Size-dependent geometrically nonlinear free vibration analysis of fractional viscoelastic nanobeams based on the nonlocal elasticity theory

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Highlights

  • Development of a nonlinear fractional nonlocal nanobeam model.

  • Geometrically nonlinear freevibration analysis of fractional viscoelastic nanobeams.

  • Using the Galerkin methodand predictor-corrector technique to solve the problem.

  • Study of influence of nonlocal parameter the nonlinear time response.

  • Study of effects oforder of fractional derivative and viscoelasticity coefficient.

Abstract

In recent decades, mathematical modeling and engineering applications of fractional-order calculus have been extensively utilized to provide efficient simulation tools in the field of solid mechanics. In this paper, a nonlinear fractional nonlocal Euler–Bernoulli beam model is established using the concept of fractional derivative and nonlocal elasticity theory to investigate the size-dependent geometrically nonlinear free vibration of fractional viscoelastic nanobeams. The non-classical fractional integro-differential Euler–Bernoulli beam model contains the nonlocal parameter, viscoelasticity coefficient and order of the fractional derivative to interpret the size effect, viscoelastic material and fractional behavior in the nanoscale fractional viscoelastic structures, respectively. In the solution procedure, the Galerkin method is employed to reduce the fractional integro-partial differential governing equation to a fractional ordinary differential equation in the time domain. Afterwards, the predictor–corrector method is used to solve the nonlinear fractional time-dependent equation. Finally, the influences of nonlocal parameter, order of fractional derivative and viscoelasticity coefficient on the nonlinear time response of fractional viscoelastic nanobeams are discussed in detail. Moreover, comparisons are made between the time responses of linear and nonlinear models.

Graphical abstract

Using a proposed nonlinear fractional nonlocal nanobeam model, Galerkin scheme and predictor–corrector method, the geometrically nonlinear free vibration of the fractional viscoelastic nanobeams is investigated.

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Introduction

Fractional-order calculus including the fractional-order derivatives and integrals has attracted considerable attention in many areas of mathematics, nonlinear science, mechanics and physics and nano-engineering [1], [2], [3], [4], [5], [6] and has been widely utilized to capture the natural and physical phenomena that cannot be predicted by classical integral and differential models. In mechanical engineering applications, the fractional calculus has been introduced to more accurately describe the viscous behavior of materials [7], the memory and properties in various materials and processes [8], [9], and fractional-order feedback in the control engineering [10], [11].

The analysis of nanostructures including nanobeams, carbon and boron-nirtide nanotubes and nanorods which are mostly applied in nano- and micro-electromechanical systems (NEMS and MEMS) and tracking their mechanical behaviors can give the truthful and promising results for designing such as devices. The experimental investigations [12], [13], [14] and atomistic and molecular dynamics simulations [15], [16], [17] have proven the size-dependence of mechanical behaviors and material properties of nano-scale structures. Moreover, due to being scale-free, the classical continuum theory is unable to accurately detect the static and dynamic mechanical behaviors of nano- and micro-structures. Therefore, a wide range of non-classical continuum theories including nonlocal elasticity theory [18], surface stress elasticity theory [19], [20], strain gradient elasticity theory [21], modified couple stress theory [22] and modified strain gradient theory [23] are extended to take size effects into account. In most recent studies, the aforementioned non-classical theories have been applied to broaden the size-dependent continuum beam, plate and shell models for analysis of mechanical behavior of micro- and nano-scale structures [24], [25], [26], [27], [28], [29], [30].

One widely utilized size-dependent theory is the nonlocal elasticity theory proposed by Eringen [31], [32], [33]. Recently, this theory has been used to investigate the mechanical behavior of viscoelastic and fractional nanostructures. For instance, according to the Kelvin–Voigt and viscoelastic models, velocity-dependent external damping and nonlocal elasticity theory, Lei et al. [34] proposed a nonlocal viscoelastic Euler–Bernoulli beam model to study the dynamic characteristics of damped viscoelastic nanobeams. A theoretical analysis was carried out by Sumelka et al. [35] to investigate the free axial vibration of nanorods using the nonlocal elasticity theory and fractional continuum mechanics. To analyze the free bending vibration of carbon nanotubes, Lei et al. [6] tried to establish the governing equations and boundary conditions of viscoelastic Timoshenko nanobeams by means of the Kelvin–Voigt viscoelastic model, velocity-dependent external damping and nonlocal Timoshenko beam theory. Recently, Karličić et al. [36] studied the free transverse vibration of nonlocal viscoelastic orthotropic multi-nanoplate system embedded in a viscoelastic medium using the nonlocal elasticity theory and modified Kelvin–Voigt viscoelastic constitutive relation.

This study is intended to analyze the geometrically nonlinear free vibration of fractional viscoelastic nanobeams in the framework of Euler–Bernoulli beam theory, nonlocal elasticity theory and von Kármán geometric nonlinearity. Firstly, a nonlinear fractional nonlocal Euler–Bernoulli beam model for the factional viscoelastic nanobeams is developed using the concept of fractional derivative and nonlocal elasticity theory. Then, the fractional integro-partial differential governing equation is reduced to a fractional ordinary differential equation in the time domain via the Galerkin method. Afterwards, the resulting governing equation is solved by means of the predictor–corrector method.

Section snippets

Mathematical formulation of fractional viscoelastic nanobeams

A fractional viscoelastic nanobeam of length L, width b and thickness h defined in the Cartesian coordinate system is considered. The nonlocal elasticity theory of Eringen [31], [32], [33] and fractional Kelvin–Voigt viscoelastic theory [37] are used to model the fractional viscoelastic nanobeams.

For the selected fractional viscoelastic nanobeam, the thickness-to-length ratio is small enough to neglect the effects of shear deformation and rotary inertia. Therefore, the Euler–Bernoulli beam

Solution procedure

To investigate the geometrically nonlinear free vibration of fractional viscoelastic nanobeams, firstly, the Galerkin method is used to convert the fractional integro-partial differential governing equation into a fractional-ordinary differential equation. Then, the obtained equation is arranged in a more effective state-space form. Finally, the predictor–corrector method is used to solve the set of nonlinear fractional time-dependent equations and obtain the time response of fractional

Results and discussion

In this section, numerical results are presented for geometrically nonlinear free vibration of fractional viscoelastic nanobeams. The study is carried out to examine the influences of different parameters such as nonlocal parameter, order of fractional derivative and viscoelasticity coefficient on the nonlinear time response curves. To perform this, a carbon nanotube with Young's modulus E=1TPa, mass density ρ=2.24g/cm3, diameter d=1.1nm and thickness h=0.342nm is considered [6]. By examining

Concluding remarks

In this study, based on the nonlinear fractional nonlocal Euler–Bernoulli beam model developed using the concept of fractional derivative and nonlocal elasticity theory, the size-dependent geometrically nonlinear free vibration of fractional viscoelastic nanobeams was investigated. The nonlocal parameter, viscoelasticity coefficient and order of fractional derivative were considered in the proposed non-classical fractional integro-differential Euler–Bernoulli beam model to interpret the size

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