 40: 51–88
Libai A. and Simmonds J.G. (1998). The Nonlinear Theory of Elastic Shells, 2nd edn. Cambridge University Press, Cambridge
Lurie A.I. (2005). Theory of Elasticity. Foundations of Engineering Mechanics. Springer, Berlin
Mindlin R.D. (1951). Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. Trans. ASME J. Appl. Mech. 18: 31–38
Naghdi, P.: The theory of plates and shells. In: Flügge, S. (ed.) Handbuch der Physik, vol. VIa/2, pp. 425–640. Springer, Heidelberg (1972)
Nye J.F. (2000). Physical Properties of Crystals. Oxford Science Publications, Clarendon, Oxford
Reissner E. (1944). On the theory of bending of elastic plates. J. Math. Phys. 23: 184–194
Reissner E. (1945). The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech. 12(11): A69–A77
Reissner E. (1947). On bending of elastic plates. Q. Appl. Math. 5: 55–68
Reissner E. (1985). Reflection on the theory of elastic plates. Appl. Mech. Rev. 38(11): 1453–1464
Rothert, H.: Direkte Theorie von Linien- und Flächentragwerken bei viskoelastischen Werkstoffverhalten. Techn.-Wiss. Mitteilungen des Instituts für Konstruktiven Ingenieurbaus 73-2, Ruhr-Universität, Bochum (1973)
Stoer J. and Bulirsch R. (1980). Introduction to Numerical Analysis. Springer, New York
Timoshenko S.P. and Woinowsky-Krieger S. (1985). Theory of Plates and Shells. McGraw Hill, New York
Truesdell C. (1964). Die Entwicklung des Drallsatzes. ZAMM 44(4/5): 149–158
Zhilin P.A. (1976). Mechanics of deformable directed surfaces. Int. J. Solids Struc. 12: 635–648
Zhilin, P.A.: Applied mechanics. Foundations of the theory of shells (in Russian). St Petersburg State Polytechnical University (2007)
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The second author was supported by DFG grant 436RUS17/21/07.
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Altenbach, H., Eremeyev, V.A. Direct approach-based analysis of plates composed of functionally graded materials. Arch Appl Mech 78, 775–794 (2008). https://doi.org/10.1007/s00419-007-0192-3
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DOI: https://doi.org/10.1007/s00419-007-0192-3