Vibration Characteristics of Functionally Graded Fractional Derivative Viscoelastic Fluid–Conveying Pipe with Initial Geometric Defects

Abstract

A viscoelastic constitutive model constructed with fractional derivatives can accurately describe the mechanical phenomena caused by materials with time-dependent and memory properties. Using this model, the vibration characteristics and stability of a functionally graded fractional derivative viscoelastic fluid–conveying pipe with initial geometric defects are investigated in this study. First, the differential equation of motion for a functionally graded fractional derivative viscoelastic fluid–conveying pipe with initial geometric defects is derived by applying the Hamilton principle. Dimensionless differential equations and boundary conditions are discretized using a differential quadrature method, and numerical solutions of the equations under three boundary conditions are obtained. Further, the effects of geometric defect types, fractional order, power law exponent, and other factors on the complex frequency of this fluid-conveying pipe are discussed. The findings revealed that the geometric defects of the fluid-conveying pipe can significantly affect the complex frequency of the system. Moreover, the pipe’s damping characteristics vary with the fractional order of the viscoelastic material. In other words, the higher the vibration mode of the pipe, the more susceptible it is to the fractional order. Because the fractional derivative relates to values near the current moment and overall history, it can effectively describe the temporal effect of materials, such as polymers, synthetic rubber, and coatings. Many naturally occurring materials (including geological materials such as soil, oil, and minerals and biological materials such as muscle, bone, and blood) exhibit time-dependent properties. The deformation of such materials relates to the current stress level and their overall history. Because the fractional derivative constitutive relation can describe the dynamic properties of a structure in a wide frequency range with only a few parameters and terms, it represents a potential candidate for addressing the mechanical problems of viscoelastic materials and structures. In this study, functionally graded fractional derivative viscoelastic materials were applied to a fluid-conveying pipe with initial geometric curvature, as common in practical engineering. The effects of the initial geometric defect type, dimensionless delay time, power law exponent, and other parameters on the stability of the fluid-conveying pipe were discussed, providing some theoretical reference for the design of such fluid-conveying pipes.