Nonlinear deflection and traveling wave solution for FG-GPLs reinforced microtubes embedded in Kerr foundation and conveying magnetic fluid

Abstract

The nonlinear dynamic deformation analysis and traveling wave solution of microtubes reinforced with functionally graded (FG) graphene platelets (GPLs) embedded in Kerr foundation with three-parameter are studied. It is assumed that the composite microtubes convey a fluid and are exposed to a magnetic field. The displacement field of the microtubes is modeled using a higher-order shear deformation beam theory. Furthermore, to capture the tiny size impact, the modified couple stress theory with a single material parameter is applied. The nonlinear equations of motion of the fluid-conveying microtube are derived using Hamilton’s variational principle considering von-Karman’s strain–displacement relations and the fluid velocity potential. The material properties of the microtubes are calculated using the modified Halpin–Tsai model and the mixture rule. The GPL volume fraction is continuously varied across the thickness based on a refined cosine rule, taking into account different patterns of GPLs distribution. The partial differential equations are transformed into ordinary equations with different boundary conditions using the Galerkin technique. These equations are solved employing the fourth order Runge–Kutta method. While, as a special case, the nonlinear motion equation of Euler–Bernoulli microtube, that is surrounded by a shear foundation and conveying microfluid, is solved analytically based on Jacobi elliptic functions. Finally, a variety of numerical examples are presented to illustrate the impacts of magnetic field, elastic foundation parameters, material length scale parameter, flow velocity, Knudsen number, length-to-radius ratio, and length-to-thickness ratio on the nonlinear dynamic deflection and axial displacement waves in the FG-GPLs microtubes.