Imperfect Bifurcation and Chaos of Slightly Curved Carbon Nanotube Conveying Hot Pressurized Fluid Resting on Foundations

Abstract

Unintended slight curvature of a straight pipe and temperature variation in a pipe has been found to create uncertainties in tubes and pipes. Fluttering, divergence, and chaotic instabilities of slightly curved carbon nanotubes (SCCNT) conveying hot pressurized fluid are investigated in this paper. The SCCNT is modeled on the basis of large deformation strains. Their gradients are included in the strain energy expression and the velocity and its gradients in the kinetic energy derivation. In modeling the size effects, both the static and kinetic length scales in the energy equations were considered. Governing equation is derived using Lagrangian approach. The effects of geometric imperfection (which leads to cusp bifurcation), small length scale, and kinetic material length parameter on the static and dynamic instability characteristics of the pipes are studied. Analysis is performed using the eigenfunction expansion method. It is found that the material length scale parameter increase tends to shift instability to the lower fluid velocity while the kinematic material length parameter increase does not change the buckling point but lowers the frequency. In the nonlinear dynamic case, both the parameters lead to chaos of the nanotube beyond the critical fluid velocity. The thermal loading changes the sudden supercritical pitchfork bifurcation to cusp bifurcation. The increasing linear and nonlinear foundation stiffness leads the system to chaotic features after the critical point.