Abstract
The integro-partial differential equations governing the dynamic behavior of viscoelastic plates taking account of higher-order shear effects and finite deformations are presented. From the matrix formulas of differential quadrature, the special matrix product and the domain decoupled technique presented in this work, the nonlinear governing equations are converted into an explicit matrix form in the spatial domain. The dynamic behaviors of viscoelastic plates are numerically analyzed by introducing new variables in the time domain. The methods in nonlinear dynamics are synthetically applied to reveal plenty and complex dynamical phenomena of viscoelastic plates. The numerical convergence and comparison studies are carried out to validate the present solutions. At the same time, the influences of load and material parameters on dynamic behaviors are investigated. One can see that the system will enter into the chaotic state with a paroxysm form or quasi-periodic bifurcation with changing of parameters.
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References
Cederbaum, G., Aboudi, J., Elishakoff, I.: Dynamic instability of shear-deformable viscoelastic laminated plates by Lyapunov exponents. Int. J. Solids Struct. 28(3), 317–327 (1991)
Touati, D., Cederbaum, G.: Dynamic stability of nonlinear viscoelastic plates. Int. J. Solids Struct. 31(17), 2367–2376 (1994)
Touati, D., Cederbaum, G.: Influence of large deflections on the dynamic stability of nonlinear viscoelastic plates. Acta Mech. 113, 215–231 (1995)
Cheng, C.J., Zhang, N.H.: Variational principles on static-dynamic analysis of viscoelastic thin plates with applications. Int. J. Solids Struct. 35(33), 4491–4505 (1998)
Zhang, N.H., Cheng, C.J.: Non-linear mathematical model of viscoelastic thin plates with its applications. Comput. Meth. Appl. Mech. Eng. 165(4), 307–319 (1998)
Zhang, N.H., Cheng, C.J.: Two-mode Galerkin approach in dynamic stability. Appl. Math. Mech. 24(3), 247–255 (2003)
Sheng, D.F., Cheng, C.J.: Dynamical behaviors of nonlinear viscoelastic thick plates with damage. Int. J. Solids Struct. 41, 7287–7308 (2004)
Bert, C.W., Malik, M.: Differential quadrature method in computational mechanics: a review. Appl. Mech. Rev. 49, 1–28 (1996)
Bert, C.W., Wang, X., Striz, A.G.: Differential quadrature for static and free vibrational analyses of anisotropic plates. Int. J. Solids Struct. 30, 1737–1744 (1993)
Wang, X., Bert, C.W.: A new approach in applying differential quadrature to static and free vibrational analyses of beams and plates. J. Sound Vib. 162, 566–572 (1993)
Want, X., Bert, C.W.: Differential quadrature analysis of deflection, buckling and free vibrations of beams and rectangular plates. Comput. Struct. 48, 473–479 (1993)
Chen, W.: Differential quadrature method and its applications in engineering. Ph.D. dissertation, Shanghai Jiao Tong University, China (1996)
Chen, W., Shu, C., He, W., Zhong, T.: The applications of special matrix products to differential quadrature solution of geometrically nonlinear bending of orthotropic rectangular plates. Comput. Struct. 74, 65–76 (2000)
Liew, K.M., Han, J.B., Xiao, Z.M.: Differential quadrature method for thick symmetric cross-ply laminates with first-order shear flexibility. Int. J. Solids Struct. 33, 2647–2658 (1996)
Han, J.B., Liew, K.M.: An eight-node curvilinear differential quadrature formulation for Reissner/Mindlin plates. Comput. Meth. Appl. Mech. Eng. 141, 265–280 (1997)
Liew, K.M., Teo, T.M., Han, J.B.: Comparative accuracy of DQ and HDQ methods for three-dimensional vibration analysis of rectangular plates. Int. J. Num. Meth. Eng. 45, 1831–1848 (1999)
Teo, T.M., Liew, K.M.: Three-dimensional elasticity solutions to some orthotropic plate problems. Int. J. Solids Struct. 36, 5301–5326 (1999)
Liu, F.L., Liew, K.M.: Analysis of vibrating thick rectangular plates with mixed boundary constraints using differential quadrature element method. J. Sound Vib. 225(5), 915–934 (1999)
Liu, F.L., Liew, K.M.: Free vibration analysis of Mindlin sector plates: Numerical solutions by differential quadrature method. Comput. Meth. Appl. Mech. Eng. 177, 77–92 (1999)
Liew, K.M., Teo, T.M., Han, J.B.: Three-dimensional static solutions of rectangular plates by variant differential quadrature method. Int. J. Mech. Sci. 43, 1611–1628 (2001)
Liew, K.M., Huang, Y.Q.: Bending and buckling of thick symmetric rectangular laminates using the moving least-squares differential quadrature method. Int. J. Mech. Sci. 45, 95–114 (2003)
Liew, K.M., Yang, J., Kitipornchai, S.: Postbuckling of the piezoelectric FGM plates subjected to thermo-electro-mechanical loading. Int. J. Solids Struct. 40, 3869–3892 (2003)
Yang, J., Liew, K.M., Kitipornchai, S.: Dynamic stability of laminated FGM plates based on higher-order shear deformation theory. Comput. Mech. 33, 305–315 (2004)
Li, J.J., Cheng, C.J.: Differential quadrature method for nonlinear vibration of orthotropic plates with finite deformations and transverse shear effect. J. Sound Vib. 281, 295–309 (2005)
Malekzadeh, P., Karami, G.: Differential quadrature nonlinear analysis of skew composite plates based FSDT. Eng. Struct. 28, 1307–1318 (2006)
Karami, G., Malekzadeh, P., Mohebpour, S.R.: DQM free vibration analysis of moderately thick symmetric laminated plates with elastically restrained edges. Compos. Struct. 74, 115–125 (2006)
Malekzadeh, P., Sctoodch, A.R.: Large deformation analysis of moderately thick laminated plates on nonlinear elastic foundation by DQM. Compos. Struct. 80, 569–579 (2007)
Malekzadeh, P.: A differential quadrature nonlinear free vibration analysis of laminated composite skew thin plates. Thin-Walled Struct. 45, 237–250 (2007)
Wang, X., Gan, L., Zhang, Y.: Differential quadrature analysis of the buckling of thin rectangular plates with cosine-distributed compressive loads on two opposite sides. Adv. Eng. Softw. 39, 497–504 (2008)
Reddy, J.N.: A refined nonlinear theory of plates with transverse shear deformation. Int. J. Solids Struct. 20(9/10), 881–896 (1984)
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Sponsored by: the Major Research Plan of the National Natural Science Foundation of China (No. 90816001), the National Science Foundation for Post-doctoral Scientists of China (No. 20080440613); the Shanghai Postdoctoral Sustentation Fund, China (No. 09R21412700), the Shanghai Leading Academic Discipline Project (No. S30106).
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Li, JJ., Cheng, CJ. Differential quadrature method for analyzing nonlinear dynamic characteristics of viscoelastic plates with shear effects. Nonlinear Dyn 61, 57–70 (2010). https://doi.org/10.1007/s11071-009-9631-8
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DOI: https://doi.org/10.1007/s11071-009-9631-8