Abstract
This paper presents a study on nonlinear vibration of inhomogeneous functional plates composed of sigmoid graded metal-ceramic materials. The material properties vary continuously along the thickness direction according to a sigmoid distribution rule, which is defined by piecewise functions to ensure smooth distribution of stress among all the interfaces. The geometric nonlinearity is considered by adopting the von Kármán geometrical relations. Based on the d’Alembert’s principle, the nonlinear out-of-plane equation of motion of the plates is developed. The Galerkin method is employed to discretize the motion equation to a series of ordinary differential ones, which are subsequently analyzed via the use of the method of harmonic balance. Then, the analytical results are validated by the comparison to numerical solutions, which are obtained by using the adaptive step-size fourth-order Runge-Kutta method. The stability of the steady-state response is examined by the perturbation technique. Results show the first and third modes are both activated while the second mode is not activated for the plates under harmonic point excitation. The frequency response relationships of activated modes exhibit very complicated curves due to the nonlinear modal interaction. In addition, influences of key system parameters on nonlinear vibrational characteristics of the present inhomogeneous plates are illustrated.
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References
Koizumi M. FGM activities in Japan. Compos Part B-Eng, 1997, 28: 1–4
Wirowski A, Michalak B, Gajdzicki M. Dynamic modelling of annular plates of functionally graded structure resting on elastic heterogeneous foundation with two modules. J Mech, 2015, 31: 493–504
Gupta A, Talha M, Singh B N. Vibration characteristics of functionally graded material plate with various boundary constraints using higher order shear deformation theory. Compos Part B-Eng, 2016, 94: 64–74
Bernardo G M S, Damásio F R, Silva T A N, et al. A study on the structural behaviour of FGM plates static and free vibrations analyses. Compos Struct, 2016, 136: 124–138
Han S C, Park W T, Jung W Y. 3D graphical dynamic responses of FGM plates on Pasternak elastic foundation based on quasi-3D shear and normal deformation theory. Compos Part B-Eng, 2016, 95: 324–334
Sobhy M. An accurate shear deformation theory for vibration and buckling of FGM sandwich plates in hygrothermal environment. Int J Mech Sci, 2016, 110: 62–77
Alibeigloo A, Alizadeh M. Static and free vibration analyses of functionally graded sandwich plates using state space differential quadrature method. Eur J Mech-A/Solids, 2015, 54: 252–266
Thai H T, Nguyen T K, Vo T P, et al. Analysis of functionally graded sandwich plates using a new first-order shear deformation theory. Eur J Mech-A/Solids, 2014, 45: 211–225
Akavci S S, Tanrikulu A H. Static and free vibration analysis of functionally graded plates based on a new quasi-3D and 2D shear deformation theories. Compos Part B-Eng, 2015, 83: 203–215
Swaminathan K, Naveenkumar D T, Zenkour A M, et al. Stress, vibration and buckling analyses of FGM plates—A state-of-the-art review. Compos Struct, 2015, 120: 10–31
Alijani F, Bakhtiari-Nejad F, Amabili M. Nonlinear vibrations of FGM rectangular plates in thermal environments. Nonlinear Dyn, 2011, 66: 251–270
Wang Y Q, Zu J W. Porosity-dependent nonlinear forced vibration analysis of functionally graded piezoelectric smart material plates. Smart Mater Struct, 2017, 26: 105014
Wang Y Q, Zu J W. Vibration behaviors of functionally graded rectangular plates with porosities and moving in thermal environment. Aerospace Sci Tech, 2017, 69: 550–562
Allahverdizadeh A, Oftadeh R, Mahjoob M J, et al. Homotopy perturbation solution and periodicity analysis of nonlinear vibration of thin rectangular functionally graded plates. Acta Mech Solid Sin, 2014, 27: 210–220
Hao Y X, Zhang W, Yang J. Nonlinear oscillation of a cantilever FGM rectangular plate based on third-order plate theory and asymptotic perturbation method. Compos Part B-Eng, 2011, 42: 402–413
Hao Y X, Zhang W, Yang J. Nonlinear dynamics of a FGM plate with two clamped opposite edges and two free edges. Acta Mech Solid Sin, 2014, 27: 394–406
Ke L L, Yang J, Kitipornchai S, et al. Axisymmetric nonlinear free vibration of size-dependent functionally graded annular microplates. Compos Part B-Eng, 2013, 53: 207–217
Duc N D, Bich D H, Cong P H. Nonlinear thermal dynamic response of shear deformable FGM plates on elastic foundations. J Thermal Stresses, 2016, 39: 278–297
Cong P H, Anh V M, Duc N D. Nonlinear dynamic response of eccentrically stiffened FGM plate using Reddy’s TSDT in thermal environment. J Thermal Stresses, 2017, 40: 704–732
Duc N D, Cong P H. Nonlinear postbuckling of an eccentrically stiffened thin FGM plate resting on elastic foundations in thermal environments. Thin-Walled Struct, 2014, 75: 103–112
Duc N D, Cong P H, Quang V D. Nonlinear dynamic and vibration analysis of piezoelectric eccentrically stiffened FGM plates in thermal environment. Int J Mech Sci, 2016, 115–116: 711–722
Duc N D, Cong P H, Tuan N D, et al. Nonlinear vibration and dynamic response of imperfect eccentrically stiffened shear deformable sandwich plate with functionally graded material in thermal environment. J Sandwich Struct Mater, 2016, 18: 445–473
Zhang W, Yang J, Hao Y. Chaotic vibrations of an orthotropic FGM rectangular plate based on third-order shear deformation theory. Nonlinear Dyn, 2010, 59: 619–660
Zhang W, Hao Y, Guo X, et al. Complicated nonlinear responses of a simply supported FGM rectangular plate under combined parametric and external excitations. Meccanica, 2012, 47: 985–1014
Yang J, Hao Y X, Zhang W, et al. Nonlinear dynamic response of a functionally graded plate with a through-width surface crack. Nonlinear Dyn, 2010, 59: 207–219
Wang Y Q, Zu J W. Nonlinear steady-state responses of longitudinally traveling functionally graded material plates in contact with liquid. Compos Struct, 2017, 164: 130–144
Chi S H, Chung Y L. Mechanical behavior of functionally graded material plates under transverse load—Part I: Analysis. Int J Solids Struct, 2006, 43: 3657–3674
Chi S H, Chung Y L. Cracking in sigmoid functionally graded coating. J Mech, 2002, 18: 41–53
Wang Y Q, Zu J W. Nonlinear dynamics of a translational FGM plate with strong mode interaction. Int J Struct Stabil Dyn, 2018, 18: 1850031
Wang Y Q, Zu J W. Nonlinear dynamic thermoelastic response of rectangular FGM plates with longitudinal velocity. Compos Part BEng, 2017, 117: 74–88
Wang Y Q. Electro-mechanical vibration analysis of functionally graded piezoelectric porous plates in the translation state. Acta Astronaut, 2018, 143: 263–271
Yang X D, Chen L Q, Zu J W. Vibrations and stability of an axially moving rectangular composite plate. J Appl Mech, 2011, 78: 011018
Wolfram S. The Mathematica Book. Cambridge: Cambridge University Press, 1999
Amabili M. Nonlinear vibrations of rectangular plates with different boundary conditions: Theory and experiments. Comput Struct, 2004, 82: 2587–2605
Wang Y Q. Nonlinear vibration of a rotating laminated composite circular cylindrical shell: Traveling wave vibration. Nonlinear Dyn, 2014, 77: 1693–1707
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Wang, Y., Zu, J.W. Nonlinear dynamic behavior of inhomogeneous functional plates composed of sigmoid graded metal-ceramic materials. Sci. China Technol. Sci. 61, 1654–1665 (2018). https://doi.org/10.1007/s11431-017-9167-9
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DOI: https://doi.org/10.1007/s11431-017-9167-9