Variational Principles for Coupled Boron Nitride Nanotubes Undergoing Vibrations, Including Piezoelastic and Surface Effects

Abstract

A variational formulation and variationally consistent boundary conditions were derived for a coupled system of two boron nitride nanotubes (BNNTs), with the piezoelectric and surface effects taken into account in the formulation. The coupling between the nanotubes was defined in terms of Winkler and Pasternak interlayers. The equations governing the vibrations of the coupled system were expressed as a system of four partial differential equations based on nonlocal elastic theory. After deriving the variational principle for the double BNNT system, Hamilton’s principle was expressed in terms of potential and kinetic energies. Next, the differential equations for the free vibration case were presented and the variational form for this case was derived. The Rayleigh quotient was formulated for the vibration frequency, which indicated that piezoelectric and surface effects led to higher vibration frequencies. Next, the variationally consistent boundary conditions were formulated in terms of moment and shear force expressions. It was observed that the presence of the Pasternak interlayer between the nanotubes led to coupled boundary conditions when a shear force and/or a moment was specified at the boundaries.

1. Introduction

Currently, nanoscale components are extensively used in several applications involving nano- and micro-electromechanical systems (NEMS and MEMS) [1]. One reason for their wide use is the improvement in the material properties when the scale of the material is reduced to a nanosize, as noted in [2]. A number of theories have been developed to study nanoscale components, such as nonlocal elasticity, strain gradient theory and couple stress theory. These theories and their applications to nanostructures have been the subject of a number of review articles [3,4,5,6]. Among the various approaches, nonclassical continuum mechanics has been extensively used in the study of nanobeams, nanoplates, carbon nanotubes (CNTs) and boron nitride nanotubes (BNNTs), as detailed in the review articles [7,8]. Concerning nanotubes, it is noted that the paper by James [9] gives a detailed study and definition of objective molecular structures and notes that carbon nanotubes—as well as a number of other carbon-based structures, such as a single graphite sheet and diamonds—are objective molecular structures. It is noted that significant simplification can be achieved for first-principles calculations of the energy and equilibrium equations of objective structures. The study gives an application of the objective structures to the bending and torsion of nanoscale beams.

Boron nitride nanotubes are similar to carbon nanotubes, with carbon atoms replaced by alternating boron (B) and nitrogen (N) atoms. They are known to have a high elastic modulus, high thermal conductivity, low density and a high level of structural stability as well as chemical inertness and high heat resistance. Several studies have involved the constitutive modelling of graphene and hexagonal boron nitride, which are used in a large number of nanotechnology applications. In particular, the constitutive modelling of monolayer graphene and hexagonal boron nanotubes was studied in Sfyris et al. [10], with the electro-magneto-mechanical properties of monolayer graphene presented in Sfyris et al. [11]. The determination of the constitutive constants of monolayer hexagonal boron nitride was the subject of the paper by Zhang and Yang [12]. The constitutive modeling of three-dimensional multilattices was the subject of the work by Sfyris and Sfyris [13]. Constitutive constants for hexagonal boron nitride were determined in [12]. The electrical properties of graphene and hexagonal boron nitride were the subject of the paper by Wang et al. [14]. The paper by Wang et al. [15] provided further discussion on the properties and applications of graphene and hexagonal boron nitride. These studies indicate that the constitutive modeling of graphene and hexagonal boron nitride differ due to the fact that their constitutive constants differ.

BNNTs have been the subject of a number of studies, including the review articles [8,16], providing information on the properties and applications of BNNTs. Their unique properties have led to their use in several applications and are the subject of studies in [17,18,19]. A detailed study of BNNTs involving their properties and applications is given in Tamilkovan and Arumugam [20]. The thermal properties of hexagonal boron nitrite and BNNTs were the subject of the work by Seo et al. [21]. An important property of BNNTs is their piezoelectric characteristics, which make them suitable for use in several nanotechnology applications, as noted in [22,23].

BNNT applications include sensors and resonators, where their piezoelectric properties can be used to great effect. The use of BNNTs in bio-sensors and nano-sensors was the subject of the studies in [24,25] and in nano-resonators in [26,27]. Further applications of BNNTs in energy harvesting, nanoelectronics and biomedicine have been discussed in the recent articles by Zhang et al. [28], Jakubinek et al. [29] and Adel et al. [30]. Recent review articles [8,20,31,32] discuss further applications of boron nanotubes in water purification membranes, biological applications and optical devices.

An important phenomenon with nanosized structures is the so-called surface effect, which becomes significant in nanostructures due to a high surface-to-bulk ratio, which affects the material properties. The surface effect leads to a higher elastic modulus compared with the elastic modulus values applicable in local continuum mechanics. This phenomenon led to the development of the surface stress elasticity theory, as formulated by Gurtin and Murdoch [33,34], leading to size-dependent nonlocal continuum mechanics. Based on this theory, the surface layer of the nanoscale material is defined as a two-dimensional membrane with zero thickness and a constitutive property different from the bulk material, as noted in [35]. Due to its importance in nanoscale applications, surface elasticity was extensively studied in the context of nonlocal stress and deformation in [36,37]. It was observed in [38,39,40] that the surface effect strongly influenced the nonlocal stress and deformation properties of nanoscale structures. The surface effect in the case of boron nitride nanotubes undergoing vibrations was studied in [41].

The piezoelectric properties of BNNTs have been the subject of several studies due to their importance in a number of applications. Some recent work on this topic can be found in [42,43,44,45], where molecular dynamics were employed in [42,43], atomistic finite elements were employed in [44] and defect engineering was employed in [45]. The combined effect of piezoelasticity and surface elasticity, as exhibited by BNNTs, was the subject of the work by Ghorbanpour Arani and Roudbari [46] in the context of the vibrations of a coupled BNNT system.

The piezoelectric properties of BNNTs have been effectively used by developing BNNT-reinforced piezoelectric nanocomposites. The subject has been extensively studied in a number of publications, which include [22,23,47]. The buckling and vibration of a BNNT-reinforced piezoelectric plate were studied in [47], based on modified couple stress theory. Multifunctional electroactive nanocomposites reinforced with piezoelectric BNNTs were studied in [22]. The vibrations of functionally graded nanocomposite plates subject to thermal and mechanical loads were studied in [48], with a plate reinforced by single-walled hollow BNNTs. The piezoelectric properties of zigzag and armchair boron nitride nanotubes were the subject of the study by Adel et al. [49].

Single-walled BNNTs (SWBNNTs) have been the subject of a number of studies. Linear and nonlinear vibration studies of single-walled BNNTs include the works [50,51,52]. The effects of surface stress and flow velocity on wave propagation along single-walled BNNTs were studied in [53]. The bending of SWBNNTs was the subject of [54], with a formulation based on Euler–Bernoulli beam theory. The vibrations of bilayer hexagonal boron nitride were investigated in [55], with the study taking the effect of different stacking sequences into account.

An extension of single-walled nanotubes is the double-walled nanotube, whose properties were studied in [56,57] in the case of carbon nanotubes. Double-walled BNNTs (DWBNNTs) have been the subject of a number of publications. Wave propagation along DWBNNTs and triple-walled BNNTs conveying fluid were the subjects of [58,59]. Nonlinear vibrations of DWBNNTs were studied in [60]. One of the applications of DWBNNTs is their use for conveying fluid. The results of the vibrations and wave propagation of DWBNNTs conveying fluid were presented in [61,62].

Many technological applications of nanotubes involve complex nanosystems, which involve double single-walled nanotubes. The mechanics of a system of double nanotubes differ from those of double-walled nanotubes due to the presence of an interlayer between the nanotubes affecting the mechanics of the problem. Recent studies involving the vibrations of double single-walled BNNTs include the papers [46,61]. Wave propagation in the case of a coupled double BNNT system conveying fluid was the subject of the study by Ghobanpour-Arani et al. [62]. The nonlinear vibrations of viscoelastically coupled boron nitride nanotube-reinforced composite micro-tubes were studied in [63].

The objective of the present study was to formulate the variational principles and determine the variationally consistent boundary conditions for a double single-walled BNNT system undergoing forced and free vibrations. The double BNNT system was connected by a Winkler–Pasternak interlayer. The constitutive equations of the system included the piezoelectric and surface effects, expressed as a system of four partial differential equations. The formulation was based on the nonlocal theory of elasticity, as formulated in Eringen [64,65]. After deriving the variational expression for the system of differential equations, Hamilton’s principle was formulated. In the case of a freely vibrating BNNT system, the Rayleigh quotient for the vibration frequency was derived. The Rayleigh quotient indicated that the presence of piezoelectric and surface effects led to an increase in the frequency. Next, the natural and geometric boundary conditions were derived and the expressions for shear force and moment were given. Moment and shear force expressions were obtained as part of the derivation of the variationally consistent boundary conditions. A number of methods for solutions use the variational form of a given problem to obtain approximate solutions, with the Rayleigh–Ritz method being extensively used for this purpose.

Concerning the variational formulation, it is noted that the work by Steigmann and Ogden [66] involved the derivation of energy-minimizing configurations for a surface interacting with a substrate material. The work was based on a variational approach to obtain the canonical form of the constitutive equations. In the present work, constitutive equations were defined for the physical problem, involving the vibrations of a coupled boron nanotube system. The applicable variational expressions were then derived. A recent work by Ye [67] studied the Steigmann–Ogden interface effect concerning the elastic properties of nanoparticle-reinforced composites. Another recent work [68] involved the application of the Steigmann–Ogden surface model to inclusion problems.

A variational formulation of the time-dependent case was obtained using a semi-inverse method to derive the variational expression for the coupling terms involving the Winkler and Pasternak interlayers. This was followed by the expression for Hamilton’s principle in terms of kinetic and potential energies. Next, the variational principle for the time-independent case was given and the Rayleigh quotient for the frequency was derived. This was followed by the derivation of the applicable natural and geometric boundary conditions. Further examples of the variational principles and the related boundary conditions involve multi-walled carbon nanotubes undergoing buckling [69] and vibration [70]. Variational principles involving the vibrations of a coupled Rayleigh beam system are given in [71] and the vibrations of coupled single-walled carbon nanotubes subject to a magnetic field are given in [72]. The present study extends these results to coupled boron nitride nanotubes undergoing vibrations, with the piezoelectric and surface effects included in the formulation. These effects in combination with boron nitride nanotubes have not been the subject of previous studies involving variational formulations. As such, the present work extends the variational results to boron nitride nanotubes and takes piezoelectric and surface effects into account.

2. Physical Problem

Each nanotube of the double BNNT system was modeled as a Euler–Bernoulli beam, with the interlayer between the BNNTs modelled as an elastic Winkler–Pasternak layer, as shown in Figure 1, where $p_i\left(x,t\right)$ indicates the external forces acting on the BNNTs.

The lateral and shear force $F(w_1,w_2)$ between the nanotubes can be expressed as

$${F\left(w_1,w_2\right)=K}_w\left(w_1-w_2\right)-G_p\left({\displaystyle \frac{{\partial }^2{w}_1}{\partial x^2}}-{\displaystyle \frac{{\partial }^2{w}_2}{\partial x^2}}\right)$$

(1)

where $K_w$ is the Winkler foundation modulus and $G_p$ is the shear modulus of the Pasternak interlayer. The length of each nanotubes is $L$, with the BNNTs subject to the axial force $N_x=N_E+N_{se}$, which is the summation of the electric $N_E$ and surface effect $N_{se}$ loads [42]. $N_{Se}$ is given by

$$N_{Se}=4{\tau }_{0}R$$

(2)

where ${\tau }_0$ is the residual surface tension per unit length and $R$ is the radius of the nanotubes, as shown in Figure 2.

The stiffness of the $i\mathrm{th}$ BNNT can be expressed as

$${\left(EI\right)}_i^{\ast }=E_i{I}_i+\pi E_s{R}^3$$

(3)

where $E_i$ is Young’s modulus and $I_i$ is the moment of inertia of the $i\mathrm{th}$ nanotube. In Equation (3), $E_s$ is given by

$$E_s=2{\mu }_s+{\eta }_s$$

(4)

where ${\mu }_s$ and ${\eta }_s$ are the surface Lame constants [42]. Displacements of BNNTs in the longitudinal direction are denoted as $u_1\left(x,t\right)$ and $u_2\left(x,t\right)$ and in the transverse direction as $w_1\left(x,t\right)$ and $w_2\left(x,t\right)$. The governing equations, based on the nonlocal elasticity theory, can be expressed as follows [46]:

$$D_1\left(u_1\right)=L_1\left(u_1\right)=0$$

(5)

$$D_2\left(u_2\right)=L_2\left(u_2\right)=0$$

(6)

$$D_3\left(w_1,w_2\right)=M_1\left(w_1\right)+K_W\left(w_1,w_2\right)+K_P\left(w_1,w_2\right)+f_1\left(x,t\right)=0$$

(7)

$$D_4\left(w_1,w_2 \right)=M_2\left(w_2\right)-K_W\left(w_1,w_2\right)-K_P\left(w_1,w_2\right)+f_2\left(x,t\right)=0$$

(8)

where $f_i\left(x,t\right)$ is the input function involving the forced vibrations case. This is given by

$$f_i\left(x,t\right)=p_i\left(x,t\right)-\mu {\displaystyle \frac{{\partial }^2{p}_i\left(x,t\right)}{\partial x^2}}$$

(9)

where $p_i\left(x,t\right)$ denotes the forcing function acting on the $i\mathrm{th}$ BNNT, as shown in Figure 1. In Equations (7) and (8), the function $K_W\left(w_1,w_2\right)$ represents the effect of the Winkler interlayer and the function $K_P\left(w_1,w_2\right)$ represents the effect of the Pasternak shear interlayer between the nanotubes, as shown in Figure 1. The time domain is given by $t_1\le t\le t_2$. In Equation (9), $\mu ={\left(e_{0}a\right)}^2$ is the nonlocal parameter based on the nonlocal theory of elasticity [65,66]. The differential operators $L_1\left(u_1\right)$, $L_2\left(u_2\right), M_1\left(w_1\right)$, $M_2\left(w_2\right)$, $K_W\left(w_1,w_2\right)$ and $K_P\left(w_1,w_2\right)$ in Equations (5)–(8) are defined as follows [46]:

$$L_i\left(u_i\right)=\left(E_i{A}_i+{\displaystyle \frac{e}{\chi }}\right){\displaystyle \frac{{\partial }^2{u}_i}{\partial x^2}}-\rho A_i{\displaystyle \frac{{\partial }^2{u}_i}{\partial t^2}}+\mu \rho A_i{\displaystyle \frac{{\partial }^4{u}_i}{\partial x^2\partial t^2}}$$

(10)

$$M_i\left(w_i\right)={\left(EI\right)}_i^{\ast }{\displaystyle \frac{{\partial }^4{w}_i}{\partial x^4}}-N_x{\displaystyle \frac{{\partial }^2{w}_i}{\partial x^2}}+\mu N_x{\displaystyle \frac{{\partial }^4{w}_i}{\partial x^4}}+\rho A_i{\displaystyle \frac{{\partial }^2{w}_i}{\partial t^2}}-\mu \rho A_i{\displaystyle \frac{{\partial }^4{w}_i}{\partial x^2\partial t^2}}$$

(11)

$$K_W\left(w_1,w_2\right)=k_w\left(w_1-w_2\right)-\mu k_w\left({\displaystyle \frac{{\partial }^2{w}_1}{\partial x^2}}-{\displaystyle \frac{{\partial }^2{w}_2}{\partial x^2}}\right)$$

(12)

$$K_P\left(w_1,w_2\right)=-G_p\left({\displaystyle \frac{{\partial }^2{w}_1}{\partial x^2}}-{\displaystyle \frac{{\partial }^2{w}_2}{\partial x^2}}\right)+{\mu G}_p\left({\displaystyle \frac{{\partial }^4{w}_1}{\partial x^4}}-{\displaystyle \frac{{\partial }^4{w}_2}{\partial x^4}}\right)$$

(13)

where $i=1, 2$. In Equation (12), $k_w$ is the Winkler interlayer constant and, in Equation (13), $G_p$ is the Pasternak interlayer constant. In Equation (10), $e$ and $\chi$ are the piezoelectric and the dielectric constants [19,38]. The densities and cross-sectional areas of the nanotubes are given by $\rho$ and $A_i$, respectively. In Equation (11), $N_x$ is the summation of the electric and surface effect loads, which is

$$N_x=N_E+N_{Se}$$

(14)

where $N_{Se}$ is given by Equation (2) [42].

3. Variational Formulation

The variational principle for the system of differential Equations (5)–(8) was derived by introducing the functionals $V_i\left(u_i\right), i=\mathrm{1,2}$ and $V_i\left(w_1,w_2\right), i=\mathrm{3,4}$, such that

$$V\left({u_1, u_2, w}_1,w_2\right)=V_1\left(u_1\right)+V_2\left(u_2\right)+V_3\left(w_1,w_2\right)+V_4\left(w_2\right)$$

(15)

where $V\left({u_1, u_2, w}_1,w_2\right)$ is the variational functional to be determined. Differential Equations (5) and (6) involve only the functions $u_1\left(x,t\right)$ and $u_2\left(x,t\right)$. As such, the applicable variational functional only involves these two functions. In this case, variational functionals $V_i\left(u_i\right)$, $i=1, 2$ are defined as follows:

$$V_i\left(u_i\right)={\displaystyle \frac{1}{2}}{\int }_{t_1}^{t_2}{\int }_0^L\left[-\left(E_i{A}_i+{\displaystyle \frac{e}{\chi }}\right){\left({\displaystyle \frac{\partial u_i}{\partial x}}\right)}^2+{\rho }_i{A}_i{\left({\displaystyle \frac{\partial u_i}{\partial t}}\right)}^2+\mu \rho A_i{\left({\displaystyle \frac{{\partial }^2{u}_i}{\partial x \partial t}}\right)}^2\right]dx dt , i=\mathrm{1,2}$$

(16)

In the case of functions $w_1(x,t)$ and $w_2(x,t)$, the applicable differential Equations (7) and (8) have the forcing functions $f_i\left(x,t\right)$ as well as the interlayer functions $K_W\left(w_1,w_2\right)$ and $K_P\left(w_1,w_2\right)$. The next step is the formulation of the variational expressions involving the forcing functions $f_i\left(x,t\right)$. First, the functionals $V_3\left(w_1,w_2\right)$ and $V_4\left(w_2\right)$ are defined as follows:

$$V_3\left(w_1,w_2\right)={\mathsf{\Phi }}_1\left(w_1\right)+{\mathsf{{\rm Y}}}_1\left(w_1\right)+{\int }_{t_1}^{t_2}{\int }_0^L{F(w}_1,w_2) dx dt$$

(17)

$$V_4\left(w_2\right)={\mathsf{\Phi }}_2\left(w_2\right)+{\mathsf{{\rm Y}}}_2\left(w_2\right)$$

(18)

where ${\mathsf{\Phi }}_i\left(w_i\right)$ and ${\mathsf{{\rm Y}}}_i\left(w_i\right)$ are given by

$${\mathsf{\Phi }}_i\left(w_i\right)={\displaystyle \frac{1}{2}}{\int }_{t_1}^{t_2}{\int }_0^L\left[{\left(EI\right)}_i^{\ast }{\left({\displaystyle \frac{{\partial }^2{w}_i}{\partial x^2}}\right)}^2+N_x{\left({\displaystyle \frac{\partial w_i}{\partial x}}\right)}^2+\mu N_x{\left({\displaystyle \frac{{\partial }^2{w}_i}{\partial x^2}}\right)}^2-{\rho }_i{A}_i{\left({\displaystyle \frac{\partial w_i}{\partial t}}\right)}^2-\mu \rho A_i{\left({\displaystyle \frac{{\partial }^2{w}_i}{\partial x \partial t}}\right)}^2\right]dx dt$$

(19)

$${\mathsf{{\rm Y}}}_i\left(w_i\right)={\int }_{t_1}^{t_2}{\int }_0^L\left(f_i\left(x,t\right)-\mu {\displaystyle \frac{{\partial }^2{f}_i\left(x,t\right)}{\partial x^2}}\right)w_i dx dt$$

(20)

Equations (19) and (20) correspond with the variational forms of the differential Equations (7) and (8), respectively, with the coupling terms (12) and (13) excluded, which involve the expressions for the interlayer functions. Next, the variational forms of the expressions due to the Winkler and Pasternak interlayers between the nanotubes were determined. The functional ${F(w}_1,w_2)$ in Equation (17) has to be defined such that differential Equations (7) and (8) correspond with the Euler–Lagrange equations of $V\left(w_1,w_2\right)$ given by Equation (15). The Euler–Lagrange equations of the functional (17) are

$$L_1\left(w_1\right)+Q_1\left(w_1\right)+{\displaystyle \frac{\delta F}{\delta w_1}}=0$$

(21)

$$L_2\left(w_2\right)+Q_2(w_2)+{\displaystyle \frac{\delta F}{\delta w_2}}=0$$

(22)

where ${\displaystyle \frac{\delta F}{\delta w_i}}$ is the variational derivative of ${F(w}_1,w_2)$. The variational derivative of ${F(w}_1,w_2)$ has to satisfy the following coupling equations:

$${\displaystyle \frac{\delta F}{\delta w_1}}=K_w\Delta w_{12}-\mu K_w{\displaystyle \frac{{\partial }^2\Delta w_{12}}{\partial x^2}}-G_p{\displaystyle \frac{{\partial }^2\Delta w_{12}}{\partial x^2}}+{\mu G}_p{\displaystyle \frac{{\partial }^4\Delta w_{12}}{\partial x^4}}$$

(23)

$${\displaystyle \frac{\delta F}{\delta w_2}}={-K}_w\Delta w_{12}+\mu K_w{\displaystyle \frac{{\partial }^2\Delta w_{12}}{\partial x^2}}+G_p{\displaystyle \frac{{\partial }^2\Delta w_{12}}{\partial x^2}}-{\mu G}_p{\displaystyle \frac{{\partial }^4\Delta w_{12}}{\partial x^4}}$$

(24)

where $\Delta w_{12}=w_1-w_2.$ To determine the functional $F\left(w_1,w_2\right)$, Equations (7) and (8) were compared with Equations (23) and (24). This comparison yielded the functional $F\left(w_1,w_2\right)$ as follows:

$$F\left(w_1,w_2\right)={\displaystyle \frac{1}{2}}{K}_w{(\Delta w_{12})}^2+{\displaystyle \frac{1}{2}}\mu K_w{\left({\displaystyle \frac{\partial (\Delta w_{12})}{\partial x}}\right)}^2+{\displaystyle \frac{1}{2}}{G}_p{\left({\displaystyle \frac{\partial (\Delta w_{12})}{\partial x}}\right)}^2+{\displaystyle \frac{1}{2}}\mu G_p{\left({\displaystyle \frac{{\partial }^2(\Delta w_{12})}{\partial x^2}}\right)}^2$$

(25)

It was noted that applying the variational operations defined by ${\displaystyle \frac{\delta F}{\delta w_1}}$ and ${\displaystyle \frac{\delta F}{\delta w_2}}$ to the function $F\left(w_1,w_2\right)$ in Equation (25) produced Equations (23) and (24). It was observed that the Euler–Lagrange equations of the variational functional $V\left(w_1,w_2\right)$ given by Equation (15) corresponded with the governing Equations (5)–(8), with ${\mathsf{\Phi }}_i\left(w_i\right)$ and ${\mathsf{{\rm Y}}}_i\left(w_i\right)$ defined by Equations (19) and (20) and ${F(w}_1,w_2)$ given by Equation (25). This completed the derivation of the variational principle for the forced vibration case of the coupled double boron nanotubes.

4. Hamilton’s Principle

Hamilton’s principle is extensively used in the derivation of equations governing a number of physical phenomena. It also allows the formulation of analytical and numerical approximations for the solution of a large number of mechanics problems, as noted in [73,74]. For the present case, Hamilton’s principle was given by

$${\int }_{t_1}^{t_2}\left[\delta KE\left(t\right)-\left(\delta W_E\left(t\right)+\delta {PE}_1\left(t\right)+\delta {PE}_2\left(t\right)\right)\right]dt=0$$

(26)

Functionals $KE\left(t\right)$, $W_E\left(t\right)$, ${PE}_1\left(t\right)$ and ${PE}_2\left(t\right)$ in Equation (26) are defined as follows:

$$KE\left(t\right)={\displaystyle \frac{1}{2}}\sum _{i=1}^2{\int }_0^L\left[{\rho }_i{A}_i{\left({\displaystyle \frac{\partial u_i}{\partial t}}\right)}^2+\mu \rho A_i{\left({\displaystyle \frac{{\partial }^2{u}_i}{\partial x \partial t}}\right)}^2+{\rho }_i{A}_i{\left({\displaystyle \frac{\partial w_i}{\partial t}}\right)}^2+\mu \rho A_i{\left({\displaystyle \frac{{\partial }^2{w}_i}{\partial x \partial t}}\right)}^2\right]dx$$

(27)

$$W_E\left(t\right)=\sum _{i=1}^2{\int }_0^L\left(f_i\left(x,t\right)-\mu {\displaystyle \frac{{\partial }^2{f}_i\left(x,t\right)}{\partial x^2}}\right)w_{i}dx$$

(28)

$${PE}_1\left(t\right)={\displaystyle \frac{1}{2}}\sum _{i=1}^2{\int }_0^L\left[\left(E_i{A}_i+{\displaystyle \frac{e}{\chi }}\right){\left({\displaystyle \frac{\partial u_i}{\partial x}}\right)}^2+{\left(EI\right)}_i^{\ast }{\left({\displaystyle \frac{{\partial }^2{w}_i}{\partial x^2}}\right)}^2+N_x{\left({\displaystyle \frac{\partial w_i}{\partial x}}\right)}^2+\mu N_x{\left({\displaystyle \frac{{\partial }^2{w}_i}{\partial x^2}}\right)}^2\right]dx$$

(29)

$${PE}_2\left(t\right)={\displaystyle \frac{1}{2}}{\int }_0^L\left[K_w{(\Delta w_{12})}^2+\mu K_w{\left({\displaystyle \frac{\partial (\Delta w_{12})}{\partial x}}\right)}^2+G_p{\left({\displaystyle \frac{\partial (\Delta w_{12})}{\partial x}}\right)}^2+\mu G_p{\left({\displaystyle \frac{{\partial }^2(\Delta w_{12})}{\partial x^2}}\right)}^2\right]dx$$

(30)

where $\Delta w_{12}=w_1-w_2$. In Equations (27)–(30), $KE\left(t\right)$ is the kinetic energy, $W_E\left(t\right)$ is the work performed by actuating forces, ${PE}_1\left(t\right)$ is the potential energy of deformation and ${PE}_2\left(t\right)$ is the potential energy due to interlayer forces between the BNNTs.

5. Free Vibrations

In the present section, a variational formulation for the time-independent case involving the free vibrations of the coupled nanotubes is presented. In the case of a freely vibrating double-walled BNNT system, the deflection functions are given by

$$u_i\left(x,t\right)=U_i{\left(x\right) e}^{i\overline{\omega }t}$$

(31)

$$w_i\left(x,t\right)=W_i{\left(x\right) e}^{i\omega t}$$

(32)

where $\overline{\omega }$ is the longitudinal natural frequency and $\omega$ is the transverse natural frequency with the subscript $i=\mathrm{1,2}$. Governing equations can be obtained by substituting Equations (31) and (32) into Equations (5)–(9) as follows:

$${ D}_{FV1}\left(U_1\right)=L_{FV1}\left(U_1\right)=0$$

(33)

$$D_{FV2}\left(U_2\right)=L_{FV2}\left(U_2\right)=0$$

(34)

$$D_{FV3}\left(W_1,W_2\right)=L_{FV3}\left(W_1\right)+K_{FV}\left(W_1,W_2\right)=0$$

(35)

$$D_{FV4}\left(W_1,W_2 \right)=L_{FV4}\left(W_2\right)-K_{FV}\left(W_1,W_2\right)=0$$

(36)

where the differential operators $L_{FVi}\left(U_i\right),i=\mathrm{1,2}$, $L_{FVi}\left(W_i\right),i=\mathrm{3,4}$ and $K_{FV}\left(W_1,W_2\right)$ are defined as follows:

$$L_{FVi}\left(U_i\right)=\left(E_i{A}_i+{\displaystyle \frac{e}{\chi }}\right){\displaystyle \frac{d^2{U}_i}{d{x}^2}}+{\overline{\omega }}^2\rho A_i{U}_i-{\overline{\omega }}^2\mu \rho A_i{\displaystyle \frac{d^2{U}_i}{d{x}^2}}$$

(37)

$$L_{FVi}\left(W_i\right)={\left(EI\right)}_i^{\ast }{\displaystyle \frac{d^4{W}_i}{d{x}^4}}-N_x{\displaystyle \frac{d^2{W}_i}{d{x}^2}}+\mu N_x{\displaystyle \frac{d^4{W}_i}{d{x}^4}}-{\omega }^2\rho A_i{W}_i+{\omega }^2\mu \rho A_i{\displaystyle \frac{d^2{W}_i}{d{x}^2}}$$

(38)

$$K_{FV}\left(W_1,W_2\right)=K_w\Delta W_{12}-\mu K_w{\displaystyle \frac{d^2(\Delta W_{12})}{d{x}^2}}-G_p{\displaystyle \frac{d^2(\Delta W_{12})}{d{x}^2}}+{\mu G}_p{\displaystyle \frac{d^4(\Delta W_{12})}{d{x}^4}}$$

(39)

where $\Delta W_{12}=W_1-W_2$. For the time-independent case, the variational functionals $V_{FVi}\left(U_i\right)$, ${\mathrm{V}}_{FVi}\left(W_i\right)$ and ${F_{FV}(W}_1,W_2)$ are defined as

$$V_{FVi}\left(U_i\right)={\displaystyle \frac{1}{2}}{\int }_0^L\left[-\left(E_i{A}_i+{\displaystyle \frac{e}{\chi }}\right){\left({\displaystyle \frac{d{U}_i}{dx}}\right)}^2+{\overline{\omega }}^2\rho A_i{U}_i^2+{\overline{\omega }}^2\mu \rho A_i{\left({\displaystyle \frac{d{U}_i}{dx}}\right)}^2\right]dx$$

(40)

$${\mathrm{V}}_{FVi}\left(W_i\right)={\displaystyle \frac{1}{2}}{\int }_0^L\left[{\left(EI\right)}_i^{\ast }{\left({\displaystyle \frac{d^2{W}_i}{d{x}^2}}\right)}^2+N_x{\left({\displaystyle \frac{d{W}_i}{dx}}\right)}^2+\mu N_x{\left({\displaystyle \frac{d^2{W}_i}{d{x}^2}}\right)}^2-{\omega }^2\rho A_i{W}_i^2-{\omega }^2\mu \rho A_i{\left({\displaystyle \frac{d{W}_i}{dx }}\right)}^2\right]dx$$

(41)

$${F_{FV}(W}_1,W_2)={\displaystyle \frac{1}{2}}{K}_w{(\Delta W_{12})}^2+{\displaystyle \frac{1}{2}}\mu K_w{\left({\displaystyle \frac{d\Delta W_{12}}{dx}}\right)}^2+{\displaystyle \frac{1}{2}}{G}_p{\left({\displaystyle \frac{d\Delta W_{12}}{dx}}\right)}^2+{\displaystyle \frac{1}{2}}\mu G_p{\left({\displaystyle \frac{d^2\Delta W_{12}}{d^{2}x}}\right)}^2$$

(42)

where $\Delta W_{12}=W_1-W_2$. Thus, the variational expression for the time-independent case is given by

$$V_{FV}\left({U_1,U_2,W}_1,W_2\right)=\sum _{i=1}^{i=2}\left[V_{FVi}\left(U_i\right)+V_{FVi}\left(W_i\right)\right]+{\int }_0^L{F_{FV}(W}_1,W_2) dx$$

(43)

Expression (43) gives the variational formulation of the problem for the time-independent case, with the Euler–Lagrange equations of Equation (43) giving the differential Equations (33)–(36). The Rayleigh quotients for the vibration frequencies are given by

$${\overline{\omega }}^2=\mathrm{m}\mathrm{i}\mathrm{n}{\displaystyle \frac{{\int }_0^L\left[\left(E_i{A}_i+\frac{e}{\chi }\right){\left(\frac{d{U}_i}{dx}\right)}^2\right]dx}{{\int }_0^L\left[\rho A_i{U}_i^2+\mu \rho A_i{\left(\frac{d{U}_i}{dx}\right)}^2\right]dx}}$$

(44)

$${\omega }^2=\mathrm{m}\mathrm{i}\mathrm{n}{\displaystyle \frac{\sum _{i=1}^2{\int }_0^L\left[{\left(EI\right)}_i^{\ast }{\left(\frac{d^2{W}_i}{d{x}^2}\right)}^2+N_x{\left(\frac{d{W}_i}{dx}\right)}^2+\mu N_x{\left(\frac{d^2{W}_i}{d{x}^2}\right)}^2\right]dx+{\int }_0^L{F_{FV}(W}_1,W_2) dx}{{\int }_0^L\left[\rho A_i{W}_i^2+\mu \rho A_i{\left(\frac{d{W}_i}{dx }\right)}^2\right]dx}}$$

(45)

where ${F_{FV}(W}_1,W_2)$ is given by Equation (42). It was observed from Equation (45) that the effect of $N_x$, which refers to the sum of the electric and surface forces given by Equation (14), was to increase the frequency. Similarly, the effect of Winkler and Pasternak foundations, as given by the term ${\int }_0^L{F_{FV}(W}_1,W_2) dx$ in Equation (45), was also to increase the frequency.

6. Boundary Conditions

Next, natural and geometric boundary conditions were derived for the freely vibrating double BNNT system. The variations of $V_{FV}\left(W_1,W_2\right)$ with respect to $W_i$ are given by

$${\delta }_{W_1}{V}_{FV}\left(W_1, W_2\right)={\delta }_{W_1}{V}_{FV1}+{\delta }_{W_1}{V}_{FV2}={\int }_0^L{D}_{FV3}\left(W_1,W_2\right)\delta W_1 dx+\partial {\mathsf{\Omega }}_1\left(0,L\right)$$

(46)

$${\delta }_{W_2}{V}_{FV}\left(W_1, W_2\right)={\delta }_{W_2}{V}_{FV1}+{\delta }_{W_2}{V}_{FV2}={\int }_0^L{D}_{FV4}\left(W_1,W_2\right)\delta W_2 dx+\partial {\mathsf{\Omega }}_2\left(0,L\right)$$

(47)

where $\partial {\mathsf{\Omega }}_i\left(0,L\right)$ is the boundary term with $i=\mathrm{1,2}$. This calculation gives

$${\delta }_{W_i}{V}_{FVi}\left(W_1,W_2\right)={\int }_0^L\left[{\displaystyle \frac{d{V}_{FVi}}{d{W}_i}}-{\displaystyle \frac{d}{dx}}\left({\displaystyle \frac{d{V}_{FVi}}{d{W}_{ix}}}\right)+{\displaystyle \frac{d^2}{d^{2}x}}\left({\displaystyle \frac{d{V}_{FVi}}{d{W}_{ixx}}}\right)\right]\delta W_i dx+\partial {\mathsf{\Omega }}_{FVi}\left(0,L\right)$$

(48)

with $i=1, 2$. Derivations of the variational expression (41) and the corresponding boundary conditions are given next. We noted that

$${\int }_0^L\left({\left(EI\right)}_i^{\ast }+\mu N_x\right){\displaystyle \frac{d^4{W}_i}{d{x}^4}}\delta W_i dx=B_{1iFV}\left(W_i,\delta W_i\right)+\delta \left[{\displaystyle \frac{1}{2}}{\int }_0^L\left({\left(EI\right)}_i^{\ast }+\mu N_x\right){\left({\displaystyle \frac{d^2{W}_i}{d{x}^2}}\right)}^{2}dx\right]$$

(49)

where

$$B_{1FVi}\left(W_i,\delta W_i\right)={\left.\left(\left({\left(EI\right)}_i^{\ast }+\mu N_x\right){\displaystyle \frac{d^3{W}_i}{d{x}^3}}{\delta W}_i\right)\right|}_{x=0}^{x=L}-{\left(\left({\left(EI\right)}_i^{\ast }+\mu N_x\right){\displaystyle \frac{d^2{W}_i}{d{x}^2}}\delta \left({\displaystyle \frac{d{W}_i}{dx}}\right)\right)⌋}_{x=0}^{x=L}$$

(50)

Similarly,

$${\int }_0^L\left({-N}_x+{\omega }^2\mu \rho A_i\right){\displaystyle \frac{d^2{W}_i}{d{x}^2}}\delta W_i dx=B_{2FVi}\left(W_i,\delta W_i\right)+\delta \left[{\displaystyle \frac{1}{2}}{\int }_0^L\left(N_x-{\omega }^2\mu \rho A_i\right){\left({\displaystyle \frac{{dW}_i}{dx}}\right)}^{2}dx\right]$$

(51)

where

$$B_{2FVi}\left(W_i,\delta W_i\right)={\left.\left(\left({-N}_x+{\omega }^2\mu \rho A_i\right){\displaystyle \frac{d{W}_i}{dx}}{\delta W}_i\right)\right|}_{x=0}^{x=L}$$

(52)

Next, the coupled expression Equation (42) containing both $W_1$ and $W_2$ due to the interlayer between the nanotubes was studied. We noted that

$${\int }_0^L\left(-\mu K_w-G_p\right){\displaystyle \frac{d^2\left(\Delta W_{12}\right)}{d{x}^2}}\delta \left(\Delta W_{12}\right) dx=B_{3FVi}\left(W_i,\delta W_i\right)+\delta \left[{\displaystyle \frac{1}{2}}{\int }_0^L{\left(\mu K_w+G_p\right)\left({\displaystyle \frac{d\left(\Delta W_{12}\right)}{dx}}\right)}^{2}dx\right]$$

(53)

where $\Delta W_{12}=W_1-W_2$ and

$$B_{3FVi}\left(W_i,\delta W_i\right)={\left.\left(\left(-\mu K_w-G_p\right){\displaystyle \frac{d\left(\Delta W_{12}\right)}{dx}}\delta (\Delta W_{12})\right)\right|}_{x=0}^{x=L}$$

(54)

The last term in the coupled expression (42) due to the Pasternak interlayer was integrated after multiplying by $\delta \left(\Delta W_{12}\right)$ to obtain

$${\int }_0^L{\mu G}_p{\displaystyle \frac{d^4\left(\Delta W_{12}\right)}{d{x}^4}}\delta \left(\Delta W_{12}\right)dx=B_{4FVi}\left(W_i,\delta W_i\right)+\delta \left[{\displaystyle \frac{1}{2}}{\int }_0^L{\mu G}_p{\left({\displaystyle \frac{d^2\left(\Delta W_{12}\right)}{d{x}^2}}\right)}^{2}dx\right]$$

(55)

where the boundary term $B_{4FVi}\left(W_i,\delta W_i\right)$, $i=1, 2$, is given by

$$B_{4FVi}\left(W_i,\delta W_i\right)={\left.\left({\mu G}_p{\displaystyle \frac{d^3\left(\Delta W_{12}\right)}{d{x}^3}}\delta \left(W_1-W_2\right)\right)\right|}_{x=0}^{x=L}-{\left({\mu G}_p{\displaystyle \frac{d^2\left(\Delta W_{12}\right)}{d{x}^2}}\delta \left({\displaystyle \frac{d\left(\Delta W_{12}\right)}{dx}}\right)\right)⌋}_{x=0}^{x=L}$$

(56)

The boundary conditions could then be formulated based on Equations (50), (52), (54) and (56), in terms of shear force and moment at $x=0$ and $x=L$. The boundary conditions of the freely vibrating double BNNT system were given by

$$W_i \mathrm{o}\mathrm{r} Q_i\left(x\right)=\left[{\left(EI\right)}_i^{\ast }+\mu N_x\right]{\displaystyle \frac{d^3{W}_i}{d{x}^3}}+\left({-N}_x+{\omega }^2\mu \rho A_i\right){\displaystyle \frac{d{W}_i}{dx}}-\left(\mu K_w+G_p\right){\displaystyle \frac{d\left(\Delta W_{12}\right)}{dx}}{-\mu G}_p{\displaystyle \frac{d^2\left(\Delta W_{12}\right)}{d{x}^2}}$$

(57)

$$W_{ix} \mathrm{o}\mathrm{r} M_i\left(x\right)=-\left[{\left(EI\right)}_i^{\ast }+\mu N_x\right]{\displaystyle \frac{d^2{W}_i}{d{x}^2}}+{\mu G}_p{\displaystyle \frac{d^2\left(\Delta W_{12}\right)}{d{x}^2}} \mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{d} (58)$$

(58)

with $i=1, 2.$ The expression for $Q_i\left(x\right)$ given by Equation (57) corresponded with the shear force and the expression for $M_i\left(x\right)$ given by Equation (58) corresponded with the bending moment for the coupled BNNT system at the boundaries $x=0$ and $x=L$. The boundary conditions (57) and (58) corresponding with the shear force $Q_i\left(x\right)$ and the bending moment $M_i\left(x\right)$ constituted the natural boundary conditions of the coupled BNNT system.

7. Conclusions

Variational formulations for a vibrating double boron nitrate nanotube system have been given for time-dependent and time-independent cases, taking the piezoelectric and surface effects into account in the constitutive formulation. The time-dependent case referred to the forced vibrations and the time-independent case referred to the free vibrations of the coupled BNNT system. The interlayer between the BNNTs was expressed as a combination of Winkler and Pasternak foundations. The constitutive relations were based on nonlocal Euler–Bernoulli beam theory. The formulation for the time-dependent case was presented first in the form of a system of four partial differential equations. The problem formulation led to a coupled system of two differential equations in term of the displacement in the longitudinal direction and to a system of two differential equations in terms of the displacement in the transverse direction. The variational expression involving the system of four time-dependent differential equations was derived using a semi-inverse approach. The semi-inverse approach was implemented in order to formulate the variational expression involving the interlayer coupling terms. Based on the variational formulation of the problem, Hamilton’s principle was derived.

Next, a time-independent case corresponding with the free vibrations of the coupled BNNT nanotube system was formulated by setting the forcing functions to zero. Furthermore, the longitudinal natural frequency and the transverse natural frequency were introduced. The variational formulation for this case was derived, followed by the derivation of the Rayleigh quotients for the longitudinal natural frequency and the transverse natural frequency. The Rayleigh quotient for the transverse frequency indicated that piezoelectric and surface effects led to an increase in the vibration frequency. Furthermore, it was observed that the Winkler and Pasternak interlayers led to an increase in the transverse frequency.

Next, the boundary conditions were formulated and the moment and shear force expressions were derived for the time-independent case. This allowed the implementation of different boundary conditions in the study of the present problem, such as the natural boundary conditions and/or geometric boundary conditions, allowing the solution of problems with various boundary conditions. Several solution methods were based on the variational formulations of the problems and, as such, the results presented in the present study are useful for the implementation of these methods.