Nonlinear dynamic analysis of FG micro-pipes conveying fluid based on strain gradient theory

In this article, an analytical solution is presented for the size dependent nonlinear vibration behavior of micro-pipes conveying fluid made of functionally graded materials (FGMs). On the basis of the Euler–Bernoulli beam model, the strain gradient theory and von Kármán geometric nonlinearity, the mathematical formulations are developed in terms of three length scale parameters. The material properties of the functionally graded (FG) micro-pipes vary continuously across the thickness according to the power law distribution. The Hamilton’s principle is employed to obtain the differential equation of motion and the corresponding boundary conditions. Without loss of generality, simply supported pipes are considered. The governing equation is written in the form of duffing equation by using Galerkin method. Subsequently, a powerful analytical technique called the homotopy analysis method (HAM) is employed to determine the explicit expressions for nonlinear fundamental frequency for different fluid velocities and power law gradient indices. Comprehensive comparison studies between linear and nonlinear theories using the strain gradient, the couple stress and classical theories are conducted. The results show that the length scale parameter and the FG power law index have significant effect on the fundamental frequency of the FG micro-pipes and the fluid critical velocity.