Nonlinear vibration of fractional viscoelastic piezoelectric nanobeams based on nonlocal theory

This paper investigates the nonlinear vibrations of fractional viscoelastic piezoelectric nanobeams based on nonlocal theory and Euler–Bernoulli beam theory. A nonlinear fractional nonlocal Euler–Bernoulli beam model is established, incorporating the concept of fractional derivatives, considering that piezoelectric nanobeams are subjected to both applied voltage and uniform temperature conditions. The nonlinear governing equations and boundary conditions are derived through Hamilton's principle. During the solution process, the fractional integral–partial differential governing equation is initially transformed into a time-domain fractional-order ordinary differential equation using the Galerkin method. Subsequently, the resulting nonlinear time-varying equation of fractional order is addressed using a predictive correction method. Eventually, a detailed analysis is presented, examining the effect of nonlocal parameters, fractional derivatives, viscoelastic coefficients, and applied voltages have an influence on the nonlinear time response of beams. Our findings indicate that there exists a correlation between the fractional order and the nonlinear vibration behavior of viscoelastic piezoelectric nanobeams. Specifically, the system damping increases with rising fractional orders. Therefore, it is crucial to account for considering the influence of fractional order when investigating materials exhibiting viscoelastic characteristics. Additionally, both nonlocal parameters and piezoelectric properties play a significant role in shaping their nonlinear vibration behavior.