Nonlinear magneto-thermo-elastic vibration of mass sensor armchair carbon nanotube resting on an elastic substrate

Abstract The present paper aims at studying the nonlinear ultrasonic waves in a magneto-thermo-elastic armchair single-walled (SW) carbon nanotube (CNT) with mass sensors resting on a polymer substrate. The analytical formulation accounts for small scale effects based on the Eringen’s nonlocal elasticity theory. The mathematical model and its differential equations are solved theoretically in terms of dimensionless frequencies while assuming a nonlinear Winkler-Pasternak-type foundation. The solution is obtained by means of ultrasonic wave dispersion relations. A parametric work is carried out to check for the effect of the nonlocal scaling parameter, together with the magneto-mechanical loadings, the foundation parameters, the attached mass, boundary conditions and geometries, on the dimensionless frequency of nanotubes. The sensitivity of the mechanical response of nanotubes investigated herein, could be of great interest for design purposes in nano-engineering systems and devices.


Introduction
Magneto-thermo-elastic armchair single-walled (SW) carbon nanotubes (CNTs) represent an advanced material largely adopted in many nanostructural applications. The application of MnO 2 -FeTiO 3 (MFT)-based nanocomposites in a polymer matrix, as surrounding medium, has attracted the attention of the scientific community, due to the increased efficiency of embedded nanostructures [1]. Among nonlocal continuum theories, the nonlocal Euler-Bernoulli and Timoshenko beam models enable a satisfactory size-dependent assessment of CNTs, see Refs. [2,3]. Some pioneering studies on the topic were presented by Eringen [4][5][6], who proposed some nonlocal continuum theories that were validated for different nanomaterials. Wang et al. [7] applied some nonlocal continuum models to investigate the size-dependent buckling response and vibration modes of CNTs.
Fang [8] applied a nonlocal elasticity theory to study the nonlinear free vibration of double walled CNTs. He found that the surrounding elastic medium plays a key role in the nonlinear propagation and amplitudes development. Saadatnia and Esmailzadeh [9] performed a systematic study of the nonlinear harmonic vibration of a piezoelectric-layered nanotube conveying fluid flow, while verifying the effect of the small scale parameters on the frequency response of the system in presence of fluid. In a further work, Askari and Esmailzadeh [10] investigated the forced vibration of fluid conveying CNTs, including the thermal effects and nonlinear foundations. Gheshlaghi and Hasheminejad [11] investigated the nonlinear vibrational behaviour of homogenous nanobeams and its dependence on the surface properties.
Sadeghi-Goughari et al. [12] studied the vibration response of a CNT conveying magnetic fluid under a longitudinal magnetic field. Similarly, Ya-Xin Zhena et al. [13] modeled the free vibration behaviour of viscoelastic nanotubes under a longitudinal magnetic field and verified the nonlocal parameter-dependence of the natural frequencies. Dai et al. [14] explored theoretically the postbuckling properties of nonlocal nanobeams in a longitudinal magnetic field. Ebrahimi and Barati [15] investigated the wave propagation in nonlocal porous multiphase nano-crystalline nanobeams under a longitudinal magnetic field effect. They found that wave frequencies and phase velocities may increase or decrease with reduced inhomogeneity magnitudes. Li et al. [16] applied a nonlocal strain gradient theory to study the wave propagation in viscoelastic single-walled CNTs under a magnetic field, while checking for the sensitivity of the response to the surface properties and damping parameters. Arani et al. [17] discussed the longitudinal magnetic field effect on the wave propagation of fluid-conveyed SWCNTs using the Knudsen number and surface considerations. Zhang et al. [18] investigated the vibration of horn-shaped SWC-NTs embedded in a viscoelastic medium under a longitudinal magnetic field. A two-scale coefficient model was developed by Güven [19] to study the propagation of longitudinal stress waves under a longitudinal magnetic field in a unified nonlocal elasticity context. Wang [20] validated the nonlocal elastic shell model for the study of longitudinal waves in SWCNTs and found that the microstructure and the coupling effect of the longitudinal wave and radial motion play a key role in the dispersion of waves. Azarboni [21] explored the magneto-thermal primary frequency response of CNTs, including the surface effect, under different boundary conditions. They inferred that an increased longitudinal magnetic field moves the backward jumping at higher excitation amplitudes, for different boundary conditions. Pradhan and Phadikar [22] applied a nonlocal continuum model to analyse the size-dependent vibration of embedded multi-layered graphene sheets.
The thermal buckling properties of zigzag SWCNTs were investigated by Semmah et al. [23], using a refined nonlocal model, while checking their dependence on the scale effect and chirality. Naceri et al. [24] investigated numerically the wave propagation in armchair SWCNTs under thermal conditions. Baghdadi et al. [25] proposed a nonlocal parabolic beam theory to analyze the thermomechanical vibration properties of armchair and zigzag SWCNTs. Based on a wide systematic numerical campaign, the authors verified a meaningful dependence of the natural frequencies on the temperature variation and chirality of armchair and zigzag CNTs. In a similar directions moves the work by Benzair et al. [26], where a nonlocal Timoshenko beam theory was employed to analyze the thermal sensitivity of SWCNTs, whose vibration response was compared to predictions based on a nonlocal Euler beam model. For further studies on the coupled vibration of CNTs based on a nonlocal elasticity, the reader is referred to [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45].
In recent years, different studies on nanoscale structures coupled with mass sensors have been performed in literature [46][47][48][49][50][51][52][53][54]. More specifically, Wu et al. [46] analysed the resonance frequency of a SWCNT via a contin-uum mechanics-based finite element method (FEM). Li and Chou [47] modeled the CNT nano-mass sensors with a molecular structural mechanics method. Another similar study was presented by Barati and Shahverdi [48] to examine the frequency shift behavior of plate-type nano-mass sensors made of FG nanostructures. The potential of SWC-NTs as mass sensors was recently investigated by Chowdhury et al. [49] based on a continuum mechanics theory.
Arda and Aydogdu [50] analysed the vibration response of CNT mass sensors considering its sensitivity to the mass and stiffness ratios, along with the position of the detected elastic mass and nonlocal parameters. Similarly, Liu and Lyu [51] proposed a nonlocal strain gradient plate theory to model a novel nanoscale mass sensor made of a smart functionally-graded (FG) magneto-electroelastic nanofilm integrated with graphene skins. In the further work by Lee et al. [52], the frequency equation of CNTbased cantilever sensors with an attached mass was derived theoretically using a nonlocal elasticity theory. Moreover, Nematollahi et al. [53] solved theoretically an inverse problem to determine the fluid velocity and mass ratio of a piezoelectric nanotube conveying fluid flow, according to the Eringen's nonlocal elasticity approach and the Euler-Bernoulli beam theory. Based on the same assumptions, Aydogdu and Filiz [54] performed a parametric analysis of the axial small-scale vibration behaviour of SW-CNT-based mass sensors, considering the nonlocal effect together with the length of CNTs and the attached mass.
Based on the available literature, however, it seems that there is a general lack of studies focusing on the nonlinear ultrasonic waves in a magneto-thermo-elastic armchair SW-CNTs with mass sensors resting on polymer matrix. This is here proposed and analysed by means of a nonlocal Euler beam theory, in agreement with the Eringen's assumptions, in order to account for small scale effects. Thus, we determine the solutions of the ultrasonic wave dispersion relations governing the problem, whereby a parametric study is carried out systematically to check for the influence of the magneto-electro-mechanical loading, attached mass, nonlocal parameter, and aspect ratio, on the deflection properties of nanotubes.

Eringen nonlocal theory of elasticity
Based on the Eringen's nonlocal theory of elasticity, we assume that the stress state at a reference point x in the body depends not only on the strain state at the same point x, but also on the strain states at all other points X ′ of the body. The general form of the constitutive equations in a nonlocal form contains an integral over the entire region of interest. The integral contains a non-local kernel function, which describes the relative influence of the strain field at various locations on the stress state at a given location. The constitutive equations of linear, homogeneous, isotropic, nonlocal elastic solid with zero body forces, are given by Eringen [4][5][6] as follows Eq. (1) is the equilibrium equation, where σ ij,i , ρ, f j , u j refer to the stress tensor, mass density, body force density and displacement vector at a reference point x in the body, respectively, at time t. Eq. (3) is the classical constitutive relation where σ c ij (X ′ ) is the classical stress tensor at any point X ′ in the body, which is related to the linear strain tensor e ij (X ′ ) at the same point. Eq. (4) refers to the classical kinematic relations. The kernel function π( ⃒ ⃒ X − X ′ ⃒ ⃒ , τ) is the attenuation function which includes the nonlocal effect in the constitutive equations. The volume integral in Eq. (2) is defined over the volume V of the body. The nonlocal modulus in Eq. (2) has dimensions (lengh) −3 and it depends on a characteristic length (i.e. lattice parameter, size of grain, granular distance, etc.) and on the external characteristic length of the system l (i.e. wavelength, crack length, size or dimensions of sample, etc.). Therefore, the nonlocal modulus can be written in the following form where e 0 a is a constant parameter that must be determined for each material, independently. Then, the integropartial differential Eq. (2) for a non-local elasticity problem can be simplified into a partial differential equation, as follows where C ijkl is the elastic modulus tensor of a classical isotropic elasticity, and e ij is the strain tensor; ∇ 2 denotes the second-order spatial gradient applied on the stress tensor σ ij , and τ = eo a/l. In agreement with findings by Eringen, we can consider eo = 0.39 by matching the dispersion curves via a nonlocal theory for place waves and born-Karman model of lattice dynamics at the end of the Brillouin zone (ka = π), where a is the distance between atoms, and k is the wavenumber in a phonon analysis. On the other hand, Eringen [5] assumed eo = 0.31 in his study for a Rayleigh surface wave, via a nonlocal continuum mechanics and lattice dynamics.

Atomic structure of CNTs
CNTs are tubes formed by rolling a graphene sheet about the ⃗ T vector. A vector perpendicular to the ⃗ T is the chiral vector denoted by ⃗ C h . The chiral vector and corresponding chiral angle define the type of CNT. Zigzag, armchair and chiral ⃗ C h can be expressed with respect to the base vectors ⃗ a 1 and ⃗ a 2 as follows where n and m are indices of translation that define the structure around the circumference. Figure 1 descripts the lattice of transition (n, m) along with the base vectors ⃗ a 1 and ⃗ a 2 . If the indices of translation are such that m = 0 and m = n, the CNT corresponds to a zigzag or armchair structure, respectively. The diameter of an armchair SW CNT for m = n is defined as [55] Based on the connection between molecular mechanics and solid mechanics, Wu et al. [56] developed an energyequivalent model for studying the mechanical properties of SW CNT. Using the same method, the equivalent Young's modulus for an armchair nanotube is expressed as where K and C are the force constants; t is the thickness; parameters 21 and 22 are defined as By assuming n → ∞, the Young's modulus for a graphite sheet becomes as

Basic equations of the problem for a magnetic field force
CNTs appear in a hollow structure formed by covalently bonded carbon atoms, which can be imagined as a rectangular graphite sheet rolled from one side of its longest edge, to form a cylindrical tube (see Figure 2). A cylindrical coordinate system (X, θ, Z) is shown in Figure 3, where X refers to the longitudinal direction of the shell, θ defines its circumferential direction, and Z corresponds to the radial direction. This means that the surface defined by Z = 0 is set on the middle surface of the shell. By assuming that the magnetic permeability, η, of CNTs equals the magnetic permeability of the surrounding medium, we can write the Maxwell equation as follows where f ,s, ε and U represent the current density, strength vectors of electric field, disturbing vectors of magnetic pling a longitudinal magnetic field vector H(Hx,0, 0) on the ith layer CNT with the cylindrical coordinate (R, θ, Z) and the displacement vector (11), we get the following relations The Lorentz force q induced by a longitudinal magnetic field can be written as where q X , q θ and q Z express the Lorentz force along the X, θ and Z direction as follows ( Figure 3) The external force qmag consists of the Lorentz force q Z due to the longitudinal magnetic field, and the distributed transverse force Fs due to the effect of surface tension, namely where the Lorentz force q Z is defined by Eq. (12f), [16], and the distributed transverse force Fs can be defined as [51] Fs = Here, η is the magnetic permeability, Hx is the component of the longitudinal magnetic field vector exerted on the SWCNTs in the x direction, and Hs is a constant, defined as where µ denotes the residual surface tension. The term q Z is the magnetic force per unit length due to Lorentz force exerted on the tube in z-direction, where the effect bending stiffness EI is defined as [57,58] EI * = EI + Qs Es In the last relation, Qs = π 8 (d + h) 3 , Es denotes the surface Young's modulus, h is the effective thickness of SWCNTs, and d is defined by Eq. (8).

Euler Bernoulli beam theory (EBT) based on nonlocal relations
The partial differential equation governing the free vibration of a nanotube under a thermal and Lorentz force can be expressed as where f (x) is the interaction pressure per unit axial length between the nanotube and the surrounding elastic medium; A is the cross section of CNT, and where f is the non-dimensional frequency parameter, α is the mass ratio (attached mass/mass of CNT), m is the mass per unit length according to a nonlocal elasticity approach.
The resultant shear force ∏︀ on the cross section of the nanotube is defined as ∏︁ = ∂M ∂X (19) where N t denotes the temperature-dependent axial force with thermal expansion coefficient α. This constant force is defined as [59] N t = −EAαT (20) where T is the temperature variable. The longitudinal magnetic flux due to a Lorentz force exerted on the tube in Zdirection is represented by q Z , which is defined as being Hx the magnetic field strength and η the magnetic permeability. For the Euler-beam theory, the resultant bending moment M in Eq. (19) reads where σxx is the nonlocal axial stress defined by the nonlocal continuum theory. The constitutive Eq. (6) for a homogeneous isotropic elastic solid in a non-local form for one-dimensional nanotube gets the following form where E is the Young's modulus of the tube, εxx is the axial strain, (e0a) is a nonlocal parameter; a is the internal characteristic length. The temperature-dependent nonlocal relations in Eq. (23) can be rewritten as According to a Euler -Bernoulli beam model, the axial strain εxx for small deflection is defined as where z is the transverse coordinate in the positive direction of deflection. By combining Eqs. (24)-(25), with Eq. (22), the bending moment M can be expressed as where I = ∫︀ A z 2 dA is the moment of inertia. By substituting Eq. (26) into Eq. (17), the nonlocal bending moment M and shear force ∏︀ become For the transverse vibration of the nanostructure under a distributed pressure and thermal interaction with the surrounding polymer elastic medium, the equation of motion (17) takes the following form The pressure per unit axial length acting on the nanotube surface due to the surrounding elastic medium, can be described by a Winkler-Pasternak type model [59] f (x) = −(−Kw + Kp + K nl ) where Kw, Kp, K nl refer to the Winkler, Pasternak and nonlinear constant foundation. The introduction of Eq. (30) into Eq. (29) yields

Ultrasonic wave solution
Eq. (29) can be transformed in a frequency domain using the Fourier transformation [60] Y( whereŶ is the amplitude of the wave motion, j = √ −1, k is the wave number, ωn the circular frequency of a sampling point, and N is the Nyquist frequency. The sampling rate and the number of sampling points should be sufficiently large to reach a good resolution for both high and low frequencies, respectively. By substitution of Eq. (32) into Eq. (30), we get ]︃ e iωn t = 0 This equation must be satisfied for each N and hence can be written as the ordinary differential equation in a single variable X as ]︃ e iωn x = 0 The last equation can be rewritten in terms of the single variable X, reducing as follows The dimensionless variables are defined as A non-trivial solution of the wave amplitude Y implies that Thus, by substituting Eq. (38) into Eq. (37), we get the following equation which represents the characteristic equation for a continuum structure (ECS) coupled with a surrounding medium of SWCNT.

Boundary conditions
In the following we provide an analytical solution of the governing vibration equations for a simply-supported (S-S) and clamped-clamped (C-C) nanobeam.

Simply supported SWCNT
For a S-S SWCNT we have to enforce the following conditions at (X) = (0, L)

Clamped -Clamped SWCNT
Let us now assume a C-C SWCNT subjected to an axial compressive load. In this case, we have to enforce the following boundary conditions

Numerical results and discussion
In this section we analyse the nonlinear vibration of a SWCNT embedded in a polymer matrix subjected to a magneto-thermo-elastic force. The geometrical and material properties assumed within the numerical investigation is shown in Table 1, whose results are summarized in Tables 2-4 in terms of natural frequencies for different nonlocal constants and foundations parameters. Based on results in these tables, it is worth noticing that frequencies decrease for an increasing value of the nonlocal parameter, for both C-C and S-S boundary conditions. It is also visible that the magnitude of frequency increases, as the foun-     Table 5. From results, it is observed that as the thermal parameter grows, the frequency increases but the small scale effects reduces the values of frequency for both boundary conditions. In Table 6, the natural frequency values are reported for different densities and Poisson's ratios. As listed in Table 6, the density and Poisson's ratio variation does not affect significantly the natural frequency (below 2%). Table 7 also presents the numerical results from a comparative study in terms of maximum transverse deflection for a C-C CNT accounting (or not) possible surface effects. Our results are in reasonable agreement with predictions from the literature, which confirms the accuracy of the proposed formulation.    In Figures 4 and 5 we show the dimensionless frequency versus the dimensionless amplitude for various non-local parameters, while assuming L/h = 10-20, V = 0, Kp = 20, ∆T = 20 ∘ and α = 0.5. Based on the results, it is found that the dimensionless frequency increases with the amplitude. When the nonlocal parameter increases, the frequency magnitude decreases, thus revealing the importance of considering nonlocalities within such vibration problems. A softening frequency behaviour is also observed for an increased structural slenderness, as shown in Figure 5. A similar systematic investigation is, thus, repeated for different Winkler parameters, while keeping L/h = 10, V = 0, α = 0.5, Kp = 20, ∆T = 20 ∘ and 30 ∘ , as plotted in Figures 6 and 7, respectively. All the plots in both figures, increase monotonically, where an increased Win-kler's parameter yields higher frequencies for each fixed dimensionless amplitude. This corresponds to an overall hardening behaviour of the structural stiffness of the nanotube. The frequency variation of the nanotube is presented in Figures 8 and 9, in dimensionless form, with respect to the amplitude for a varying voltage level, under the following assumptions: L/h = 10-20, α = 0.5, Kp = Kw = 20-30 and ∆T = 20 ∘ . It is found that an increased amplitude gets higher frequencies, whose variation is strictly dependent on the applied voltage. An axial tensile or compressive force is produced within the nanotube by applying a positive or negative voltage, respectively. It is also found that the dimensionless frequency is slightly dependent on  (20,20), respectively. It is seen that the frequency parameter decreases as the attached mass ratio values increases. It should be also pointed out that thermal and hygro-thermal environment degrades the plate stiffness for an increased mass ratio. Moreover, it is found that the effects of a varying temperature and moisture concentration, are more pronounced at lower mass ratios. Generally, by attaching a particle on a SWCNT, the total mass of the system amplifies although the stiffness remains unchanged, with a consequent decreased frequency. As far as the nonlocal effect is concerned, we can finally observe that an increasing nonlocal parameter decreases the frequency response of the structure, along with its stiffness.

Conclusion
This paper studies the nonlinear magneto-thermo-elastic waves in an armchair SWCNT resting on a polymer matrix via an Euler beam theory and the Eringen's nonlocal elas-ticity assumptions to account for small scale effects. The ultrasonic wave dispersion problem is defined and solved theoretically, where a parametric investigation checks for the influence of the magneto-electro-mechanical loading, nonlocal parameter, and aspect ratio, on the deflection response of nanotubes. Based on the results from the systematic study, the main conclusions can be summarized as follows: • The nonlocal scaling constant amplifies the natural frequencies of CNTs. • An increased foundation parameter enhances the stiffness and frequency of the nanostructure, especially when assuming clamped boundary conditions. • Armchair CNTs embedded in an elastic foundation, feature an improved dynamic behavior. Moreover, the Pasternak parameter affects the structural response more significantly than a Winkler-type foundation. • The natural frequency variation between local and non local boundary conditions, maintains below 1%, in presence of thermal coefficients. • The vibration modes of SWCNT are softened by an attached mass and for small length scale values. • The results from this work can be useful benchmarks for the study and design of nanodevices that make use of the wave propagation properties of armchair SWCNTs embedded on a polymer matrix, i.e. nano-oscillators, micro wave absorbing, nanoelectron technology and nano-electro-magnetomechanical systems (NEMMS).