Abstract
A nonlinear vibration analysis of a simply supported functionally graded rectangular plate with a through-width surface crack is presented in this paper. The plate is subjected to a transverse excitation force. Material properties are graded in the thickness direction according to exponential distributions. The cracked plate is treated as an assembly of two sub-plates connected by a rotational spring at the cracked section whose stiffness is calculated through stress intensity factor. Based on Reddy’s third-order shear deformation plate theory, the nonlinear governing equations of motion for the FGM plate are derived by using the Hamilton’s principle. The deflection of each sub-plate is assumed to be a combination of the first two mode shape functions with unknown constants to be determined from boundary and compatibility conditions. The Galerkin’s method is then utilized to convert the governing equations to a two-degree-of-freedom nonlinear system including quadratic and cubic nonlinear terms under the external excitation, which is numerically solved to obtain the nonlinear responses of cracked FGM rectangular plates. The influences of material property gradient, crack depth, crack location and plate thickness ratio on the vibration frequencies and transient response of the surface-racked FGM plate are discussed in detail through a parametric study.
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References
Ichikawa, K. (ed.): Functionally Graded Materials in the 21st Century: A Workshop on Trends and Forecasts. Kluwer Academic, Norwell (2000)
Praveen, G.N., Reddy, J.N.: Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates. Int. J. Solids Struct. 35(33), 4457–4476 (1998)
Reddy, J.N.: Analysis of functionally graded plates. Int. J. Numer. Methods Eng. 47(1–3), 663–684 (2000)
Yang, J., Kitipomchai, S., Liew, K.M.: Large-amplitude vibration of thermo-electromechanically stressed FGM laminated plates. Comput. Methods Appl. Mech. Eng. 192(35–36), 861–3885 (2003)
Huang, X.L., Shen, H.S.: Nonlinear vibration and dynamic response of functionally graded plates in thermal environments. Int. J. Solids Struct. 41(9–10), 2403–2407 (2004)
Chen, C.S.: Nonlinear vibration of a shear deformable functionally graded plate. Compos. Struct. 68(3), 295–302 (2005)
Kitipornchai, S., Yang, J., Liew, K.M.: Semi-analytical solution for nonlinear vibration of laminated FGM plates with geometric imperfections. Int. J. Solids Struct. 41(9–10), 2235–2257 (2004)
Yang, J., Huang, X.L.: Nonlinear transient response of functionally graded plates with general imperfections in thermal environments. Comput. Methods Appl. Mech. Eng. 196(25–28), 2619–2630 (2007)
Dimarogonas, A.D.: Vibration of cracked structures: a state-of-the-art review. Eng. Fract. Mech. 55(5), 831–857 (1996)
Jin, Z.H., Noda, N.: Crack–tip singular fields in nonhomogeneous materials. ASME J. Appl. Mech. 61, 738–740 (1994)
Erdogan, F., Wu, B.H.: The surface crack problem for a plate with functionally graded properties. ASME J. Appl. Mech. 64(3), 449–456 (1997)
Jin, Z.H., Paulino, G.H.: Transient thermal stress analysis of an edge crack in a functionally graded material. Int. J. Fract. 107(1), 73–98 (2001)
Delale, F., Erdogan, F.: The crack problem for a non-homogeneous plane. ASME J. Appl. Mech. 50(3), 609–614 (1983)
Abanto-Bueno, J., Lambros, J.: Parameters controlling fracture resistance in functionally graded materials under mode I loading. Int. J. Solids Struct. 43(13), 3920–3939 (2006)
Wang, B.L., Noda, N.: Thermally induced fracture of a smart functionally graded composite structure. Theor. Appl. Fract. Mech. 35(2), 93–109 (2001)
Guo, L.C., Wu, L.Z., Zeng, T., Ma, L.: The dynamic fracture behavior of a functionally graded coating–substrate system. Compos. Struct. 64(3–4), 433–441 (2004)
Sridhar, R., Chakraborty, A., Gopalakrishnan, S.: Wave propagation analysis in anisotropic and inhomogeneous uncracked and cracked structures using pseudospectral finite element method. Int. J. Solids Struct. 43(16), 4997–5031 (2006)
Birman, V., Byrd, L.W.: Vibration of damaged cantilevered beams manufactured from functionally graded materials. AIAA J. 45, 2747–2757 (2007)
Yang, J., Chen, Y.: Free vibration and buckling analyses of functionally graded beams with edge cracks. Compos. Struct. 83(1), 48–60 (2008)
Yang, J., Chen, Y., Xiang, Y., Jia, X.L.: Free and forced vibration of cracked inhomogeneous beams under an axial force and a moving load. J. Sound Vib. 312(1–2), 166–181 (2008)
Ke, L.L., Yang, J., Kitipornchai, S., Xiang, Y.: Flexural vibration and elastic buckling of a cracked Timoshenko beam made of functionally graded materials. Mech. Adv. Mater. Struct. (2009, in press)
Reddy, J.N.: Mechanics of Laminated Composite Plates and Shells: Theory and Analysis. CRC Press, New York (2004)
Broek, D.: Elementary Engineering Fracture Mechanics. Nijhoff, Dordrecht (1986)
Bhimaraddi, A.: Large-amplitude vibrations of imperfect antisymmetric angle-ply laminated plates. J. Sound Vib. 162(3), 457–470 (1997)
Nosir, A., Reddy, J.N.: A study of non-linear dynamic equations of higher-order deformation plate theories. Int. J. Non-Linear Mech. 26(2), 233–249 (1991)
Yang, J., Kitipomchai, S., Liew, K.M.: Non-linear analysis of the thermo-electromechanical behaviour of shear deformable FGM plates with piezoelectric actuators. Int. J. Numer. Methods Eng. 59(12), 1605–1632 (2004)
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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 59, 207–219 (2010). https://doi.org/10.1007/s11071-009-9533-9
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DOI: https://doi.org/10.1007/s11071-009-9533-9