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Research Papers

Study of Small Scale Effects on the Nonlinear Vibration Response of Functionally Graded Timoshenko Microbeams Based on the Strain Gradient Theory

[+] Author and Article Information
R. Ansari1

Department of Mechanical Engineering,  University of Guilan, P.O. Box 3756,Rasht, Iranr_ansari@guilan.ac.ir

R. Gholami, S. Sahmani

Department of Mechanical Engineering,  University of Guilan, P.O. Box 3756,Rasht, Iran

1

Corresponding author.

J. Comput. Nonlinear Dynam 7(3), 031009 (Apr 03, 2012) (9 pages) doi:10.1115/1.4006040 History: Received July 01, 2011; Revised January 19, 2012; Accepted February 07, 2012; Published April 03, 2012

In the current study, the nonlinear free vibration behavior of microbeams made of functionally graded materials (FGMs) is investigated based on the strain gradient elasticity theory and von Karman geometric nonlinearity. The nonclassical beam model is developed in the context of the Timoshenko beam theory which contains material length scale parameters to take the size effect into account. The model can reduce to the beam models based on the modified couple stress theory (MCST) and the classical beam theory (CBT) if two or all material length scale parameters are taken to be zero, respectively. The power low function is considered to describe the volume fraction of the ceramic and metal phases of the FGM microbeams. On the basis of Hamilton’s principle, the higher-order governing differential equations are obtained which are discretized along with different boundary conditions using the generalized differential quadrature method. The dimensionless linear and nonlinear frequencies of microbeams with various values of material property gradient index are calculated and compared with those obtained based on the MCST and an excellent agreement is found. Moreover, comparisons between the various beam models on the basis of linear and nonlinear types of strain gradient theory (SGT) and MCST are presented and it is observed that the difference between the frequencies obtained by the SGT and MCST is more significant for lower values of dimensionless length scale parameter.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

Schematic of a FGM Timoshenko microbeam: kinematic parameters, coordinate system, geometry and loading

Grahic Jump Location
Figure 2

Effect of the material property gradient index n on the nonlinear frequency ratio ωNL/ωNLωLωL of FGM microbeams with h/hll=2 and L/Lhh=12: (a) C–C; and (b) SS–SS

Grahic Jump Location
Figure 3

Effect of the dimensionless length scale parameter h/hll on the nonlinear frequency ratio ωNL/ωNLωLωL of FGM microbeams with k=2 and L/Lhh=12: (a) C–C; and (b) SS–SS

Grahic Jump Location
Figure 4

Effect of slenderness ratio L/Lhh on the nonlinear frequency ratio ωNL/ωNLωLωL of FGM microbeams with h/hll=3 and k=2: (a) C–C; and (b) SS–SS

Grahic Jump Location
Figure 5

Comparison of the different theories: effect of the dimensionless length scale parameter h/hll on the frequency of the C–C FGM microbeams with k=2 and L/Lhh=12

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