Nanomechanical sensors based on elastically supported double-walled carbon nanotubes

Applied Mathematics and Computation 270 (2015) 216–241

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Nanomechanical sensors based on elastically supported double-walled carbon nanotubes Keivan Kiani∗ Department of Civil Engineering, K.N. Toosi University of Technology, P.O. Box 15875-4416, Valiasr Ave., Tehran, Iran

a r t i c l e

i n f o

Keywords: Nanomechanical sensor Double-walled carbon nanotubes (DWCNTs) Reproducing kernel particle method (RKPM) Nonlocal Rayleigh beam theory Nonlocal Timoshenko beam theory Nonlocal higher-order beam theory

a b s t r a c t Carbon nanotubes are anticipated to have potential applications in nanosensor technology; however, their usage under various boundary conditions has not been thoroughly revealed. In this article, we are seeking for appropriate numerical models to bridge such a scientific gap for double-walled carbon nanotubes (DWCNTs) as nanomechanical sensors. Some nonlocal beam models are developed for exploring the vibration performance of embedded DWCNTs in an elastic matrix due to the arbitrarily added nanoparticles. The nonlocal continuum theory of Eringen is employed, and the governing equations of each model are constructed by considering the lateral and rotary inertial effects of the attached nanoparticles. Since examining the problem for a wide range of boundary conditions is of particular interest, an effective meshless method is exploited. For the proposed numerical models, a comparison study along with a convergence check is carried out and reasonably good agreements are achieved. The key factor in mechanical performance of DWCNTs for sensing the nanoparticles is the alteration of their natural flexural frequencies. A fairly conclusive study is then conducted to determine the influences of the crucial factors on the frequency shift of DWCNTs. The obtained results explain the potential applications of DWCNTs as mass nanosensors for a diverse range of boundary conditions. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Thanks to the exceptional properties of carbon nanotubes (CNTs), numerous potential applications for them have been proposed and examined during recent years. In combination of physical and chemical properties, CNTs could be applicable in various branches of engineering sciences, such as transistors and emitters [1–3], thermal acoustic instruments [4], composite materials for wind turbine, bullet-proof vests, body armors, chemical/biochemical sensors, and strain and mechanical sensors [5–15]. From applied mechanics point of view, the latter application for a special class of CNTs, namely double-walled carbon nanotubes (DWCNTs), is of concern in this study. A nanosensor is a nanodevice that acts in response to a physical, chemical or biological parameter and converts its response into a signal or an output. The market drivers, engineering and scientific communities relevant to sensors point toward progress of nanosensors constructed from constituents for nanoscale sampling, pre-concentrating, and signal analysis. On account of the high ratio of the elasticity modulus to the mass density of CNTs, they have been increasingly paid attention to for the last abovementioned task. There exist evidences indicating that the fundamental frequencies of multi-walled carbon nanotubes

∗

Tel: +98 21 88779473; fax: +98 21 88779476. E-mail address: [email protected], [email protected]

http://dx.doi.org/10.1016/j.amc.2015.07.114 0096-3003/© 2015 Elsevier Inc. All rights reserved.

K. Kiani / Applied Mathematics and Computation 270 (2015) 216–241

217

(MWCNTs) are in the range of several gigahertz [16–18] or even terahertz [19–21]. This fact has brought CNTs as a new generation of oscillators. Based as mass nanosensors, with outstanding sensing of nano-sized objects, for an extensive range of applications. A DWCNT consists of two coaxial cylindrically rolled graphene sheets in the vicinity of each other. The atoms of two neighboring walls are interacted with each other due to van der Waals forces. Such attractive forces keep tightly two walls. When a nanoparticle becomes close to the outer surface of a DWCNT, it could be absorbed by its outermost tube due to the existing physical attraction forces. Therefore, the resonant frequencies of the DWCNT change according to the mass weights of the attached nanoparticles as well as the position(s) of the contact point(s). Additionally, some structural aspects and environmental factors associated with the DWCNT, such as ratio of length to diameter, ratio of diameter to the effective thickness of the tube wall, existing initially axial forces, stiffness of the surrounding environment, and boundary conditions of two ends of each tube are among verity parameters influencing on the alteration of the frequency change of the DWCNT due to addition of nanoparticles. For optimal design as well as characterization of vibration behavior of DWCNTs based nanomechanical sensor, a true understanding of the effects of the influential parameters on the free dynamic response of such nanosensors would be helpful. In this study, it is attempted to explore the influences of various factors on the change of fundamental frequency of the understudy nanostructure-based nanomechanical sensor. Before brief reviewing of the related articles, it is emphasized that understanding the true mechanism of the lateral vibration of DWCNTs is a preliminary step in understanding characterization of the dynamic analysis of them as nanomechanical sensors. Until now, free and forced vibrations of DWCNTs have been studied by many researchers [22–28] and an inclusive knowledge regarding such phenomena has been provided. Concerning frequency analysis of CNTs as mass nanosensors, Li and Chou [29] investigated vibration behavior of both single- and doubled-walled carbon nanotubes using the molecular-structural-mechanics method. The obtained results displayed that the predicted resonant frequencies of the atomistic-based model are averagely 50% higher than those of the classical continuum-based shell model. Elishakoff et al. [30] studied free dynamic response of a cantilevered DWCNT with an attached bacterium at the tip of the innermost or outermost tube as a biosensor. By considering both the lateral and rotational inertial effects of the attached bacteria, classical Euler–Bernoulli beam theory was adopted for the frequency shift of the nanosensor using the finite difference method. Using the transfer function method (TFM), Shen et al. [31] examined the vibration of DWCNTs with an attached mass as nanomechanical sensors employing the nonlocal Euler-Bernoulli beam model. The undertaken work was restricted to the fully clamped (i.e., bridged) DWCNT with an attached nanoparticle undergoing initial axial forces in both tubes. The effects of initial axial force as well as weight and location of the attached nanoparticle on the frequency shift of the DWCNT were addressed. In another work, Shen et al. [32] studied vibration behavior of DWCNT-based mass sensor via TFM when the mass was attached to the tip of the innermost tube. The lengths of the tubes were not the same, but both tubes were modeled according to the nonlocal Timoshenko beam model. As is seen in the literature, the mechanical aspects of slender and stocky DWCNTs with different boundary conditions as nanomechanical sensors have not been thoroughly realized. Possibly, it is related to the complexities that appear in developing analytical solutions for the governing equations. According to the literature, application of analytical approaches to the problem at hand is restricted to DWCNTs as nanomechanical sensors with simply supported ends or fully clamped boundary conditions. In the lack of analytical solutions, efficient numerical methods would be a good alternative. The reproducing kernel particle method (RKPM) is a renowned approach of meshless schemes’ family. This methodology was developed by Liu et al. [33,34] to treat shortcomings of the smooth particle hydrodynamics (SPH) method. The major feature of this method with respect to SPH is the incorporation of correction functions into the formulations of the shape functions. By this strategy, the performance and smoothness of RKPM in regions near to the boundaries of the problem are highly improved. In this newly developed method, the spatial domain of the problem is discretized by appropriate shape functions associated with the particles of the RKPM. Based on the completeness condition, the shape functions of the RKPM’s particles are numerically evaluated at the desired points according to the dilation parameter as well as the chosen window and base functions. Higher-order base functions as well as window functions yield higher-order shape functions. However, increasing the order of the base function can significantly magnify the cost of the shape functions’ generation; besides, the accuracy of the generated shape functions would lessen due to a large amount of calculations. The suitably higher-order shape functions provide RKPM as an effective approach for those problems suffer from discontinuity, sharp variation of fields, or governing equations with higher-order spatial derivatives. RKPM has been applied to many engineering problems such as structural dynamics, large deformation solids, fluid mechanics, microelectro-mechanical systems, single-walled carbon nanotubes, and nanowires [33,35–44]. Herein, we adopt RKPM to discretize the unknown fields pertinent to the proposed nonlocal models for mechanical modeling of DWCNTs as mass sensors. The potential application of DWCNTs as mass nanosensors is examined in the framework of nonlocal continuum theory of Eringen. In doing so, the nanostructure with the attached masses is modeled by using the nonlocal Rayleigh beam theory (NRBT), nonlocal Timoshenko beam theory (NTBT), and nonlocal higher-order beam theory (NHOBT). Both the translational and rotary inertial effects of the attached nanoparticles are incorporated into the governing equations of the proposed models. As will be shown, finding explicit solutions to the equations of motion of the elastically supported DWCNTbased mass sensor is a very difficult task. Therefore, RKPM is implemented to determine the unknown fields of each model. By solving the set of eigenvalue equations of each model, the natural frequencies of the elastically supported DWCNT with arbitrarily attached nanoparticles are determined. The most important feature of addition of nanoparticles to a DWCNT is the alteration of its natural frequencies. Hence, investigation of the influential factors on the change of the natural frequencies of the DWCNT would be vital for optimal design of them as nanosensors. A moderately comprehensive parametric study is then conducted to reveal the role of various factors on the frequency shift of the DWCNT due to the addition of nanoparticles. Through such studies,

218

K. Kiani / Applied Mathematics and Computation 270 (2015) 216–241 xpN

p

xpi

Kz (x=lb)

xp1

K (x=0) z

1

Mpi

M

1

p1

MpN

p

Cv/2

Ky (x=0)

Ky (x=lb) 1

x

1

C /2 v

y

Kt

Ky (x=0) 2

K (x=l ) y

Kr

2

b

K (x=l )

Kz (x=0)

z

2

2

b

l

b

z Fig. 1. Schematic representation of an elastically supported DWCNT with attached nanoparticles as a nanomechanical sensor.

the capabilities of the proposed nonlocal models in predicting the frequency shift of DWCNTs are explored for a wide range of boundary conditions. 2. Details of a DWCNT as a nanomechanical sensor Consider an equivalent continuum structure (ECS) associated with the DWCNT as shown in Fig. 1. The undeformed ECS comprises of two coaxial cylindrical tubes of mean radii rmi , walls’ thickness tb , length lb , cross-sectional areas Abi , elasticity modulus Ebi , and elasticity shear modulus Gbi ; i = 1, 2. The innermost and outermost tubes interact with each other due to the vdW forces between their atoms. Such interaction forces between two adjacent tubes are commonly modeled by a continuous spring system of constant Cv [26] (see Fig. 1). Assume that Np nanoparticles of mass M pi and mass moment of inertia I pi have been attached to the outermost tube of the DWCNT at the locations denoted by x pi ; i = 1, . . . , N p . The main effect of addition of such masses is the change of the natural frequency or the frequency shift of the DWCNT. Herein, dynamic interactions between the added nanoparticles and the DWCNTs have been ignored in the formulations of the proposed models. In fact, it is assumed that all nanoparticles have been tightly attached to the nanostructure. However, for a more realistic modeling of a system of nanoparticles and the DWCNT, considering the existing vdW forces between the nanoparticles and the nanotubes would lead to more accurate results. Role of the vdW in sensing of nanoparticles by DWCNTs can be considered as an important subject for future works. For examining a wide range of boundary conditions, the ends of the innermost and outermost tubes have been connected to the transverse and rotary springs with constants Kzi and Kyi ; i = 1, 2, respectively. The interaction of the DWCNT with its surrounding matrix is modeled by continuous transverse and rotary springs with constants Kt and Kr , respectively (see Fig. 1). The initially axial forces within the tubes are represented by Nbi . The main objective of this work is to determine the amount of the frequency shift of the DWCNT under various boundary conditions due to the addition of arbitrarily distributed nanoparticles. 3. An introduction to one-dimensional RKPM In this part, a brief introduction to the one-dimensional RKPM is given. The construction of RKPM shape functions as well as their first, second, and third derivatives are explained. Subsequently, implementation of RKPM for solving the partial differential equations pertinent to the problem at hand based on NRBT, NTBT, and NHOBT is provided in upcoming parts. According to the works of Liu and his coworkers [33,34,38], a continuous approximation of an arbitrarily one-dimensional field u(x), could be expressed by

ua (x) =





φa∗ (x; x − s) u(s) ds,

(1)

where  represents the one-dimensional spatial domain, and φa∗ (x; x − s) denotes the modified kernel function which is expressed as,

φa∗ (x; x − s) = φa (x − s) c(x; x − s); φa (x − s) =

1 a

φ

x − s a

,

(2)

where c(x; x − s) is the correction function, φa (x − s) is the kernel or window function which is characterized by the choice of both the dilation parameter, a, and the function φ . The dilation parameter controls the support domain of the kernel function. The main task of the correction function is to reduce the difficulties raised from finite domain effect. Thereby, the generated errors throughout the computational domain would be reduced, particularly in those regions close to the boundaries. The correction function is commonly stated as:

c(x; x − s) =

N  i=0

bi (x)(x − s)i = HT (x − s)b(x),

(3)

K. Kiani / Applied Mathematics and Computation 270 (2015) 216–241

219

where HT is the base function, and b contains unknown coefficients,

HT = [1, x, x2 , . . . , xN ], bT (x) = [b0 (x), b1 (x), . . . , bN (x)],

(4)

in view of Eq. (2), by substituting Eq. (3) in to Eq. (1), one can obtain

ua (x) =

∞ N   (−1)n bk (x) mn+k (x) u(n) (x), n!

(5)

k=0 n=0

where u(0) (x) = u(x), u(n) = ∂∂ xnu ; n > 1, and mn (x) represents the moment function of order n, which is defined by, n

mn (x) =



(x − s)n φa (x − s) ds.



(6)

In order to satisfy the N-th order completeness condition for ua (x), N 

bk (x)mk (x) = 1,

k=0 N 

bk (x)mn+k (x) = 0; n ≥ 1,

(7)

k=0

through solving the set of linear equations in Eq. (7) for bk (x),

b(x) = M−1 (x) H(0),

(8)

where M, the moment matrix, is as,

⎛ m (x) 0 ⎜ m1 (x) ⎜ M(x) = ⎜ . ⎝ .. mN (x)

m1 (x)

...

m2 (x)

...

.. .

..

MN+1 (x)

...

.

⎞

mN (x)

mN+1 (x) ⎟

⎟ ⎟, ⎠

.. .

(9)

m2N (x)

by substituting Eq. (9) into Eq. (3), and introducing Eqs. (2) and (3) to Eq. (1), the modified kernel function of RKPM is obtained as,

φa∗ (x; x − s) = HT (x − s) M−1 (x) H(0) φa (x − s).

(10)

As is seen in Eq. (1), the approximate function is expressed continuously in terms of modified function. For the sake of numerical analysis, the discretized form of the approximate function is required. By using the trapezoidal rule, ua (x) can be rewritten as,

ua (x) =

NP 

φI (x) uI ,

(11)

I=1

where NP is the total number of RKPM’s particles, uI is the nodal parameter value of the Ith RKPM’s particle, and φ I (x) is the shape function of the Ith particle,

φI (x) = φ ∗ (x; x − xI )xI ,

(12)

where xI is the length of the one-dimensional subdomain associated with the I-th particle. For example, in the case of unilb lb , and xi = NP ; 2 ≤ i ≤ NP − 1. The first, second, and third formly distributed particles in a domain of length lb , x1 = xN = 2NP derivatives of the RKPM’s shape functions are evaluated as,

⎛

HT (x − s)M−1 (x)φa,x (x; x − xI )

⎞

φI,x (x) = ⎝ +HT,x (x − s)M−1 (x)φa (x; x − xI )⎠H(0) xI , +HT (x − s)M−1 ,x (x)φa (x; x − xI ) ⎞ ⎛ T H (x − s)M−1 (x)φa,xx (x; x − xI ) + HT,xx (x − s)M−1 (x)φa (x; x − xI ) T −1 ⎠H(0)xI , φI,xx (x) = ⎝ +HT (x − s)M−1 ,xx (x)φa (x; x − xI ) + 2H,x (x − s)M (x)φa,x (x; x − xI ) T −1 T −1 +2H,x (x − s)M,x (x)φa (x; x − xI ) + 2H (x − s)M,x (x)φa,x (x; x − xI )

(13a)

(13b)

220

K. Kiani / Applied Mathematics and Computation 270 (2015) 216–241

⎞ (x; x − xI ) + HT,xxx (x − s)M−1 (x)φa (x; x − xI ) T −1 ⎟ +HT (x − s)M−1 ,xxx (x)φa (x; x − xI ) + 3H,x (x − s)M (x)φa,xx (x; x − xI ) ⎟ T −1 T −1 +3H (x − s)M,x (x)φa,xx (x; x − xI ) + 3H,xx (x − s)M (x)φa,x (x; x − xI )⎟ ⎟H(0)xI . +3HT (x − s)M−1 (x)φa (x; x − xI ) + 3HT (x − s)M−1 (x)φa,x (x; x − xI ) ⎠

⎛HT (x − s)M−1 (x)φ

⎜ ⎜ φI,xxx (x) = ⎜ ⎜ ⎝

,xx

a,xxx

,x

(13c)

,x

T −1 +3HT,x (x − s)M−1 ,xx (x)φa (x; x − xI ) + 6H,x (x − s)M,x (x)φa,x (x; x − xI )

−1 −1 Using the relation M(x) M−1 (x) = I, where I is the identity matrix, M−1 ,x (x), M,xx (x), and M,xxx (x) are calculated as follows: −1 M−1 M,x M−1 , ,x = −M −1 M−1 M,xx M−1 + 2M−1 M,x M−1 M,x M−1 , ,xx = −M −1 −1 M−1 + 3M−1 M,xx M−1 M,x M−1 + 2M−1 M,xx M−1 M,xx M−1 − 4M−1 M,xx M−1 M,x M−1 M,x M−1 ,xxx = −M M,xxx M

+ M−1 M,x M−1 M,xx M−1 − 2M−1 M,x M−1 M,x M−1 M,x M−1 ,

(14)

by using the discretized form of Eq. (6), the first, second, and third derivatives of the elements of the moment matrix could be readily calculated as,

mn (x) =

NP 

(x − xi )n φa (x − xi ) xi ,

(15a)

i=1

NP

φa,x (x − xi ) xi ; n=0 , n−1 ((x − xi )φa,x (x − xi ) + nφa (x − xi )) xi ; n ≥ 1 i=1 (x − xi ) ⎧ NP n=0 i=1 φa,xx (x − xi ) xi ; ⎪ ⎪ ⎨ NP (( x − x )φ ( x; x − x ) + 2 φ ( x; x − x )) x ; n =1 a,xx a,x i i i i , mn,xx (x) = i=1 n NP n−1 ⎪ x − xi ) + 2n(x − xi ) φa,x (x; x − xi ) ⎪ ⎩ i=1 ((x − xi ) φa,xx (x; n−2 + n(n − 1)(x − xi ) φa (x − xi )) xi ; n≥2 ⎧ NP n=0 ⎪ i=1 φa,xxx (x − xi ) xi ; ⎪ ⎪

NP ⎪ ⎪ (( x − x )φ ( x; x − x ) + 3 φ ( x; x − x )) x ; n =1 a,xxx a,xx i i i i i=1 ⎪ ⎪

NP ⎪ 2 ⎪ ⎨ i=1 ((x − xi ) φa,xxx (x; x − xi ) + n(n + 1)φa,x (x; x − xi ) + 2(n + 1)(x − xi )φa,xx (x; x − xi )) xi ; n = 2. mn,xxx (x) = ⎪

NP n n−1 ⎪ ⎪ φa,xx (x; x − xi ) ⎪ i=1 ((x − xi ) φa,xxx (x; x − xi ) + 3n(x − xi ) ⎪ n−2 ⎪ ⎪ + 3n ( n − 1 )( x − x ) φ ( x; x − x ) a,x i i ⎪ ⎪ ⎩ n−3 + n(n − 1)(n − 2)(x − xi ) φa (x − xi )) xi ; n≥3 mn,x (x) =

i=1

NP

(15b)

(15c)

(15d)

In this article, the linear base function, HT (x) = [1, x], and the exponential window function, φa (x) = exp (−( αx ) ); α = 0.3; a = 3.2, are employed for construction of shape functions of RKPM. As will be seen, the obtained nonlocal equations of motion of the DWCNT include higher-order derivatives of deflection field. As a result, application of RKPM shape functions would be more beneficial with respect to the finite element method due to the employment of shape functions with higher-order continuity. 2

4. Modeling of DWCNT-based mass sensors employing the NRBT 4.1. Nonlocal equations of motion The kinetic energy, TR (t), and elastic strain energy, UR (t), of the DWCNT with attached nanoparticles based on the NRBT are stated as follows

⎛





2



⎞

2 

R R 2  1  lb ρbi Abi w˙ i (x, t ) + Ibi w˙ i,x (x, t ) R  ⎝ T (t ) =  i N p 2 0 + 1−(−1) M w˙ R (x, t )δ(x − x

2 2 ⎠dx,  R ˙ ) + I ( x, t )δ( x − x ) w p p j j 2 i i,x j=1  nl R 2 ⎞  R ⎛ R R −wi,xx (x, t ) Mb (x, t ) + (i − 1)Cv w2 (x, t ) − w1 (x, t ) i   2    ⎟  1  lb ⎜ ⎜+Nbi (wRi,x (x, t ))2 + (i − 1) Kt wRi (x, t ) 2 + Kr wRi,x (x, t ) 2 ⎟dx, U R (t ) = ⎝ ⎠ 2  0  2  2  i=1

δ(x − xk ) + 2k=1 Kzi (xk ) wRi (x, t ) + Kyi (xk ) wRi,x (x, t ) pj

i=1

(16a)

pj

(16b)

where wR1 /wR2 and (Mbnl )R /(Mbnl )R respectively represent the lateral displacement and the nonlocal bending moment of the in1

2

nermost/outermost tube, and δ denotes the one-dimensional Dirac delta function. According to the nonlocal continuum theory

K. Kiani / Applied Mathematics and Computation 270 (2015) 216–241

221

of Eringen [45,46], the nonlocal bending moments within the innermost and outermost tubes modeled on the basis of the NRBT are given by



Mbnli

R



− (e0 a)2 Mbnli

R

,xx

= −Ebi Ibi wRi,xx ; i = 1, 2,

(17)

where a is an internal characteristic length. The value of e0 is determined by adjusting the dispersion curves of the nonlocal model with those of an appropriate atomic model. The parameter e0 a is called small-scale parameter. Using Hamilton’s princit ple, 0 (δ T R − δU R ) dt = 0, the equation of motion describing free vibration of a DWCNT-based mass resonator which is embedded in an elastic matrix, is obtained as Np       R ρb1 Ab1 w¨ R1 − Ib1 w¨ R1,xx + M pi w¨ R1 − Ipi w¨ R1,xx δ(x − x pi ) − Mbnl1 + Cv (wR1 − wR2 ) ,xx i=1



− Nb1 wR1,x

 ,x

+

2  

Kz1 (xk )wR1 − Ky1 (xk )wR1,xx



δ(x − xk ) = 0,

(18a)

k=1

   R   ρb2 Ab2 w¨ R2 − Ib2 w¨ R2,xx − Mbnl2 ,xx − Cv (wR1 − wR2 ) − Nb2 wR2,x ,x + Kt wR2 − Kr wR2,xx +

2  

Kz2 (xk )wR2 − Ky2 (xk )wR2,xx



δ(x − xk ) = 0,

(18b)

k=1

by combining Eq. (17) with Eqs. (18) and (18b), the nonlocal equations of motion for transverse vibration of a DWCNT with attached nanoparticles as a function of deflection fields of the innermost and outermost tubes are obtained as in the following form Np  

M pi w¨ R1 − Ipi w¨ R1,xx

i=1



δ(x − x pi ) − (e0 a)2





Np  







M pi w¨ R1 δ(x − x pi )

i=1

+ ρb1 Ab1 w¨ R1 − (e0 a)2 w¨ R1,xx − ρb1 Ib1 w¨ R1,xx − (e0 a)2 w¨ R1,xxxx

,xx



+ Cv wR1 − wR2 − (e0 a)2 (wR1,xx − wR2,xx ) + Eb1 Ib1 wR1,xxxx − Nb1 w¨ R1,xx − (e0 a)2 w¨ R1,xxxx +

2  



,xx







 



− Ipi w¨ R1,xx δ(x − x pi )



 



Kz1 (xk ) wR1 δ(x − xk ) − (e0 a)2 wR1 δ(x − xk )

,xx

k=1

    − Ky1 (xk ) wR1,xx δ(x − xk ) − (e0 a)2 wR1,xx δ(x − xk ) = 0, ,xx

(19a)

      ρb2 Ab2 w¨ R2 − (e0 a)2 w¨ R2,xx − ρb2 Ib2 w¨ R2,xx − (e0 a)2 w¨ R2,xxxx − Cv wR1 − wR2 − (e0 a)2 (wR1,xx − wR2,xx )       + Eb2 Ib2 wR2,xxxx − Nb2 w¨ R2,xx − (e0 a)2 w¨ R2,xxxx + Kt wR2 − (e0 a)2 wR2,xx − Kr wR2,xx − (e0 a)2 wR2,xxxx  2      + Kz2 (xk ) wR2 δ(x − xk ) − (e0 a)2 wR2 δ(x − xk ) ,xx k=1



 



− Ky2 (xk ) wR2,xx δ(x − xk ) − (e0 a)2 wR2,xx δ(x − xk )

,xx

= 0.

(19b)

For more generalization, the following dimensionless parameters are introduced

ξ

x 1 = ,τ = 2 lb lb



Eb1 Ib1

ρb 1 A b 1

t,

(20)

where ξ is the dimensionless coordinate, wRi is the dimensionless deflection field of the ith nanotube, τ is the dimensionless time parameter, μ is the dimensionless small-scale parameter, rb1 denotes the gyration radius of the innermost tube, and λ1 is the slenderness ratio of the innermost tube. Therefore, the non-dimensional equations of motion of the DWCNT-based mass sensor, introduced by Eqs. (19a) and (19b), are derived as follows





w1,τ τ − μ2 w1,τ τ ξ ξ − λ−2 w1,τ τ ξ ξ − μ2 w1,τ τ ξ ξ ξ ξ + 1 R

R

R

R

Np  i=1

Np

−



λ−2 pi





w1,τ τ δ(ξ − ξ pi ) R

i=1

R

+ C v w1 − w2 − μ2 (w1,ξ ξ R

R

R



,ξ ξ



R





− μ2 w1,τ τ δ(ξ − ξ pi ) R



 

m pi w1,τ τ δ(ξ − ξ pi ) − μ2 w1,τ τ δ(ξ − ξ pi ) ,ξ ξ

,ξ ξ ξ ξ

  R  R R R R − w2,ξ ξ ) + w1,ξ ξ ξ ξ − Nb1 w1,ξ ξ − μ2 w1,ξ ξ ξ ξ

R

222

K. Kiani / Applied Mathematics and Computation 270 (2015) 216–241

+

2  



R



 

K z1 (ξk ) w1 δ(ξ − ξk ) − μ2 w1 δ(ξ − ξk ) R

R

,ξ ξ

k=1

    R − K y1 (ξk ) wR1,ξ ξ δ(ξ − ξk ) − μ2 wR1,ξ ξ δ(ξ − ξk ) = 0, ,ξ ξ      R R R R R R R 12 wR2,τ τ − μ2 wR2,τ τ ξ ξ − 22 λ−2 w2,τ τ ξ ξ − μ2 w2,τ τ ξ ξ ξ ξ − C v w1 − w2 − μ2 (w1,ξ ξ − w2,ξ ξ ) 1    R  R R R R R R R R R + 32 w2,ξ ξ ξ ξ − Nb2 w2,ξ ξ − μ2 w2,ξ ξ ξ ξ + K t w2 − μ2 w2,ξ ξ − K r w2,ξ ξ − μ2 w2,ξ ξ ξ ξ  2      R R R + K z2 (ξk ) w2 δ(ξ − ξk ) − μ2 w2 δ(ξ − ξk ) ,ξ ξ k=1



R

 



− K y2 (ξk ) wR2,ξ ξ δ(ξ − ξk ) − μ2 wR2,ξ ξ δ(ξ − ξk )

,ξ ξ

= 0,

(21a)

(21b)

where

e0 a , lb

μ=

λ1 =

Cv lb4

R

Cv =

Eb1 Ib1

R

xp j

R

,

N bi =

Kzi (xk )lb3

K zi (ξk ) =

ξp j =

lb , r b1

Eb1 Ib1

R

wi =

Nbi lb2

,

Eb1 Ib1

12 =

mp j =

ρb1 Ab1 lb

Kyi (xk )lb , Eb1 Ib1

R

K yi (ξk ) =

,

ρb 2 A b 2 ρb Ib Eb Ib , 22 = 2 2 , 32 = 2 2 , ρb 1 A b 1 ρb1 Ib1 Eb1 Ib1  Ip j Mp j l , λp j = b , rp j = ,

wRi , lb

rp j

R

Kt =

Kt lb4 Eb1 Ib1

,

Mp j

R

Kr =

Kr lb2 Eb1 Ib1

,

; i = 1, 2; j = 1, . . . , Np .

lb

(22)

Finding an analytical solution to Eqs. (21a) and (21b) is a problematic task; the analytical solutions to these equations are only available for special boundary conditions and environmental circumferences [30,31]. In the next part, an efficient numerical scheme is proposed for solving these equations. 4.2. Implementation of RKPM Both sides of Eqs. (21a) and (21b) are, respectively, multiplied by δ wR1 and δ wR2 and the sum of the resulting equations is integrated over the normalized length of the DWCNT. After taking successful integration by parts, the following expression is obtained 2  

1



0

i=1

Np  R  R  R R  1 − (−1)i   R m p j δ wi − μ2 δ wi,ξ ξ wi,τ τ + λ−2 δ wi,ξ wi,ξ τ τ + μ2 δ wRi,ξ ξ wRi,ξ ξ τ τ δ(ξ − ξ p j ) pi 2 j=1

 δ wRi,ξ wRi,ξ τ τ + μ2 δ wRi,ξ ξ wRi,ξ ξ τ τ  R   R R  R R R R R R R R R + (−1)i+1C v δ wi − μ2 δ wi,ξ ξ w1 − w2 + 32i−2 δ wi,ξ ξ wi,ξ ξ + Nbi δ wi,ξ wi,ξ + μ2 δ wi,ξ ξ wi,ξ ξ 2    R  R  R  R R R R + K zi (ξk ) δ wi − μ2 δ wi,ξ ξ wi + K yi (ξk ) δ wi,ξ wRi,ξ + μ2 δ wi,ξ ξ wRi,ξ ξ δ(ξ − ξk ) +

2i−2 1



δ

R wi



2i−2 − μ2 δ wi,ξ ξ wi,τ τ + λ−2 1 2 R

R



k=1

  R  R  R R R R R R R 2 2 + (i − 1) K t δ wi − μ δ wi,ξ ξ wi + K r δ wi,ξ wi,ξ + μ δ wi,ξ ξ wi,ξ ξ dξ = 0,

(23)

the unknown fields associated with the elastically supported DWCNT as a mass sensor are discretized in the spatial domain as

wi (ξ , τ ) = R

NPi 

φIwi (ξ )wRiI (τ ); i = 1, 2,

(24)

I=1

w

w

where NP1 /NP2 , φI 1 (ξ )/φI 2 (ξ ), and wR1I (τ )/wR2I (τ ) represent the number of RKPM particles, the RKPM shape function associated with the Ith particle, and the nodal parameter value of the Ith particle of the innermost/outermost tubes, respectively. By introducing Eq. (24) to Eq. (23), the discrete form of the equations of motion are expressed as



R w1 w1

R w1 w2

[Mb ]

[Mb ]

R w2 w1 [Mb ]

R w2 w2 [Mb ]



R

w1,τ τ R w2,τ τ





+

R w1 w1

R w1 w2

[Kb ]

[Kb ]

R w2 w1 [Kb ]

R w2 w2 [Kb ]



R

w1

R w2





=

0 0

,

(25)

where the nonzero submatrices associated with the mass and stiffness matrices of the DWCNT-based mass sensor would be R wi wi

[Mb ]IJ



=

1 0



    12i−2 φIwi − μ2 φI,wξi ξ φJwi + λ−2 22i−2 φI,wξi φJ,wξi + μ2 φI,wξi ξ φJ,wξi ξ dξ

K. Kiani / Applied Mathematics and Computation 270 (2015) 216–241

R

i wi [Kb ]w IJ

R wi w j

[Kb ]IJ

223

 ⎞ wi wi 2 wi Np m (ξ ) − μ φ (ξ ) φ (ξ ) φ i  p p p p j j J 1 − (−1) I,ξ ξ ⎠, ⎝ j I j + wi wi 2 −2 2 wi +λ p j φI,ξ (ξ p j )φJ,ξ (ξ p j ) + μ φI,ξ ξ (ξ p j )φJ,wξi ξ (ξ p j ) j=1 ⎛     ⎞  1 2i−2 φ wi φ wi + NR φ wi φ wi + μ2 φ wi φ wi + C R φ wi − μ2 φ wi φ wi v bi I I,ξ ξ J,ξ ξ I,ξ J,ξ I,ξ ξ J,ξ ξ I,ξ ξ    J ⎠dξ ⎝ 3 = R R wi wi wi wi wi 2 wi 2 wi 0 + (i − 1)K t φI − μ φI,ξ ξ φJ + (i − 1)K r φI,ξ φJ,ξ + μ φI,ξ ξ φJ,ξ ξ ⎛ R ⎞   2 K zi (ξk ) φIwi (ξk ) − μ2 φI,wξi ξ (ξk ) φJwi (ξk )  ⎝  ⎠, + R +K yi (ξk ) φI,wξi (ξk )φJ,wξi (ξk ) + μ2 φI,wξi ξ (ξk )φJ,wξi ξ (ξk ) k=1  1   R w =− C v φIwi − μ2 φI,wξi ξ φJ j dξ ; i = j, ⎛



(26a)

(26b)

(26c)

0

where i, j = 1, 2. For calculating the flexural frequencies of the DWCNT-based mass resonator, it is presumed wi (τ ) = R R R R wi0 ei τ ; i = 1, 2 where w10 and w20 are the initial nodal parameter values of the innermost and outermost tubes, respecR

tively, and ϖR denotes the dimensionless flexural frequency of the DWCNT with attached nanoparticles in which embedded in R an elastic matrix under initially axial force. By substituting these new forms of wi into Eq. (25),





−(

)

R 2

R w1 w1

R w1 w2

[Mb ]

[Mb ]

R w2 w1 [Mb ]

R w2 w2 [Mb ]





+

R w1 w1

R w1 w2

[Kb ]

[Kb ]

R w2 w1 [Kb ]

R w2 w2 [Kb ]

 

R

w10 R w20





=

0 0

,

(27)

by solving the set of eigenvalue equations in Eq. (27), the eigenvalues (i.e., dimensionless flexural frequencies) and the corresponding eigenvectors (i.e., vibration modes) of the DWCNT-based mass resonator in which embedded in an elastic matrix under initially axial force are obtained according to the NRBT. 5. Modeling of DWCNT-based mass sensors employing the NTBT In this part, the nonlocal equations of motion of transverse vibrations of DWCNT for sensing arbitrarily attached nanoparticles would be developed on the basis of the Timoshenko beam theory [47,48]. In contrast to the proposed model based on the NRBT, herein, the effect of shear deformation of the nanostructure is also included. Subsequently, RKPM would be employed for solving the resulted governing equations. 5.1. Nonlocal equations of motion In the context of the NTBT, the kinetic energy, TT (t), and the elastic strain energy, UT (t), of a DWCNT with attached nanoparticles in which embedded in an elastic matrix are given by

⎞ 2 2   + Abi w˙ Ti (x, t )   2 2 ⎠dx,  T T ˙ (x, t )δ(x − x p ) ˙ M ( x, t )δ( x − x ) + I θ w p p p i=1 j j j j 2 i i j=1  nl T  nl T 2 ⎞  T ⎛ T T T −θi,x (x, t ) Mb (x, t ) + (wi,x (x, t ) − θi (x, t )) Qb (x, t ) + Nbi wi,x (x, t ) i i   2       ⎟ 1  lb ⎜ ⎜+(i − 1) Cv wT2 (x, t ) − wT1 (x, t ) 2 + Kt wTi (x, t ) 2 + Kr θiT (x, t ) 2 ⎟dx, U T (t ) = ⎝ ⎠ 2   0     i=1

2 2 + 2k=1 Kzi (xk ) wTi (x, t )δ(x − xk ) + Kyi (xk ) θi (x, t )T δ(x − xk ) ⎛

 

˙T 2  1  lb ρbi Ibi θi (x, t ) ⎝ T T (t ) = i N p 2 0 + 1−(−1)

(28)

where wTi , θiT , (Qbnl )T , and (Mbnl )T denote the deflection field, deformation angle field, nonlocal shear force, and nonlocal bendi

i

ing moment of the ith nanotube based on the hypotheses of the NTBT. On the basis of the nonlocal continuum theory of Eringen [45,46], the nonlocal resultant shear force and the nonlocal bending moment within the ith tube are expressed by

 

Qbnli

T

Mbnli

T



− (e0 a)2 Qbnli



T

− (e0 a)2 Mbnli

,xx

T

,xx

= ksi Gbi Abi (wTi,x − θiT ), T = −Ebi Ibi θi,x ,

(29)

where ksi and Gbi denote the shear correction factor and the elasticity shear modulus of the ith nanotube, respectively, and t Gbi = Ebi /(2(1 + νi )) where ν i is Poisson’s ratio of the ith tube. By employing Hamilton’s principle, 0 (δ T T − δU T ) dt = 0, the strong form of the governing equations of an elastically supported DWCNT as a mass sensor in which embedded in an elastic matrix under initially axial force, is obtained as follows

ρb1 Ib1 θ¨1T +

Np  i=1



Ipi θ¨1T δ(x − x pi ) − Qbnl1

T



+ Mbnl1

T ,x

+

2  k=1

Ky1 (xk )θ2T δ(x − xk ) = 0,

(30a)

224

K. Kiani / Applied Mathematics and Computation 270 (2015) 216–241

Np 2  T       ρb1 Ab1 w¨ T1 − Qbnl1 ,x + M pi w¨ T1 δ(x − x pi ) − Nb1 wT2,x + Cv wT1 − wT2 + Kz1 (xk )wT2 δ(x − xk ) = 0, ,x i=1

(30b)

k=1

2    T  T ρb2 Ib2 θ¨2T − Qbnl2 + Mbnl2 ,x + Kr θ T + Ky2 (xk )θ2T δ(x − xk ) = 0,

(30c)

k=1

2   T     ρb2 Ab2 w¨ T2 − Qbnl2 ,x − Nb2 wT2,x ,x − Cv wT1 − wT2 + Kt wT + Kz2 (xk )wT2 δ(x − xk ) = 0,

(30d)

k=1

by combining Eqs. (29 a) and (29 b) with Eqs. (30a)–(30d), the nonlocal equations of motion of an elastically supported DWCNTbased mass resonator embedded in an elastic matrix in terms of displacements and deflection angles of the innermost and outermost tubes are derived as Np        T ρb1 Ib1 θ¨1T − (e0 a)2 θ¨1,xx + Ipi θ¨1T δ(x − x pi ) − (e0 a)2 θ¨1T δ(x − x pi ) ,xx i=1



 T

T − ks1 Gb1 Ab1 wT1,x − θ1 − Eb1 Ib1 θ1,xx +

2 

 Ky1

k=1

   θ1T δ(x − xk ) − (e0 a)2 θ1T δ(x − xk ) ,xx = 0,

(31a)

Np        ρb1 Ab1 w¨ T1 − (e0 a)2 w¨ T1,xx + M pi w¨ T1 δ(x − x pi ) − (e0 a)2 w¨ T1 δ(x − x pi ) ,xx i=1







T − ks1 Gb1 Ab1 wT1,xx − θ1,x − Nb1 wT1,xx − (e0 a)2 wT1,xxxx





+ Cv wT1 − wT2 − (e0 a)2 (wT1,xx − wT2,xx ) +

2 





 



Kz1 wT1 δ(x − xk ) − (e0 a)2 wT1 δ(x − xk )

k=1

,xx

= 0,

    T T ρb2 Ib2 θ¨2T − (e0 a)2 θ¨2,xx − ks2 Gb2 Ab2 wT2,x − θ2T − Eb2 Ib2 θ2,xx  2       T + Kr θ2T − (e0 a)2 θ,xx + Ky2 θ2T δ(x − xk ) − (e0 a)2 θ2T δ(x − xk ) = 0, ,xx

(31b)

(31c)

k=1

      T ρb2 Ab2 w¨ T2 − (e0 a)2 w¨ T2,xx − ks2 Gb2 Ab2 wT2,xx − θ2,x − Nb2 wT2,xx − (e0 a)2 wT2,xxxx     + Kt wT2 − (e0 a)2 wT2,xx − Cv wT1 − wT2 − (e0 a)2 (wT1,xx − wT2,xx )  2     + Kz2 wT2 δ(x − xk ) − (e0 a)2 wT2 δ(x − xk ) = 0. ,xx

(31d)

k=1

In order to study the problem in a more general context, the following dimensionless quantities are defined,

τ

1 = lb



ks1 Gb1

ρb 1

t,

η=

Eb1 Ib1 ks1 Gb1 Ab1 lb2

,

(32)

hence, the dimensionless nonlocal equations of motion of DWCNT-based mass sensors are derived as Np  T    T  T T T 2 T −2 2 T λ−2 − μ θ + λ − μ θ θ θ pi 1,τ τ 1,τ τ ξ ξ 1,τ τ 1,τ τ ξ ξ δ(ξ − ξ pi ) − w1,ξ + θ 1 − η θ 1,ξ ξ 1 i=1

+

2 

T K y1

  T   T 2 (ξk ) θ 1 δ(ξ − ξk ) − μ θ 1 δ(ξ − ξk ) = 0,

w1,τ τ − μ2 w1,τ τ ξ ξ + T

T

Np 



T



T



m pi w1,τ τ − μ2 w1,τ τ ξ ξ T

i=1 T





− Nb1 w1,ξ ξ − (e0 a)2 w1,ξ ξ ξ ξ + T

(33a)

,ξ ξ

k=1

T

2 

  T T  δ(ξ − ξ pi ) − wT1,ξ ξ + θ 1,ξ + C v wT1 − wT2 − μ2 wT1,ξ ξ − wT2,ξ ξ 

T

k=1

T

 T     T  T T T 2 2 T 2 2 T 2 T λ−2 1 2 θ 2,τ τ − μ θ 2,τ τ ξ ξ − 4 w2,ξ − θ 2 − η3 θ 2,ξ ξ + K r θ 2 − μ θ 2,ξ ξ   T   2  T T + K y2 (ξk ) θ 2 δ(ξ − ξk ) − μ2 θ 2 δ(ξ − ξk ) = 0, k=1

 

K z1 (ξk ) w1 δ(ξ − ξk ) − μ2 w1 δ(ξ − ξk )

,ξ ξ

,ξ ξ

= 0,

(33b)

(33c)

K. Kiani / Applied Mathematics and Computation 270 (2015) 216–241

225

  T      T T  12 wT2,τ τ − μ2 wT2,τ τ ξ ξ − 42 wT2,ξ ξ − θ 2,ξ − C v wT1 − wT2 − μ2 wT1,ξ ξ − wT2,ξ ξ − Nb2 wT2,ξ ξ − (e0 a)2 wT2,ξ ξ ξ ξ T





+ K t w2 − μ2 w2,ξ ξ + T

T

2 



T



 

K z2 (ξk ) w2 δ(ξ − ξk ) − μ2 w2 δ(ξ − ξk ) T

T

k=1

,ξ ξ

= 0,

(33d)

where T

wi =

wTi , lb

T

θ i = θiT , 42 =

Kzi (xk )lb , ks1 Gb1 Ab1

T

K zi (ξk ) =

T

Kr =

ks2 Gb2 Ab2 , ks1 Gb1 Ab1

Kr , ks1 Gb1 Ab1

Cv lb2

T

Cv =

ks1 Gb1 Ab1 Kt lb2

T

Kt =

ks1 Gb1 Ab1

T

K yi (ξk ) =

,

T

,

N bi =

Kyi (xk ) , ks Gb Ab lb

Nbi ; i = 1, 2. ks1 Gb1 Ab1

(34)

Generally, finding an analytical solution to Eqs. (33a)–(33d) is a very difficult job, except for some special cases [32]. In the following part, application of RKPM for spatial discretization of the unknown fields in Eqs. (33a)–(33d) is explained in some detail. 5.2. Implementation of RKPM T

T

Both sides of Eqs. (33a), (33b), (33c), and (33d) are, respectively, multiplied by δθ 1 , δ wT1 , δθ 2 , and δ wT2 . After summing up each side of the resulting relations and taking the necessary integration by parts, 2   i=1

1



0

Np   T T   T  T T T 1 − (−1)i  T 2 δθ δ(ξ − ξ p j ) θ + μ δθ θ m p j δ wi − μ2 δ wi,ξ ξ wi,τ τ + λ−2 pi i i,τ τ i,ξ i,ξ τ τ 2



2i−2 2



j=1

     T T T T T δθ θ i,τ τ + μ2 δθ i,ξ θ i,τ τ ξ − 42i−2 δθ i wTi,ξ − θ i + 12i−2 δ wTi wTi,τ τ + μ2 δ wTi,ξ wTi,ξ τ τ    T   T T T T T T T T T T T T + 42i−2 δ wi,ξ wi,ξ − θ i + (−1)i+1C v δ wi − μ2 δ wi,ξ ξ w1 − w2 + Nbi δ wi,ξ wi,ξ + μ2 δ wi,ξ ξ wi,ξ ξ  T  T  T T T T T T T T + η32i−2 δθ i,ξ θ i,ξ + (i − 1)K r δθ i − μ2 δθ i,ξ ξ θ i + (i − 1)K t δ wi − μ2 δ wi,ξ ξ wi   T T  2    T  T T T T T T 2 2 + K zi (ξk ) δ wi − μ δ wi,ξ ξ wi + K yi (ξk ) δθ i θ i + μ δθ i,ξ θ i,ξ δ(ξ − ξk ) dξ = 0, +λ

−2 1

T i

(35)

k=1

the unknown dimensionless fields associated with the DWCNT-based mass resonator in which modeled according to the NTBT, are expressed in terms of RKPM shape functions as follows T

wi

NPi 

(ξ , τ ) =

T

φIwi (ξ )wTiI (τ ), θ i (ξ , τ ) =

I=1

NPi  T φ θi (ξ )θ (τ ); i = 1, 2 iI

I

(36)

I=1

by substituting Eq. (36) into Eq. (35), the following set of ordinary differential equations (ODEs) is obtained

⎡

T θ1 θ1

T θ1 w 1

[Mb ]

T θ1 θ2

[Mb ]

⎢ ⎢ T w 1 θ1 ⎢[Mb ] ⎢ ⎢ T θ2 θ1 ⎢ [Mb ] ⎣

T w1 w1

[Mb ]

T θ2 w 1

T w 2 θ1

[Mb ]

⎡

T θ1 θ1

[Kb ]

T w 1 θ2

[Mb ]

[Mb ]

T w2 w1

[Mb ]

[Mb ]

T θ2 θ2

[Mb ]

T w 2 θ2

[Mb ]

T θ1 w 1

[Kb ]

T θ1 θ2

[Kb ]

[Kb ]

[Kb ]

[Kb ]

T θ2 w 1 [Kb ] T w2 w1

[Kb ]

T θ2 θ2 [Kb ] T w 2 θ2

T w 2 θ1

T w 1 θ2

[Kb ]

T w2 w2

[Mb ]

⎢ ⎢ T w 1 θ1 ⎢[Kb ] +⎢ ⎢ T θ2 θ1 ⎢ [Kb ] ⎣

T w1 w1

⎤

⎧ T ⎫ ⎪ ⎥⎪ ⎪ ⎪1,τ τ ⎪ ⎪ ⎪ T w1 w2 ⎥ ⎪ ⎨ T ⎥ w1,τ τ ⎬ [Mb ] ⎥ T θ2 w 2 ⎥ ⎪ T ⎥⎪ ⎪ 2,τ τ ⎪ [Mb ] ⎪ ⎪ ⎦⎪ ⎩ T ⎪ ⎭ T θ1 w 2

[Mb ]

w2,τ τ

⎤ ⎧ T⎫ ⎧ ⎫ ⎪1 ⎪ ⎥⎪ 0⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ T w1 w2 ⎥⎪ ⎨ ⎬ ⎨ T ⎬ ⎥ [Kb ] ⎥ w1 = 0 , T ⎥ T θ2 w 2 ⎥⎪ ⎪0⎪ ⎪ ⎪2 ⎪ ⎪ ⎪ [Kb ] ⎪ T⎪ ⎪ ⎩0⎭ ⎦⎪ ⎩ ⎭ T θ1 w 2

[Kb ]

T w2 w2

(37)

w2

[Kb ]

where the nonzero vectors and submatrices in Eq. (37) are as, T

i wi [Mb ]w = IJ



1 0

Np       1 − (−1)i  12i−2 φIwi φJwi + μ2 φI,wξi φJ,wξi dξ + m p j φIwi (ξ p j ) − μ2 φI,wξi ξ (ξ p j ) φJwi (ξ p j ) ,

2

j=1

(38a)

226

K. Kiani / Applied Mathematics and Computation 270 (2015) 216–241



[Mb ]θIJi θi = T

1

Np      1 − (−1)i  −2 θi θi θi θi θi θi 2i−2 2 θi 2 θi φ λ φ λ−2  φ + μ φ φ d ξ + (ξ )φ (ξ ) + μ φ (ξ )φ (ξ ) , p p p p p j j j j 1 2 I J I J I,ξ J,ξ I,ξ J,ξ j

2

0

j=1

(38b)

T

i wi [Kb ]w = IJ

⎛





⎞

42i−2 φI,wξi φJ,wξi + Nbi φI,wξi φJ,wξi + μ2 φI,wξi ξ φJ,wξi ξ +   ⎠dξ T C v φIwi − μ2 φI,wξi ξ φJwi + (i − 1)K t φIwi φJwi + μ2 φI,wξi φJ,wξi

1

⎝ T

0

+

 T

2 



T

K zi (ξk )

 φIwi (ξk )φJwi (ξk ) + μ2 φI,wξi (ξk )φJ,wξi (ξk ) ,

(38c)

k=1

i θi [Kb ]w =− IJ

T

[Kb ]θIJi wi = − T

[Kb ]θIJi θi = T



 

1 0 1 0

1



0

+

42i−2 φI,wξi φJθi dξ ,

(38d)

42i−2 φIθi φJ,wξi dξ ,

(38e)

  T 42i−2 φIθi φJθi + η32i−2 φI,θiξ φJ,θξi + (i − 1)K r φIθi φJθi + μ2 φI,θiξ φJ,θξi dξ

2 



T

K yi (ξk )

k=1 T w wj

[Kb ]IJ i



=−

1 0

T



Cv

 φIθi (ξk )φJθi (ξk ) + μ2 φI,θiξ (ξk )φJ,θξi (ξk ) ,

(38f)

 φIwi − μ2 φI,wξi ξ φJw j dξ ; i = j,

(38g)

where i, j = 1, 2. By following the same procedure mentioned in the previous section, the flexural frequencies and vibration modes of an elastically supported DWCNT-based mass sensor will be determined when it has been embedded in an elastic matrix and experiences initially axial force as well. 6. Modeling of DWCNT-based mass sensors employing the NHOBT 6.1. Nonlocal equations of motion On the basis of the NHOBT of the Reddy models [49,50], the kinetic energy, TH (t), and the elastic strain energy, UH (t), of the DWCNT-based mass sensor, embedded in an elastic medium under initially axial force are expressed as follows:

⎛ 



2





2

⎞

2 I0i w˙ H (x, t ) + I2i ψ˙ iH (x, t ) + αi2 I6i ψ˙ iH (x, t ) + w˙ Hi,x (x, t ) i,x  2   ⎟ 1  lb ⎜ ⎜−2αi I4i ψ˙ iH (x, t ) ψ˙ iH (x, t ) + w˙ Hi,x (x, t ) ⎟dx, T H (t ) = ⎝ ⎠   2 0     2 i N 2 i=1 p H ˙ H (x, t )δ(x − x p ) ˙ + 1−(2−1) M ( x, t )δ( x − x ) + I ψ w p p p j j j j i i j=1

 H 2  ⎞ H ψi,x (x, t ) Mbnli (x, t ) + Nbi wHi,x (x, t )    nl H  nl H ⎟ ⎜  ⎟ 2  lb ⎜ + ψ H (x, t ) + wH (x, t ) α P ( x, t ) + Q ( x, t ) i  i i,x b b 1 i i ,x ⎟ ⎜ H    U (t ) =      ⎜ 2 2 2 ⎟dx, H H H 2 ⎟ 0 ⎜ +(i − 1) Kr ψ H (x, t ) + K w ( x, t ) + C w ( x, t ) − w ( x, t ) t v i=1 2 2 2 1 ⎠ ⎝   H 2  H 2 

2 + k=1 Kzi (xk ) wi (x, t )δ(x − xk ) + Kyi (xk ) ψi (x, t )δ(x − xk )

(39a)

⎛

(39b)

H H H nl H nl H nl H nl H where wH 1 /w2 , ψ1 /ψ2 , (Mb ) /(Mb ) , and (Qb ) /(Qb ) represent the deflection field, the angle of deflection field, the non1

2

1

2

local bending moment, and the nonlocal resultant shear force of the innermost/outermost tube, respectively. According to the nonlocal continuum theory of Eringen [45,46], the nonlocal resultant forces within the tubes are approximated by

 H + wH ψi,x i,xx ; i = 1, 2  H       H  H 2 H Qbi + αi PbHi ,x − (e0 a)2 QbHi + αi PbHi ,x = κi ψiH + wH i,x + αi J4i ψi,xx − αi J6i ψi,xx + wi,xxx , ,xx H MbHi − (e0 a)2 MbHi ,xx = J2i ψi,x − αi J4i

where

αi = 1/(3ro2i ); i = 1, 2, κi =



Ab

Gbi (1 − 3αi z2 ) dA, i



(40)

K. Kiani / Applied Mathematics and Computation 270 (2015) 216–241

 Ini =  Jni =

Ab

Ab

227

ρbi zn dA; n = 0, 2, 4, 6, i

Ebi zn dA; n = 2, 4, 6.

(41)

i

 t  By using Hamilton’s principle (i.e., 0 δ T H (t ) − δU H (t ) dt = 0), the governing equations of elastically embedded DWCNTs for sensing of arbitrary added nanoparticles are provided by



I21 − 2α1 I41 + α12 I61



Np 

ψ¨ 1H +

Ipi ψ¨ 1H δ(x − x pi ) +



  H α12 I61 − α1 I41 w¨ H1,x + Qbnl1

i=1



 nl H

+ α1 Pb1 I01 w¨ H 1 +

Np 

,x



 nl H

− Mb1

,x

+

i=1

− Nb1 wH 1,x





,x

2 



(42a)



 H  nl H  nl H − α12 I61 w¨ H α12 I61 − α1 I41 ψ¨ 1,x 1,xx − Qb1 ,x − α1 Pb1 ,xx 

H + Cv wH 1 − w2 +

I22 − 2α2 I42 + α22 I62 +

Ky1 (xk )ψ1H δ(x − xk ) = 0,

k=1

M pi w¨ H 1 δ(x − x pi ) −





2 

2 

Kz1 (xk )wH 1 δ(x − xk ) = 0,

(42b)

k=1

   H  H  H ψ¨ 2H + α22 I62 − α2 I42 w¨ H2,x + Qbnl2 + α2 Pbnl2 ,x − Mbnl2 ,x + Kr ψ2H

Ky2 (xk )ψ1H δ(x − xk ) = 0,

(42c) (42c)

k=1

I02 w¨ H 1 −



 H  nl H  nl H − α22 I62 w¨ H α22 I62 − α2 I42 ψ¨ 2,x 2,xx − Qb2 ,x − α2 Pb2 ,xx



− Nb2 wH 2,x





,x



H H − Cv wH 1 − w2 + Kt w2 +

2 

Kz2 (xk )wH 2 δ(x − xk ) = 0,

(42d)

k=1

by combining Eqs. (40 a)–(40 b) and Eqs. (42a)–(42d), the nonlocal equations of motion of the considered DWCNT-based mass resonator in terms of deflections and deformation angles of the innermost and outermost tubes are derived in the following form



I21 − 2α1 I41 + α12 I61

Np       H ψ¨ 1H − (e0 a)2 ψ¨ 1,xx + Ipi ψ¨ 1H δ(x − x pi ) − (e0 a)2 ψ¨ 1H δ(x − x pi ) ,xx

α

2 1 I61

− α1 I41



w¨ H 1,x

− (e0 a)

2

w¨ H 1,xxx



i=1

   H α12 J61 − α1 J41 wH1,xxx − J21 − 2α1 J41 + α12 J61 ψ1,xx  2     + Ky1 (xk ) ψ1H δ(x − xk ) − (e0 a)2 ψ1H δ(x − xk ) = 0, ,xx +





−



(43a)

k=1





Np 



i=1

2 H ¨ 1,xx + I01 w¨ H 1 − (e0 a) w



 



k=1



,xx

  H  H H 2 H ¨ 1,xxxx − κ1 ψ1,x − (e0 a)2 ψ¨ 1,xxx − α12 I61 w¨ H + wH α12 I61 − α1 I41 ψ¨ 1,x 1,xx − (e0 a) w 1,xx   H  H  H  H 2 H + α12 J61 − α1 J41 ψ1,xxx + α12 J61 wH 1,xxxx + Cv w1 − w2 − (e0 a) w1,xx − w2,xx  2     H 2 + Kz1 (xk ) wH δ( x − x ) − ( e a ) w δ( x − x ) = 0, 0 k k 1 1 ,xx −





2 ¨H M pi w¨ H 1 δ(x − x pi ) − (e0 a) w 1 δ(x − x pi )



  2   H 2 H ¨ 2,xxx ψ¨ 2H − (e0 a)2 ψ¨ 2,xx + α2 I62 − α2 I42 w¨ H 2,x − (e0 a) w     H 2 − α22 J62 − α2 J42 wH 2,xxx − J22 − 2α2 J42 + α2 J62 ψ2,xx  2       H + Kr ψ2H − (e0 a)2 ψ2,xx + Ky2 (xk ) ψ2H δ(x − xk ) − (e0 a)2 ψ2H δ(x − xk ) = 0, ,xx

I22 − 2α2 I42 + α22 I62



(43b)





 H    H 2 H ¨ 2,xxxx − (e0 a)2 ψ¨ 2,xxx − α22 I62 w¨ H α22 I62 − α2 I42 ψ¨ 2,x 12xx − (e0 a) w  H   2  H 2 H − κ2 ψ2,x + wH 2,xx + α2 J62 − α2 J42 ψ2,xxx + α2 J62 w2,xxxx  H     2 H H 2 H − Cv w1 − wH + Kt wH 2 − (e0 a) w1,xx − w2,xx 2 − (e0 a) w2,xx

2 H ¨ 2,xx − I02 w¨ H 2 − (e0 a) w



k=1

(43c)

228

K. Kiani / Applied Mathematics and Computation 270 (2015) 216–241 2 

+



 



2 H Kz2 (xk ) wH 2 δ(x − xk ) − (e0 a) w2 δ(x − xk )

k=1

,xx

= 0.

(43d)

To investigate the problem in a more general context, it is convenient to work with the dimensionless governing equations. To this end, the following dimensionless quantities are introduced:



α2 I42 − α22 I62 α 2 I6 α2 J42 − α22 J62 κ2 , ϑ32 = 22 2 , ϑ42 = , ϑ52 = , 2 κ1 α1 I41 − α1 I61 α1 I61 α1 J41 − α12 J61 I − 2α2 I42 + α22 I62 J2 − 2α2 J42 + α22 J62 α2J ϑ62 = 22 62 , ϑ72 = 22 , ϑ82 = 2 , 2 α1 J61 I21 − 2α1 I41 + α1 I61 J21 − 2α1 J41 + α12 J61 κ1 l 2 α I − α2I α2I α J − α2J γ12 = 1 41 2 1 61 , γ22 = 1 621 , γ32 = 2 b , γ42 = 1 41 2 1 61 , I01 lb I01 lb α1 J61 α1 J61 4 2 κ1 I01 lb α1 I41 − α1 I61 γ62 = , γ72 = 2 , I21 − 2α1 I41 + α12 I61 α1 J61 (I21 − 2α1 I41 + α12 I61 ) ( J2 − 2α1 J41 + α12 J61 )I01 lb2 (α1 J41 − α12 J61 )I01 lb2 γ82 = 2 1 , γ92 = 2 , 2 α1 J61 (I21 − 2α1 I41 + α1 I61 ) α1 J61 (I21 − 2α1 I41 + α12 I61 ) Ip j Kyi (xk )I01 lb3 H Ipj = , K yi (ξk ) = 2 , 2 (I21 − 2α1 I41 + α1 I61 )lb α1 (I21 − 2α1 I41 + α12 I61 )J61 Kzi (xk )lb3 Kt l 4 Kr I01 lb4 H H H K zi (ξk ) = , Kt = 2 b , K r = 2 , 2 α1 J61 α1 J61 α1 (I21 − 2α1 I41 + α12 I61 )J61 τ=

H

α1 lb2

N bi =

I0 J61 t ϑ12 = 2 , I01 I01

Nbi lb2

α

2J 1 61

H

,

Cv =

ϑ22 =

Cv lb4

α

2J 1 61

H

,

wi =

wH i , lb

H

ψ i = ψiH ; i, k = 1, 2; j = 1, . . . , Np .

(44)

As a result, the dimensionless nonlocal equations of motion of an elastically supported DWCNT-based mass sensor embedded in an elastic matrix under initially axial force are obtained as a function of displacements and deformation angles as follows H

H

ψ 1,τ τ − μ2 ψ 1,ξ ξ τ τ +

Np 

I pi

  H   H H H ψ 1,τ τ δ(ξ − ξ pi ) − μ2 ψ 1,τ τ δ(ξ − ξ pi ) − γ62 (w1,τ τ ξ − μ2 w1,τ τ ξ ξ ξ ) ,ξ ξ

i=1 H

H

+ γ72 (ψ 1 + w1,ξ ) − γ82 ψ 1,ξ ξ + γ92 w1,ξ ξ ξ + H

H

2 

  H   H H 2 K y1 (ξk ) ψ 1 δ(ξ − ξk ) − μ ψ 1 δ(ξ − ξk ) = 0, ,ξ ξ

k=1

w1,τ τ − μ2 w1,τ τ ξ ξ + H

H

Np 





 

m pi w1,τ τ δ(ξ − ξ pi ) − μ2 w1,τ τ δ(ξ − ξ pi ) H

H

i=1 H

H

(45a)

,ξ ξ H

H

+ γ12 (ψ 1,τ τ ξ − μ2 ψ 1,τ τ ξ ξ ξ ) − γ22 (w1,τ τ ξ ξ − μ2 w1,τ τ ξ ξ ξ ξ ) − γ32 (ψ 1,ξ + w1,ξ ξ ) − γ42 ψ 1,ξ ξ ξ + w1,ξ ξ ξ ξ H





H

+ C v w1 − w2 − μ2 w1,ξ ξ − w2,ξ ξ H

H

H

H



H

+

2 

H



H



 

K z1 (ξk ) w1 δ(ξ − ξk ) − μ2 w1 δ(ξ − ξk ) H

H

k=1

,ξ ξ

H

= 0,

 H  H H H ϑ72 ψ 2,τ τ − μ2 ψ 2,ξ ξ τ τ − ϑ22 γ62 (wH2,τ τ ξ − μ2 wH2,τ τ ξ ξ ξ ) + ϑ42 γ72 (ψ 2 + wH2,ξ ) − ϑ82 γ82 ψ 2,ξ ξ + ϑ52 γ92 wH2,ξ ξ ξ   H    H   2 H H H 2 H 2 + K r ψ 2 − μ ψ 2,ξ ξ + K y2 (ξk ) ψ 2 δ(ξ − ξk ) − μ ψ 2 δ(ξ − ξk ) = 0, ,ξ ξ

k=1 H

(45b)

(45c)

H

ϑ12 (wH2,τ τ − μ2 wH2,τ τ ξ ξ ) + ϑ22 γ12 (ψ 2,τ τ ξ − μ2 ψ 2,τ τ ξ ξ ξ ) − ϑ32 γ22 (wH2,τ τ ξ ξ − μ2 wH2,τ τ ξ ξ ξ ξ )  H  H H H H H H H H − ϑ42 γ32 (ψ 2,ξ + w2,ξ ξ ) − ϑ52 γ42 ψ 2,ξ ξ ξ + ϑ62 w2,ξ ξ ξ ξ − C v w1 − w2 − μ2 w1,ξ ξ − w2,ξ ξ  2      H H H H H H + K t w2 − μ2 w2,ξ ξ + K z2 (ξk ) w2 δ(ξ − ξk ) − μ2 w2 δ(ξ − ξk ) = 0. ,ξ ξ

(45d)

k=1

Seeking an analytical solution to Eqs. (45a)–(45d) is a difficult task. In the following part, RKPM is employed for solving the above-mentioned coupled equations.

K. Kiani / Applied Mathematics and Computation 270 (2015) 216–241

229

6.2. Implementation of RKPM H

H

H Both sides of Eqs. (45a), (45b), (45c), and (45d) are multiplied by δψ 1 , δ wH 1 , δψ 2 , and δ w2 , respectively. The sum of the resulting expressions is then integrated over [0, 1]. After taking integration by parts, 2  

1



0

i=1

+ϑ

Np   H H   H H  H H 1 − (−1)i  H H m p j δ wi wi,τ τ + μ2 δ wi,ξ wi,ξ τ τ + I pi δψ i ψ i,τ τ + μ2 δψ i,ξ ψ i,ξ τ τ δ(ξ − ξ p j ) 2

2i−2 7



j=1

H i

ψ ψ

H i,τ τ

H



− ϑ22i−2 γ62



H

H

δψ i wHi,ξ τ τ − μ2 δψ i,ξ wHi,ξ ξ τ τ



 H H H H ψ i + wHi,ξ + ϑ82i−2 γ82 δψ i,ξ ψ i,ξ − ϑ52i−2 γ92 δψ i,ξ wHi,ξ ξ    H H  H H H H H H + ϑ12i−2 δ wi wi,τ τ + μ2 δ wi,ξ wi,ξ τ τ − ϑ22i−2 γ12 δ wi,ξ ψ i,τ τ + μ2 δ wi,ξ ξ ψ i,ξ τ τ +  H   H H  H H H H H H + ϑ32i−2 γ22 δ wi,ξ wi,ξ τ τ + μ2 δ wi,ξ ξ wi,ξ ξ τ τ + ϑ42i−2 γ32 δ wi,ξ ψ i + wi,ξ − ϑ52i−2 γ42 δ wi,ξ ξ ψ i,ξ  H  H  H  H H H H H H H H + ϑ62i−2 δ wi,ξ ξ wi,ξ ξ + (−1)i+1C v wi − μ2 wi,ξ ξ w1 − w2 + (i − 1)K r δψ i − μ2 δψ i,ξ ξ ψ i  H  H H H H H H H H + (i − 1)K t δ wi − μ2 δ wi,ξ ξ wi + Nbi δ wi,ξ wi,ξ + μ2 δ wi,ξ ξ wi,ξ ξ    H H 2    H  H H H H H H 2 2 + K zi (ξk ) δ wi − μ δ wi,ξ ξ wi + K yi (ξk ) δψ i ψ i + μ δψ i,ξ ψ i,ξ δ(ξ − ξk ) dξ = 0, H

+ ϑ42i−2 γ72 δψ i



H

+ μ2 ψ i,ξ ψ i,ξ τ τ

(46)

k=1

using RKPM, the unknown fields of the innermost and outermost tubes modeled based on the NHOBT are discretized in the spatial domain as H

wi

(ξ , τ ) =

NPi 

H

φIwi (ξ )wHiI (τ ), ψ i (ξ , τ ) =

I=1

NPi  ψi

H

φI (ξ )ψ iI (τ ); i = 1, 2

(47)

I=1

by introducing Eq. (47) to Eq. (46), the following set of ODEs is obtained

⎡

H ψ1 ψ1

H ψ1 w1

[Mb ]

[Mb ]

⎢ ⎢ H w1 ψ1 ⎢[Mb ] ⎢ ⎢ H ψ2 ψ1 ⎢[Mb ] ⎣

H w1 w1

[Mb ]

H ψ2 w1

[Mb ]

H w2 ψ1

H w2 w1

[Mb ]

⎡

[Mb ]

H ψ1 ψ1

H ψ1 w1

[Kb ]

⎢ ⎢ H w1 ψ1 ⎢[Kb ] +⎢ ⎢ H ψ2 ψ1 ⎢[Kb ] ⎣

[Kb ]

[Kb ]

[Kb ]

H w1 w1

[Kb ]

H ψ2 w1

[Kb ]

H w2 ψ1

H w2 w1

H ψ1 ψ2

[Mb ]

H w1 ψ2

[Mb ]

H ψ2 ψ2

[Mb ]

H w2 ψ2

[Mb ]

H ψ1 ψ2

[Kb ]

H w1 ψ2

[Kb ]

H ψ2 ψ2

[Kb ]

H w2 ψ2

[Kb ]

⎤

⎧ H ⎫ ⎪ ⎪ ⎥⎪ ⎪1,τ τ ⎪ ⎪ ⎪ H w1 w2 ⎥ ⎪ ⎨ H ⎥ w1,τ τ ⎬ [Mb ] ⎥ H ψ2 w2 ⎥⎪ H 2,τ τ ⎪ ⎥⎪ ⎪ [Mb ] ⎪ ⎪ ⎦⎪ ⎩ H ⎪ ⎭ H ψ1 w2

[Mb ]

H w2 w2

[Mb ]

w2,τ τ

⎤ ⎧ H⎫ ⎧ ⎫ ⎪1 ⎪ ⎥⎪ 0⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ H w1 w2 ⎥⎪ ⎨ ⎨ ⎬ H⎬ ⎥ [Kb ] ⎥ w1 = 0 , H ⎥ ψ w H 2 2 ⎪ 0⎪ ⎪ ⎪ ⎥⎪ [Kb ] ⎪2 ⎪ ⎩ ⎪ ⎭ ⎪ ⎪ ⎪ ⎦⎪ ⎩ ⎭ 0 H w2 w2 H ψ1 w2

[Kb ]

H

[Kb ]

(48)

w2

where the nonzero vectors and submatrices in Eq. (48) are as, H ψ ψi

[Mb ]IJ i

H ψ wi

[Mb ]IJ i H

 =

2

0

 =−

i wi [Mb ]w = IJ

+

Np     1 − (−1)i  ψ ψ ψ ψ ϑ72i−2 φIψi φJψi + μ2 φI,ψξi φJ,ψξi dξ + I p j φI i (ξ p j )φJ i (ξ p j ) + μ2 φI,ξi (ξ p j )φJ,ξi (ξ p j ) ,

1

1 0



1

  ϑ22i−2 γ62 φIψi φJ,wξi + μ2 φI,ψξi φJ,wξi ξ dξ ,



0

(49a)

j=1

(49b)

    ϑ12i−2 φIwi φJwi + μ2 φI,wξi φJ,wξi + ϑ32i−2 γ22 φI,wξi φJ,wξi + μ2 φI,wξi ξ φJ,wξi ξ dξ

Np   1 − (−1)i  m p j φIwi (ξ p j )φJwi (ξ p j ) + μ2 φI,wξi (ξ p j )φJ,wξi (ξ p j ) , 2

(49c)

j=1

H w ψi

[Mb ]IJ i

 =−

1 0

  ϑ22i−2 γ12 φI,wξi φJψi + μ2 φI,wξi ξ φJ,ψξi dξ ,

(49d)

230

K. Kiani / Applied Mathematics and Computation 270 (2015) 216–241 Table 1 Comparison of the predicted resonant frequencies (GHz) of the bridged DWCNT based mass sensor using NTBT and NHOBT with those of the analytical model by Shen et al. [31] for different values of the mass weight and position of the attached mass as well as various levels of initially axial force. Data of the attached mass M p1 (g) 0 10−21 10−20 10−19

1 2 3

f1T 2

f1H 3

Ref. [31]

f1T

f1H

Ref. [31]

f1T

f1H

0.3 0.5 0.3 0.5 0.3 0.5

78.4436 77.8709 77.3303 72.9821 68.9722 46.4961 39.1032

78.4395 77.8641 77.3269 72.9580 68.9578 46.2442 39.0091

73.1654 72.6214 72.1748 67.9881 64.6681 43.1659 37.0587

56.1963 55.7974 55.3714 52.4320 49.2435 34.2092 27.8303

56.2217 55.8219 55.3923 52.4545 49.2284 34.0773 27.7396

50.9507 50.5813 50.2344 47.4719 44.8689 30.8453 25.6278

40.1049 39.8265 39.5024 37.5034 35.0584 25.0160 19.7666

39.9348 39.6581 39.3289 37.3573 34.8650 24.8720 19.5881

34.1101 33.8682 33.6170 31.8577 29.9547 21.2054 17.0629

R

1

i

Ab lb2

(

i

T

+

α π

2 

H w ψi

H ψ wi

[Kb ]IJ i

H ψ ψi

[Kb ]IJ i

1 0

.

 ⎞ H ϑ42i−2 γ32 φI,wξi φJ,wξi + ϑ62i−2 φI,wξi ξ φJ,wξi ξ + Nbi φI,wξi φJ,wξi + μ2 φI,wξi ξ φJ,wξi ξ   ⎝  ⎠dξ H H φIwi − μ2 φI,wξi ξ φJwi + (i − 1)K t + C v 

H





1 0





1 0

 =

J61 I01

1

K zi (ξk )

= =

ρb1

⎛

 =

ks1 Gb

(

H 1 i 2 lb2

k=1

[Kb ]IJ i

1

0

+ H w wj

 ϑ42i−2 γ72 φIψi φJ,wξi − ϑ52i−2 γ92 φI,ψξi φJ,wξi ξ dξ ,

(49e)

(49f)

(49g)

    H ϑ42i−2 γ72 φIψi φJψi + ϑ82i−2 γ82 φI,ψξi φJ,ψξi + K r φIψi − μ2 φI,ψξi ξ φJψi dξ

2 

 =−

 φIwi (ξk )φJwi (ξk ) + μ2 φI,wξi (ξk )φJ,wξi (ξk ) ,

 ϑ42i−2 γ32 φI,wξi φJψi − ϑ52i−2 γ42 φI,wξ2ξ φJ,ψξ2 dξ ,

H

1 0



K yi (ξk )

k=1

[Kb ]IJ i

γ 0 = 0.005

Ref. [31]

fiT = 2πil b

H wi wi [Kb ]IJ

γ0 =0

ξ1

Nb Ib

γ0 = −

fiH =

γ 0 1 =−0.01

H

Cv



 φIψi (ξk ) − μ2 φI,ψξi ξ (ξk ) φJψi (ξk ).

 φIwi − μ2 φI,wξi ξ φJw j dξ ; i = j,

(49h)

(49i)

where i, j = 1, 2. By following the identical procedure stated in Section 3.2, the flexural frequencies and vibration modes of an elastically supported DWCNT-based mass sensor could be evaluated according to the hypotheses of the NHOBT. 7. Results and discussion 7.1. A comparison study Consider a bridged (fully clamped) DWCNT as a nanomechanical sensor with an attached mass. The properties of the considered DWCNT are as: lb =28 nm, rm1 =0.7 nm, rm2 =0.35 nm, tb =0.35 nm, Eb1 = Eb2 =1 TPa, ρb1 = ρb2 =1300 kg/m3 , and ν1 = ν2 =0.25. The predicted fundamental resonant frequencies of the above-mentioned nanodevice based on the proposed analytical model by Shen et al. [31] and those of the proposed NTBT and NHOBT models are provided in Table 1 in the case of e0 a = 0 nm. Using the nonlocal Euler-Bernoulli beam model, Shen et al. [31] investigated the frequency change of a bridged DWCNT due to an attached mass via the transfer function method. The results are given for various levels of initially axial force, mass weight of the attached nanoparticle, and different locations of the added mass. In RKPM analysis, 11 particles are used for discreatization of each unknown field of each tube. As is seen in Table 1, there exists a reasonable good agreement between the results of Shen et al. [31] and those based on the proposed NTBT in all case studies. Additionally, the predicted fundamental frequencies of the NHOBT are entirely lower than those of both the NTBT and the proposed model by Shen et al. [31]. The main reason of this fact is that the thickness of both the innermost and outermost tubes is comparable with their radii; thereby, the effect of shear deformation becomes important in vibration analysis of such a slender DWCNT. According to the provided results in Table 1, the fundamental frequency of the nanodevice decreases with mass weight of the attached mass, irrespective of its

K. Kiani / Applied Mathematics and Computation 270 (2015) 216–241

(c)

(b)

3.22

1

1

(a)

3.19

3.16 12

24

36

48

1

4.04

5.3

4.87

4.44 12

24

36

48

24

36

48

36

48

1.76

1.71 52

1

4.26

(d)

231

(e)

3.82 12

24

36

48

(f)

2.26

1

1

2.84

2.66

2.48 12

1.66 12

24

36

NP

48

2.13

2 12

24

NP

Fig. 2. Convergence check of the first dimensionless natural frequency of the DWCNT-based mass sensor with a mass at the midspan point according to the proposed numerical models for different boundary conditions : (a) SS, (b) CC, (c) SC, (d) Sf S, (e) Sf C, (f) CF; (M p1 = 0.5, e0 a = 1 nm, NP = NP1 + NP2 ; (…) NRBT, (−.−) NTBT, (—) NHOBT).

location and the initially axial force. Moreover, all models predict that the fundamental frequency of the DWCNT as a nanomechanical sensor magnifies as the initially tensile forces within both the innermost and outermost tubes increase. The influences of both the mass weight of the attached mass and the initially axial force on the frequency change of the nanodevice with different boundary conditions will be inclusively studied in the upcoming parts. 7.2. Convergence check of RKPM for the proposed models For making sure about the convergence of the numerical scheme for the proposed models, the predicted natural frequencies of the DWCNT-based mass sensor should be demonstrated as a function of the number of RKPM’s particles. For this purpose, consider a DWCNT with an attached mass at its midspan. The DWCNT is released from its surrounding medium and does not experience any initially axial load (i.e., Kt = Kr = Nbi = 0). The properties of the DWCNT in our analyses are as: lb = 20 nm, rm1

= 1.04 nm, rm2 = 0.7 nm, ρb1 = ρb2 = 2300 kg/m3 , Eb1 = Eb2 = 1 TPa, ν1 = ν2 = 0.25, and e0 a = 1 nm. Let introduce the new dimensionless natural frequencies of the models as:

Rn =

)

nR , ) 1

Tn = η− 4 nT ,  2 1 α1 J61 ρb1 Ab1 4 ) H H n = n . I01 Eb1 Ib1

(50)

For different boundary conditions, the predicted first dimensionless frequencies by the proposed numerical models as a function of the total number of particles of RKPM are provided in Fig. 2(a)–(f). In all presented figures in this article, the dotted, the dashed-dotted, and the solid lines are, respectively, pertinent to the predicted results for the DWCNT-based mass resonator modeled according to the NRBT, NTBT, and NHOBT. In order to evaluate the shape functions of RKPM for both the innermost and outermost tubes, cubic spline window function, and linear base function are employed and the dilation parameter of all RKPM’s particles is set equal to 3.2. As is seen in Fig. 2(a)–(f), the variation rate of the dimensionless first natural frequency of the nanosensor in terms of the total number of RKPM’s particles would lessen for all considered boundary conditions. For each boundary condition, the predicted results by various nonlocal models converge to the specified values as the number of RKPM’s particles increases. In most of the cases, the predicted results by the NTBT are close to those of the NHOBT. Furthermore, the discrepancies between the results of the NTBT and those of the NHOBT commonly decrease with the total number of RKPM’s particles. The discrepancies between the results of the NRBT and those of the nonlocal shear deformable beam theories (NSDBTs) (i.e., NTBT and NHOBT) are more obvious in the cases of CC and Sf C boundary conditions. Henceforth, NP = 22 is used for RKPM analysis unless the number of RKPM’s particles has been explicitly stated.

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K. Kiani / Applied Mathematics and Computation 270 (2015) 216–241 Table 2 Convergence check of the predicted fundamental frequency via RKPM for a simply supported DWCNT for sensing a nanoparticle at its midspan (M p1 = 0.1, lb = 100 nm, e0 a = 2 nm).

 

T 1 H 1

GM

RKPM analysis

NM1 =5

NP1 =6

NP1 =11

NP1 =16

NP1 =21

NP1 =26

NP1 =31

NP1 =36

NP1 =41

3.5030 3.5036

3.5398 3.5383

3.5058 3.5060

3.5040 3.5043

3.5036 3.5039

3.5035 3.5038

3.5034 3.5038

3.5034 3.5037

3.5033 3.5037

Table 3 [.] [.] The values of K yi and K zi for the considered boundary conditions.

[.] K zi [.] K zi [.] K yi [.] K yi

SS

CC

SC

Sf S

Sf C

CF

(0)

108

108

108

0

0

108

(1)

108

108

108

108

108

8

10

8

108

10

8

0

(0) (1)

0 0

10

8

10

8

0 10

10 8

0

0

Through comparing of the predicted dimensionless frequencies of the DWCNT-nanoparticle system by the proposed numerical models with another semi-analytical approach, convergence of the RKPM calculations is again revisited. For this purpose, consider a simply supported DWCNT in which being used for sensing a nano-object at its midspan. Based on the NTBT and NHOBT, the dimensionless fundamental frequency of the system is calculated via RKPM and the Galerkin method (GM). The procedure of application of GM to determine the fundamental frequency of the considered nanostructure is provided in Appendix A. The problem is analyzed using RKPM with NP1 =6 to 41 with step 5 as well as GM by considering the first five vibration modes of the nanostructure. The predicted dimensionless fundamental frequency by these methodologies is summarized in Table 2. As is obvious from Table 2, the discrepancies between the RKPM results and those of the GM would generally decrease as the number of RKPM’s particle magnifies. The calculations of this section reveal the high accuracy of the carried out calculations via the RKPM. Such studies ensure us regarding application of RKPM to frequency analysis of elastically supported DWCNTs as nanomechanical sensors. 7.3. Parametric analysis Let consider an ECS associated with an DWCNT with the following data: lb = 28 nm, rm1 = 0.7 nm, rm2 = 1.04 nm, tb = 0.34 nm, Eb1 = Eb2 = 1 TPa, ρb1 = ρb2 =2500 kg/m3 , and ν1 = ν2 = 0.2. Six boundary conditions represented by SS, CC, SC, Sf S, Sf C, and CF are considered in frequency analysis of DWCNTs as mass detectors. Each boundary condition is specified by two letters where the first and the second letters in order are describing the left- and right-hand support conditions of the DWCNT. The letters S, C, Sf , and F are denoting a simple, clamp, shear-free, and free support, respectively. These boundary conditions are imposed to the proposed numerical models by choosing appropriate values for dimensionless constant values of translational and rotary springs which are attached to the ends of each nanotube, as given in Table 3. In this work, addition of the nanoparticles to the DWCNT is followed up by their impacts on the alteration of the natural frequencies. Therefore, let us define the frequency shift ratio of the first vibration mode of the DWCNT as: R = 1 − 1 /1 where 1 denotes the dimensionless fundamental frequency of the DWCNT without considering the attached nanoparticles. In the following parts, the effects of the slenderness ratio, small-scale parameter, initially axial forces within the nanotubes, interaction of the DWCNT with its surrounding elastic medium, mass weight and location of the added nanoparticles on the frequency shift ratio (FSR) of the first vibration mode of the DWCNT are studied in some details. 7.3.1. Influence of slenderness ratio on the frequency shift ratio The effect of the slenderness ratio of the innermost tube on the frequency shift of the DWCNT due to an attached nanoparticle at its midspan point is of concern. In Fig. 3(a)–(f), the plots of FSR as a function of the slenderness ratio are provided for different boundary conditions as well as three levels of the mass weight of the added mass (i.e., M p1 = 0.5, 1, and 2). In the case of SS boundary conditions for both tubes for the added mass with M p1 ≤ 1 (see Fig. 3(a)), both the nonlocal shear deformable beam theories (NSDBTs) predict that the FSR increases with the slenderness ratio; however, for the attached mass with M p1 = 3, both NTBT and NHOBT show a decrease in the FSR up to λ1 ≈ 13, thereafter, the predicted FSRs by the NSDBTs magnify with the slenderness ratio. In the case of CC boundary conditions, NSDBTs show a decrease in FSR for low levels of the slenderness ratio; for M p1 = 0.5, 1, and 2, the predicted FSR by the NSDBTs would lessen with the slenderness ratio up to λ1 = 11, 14, and 16, respectively. Thereafter, the FSRs of the NSDBTs would increase with the slenderness ratio (see Fig. 3(b)). In the case of SC boundary conditions, the general trends of the FSR curves are somehow analogous to those of the CC boundary conditions for all nonlocal beam theories and mass weight of the attached nanoparticle (see Fig. 3(b) and (c)). In the cases of SS, CC, and SC

K. Kiani / Applied Mathematics and Computation 270 (2015) 216–241

(a) R R R

(c)

(d)

25

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0.19

0.04 10 0.18

40

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0.11

0.16

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40

0.04 10 0.1

25

25

0.045

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(e)

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0.14

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0.14 25

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0.09

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0.08 0.02 10 0.24

(b)

R

0.18

0.14

233

0.1 10 0.14 0.07

25 1

40

0 10

1

Fig. 3. Frequency shift ratio of the DWCNT due to an attached mass at its midspan point in terms of slenderness ratio for different boundary conditions: (a) SS, (b) CC, (c) SC, (d) Sf S, (e) Sf C, (f) CF; ((◦)M p1 = 0.5, (

)M p1 = 1, ()M p1 = 2; (…) NRBT, (−.−) NTBT, (—) NHOBT; Kt = Kr = Nbi = 0; e0 a =2 nm).

boundary conditions (see Fig. 3(a)–(c)), for all levels of the mass weight of the added nanoparticle, the NRBT overestimates the FSRs with respect to the NSDBTs. Furthermore, the predicted FSR by the NRBT decreases with the slenderness ratio. Among the studied boundary conditions for the DWCNT, the variation of the slenderness ratio has the most trivial effect on the variation of the FSRs for the case of Sf S boundary conditions (see Fig. 3(d)). In the case of a DWCNT with Sf C boundary conditions, the predicted FSRs by various nonlocal beams would reduce with the slenderness ratio (see Fig. 3(e)). Moreover, the predicted FSRs by the NRBT are the lower bonds of those of the NSDBTs. This issue also holds true in the case of CF boundary conditions; however, the FSR of the NRBT would grow with the slenderness ratio (see Fig. 3(f)). The general trend of the FSRs of the NSDBTs for the CF boundary conditions is analogous to that of Sf C ends.

7.3.2. Influence of small-scale effect parameter on the frequency shift ratio The influence of the small-scale parameter on the FSR of the DWCNT-based mass nanosensor is another interesting issue should be realized. The proposed nonlocal models enable us to explore such an important effect on the vibration behavior of the DWCNT with attached nanoparticles. Herein, a freely vibrating DWCNT with no initially axial force is considered for λ1 = 10. In Fig. 4(a)–(f), the plots of FSRs of such a DWCNT-based mass nanosensor as a function of the small-scale parameter are provided for different boundary conditions. In the case of SS boundary conditions, the predicted FSRs by the NSDBTs generally lessen with the small-scale parameter (see Fig. 4(a)); however, the NRBT predicts that the FSR would magnify as the small-scale parameter increases. Furthermore, the NRBT overestimates the FSR with respect to NSDBTs for all levels of the mass weight of the attached nanoparticle. It implies that for such a DWCNT-based mass resonator, the NRBT could not capture the true trend of the frequency shift. For a DWCNT with CC boundary conditions (see Fig. 4(b)), for the cases of M p1 = 0.5, 1, and 2, the NSDBTs predict that the FSRs would lessen with the small-scale parameter up to e0 a = 1.7, 1.4, and 1.1, respectively; for small-scale parameters greater than these values, the predicted FSRs by the NSDBTs would magnify with the small-scale parameter. For all considered levels of the mass weight of the attached nanoparticle, the predicted FSR by the NRBT would somewhat magnify with the small-scale parameter. The general trends of the FSR plots of various nonlocal models as a function of the small-scale parameter for the case of SC boundary conditions are somehow analogous to those pertinent to the CC boundary conditions (see Fig. 4(b) and (c)). For a DWCNT-based mass nanosensor with Sf S conditions, no obvious variation of the FSR in terms of the small-scale parameter is detectable for all proposed nonlocal models (see Fig. 4(d)). In the case of Sf C boundary conditions, the predicted FSRs by various nonlocal models commonly increase with the small-scale parameter (see Fig. 4(e)). Additionally, the predicted results by the NHOBT are generally between those of the NRBT and the NTBT such that the results of the NRBT are lower than those of the NSDBTs. For DWCNT-based mass sensors with CF boundary conditions, the predicted FSRs by the NSDBTs/NRBT would grow/lessen with the small-scale parameter. A scrutiny of the plotted results in Fig. 4 reveals that in the cases of SS, Sf S, Sf C, and CF boundary conditions, the discrepancies between the results of the NRBT/NTBT and those of the NHOBT would generally intensify with the small-scale parameter.

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R

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2

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0.5

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1.5

2

0.12 0 0.18

2

0.13

0 0 0.08

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2

e a (nm)

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2

0 0

e a (nm) 0

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1

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2

0.5

1

1.5

2

0.5

1

1.5

2

0.5

1

1.5

2

1

1.5

2

0.07

0.04 0 0

2

0.14

0.06 1.5

1.5

0.24

0 0 0.06

1

1

0.21

0.03 0.5

0.5

0.5

e a (nm) 0

Fig. 4. Frequency shift ratio of the DWCNT due to an attached mass at the midspan point in terms of small-scale parameter for different boundary conditions: (a) SS, (b) CC, (c) SC, (d) Sf S, (e) Sf C, (f) CF; ((◦)M p1 = 0.5, (

R

(a)

0.14

0.18

0.36

0.07

0.09

0.27

0 0 0.24

R

(b)

R

(c)

R R

(e)

R

(f)

10

20

0.12 0 0 0.2

(d)

)M p1 = 1, ()M p1 = 2; (…) NRBT, (−.−) NTBT, (—) NHOBT; Kt = Kr = Nbi = 0; λ1 = 10).

10

20

10

20

10

20

0 0 0.1

0.06 0 0.12

10

20

0.04 0 0.1

10

20

NR b1

20

0 0

10

20

0.2 0 0.4

10

20

10

20

10

20

10

20

10

20

0.3 10

20

0.2 0 0.26 0.23

10

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10

20

0.05 10

0.18 0 0.46 0.33

0.08

0.03 0 0

0 0 0.24

0.08

0.03 0 0 0.06

20

0.12

0.05 0.04 0 0.06

10

0.15

0.1

0 0 0.06

0 0 0.3

0.2 0 0.24 0.18

10 R Nb1

20

0.12 0

R Nb1

Fig. 5. Frequency shift ratio of the DWCNT due to an attached mass at the midspan point in terms of initially axial force for different boundary conditions: (a) SS, (b) CC, (c) SC, (d) Sf S, (e) Sf C, (f) CF; ((◦)M p1 = 0.5, (

)M p1 = 1, ()M p1 = 4; (…) NRBT, (−.−) NTBT, (—) NHOBT; Kt = Kr = 0; λ1 =10, e0 a =2 nm).

7.3.3. Influence of initially axial force on the frequency shift ratio Another important parametric study is carried out to determine the effect of the initially axial force on the frequency shift mechanisms of a DWCNT due to an attached nanoparticle at its midspan point. The plots of FSR in terms of dimensionless initially axial forces within the innermost and outermost tubes are provided in Fig. 5(a)–(f) for different boundary conditions. The demonstrated results are given for freely vibrant DWCNTs with λ1 = 10 and e0 a = 2 nm. For a DWCNT-based mass sensor with SS boundary conditions, no obvious variation of the predicted FSR by the NRBT as a function of initially axial forces is

K. Kiani / Applied Mathematics and Computation 270 (2015) 216–241

R

(a)

R

(b)

R

(c)

R

(d)

R

(e)

R

(f)

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0.28

0.28

0.07

0.14

0.14

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10

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30

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30

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0.1

0.17

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30

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R K t

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0.33 10

20

30

0.28 0 0.32 0.25

20

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10

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0.12 0 0.22 0.17

10

20

30

0.04 10

0 0 0.38

235

0.12 0 0.16 0.08

10

20

KtR

30

0 0

10

KtR

Fig. 6. Frequency shift ratio of the DWCNT due to an attached mass at the midspan point in terms of lateral stiffness of the surrounding medium for different boundary conditions: (a) SS, (b) CC, (c) SC, (d) Sf S, (e) Sf C, (f) CF; ((◦)M p1 = 0.5, ( e0 a =2 nm).

)M p1 = 1, ()M p1 = 2; (…) NRBT, (−.−) NTBT, (—) NHOBT; Kr = Nbi = 0; λ1 =10,

observed for the considered range of the mass weight of the attached nanoparticle (see Fig. 5(a)). The same trend is also valid for the predicted FSRs by the NSDBTs for M p1 ≤ 1. For M p1 > 0.5, the predicted FSR by the NTBT as well as the NHOBT would obviously reduce with initially axial force up to Nbi ≈ 1.6. For Nbi > 1.6, the variation of the initially axial forces has a trivial effect on the variation of the FSR by the NTBT. Furthermore, the predicted FSRs by the NHOBT as well as their discrepancies with those of the NTBT would magnify with the initially axial forces for Nbi > 1.6. For all levels of the initially axial force, the NRBT overestimates the FSRs with respect to those of the NSDBTs. For the understudy DWCNT with SC/CC boundary conditions and an attached mass at its midspan, for low values of the mass weight of the attached nanoparticle (i.e., M p1 < 0.5), the predicted FSRs by the NSDBTs slightly decrease with the initially tension force for low levels of the initially axial forces (see Fig. 5(b) and (c)). For both CC and SC boundary conditions, the rate of decrease of the predicted FSRs by the NTBT would lessen as the initially axial forces within the nanotubes increase. For such boundary conditions, the predicted FSR curves by the NHOBT take their minimum levels at Nbi ≈ 5.6. In the case of the Sf S boundary conditions, the variation of the initially axial forces has a trivial effect on the variation of the predicted FSRs by the proposed nonlocal models (see Fig. 5(d)). Such a fact is also true for the depicted results for DWCNTs with Sf C boundary conditions and M p1 < 1 (see Fig. 5(e)). In the case of Sf S boundary conditions, the predicted results by the NRBT and the NHOBT are, respectively, the lower-bond and higher-bond levels of those of the NTBT. For DWCNTs-based mass sensor with Sf C conditions and M p1 = 5, the predicted FSRs by the NSDBTs decrease with the initially axial force; however, the amount of decrease of FSR is followed with a lower rate for higher levels of initially axial force (see Fig. 5(e)). In the case of CF boundary conditions, the predicted results by the NSDBTs show that the variation of the initially axial force has a small influence on the variation of the FSR. On the contrary, the NRBT displays that the FSR generally increases with the initially axial force (see Fig. 5(f)). 7.3.4. Influence of transverse and rotational stiffness of the surrounding medium on the frequency shift ratio Equally important is to find out the influences of the lateral and rotational stiffness of the surrounding elastic medium on the flexural frequency shift of the DWCNT due to the attached nanoparticles. For this purpose, the predicted FSRs by the proposed models as a function of lateral and rotational stiffness of the surrounding medium are plotted in Figs. 6(a)–(f) and 7(a)–(f) for different boundary conditions. The depicted results are given for a DWCNT with λ1 = 10 and e0 a = 2 nm when it experiences no initially axial forces. Concerning the effect of the lateral stiffness of the surrounding medium on the frequency shift, the NRBT overestimates the FSR with respect to the NSDBTs for SS, CC, SC, and Sf S boundary conditions. For the DWCNT-based mass sensor with Sf C and CF boundary conditions, the NRBT generally underestimates the FSR with respect to the NSDBTs. In the case of SS conditions, the discrepancies between the results of the NTBT and those of the NHOBT commonly magnify with lateral stiffness and mass weight of the added nanoparticle for M p1 ≤ 1 (see Fig. 6(a)). For M p1 = 3, the discrepancies between the results of the NRBT/NTBT and those of the NHOBT generally decrease as the effect of the lateral stiffness of the surrounding medium becomes highlighted. In the case of CC boundary conditions, the discrepancies between the results of the NRBT/NTBT

236

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(b)

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10

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0.1 0 0

0 0 0.44

0.14 0 0.3 0.15

10

KrR

20

30

KrR

40

50

0 0

KrR

Fig. 7. Frequency shift ratio of the DWCNT due to an attached mass at the midspan point in terms of rotational stiffness of the surrounding medium for different boundary conditions: (a) SS, (b) CC, (c) SC, (d) Sf S, (e) Sf C, (f) CF; ((◦)M p1 = 0.5, ( e0 a =2 nm).

)M p1 = 1, ()M p1 = 2; (…) NRBT, (−.−) NTBT, (—) NHOBT; Kt = Nbi = 0; λ1 =10,

and those of the NHOBT would reduce with the slenderness ratio and the mass weight of the attached mass (see Fig. 6(b)). The depicted results of the NTBT and those of the NHOBT are commonly in line and close to each other. For all considered levels of the lateral stiffness of the surrounding medium, the NTBT could capture the results of the NHOBT with relative error lower than 30%, 8%, and 5% for M p1 = 0.5, 1, and 2, respectively. In the case of a DWCNT with SC boundary conditions, the NTBT could capture the predicted results by the NHOBT for M p1 = 0.5, 1, and 2 with relative error lower than 18, 30, and 10, respectively (see Fig. 6(c)). Generally, higher the mass weight of the attached nanoparticle, lower the discrepancies between the results of the NRBT and those of the NHOBT. Such discrepancies would also lessen as the effect of the lateral stiffness of the surrounding medium becomes highlighted. For a DWCNT-based mass sensor under Sf S conditions, the discrepancies between the predicted results by the NRBT/NTBT and those of the NHOBT would commonly magnify with the lateral stiffness of the surrounding medium (see Fig. 6(d)). Such discrepancies reduce as the mass weight of the attached nanoparticle increases. This issue is also true for DWCNTs-based mass sensors with Sf C and CF boundary conditions (see Fig. 6(e) and (f)). Regarding the influence of the rotational stiffness of the surrounding medium on the frequency shift of the DWCNT, according to Fig. 7(a)–(f), except the Sf C and CF boundary conditions, the predicted FSRs by the NRBT and the NTBT are overestimated with respect to those of the NHOBT. For all studied boundary conditions, the discrepancies between the results of the NRBT and those of the NHOBT increase as the rotational interaction of the DWCNT with its surrounding environment intensifies. Among the SS, CC, and SC boundary conditions, except the SS case with M p1 = 3, the discrepancies between the results of the NTBT and those of the NHOBT magnify with the rotational stiffness of the surrounding medium. In the cases of Sf S, Sf C, and CF boundary conditions, the NTBT could predict the results of the NHOBT with relative error lower than 14% , for all considered levels of the rotational stiffness of the surrounding medium. For such boundary conditions, the discrepancies between the results of the NTBT and those of the NHOBT generally reduce as the effect of the rotational stiffness of the surrounding medium increases. 7.3.5. Influence of mass weight of the attached nanoparticles on the frequency shift ratio The effect of the mass weight of the attached nanoparticle on the frequency shift of the resonant frequency of the DWCNT as a mass sensor is another interesting subject to be investigated on the basis of the nonlocal continuum mechanics using proposed numerical models. In doing so, the FSR plots of the first vibration mode of the DWCNT with an attached mass to its midspan point are provided in Fig. 8(a)–(f) for different boundary conditions. These results are presented for a freely vibrating DWCNT with λ1 = 10 (i.e., Kt = Kr = 0) when it is initially at rest (i.e., Nbi = 0). For all studied boundary conditions, all the proposed nonlocal beam models predict that the FSR increases with the mass weight of the attached nanoparticle. For SS, CC, SC, and Sf S boundary conditions, the predicted results by the NTBT are generally between those of the NRBT and the NHOBT such that the NRBT overestimates the FSR with respect to the NSDBTs (see Fig. 8(a)–(d)). For Sf C and CF boundary conditions, the NRBT usually underestimates the FSR in compare to other proposed models and the predicted results by the NHOBT are somehow between those of the NRBT and the NTBT for all considered mass weight of the attached nanoparticle. As a general result, the

K. Kiani / Applied Mathematics and Computation 270 (2015) 216–241

R

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0.32

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0.18

0.18

(e)

R

2

0.21

(d)

(f)

1

237

1

2

M

p1

3

0 0

1

M

p1

Fig. 8. Frequency shift ratio of the DWCNT due to an attached mass at the midspan point in terms of its mass weight for different boundary conditions: (a) SS, (b) CC, (c) SC, (d) Sf S, (e) Sf C, (f) CF; ((◦)e0 a = 0 nm, (

)e0 a = 1 nm, ()e0 a = 2 nm; (…) NRBT, (−.−) NTBT, (—) NHOBT; Kt = Kr = Nbi = 0; λ1 =10).

discrepancies between the results of the NRBT/NTBT and those of the NHOBT commonly increase with the small-scale parameter. Furthermore, such discrepancies would lessen with the mass weight of the attached nanoparticle. The sensitivity of the FSRs of the DWCNT with Sf S and CF boundary conditions to the mass weight of the attached nanoparticle is less than other studied boundary conditions. Additionally, a DWCNT with CC and SC boundary conditions commonly display a higher rate of change of FSR as a function of the mass weight of the attached nanoparticle. Maybe someone would be interested in the effects of the lateral and rotary inertia of the attached nanoparticles on the frequency shift ratio of the nanosystem. Since the attached nanoparticles have been modeled by pointed inertial loads, the natural frequencies of the nanosystem would decrease by consideration of the lateral and rotary inertia of the attached nanoparticles. This fact can be easily researched via Eqs.(26a), (38a), and (49a) when the host nanostructure has been modeled by the NRBT, NTBT, and NHOBT, respectively. Therefore, FSR of the nanostructure would generally increase by taking into account the abovementioned inertial effects of the attached nanoparticles. 7.3.6. Influence of number of the attached nanoparticles on the frequency shift ratio The variation of the FSR of the DWCNT due to the addition of nanoparticles as a function of the number of attached nanoparticles is of particular interest. To this end, the predicted FSRs associated with the first vibration mode are demonstrated in Fig. 9(a)–(f). These results are presented for different boundary conditions with various levels of the mass weight of the attached nanoparticles. It is assumed that the nanoparticles have been attached to the DWCNT at the locations ξi = i/(N p + 1). The understudy DWCNT has λ1 = 10 and e0 a = 1 nm. As it is obvious from Fig. 9(a)–(f), the predicted FSRs by the proposed models generally magnify with the number of attached nanoparticles for all studied boundary conditions. In the case of SS boundary conditions, the NTBT could capture the results of the NHOBT with relative error lower than 23% for all numbers of attached nanoparticles with M p1 = 0.05 (see Fig. 9(a)). For N p ≤ 5, the discrepancies between the predicted results by the NRBT or NTBT and those of the NHOBT would generally magnify as the number of attached nanoparticles increases. For N p > 5, such discrepancies fairly remain unchanged. The discrepancies between the results of the NRBT and those of the NSDBTs are more apparent for low numbers of the attached nanoparticles for a DWCNT with SS or CC or SC boundary conditions (see Fig. 9(a)–(c)). This matter indicates that the NRBT would not be a trustable model for such conditions. According to the plotted results in Fig. 9(b) and (c) for the case of M p1 = 0.05, the NTBT could capture the results of the NHOBT with relative error lower than 18% 21% for CC and SC boundary conditions, respectively. A close scrutiny also reveals that there exists a slight discrepancy between the results of the NTBT and those of the NHOBT for low numbers of attached nanoparticles. In the case of a DWCNT-based mass sensor with Sf S boundary conditions, the NRBT overestimates the results of the NHOBT with relative error about 20% to 100% for all considered numbers of attached nanoparticles (see Fig. 9(d)). However, the predicted results by the NTBT are generally closer to those of the NHOBT, and this model could capture the results of the NHOBT with relative error lower than 25% for M p1 ≥ 0.03. In the case of a DWCNT with Sf C boundary conditions, both NRBT and NTBT overestimate the predicted FSRs by the NHOBT (see Fig. 9(e)). As the number of attached nanoparticles increases, the discrepancies between the results of the NTBT and those of the NHOBT usually decrease.

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K. Kiani / Applied Mathematics and Computation 270 (2015) 216–241

R

(a)

R

(b)

R

(c)

R

(d)

R

(e)

R

(f)

0.04

0.06

0.08

0.02

0.03

0.04

0 1 0.08

0 1 0.08

4

7

10

0.04

0.04

0 1 0.06

0 1 0.08

4

7

10

0.03

0.04

0 1 0.02

0 1 0.04

4

7

10

0.01

0.02

0 1 0.03

0 1 0.06

4

7

10

0.015 0 1 0.03

7

10

4

7

10

0 1 0.06

4

7

10

4

7

N

p

10

4

7

10

0 1 0.08

4

7

10

4

7

10

4

7

10

4

7

10

7

10

0.04 4

7

10

0 1 0.06 0.03

4

7

10

0 1 0.06 0.03

4

7

10

0 1 0.06 0.03

0.03 0 1

0 1 0.1 0.05

0.03

0.015 0 1

4

4

7

Np

10

0 1

4

Np

Fig. 9. Frequency shift ratio of the DWCNT due to attached nanoparticles in terms of the number of nanoparticles for different boundary conditions: (a) SS, (b) CC, (c) SC, (d) Sf S, (e) Sf C, (f) CF; ((◦)M p1 = 0.01, (

)M p1 = 0.03, ()M p1 = 0.05; (…) NRBT, (−.−) NTBT, (—) NHOBT; Kr = Kt = Nbi = 0; λ1 =10, e0 a =1 nm).

In the case of a DWCNT with CF boundary conditions, the NRBT generally overestimates the results of the NSDBTs, particularly for low numbers of the attached nanoparticles (see Fig. 9(f)). The discrepancies between the results of the NRBT and those of the NHOBT would lessen as the number of the attached nanoparticles increases. 7.3.7. A discussion on the limitations of the proposed models Herein, it should be emphasized that the suggested models have some limitations in which explained in the following: (i) these models only consider the shift in frequency of the lateral deformation (i.e., both shear and flexural frequencies). To capture the shift in torsional frequencies due to the addition of nanoparticles, appropriate nonlocal shell models should be implemented. (ii) The suggested models have no sense to the viscosity of the surrounding medium and supports. By consideration of a viscoPasternak foundation in modeling of the surrounding medium and supports, the environment of the nanostructure is modeled more rationally. (iii) Herein, interactions between the attached nanoparticles and the DWCNTs have been ignored in the formulations of the proposed models. In fact, it is assumed that all nanoparticles have been tightly attached to the nanostructure. However, for a more realistic modeling of a system of nanoparticles and the DWCNT, considering the existing vdW forces between the nanoparticles and the nanotubes would lead to more accurate results. Role of the vdW force in sensing of nanoparticles by DWCNTs can be considered as an important subject for future works. (iv) Each nanoparticle has been attached to the outer surface of the DWCNT at a small portion of its length such that the length of the contact zone is approximately negligible in compare to the length of the DWCNT. It implies that the nanoparticles are modeled by inertial pointed loads which are defined by the Dirac delta function in the formulations of the problem. If the length and lateral stiffness of the attached colony of nanoparticles would be comparable with those of the DWCNT, the structure of the attached nanoparticle should be appropriately modeled and the present formulations are not applicable in such a case. Most of these limitations depict the scientific gaps in the field of the mechanical behavior of nanosensors. These guidelines should be carefully paid attention to by the nanotechnology community and more explorations on these topics will surely increase our knowledge about their near to exact mechanisms of sensing of nano-objects. 8. Conclusions From applied mechanics point of view, the possibility of exploiting DWCNTs as mass sensors is studied in the context of nonlocal continuum mechanics of Eringen for a wide range of boundary conditions. The DWCNT is modeled using doubly continuumbased tubes. The deformation fields of each tube are represented according to the nonlocal Rayleigh, Timoshenko, and higherorder beam theories. The interaction between the atoms of the innermost and outermost tubes due to the van der Waals forces is modeled by a continuous lateral spring connecting the tubes. The attached nanoparticles are considered as rigid bodies accounting for their lateral and rotary inertial effects. Based on different beam theories, the free vibration equations of motion of the DWCNT-based mass sensors are derived using Hamilton’s principle and a meshless technique. The main physical feature of addition of nanoparticles to the DWCNT is the change of its natural frequencies. Therefore, investigation the influential factors

K. Kiani / Applied Mathematics and Computation 270 (2015) 216–241

239

on the resonant frequency of the DWCNT would be of great importance for optimal design of DWCNTs as mass nanosensors. The effects of slenderness ratio, small-scale parameter, initially axial force, stiffness of the surrounding medium, mass weight and number of the attached nanoparticles on the frequency shift of DWCNTs are examined in some detail for different boundary conditions. The main obtained results are summarized in the following: 1. By an increase of the slenderness ratio of the DWCNT, the discrepancies between the predicted FSRs by the proposed models would decrease. In fact, the ratio of the shear strain energy to the total strain energy of the nanosystem would reduce as the slenderness ratio increases. Further studies shows that the trend of the plots of the FSRs as a function of slenderness ratio depends on the chosen nonlocal beam model and the boundary condition of the DWCNT. 2. In the cases of SS, CC, and SC boundary conditions, the NSDBTs predict that the FSRs of the DWCNT would commonly reduce as the small-scale parameter grows; however, the predicted FSRs by the NSDBTs in the cases of the Sf C and CF boundary conditions generally magnify with the small-scale parameter. Since the considered DWCNT is very stocky, the obtained results by the NRBT are not in line with those of the NSDBTs. 3. Except the CF boundary condition, the NSDBTs predict that the FSR of the nanosystem would grow as the lateral stiffness of the surrounding medium increases. By increasing of the rotational stiffness of the surrounding medium of the DWCNT not only the obtained FSRs by the NSDBTs would magnify, but also the discrepancies between the results of the NRBT and those of the NSDBTs increase. Generally, the FSRs of the NSDBTs are overestimated by the NRBT. This issue is mainly related to this fact that the shear deformation is not incorporated into the formulations of the NRBT. 4. For all studied boundary conditions, all proposed models display that the FSRs of the nanomechanical sensor of our concern would magnify as the mass weight or the number of the attached nanoparticle increases. Such an issue is related to an increase of both transverse and rotary inertia of the attached nanoparticles. Additionally, the predicted FSRs by the NRBT are generally greater than those of the NTBT and NHOBT. The proposed models in this work can be employed as a pivotal step for nanomechanical sensing of arbitrary attached nanoparticles by multi-walled carbon nanotubes. Additionally, the proposed nonlocal numerical models can be efficiently used for potential application of ensembles of single-walled carbon nanotubes or DWCNTs as mass sensors. Such interesting topics can be regarded for future works. Appendix A. Frequency analysis of simply supported DWCNTs as nanomechanical sensors via GM In this part, application of GM for frequency analysis of simply supported DWCNTs with attached nano-objects will be explained in some detail. For solving the problem via GM, the performed RKPM’s calculations based on the NTBT and NHOBT are appropriately revisited. Thereafter, the mass and stiffness matrices of the problem for the proposed nonlocal shear deformable beam theories are reconstructed. A1. The GM calculations based on the NTBT In order to analyze the problem by GM, the dimensionless displacement fields of the innermost and outermost tubes are expressed in terms of admissible shape functions as follows: T

wi

(ξ , τ ) =

NMi 

T

φkwi (ξ )wTik (τ ), θ i (ξ , τ ) =

k=1

w

NMi  T φ θi (ξ )θ (τ ); i = 1, 2

(A.1)

ik

k

k=1

θ

where φk i and φk i respectively denote the kth mode shape function associated with the deflection and rotational fields of the ith tube, and NMi is the number of vibration modes pertinent to the ith tube. In the case of simply supported innermost and √ √ w θ outermost tubes, such functions are expressed as: φk i (ξ ) = 2 sin (kπ ξ ) and φk i (ξ ) = 2 cos (kπ ξ ). By substituting Eq. (A.1) into Eq. (35), the nonzero matrices in Eq. (37) are evaluated as, T

i wi [Mb ]w = 12i−2 (1 + kl (μπ )2 )δkl + (1 − (−1)i ) kl

Np 

m p j (1 + (μπ k)2 ) sin (kπ ξ p j ) sin (l π ξ p j ),

(A.2a)

j=1

[Mb ]θkli θi = T

2i−2 λ−2 (1 + kl (μπ )2 )δkl + (1 − (−1)i ) 1 2

Np 

λ−2 p j ( cos (kπ ξ p j ) cos (l π ξ p j )

j=1

+ kl (μπ )2 sin (kπ ξ p j ) sin (l π ξ p j )), T





i wi [Kb ]w = kl π 2 42i−2 + kl 1 + (μπ )2 kl



T

(A.2b) T

T

Nbi kl + C v + (i − 1)K t



δkl ,

(A.2c)

i θi [Kb ]w = −kπ δkl , kl

(A.2d)

[Kb ]θkli wi = −l π δkl ,

(A.2e)

T

T

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K. Kiani / Applied Mathematics and Computation 270 (2015) 216–241

[Kb ]θkli θi = T

T w wj

[Kb ]kli



 T 42i−2 + η32i−2 kl π 2 + kl (1 + (μπ )2 )(i − 1)K r δkl ,

(A.2f)

T

= −C v (1 + (μπ k)2 )δkl ; i = j,

(A.2g)

where δ kl represents the Kronecker delta. Subsequently, by following the procedure mentioned in Section 4.2, the natural frequencies of the system would be readily determined. A2. The GM calculations based on the NHOBT For a simply supported DWCNT in which modeled based on the NHOBT, the dimensionless displacements are expressed as: H

wi

(ξ , τ ) =

NMi 

H

φkwi (ξ )wHik (τ ), ψ i (ξ , τ ) =

k=1

NMi  ψi

H

φk (ξ )ψ ik (τ ); i = 1, 2

(A.3)

k=1

√ √ w ψ where φk i (ξ ) = 2 sin (kπ ξ ) and φk i (ξ ) = 2 cos (kπ ξ ). Now, by introducing Eq. (A.3) to Eq. (46), the mass and stiffness matrices in Eq. (48) are obtained as, H ψi ψi

[Mb ]kl

= ϑ72i−2 (1 + kl (μπ )2 )δkl + (1 − (−1)i )

Np 

I p j ( cos (kπ ξ p j ) cos (l π ξ p j )

j=1

+ kl (μπ )2 sin (kπ ξ p j ) sin (l π ξ p j )), H ψi wi

[Mb ]kl

(A.4a)

= −ϑ22i−2 γ62 kπ (1 + (μπ l )2 )δkl ,



H

i wi [Mb ]w = (1 + (μπ kl )2 ) kl

(A.4b)

Np   ϑ12i−2 + ϑ32i−2 (π kl γ2 )2 δkl + (1 − (−1)i ) m p j ( sin (kπ ξ p j ) sin (l π ξ p j ) j=1

+ kl (μπ ) cos (kπ ξ p j ) cos (l π ξ p j )), 2

H w ψi

[Mb ]kli



= −ϑ22i−2 γ12 kπ 1 + (μπ )2 kl

H

i wi [Kb ]w = kl

H w ψi

[Kb ]kli

H ψi wi

[Kb ]kl

H ψi ψi

[Kb ]kl

H wi w j

[Kb ]kl



δkl ,

  H H H ϑ42i−2 γ32 π 2 kl + ϑ62i−2 π 4 k2 l 2 + (1 + (μπ )2 ) kl π 2 Nbi + (i − 1)K t + C v δkl ,

= kπ = lπ =







(A.4c) (A.4d) (A.4e)

 ϑ42i−2 γ32 − kl π 2 ϑ52i−2 γ42 δkl ,

(A.4f)

 ϑ42i−2 γ72 − kl π 2 ϑ52i−2 γ92 δkl ,

(A.4g)



 H ϑ42i−2 γ72 + kl π 2 ϑ82i−2 γ82 + K r (1 + kl (μπ )2 ) δkl , H

= −C v (1 + kl (μπ )2 )δkl ; i = j.

(A.4h) (A.4i)

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