Size-dependent nonlinear vibration of functionally graded composite micro-beams reinforced by carbon nanotubes with piezoelectric layers in thermal environments

Nonlinear free vibration characteristics of functionally graded (FG) composite micro-beams reinforced by carbon nanotubes (CNTs) with piezoelectric layers in thermal environment are investigated in this work. The uniform distribution and four nonuniform distribution types of the CNTs reinforcements are examined. The equations of motion are derived based on the Euler–Bernoulli beam theory with von Karman’s assumption, the nonlocal strain gradient theory and the Hamilton’s principle. The approximate nonlinear frequencies of FG-carbon nanotube-reinforced composite (CNTRC) micro-beams for simply supported and clamped–clamped boundary conditions are obtained by using two analytical methods including the equivalent linearization method and the energy balance method. The accuracy of the obtained results has been verified. The influences of the nonlocal parameter, material length scale parameter, geometric property of micro-beam, temperature change, applied voltage, distribution pattern and volume fraction of the CNTs on the nonlinear free vibration behaviors of the FG-CNTRC micro-beams are performed and discussed in detail.