Nonlinear size-dependent bending and forced vibration of internal flow-inducing pre- and post-buckled FG nanotubes

Highlights

• A higher-order size-dependent beam model coupled with surface energy is established.

• Nonlinear bending and principal resonance of fluid-conveying nanotubes in pre- and post-buckling states are studied.

• An analytical method for nonlinear principal resonance analysis is developed.

• The dual influences of structural size-dependent effects on bending and forced vibration are revealed.

• The responses are more sensitive to nano effects when the flow velocity is close to the critical buckling velocity.

Abstract

Fluid-conveying nanotubes are important components in nano-electromechanical systems requiring strict static and dynamic stiffness designs. The nonlinear bending and forced vibration of the fluid-conveying functionally graded nanotubes in pre- and post-buckling states are investigated. Based on Zhang–Fu’s refined displacement field, a comprehensive size-dependent nanotube model is established in which the nonlocal stress, strain gradient and surface energy effect are coupled in the constructive model. The nanotube is assumed simply supported at both ends and subjected to transverse uniform static or harmonic load. The two-step perturbation technique is extended to obtain the post-buckling equilibrium paths, providing the initial configurations for bending and resonance analysis. According to the two symmetric initial bifurcation paths, the bending load–deflection explicit relation is obtained by the two-step perturbation method again. A combination of two-step perturbation and modified Lindstedt–Poincaré method is developed to obtain the approximate analytical solution of the strongly nonlinear forced vibration. Parameter investigations are conducted to discuss the different size-dependent effects, flow velocity and material distributions on the nonlinear behaviors. Results reveal the dual influences of structural size-dependent effects and indicate that the response is more sensitive to the size-dependent effects when the flow velocity is close to the critical buckling velocity.

Introduction

The wider applications of fluid-conveying nanotubes in nano-electromechanical systems (NEMS) put forward higher requirements for the linear and nonlinear analyses of mechanical behaviors for structural designs [1], [2]. On the one hand, there are internal flow-inducing static and dynamic instability modes such as buckling and self-excited vibration under different support conditions. On the other hand, the combined action of the flow-inducing deformation and the coupling effect of external force, heat, electricity and magnetic field on the tube increases the complexity of the problem [3], [4]. Further, the existence of size-dependent nano-effects makes the performance prediction of nanotubes more complicated. Due to the difficulty of experiment and the high cost of simulation, it is of significance to study the theoretical modeling and calculation method of the mechanical behaviors of the fluid-conveying nanotubes [5], [6].

As is known, with the decreasing of the characteristic scale, substances including fluids and solids exhibit different physical properties from those at the macro scale, which is called size dependence. As to the nanofluid, the density of the fluid is very low, and the flow will show a slightly different phenomenon from the general flow, which is mainly manifested in the velocity slip and temperature jump near the boundary. As to the nanomaterials, due to the high proportion of atoms in grain boundaries, nanomaterials present a new structural state between crystalline and amorphous states, resulting in many size-dependent properties. The size dependence inside the material is mainly reflected in that the strain gradient and the stress at each point of the whole body will affect the stress at a point. Therefore, nonlocal theory, strain gradient theory and nonlocal strain gradient theory and several higher-order theories were developed to characterize these small-scale properties [7], [8], [9], [10], [11], [12]. On the other hand, the decrease of the characteristic scale leads to large specific surface areas of nanomaterials. Therefore, surface and interface properties (such as surface tension and surface stress) have important influences on the physical properties of nanostructured materials, promoting the development of the theory of surface elasticity [13], [14]. Subsequently, many scholars applied these non-classical theories to study the static and dynamic performances of the nanotubes [15], [16]. Mindlin’s strain gradient theory was employed to free vibration and instability studies of nano shells and it was found that the results are greatly different from the classical theories at small values of dimensionless length scale parameters [17]. The effects of small size on the natural frequency and stability boundaries and flutter vibrations of cantilever nanotubes conveying fluid were also studied [18], [19]. It was revealed that the fluid slip boundary condition, increasing nonlocal parameter and decreasing strain gradient parameter tend to decrease the vibrational frequencies and the critical flow velocity. Liu et al. [20] found that the non-local stress can both change the stiffness of the nanotube itself and modify the softening effect of inside fluid on the whole nanotube conveying-fluid system. The influence of nonlocal and strain gradient parameters on the wave propagation in nanotubes conveying fluid were studied [21], [22]. It was found that the small-scale material parameters have significant effects on the dispersion relationship at high wave number. Cheng et al. [23] found that the nonlocal parameter has a significant influence on the second-mode divergence, which causes a great decrease on the second-mode divergence’s velocity. Amiri [24] revealed that the surface effect makes the phase velocity higher and is more considerable for a propagating wave with higher wave number. Subsequently, the nano-effects on the linear buckling and natural frequency of the fluid-conveying nanotubes were comprehensively revealed by establishing the nonlocal strain gradient model incorporating of surface energy [25], [26].

In recent years, the influence of nano effects and structural effects on the nonlinear static and dynamic behaviors of the fluid-conveying nanotubes have also been focused on and studied for certain functionally structural design requirements [27], [28]. Kiani [29] revealed the necessity to perform the nonlinear analysis when the nanotube is traversed by a moving nanoparticle with high levels of the mass weight and velocity. Bakhtiari et al. [30] pointed out that the nonlinear responses of a deep curved beam are highly affected by the geometrical characteristics and the micro interactions. Wang et al. [31] found that the surface stress influences the nonlinear vibration characteristics of fluid-conveying nano-shells, which becomes more considerable with the decrease of the wall thickness. For the purpose of nonlinear analysis, the transverse shear deformation usually requires to be considered. Hence, many scholars tried to couple different nano theories with higher-order beam models, and establish different size dependent shear deformation models. Based on the modified couple stress theory, a size-dependent nonlinear model for electrically actuated microcantilever-based MEMS was developed by Dai and Wang [32]. The nonlocal elasticity equations of Eringen were incorporated into the various classical beam theories namely to consider the size-effects on the vibration analysis of single-walled carbon nanotubes [33]. Based on the shear deformation beam and modified strain gradient theory, novel size-dependent beam models were developed for the static bending and buckling responses of microbeams and revealed that an increase in material length scale parameter-to-diameter ratio leads to the decrease in the non-dimensional deflections [34], [35]. Subsequently, the enhanced Eringen’s differential model was extended in case of static solution with singularity function for various loading types and locations [36]. A nonlinear size-dependent fluid–structure interaction model was developed for the chaotic motion of nanofluid-conveying nanotubes subject to an external excitation [37]. Wang et al. [38] established a nonlinear theoretical model that succeeded in predicting the post-instability nonlinear dynamics of fluid-conveying micro-cantilever. Ghayesh et al. [39] applied a time-integration technique to perform a chaos analysis for the nonlinear coupled dynamics of nanotubes conveying pulsatile fluid in order to tailor the system parameters to avoid chaos. The statistical investigation was done of the dynamic response of the carbon nanotubes conveying multi-phase flow by the classical Runge–Kutta method and Monte–Carlo simulation [40]. A hull iteration method was proposed to perform the uncertainty propagation in fluid-conveying carbon nanotube system under multi-physical fields [41]. Several approximate analytical methods were used to study the nonlinear vibration of nanotubes conveying fluid, such as the multiple times scale method, Galerkin discretization and the multidimensional Lindstedt–Poincaré​ method [42], [43], [44], [45]. However, these methods are more suitable for solving the governing equations derived from the classical engineering beam theory, but are difficult or inefficient to be applied to the governing equations established from the higher-order beam models with multiple generalized displacements.

From the above literature review, it can be concluded that both the nano-effects and the flow inducing geometric nonlinearity have great influences on the mechanical responses of the nanotubes under external load or excitation. To study these influences, it is necessary to establish a comprehensive size dependent shear deformation nanotube model and develop the corresponding computation method to predict the nonlinear static and dynamic behaviors of the fluid-conveying nanotubes. To the author’s knowledge, there are few studies on establishing higher-order size dependent beam models and the matching analytical methods for nonlinear analysis of fluid-conveying nanotubes. Also, there are few researches on nonlinear bending and forced vibration of flow-inducing pre- and post-buckled nanotubes. This paper studies the nonlinear static and dynamic behaviors of nanotubes under the combined action of internal fluid flow-inducing deformation and external load by establishing a higher-order theoretical model and the analytical methods suitable for strongly nonlinear problems.

The rest of this article is organized as follows. In Section 2, the theoretical formulas are given to establish the displacement field, Von Karman displacement–strain relations, the models of size-dependent effects of both solid and fluid, and material constitutive relationship. Subsequently, the governing equations are derived by using the Hamilton’s variational principle. In Section 3, a new two-step perturbation scheme is utilized to obtain the post-buckling equilibrium path as the initial tube configuration for bending and vibration analysis. Subsequently, a two-step perturbation method is used again to obtain the load–deflection curves for bending. Further, a novel combination of two-step perturbation and modified Lindstedt–Poincaré (MLP) method is employed to obtain the approximate analytical solutions to the amplitude–frequency bifurcation equations for principal resonance. In Section 4, numerical examples are given to validate the correctness of the theoretical models and computation methods, and the parameter analysis is carried out. According to the explicit closed form solutions, the influences of flow velocity, material gradient distribution and different size-dependent effects on the responses are discussed. The paper contains a conclusion in Section 5.