Keywords: viscoelastic, Mindlin plate, buckling, rectangular plate, dynamic relaxation, higher-order shear deformation.

With increasing application of composite structures with polymeric matrix, the design and analysis of such structures also is growing. Fiber-reinforced laminated composite beams, plates and shells with epoxy resin matrix behave mostly elastically up to their point of fracture. On the other hand such structures as mentioned above with polymeric matrix have inelastic behaviour even in room temperatures. Consequently, it is more appropriate to first analyze a rectangular plate made of viscoelastic material and then add the reinforcing elements.

The study of viscoelastic plates under dynamic loading has been carried out in various forms [1]. The stability of a viscoelstic plate with non-linear integrodifferential equations under dynamic loading is considered in [2]. The plate is assumed to be thin but with non-linear in-plane strain-displacement relations. In [3], the postbuckling behaviour of viscoelastic laminated plates taking into account the shear-deformations is presented. The higher-shear deformation theory, as originally developed by Reddy [4], is applied. The effects of viscoelastic behaviour are shown, and the results for lower order theories are compared with the results obtained using the higher order theory. In [5], the dynamic stability of viscoelastic laminated plates, subjected to a harmonic in-plane excitation is considered. The viscoelastic behaviour is caused by the polymeric matrix of the fiber-reinforced laminated plate. The Boltzmann viscoelastic material model is used in the stress-strain relations of the laminated plate. This leads to an integro-differential equation of motion, obtained within the first-order shear deformation theory. The free vibration analysis of rectangular viscoelastic plates with simply supported boundary conditions is presented in [6]. The same author has presented the free vibration analysis results of viscoelastic plates, this time, using a higher-order shear deformation theory [7]. However, the so called simple higher-order plate theory is employed where two terms are added to the displacement field equations in the plane of the plate and the deflections are taken as in the first-order shear deformation theory. An exact series form of solution is presented. The chaotic dynamic analysis of viscoelastic plates is presented in [8]. The dynamic buckling of viscoelastic plates with large deflection is investigated in [10] by using chaotic and fractal theory. The material behaviour is given in terms of the Boltzmann superposition principal. In [9], the post buckling analysis of imperfect non-linear viscoelastic cylindrical panels is presented. The material in modeled according to the Schapery representation of non-linear viscoelasticity. They have developed solutions to calculate the growth of the initial imperfection in time by using the Donnell equilibrium equations for geometrically non-linear cylindrical panels.

In this paper the von Karman type of equilibrium equations taking into account the effect of shear deformations are used. The strain-displacement relations for the plate with large deflections, originally developed by Sander [10] are applied. The material behaviour is given in terms of the Boltzmann superposition principle. The Dynamic Relaxation (DR) numerical method is used to analyse the buckling behaviour of viscoelastic rectangular plates. Finally, numerical results are graphically presented for plate centre deflections versus time and different plate theories are compared. Unfortunately due to unavailability of viscoelastic plate results, elastic plate buckling problem is solved and the buckling loads are compared with those reported in [11]. The correlations are very satisfactory.

1

F.B. Badalov, Kh. Eshmatov, U.I. Akbarov, "Stability of a viscoelastic plate under dynamic loading", Translated from Prikladnaya Mekhanika, 27(9), 892, (1991). doi:10.1007/BF00887982

2

I.A. Kiiko, Sh.I. Shanazarov, "Dynamic buckling of viscoelastic circular and annular plates", Izv. Akad. Nauk UzSSR, Ser. Tekh. Nauk, No. 2, 35-40, (1986).

3

D. Shalev, J. Aboudi, "Postbuckling analysis of viscoelastic laminated plates using higher-order theory", Int. J. Solids Structures, 27(14), 1747-1755, (1991). doi:10.1016/0020-7683(91)90010-D

4

J.N. Reddy, "A simple high-order theory for laminated composite plates, J. Appl. Mech. 45, 745-752, (1984a).

5

G. Cederbaum, J. Aboudi, "Dynamic instability of shear-deformable viscoelastic laminated plates by Lyapunov Exponents" Int. J. Solids Structures, 28(3), 317-327. (1991). doi:10.1016/0020-7683(91)90196-M

6

S.D. Yu, "Free vibration of rectangular viscoelastic plates with simply supported boundary conditions", Presented at DYCONS99, Vibration of Continuous Systems, Ottawa, Ontario, Canada, (1999).

7

S.D. Yu, "Free vibration analysis of viscoelastic plates using a higher-order plate theory", Private Communication, Department of Mechanical Engineering, Ryerson Polytechnic University, 350 Victoria St., Toronto, Ontario, Canada M5B 2K3.

8

Y.X. Sun, S.Y. Zhang, "Chaotic dynamic analysis of viscoelastic plates", Int. J. Mech. Sciences, 43, 1195-1208, (2001). doi:10.1016/S0020-7403(00)00062-X

9

G. Cederbaum, D. Touati, "Postbuckling analysis of imperfect non-linear viscoelastic cylindrical panels", Int. J. Non-Linear Mechanics, 37, 757-762, (2002). doi:10.1016/S0020-7462(01)00097-X

10

J.T. Sanders, "Nonlinear theories for thin shells", Q. Appl. Math., 21(1), 21-36, (1963).

11

S.E. Swartz, R.J. O'Neill, "Linear elastic buckling of plates subjected to combined loads", Thin-Walled Structures 21,1-15,(1995). doi:10.1016/0263-8231(94)P4389-R

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