Nonlinear thermal vibration of functionally graded non-uniform and imperfect micro-tube including the porosity in the thermal environment for different cross-section

Abstract

In producing small-scale structures, having nonuniform geometrics and voids as the generation defect is the inevitable problem. However, in the mathematical simulations, the researchers have been more attentive to the perfect and uniform structures because of simple modeling. This article investigates the nonlinear free vibrational behavior of truncated conical imperfect functionally graded (FG) micro-scale tubes, including the porosity, including the various cross-section functions. The modified couple stress theory and Euler-Bernoulli beam theory coupled with the nonlinear Von-Kármán theory are employed based on Hamilton's principles to derive the general equation of motion and related boundary conditions. The cross-section is assumed on the basis of four different functions, involving the uniform section, linear section, convex section, and exponential section. The temperature-dependent materials were combined by ceramic and metal phases along the tube radius that this combination made a functionally graded tube. Furthermore, the nonlinear derived equations are finally solved via the generalized differential quadrature method (GDQM) as the numerical approach coupled with the iteration technique. Since thermal fins, fluid flow diffuser, fluid flow nozzle, etc., are designed for specific purposes with the non-uniform cross-section, the presented results have an excellent sight of developing and designing macro/- and micro-electromechanical systems (MEMS).