Postbuckling and nonlinear free vibration of FG nanobeams

What is it about?

Vibrations of micro-/nano-beams which are subjected to initial stresses due to mismatch between different materials or thermal stresses are important in some devices. The present study is an attempt to present nonlinear free vibration of simply supported size dependent FG nano-beams resting on elastic foundation and under pre compressive axial force. It is assumed that the material properties of FGMs are graded in the thickness direction. The partial differential equation of motion, which is simplified into an ordinary differential equation using the Galerkin method, is derived based on Euler-Bernoulli beam theory, von-Karman geometric nonlinearity and Eringen's nonlocal elasticity theory. The final ordinary differential equation is solved using the variational iteration method. The effects of geometrical parameters, small scale parameter, elastic coefficient of foundation, pre compressive axial force and neutral axis location on dimensionless nonlinear natural frequencies are investigated. In this study, buckling and post-buckling behavior of FG nano-beams and the effect of neutral axis location on buckling behavior are investigated as well. Results show that the effects of small scale on FG nano-beams frequencies change with the aspect ratio, the values of radius of gyration and the values of compressive axial force. It is also found that the influence of neutral axis location on the nonlinear fundamental frequency of pre-stressed FG nano-beams is more than that of pre-stressed FG nano-beams resting on elastic foundation.

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Why is it important?

Our findings show that: 1-with an increase in the stiffness of elastic foundation, not only the load-bearing capacity of FG nano-beams but also the effect of small scale on decrease the critical buckling force increases. 2- although stiffness of elastic foundation has a significant effect on the dimensionless fundamental frequency of FG nano-beams, their mode shapes won't vary if pre axial compressive force is not large enough. 3- the values of compressive axial force significantly affect the relationship between the nonlinear to linear fundamental frequency ratio and small scale parameter in a way that with a decrease in compressive axial force the small scale loses its importance. 4- the effect of neutral axis location on the nonlinear fundamental frequency of pre-stressed FG nano-beams is more than that of pre-stressed FG nano-beams resting on elastic foundation.