In the current paper, a size-dependent differential quadrature element model using the higher-order Love-Bishop rod theory in conjunction with the nonlocal strain gradient theory for axial vibration analysis in the embedded functionally graded (FG) carbon nanotube (CNT) reinforced composite microrods with varying cross-section is presented for the first time. The CNT agglomeration effect is considered applying the two-parameter Eshelby-Mori-Tanaka homogenization scheme. The differential equation form of nonlocal strain gradient theory is employed to take into account the size-dependent effects. Based on the Hamilton’s principle, the governing differential equation of the model is derived and solved using differential quadrature element method (DQEM). The higher-order variationally consistent boundary conditions are obtained using the weighted residual approach. The numerical results indicate that the present model is simple and it can be successfully employed for analysis of the small-sized structures. Moreover, the effects of small-scale parameters, the CNT distribution’s patterns, the CNT agglomeration parameters, the non-uniformity of the cross section and the stiffness of the surrounding medium on the frequency are studied and discussed in details.

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