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Transverse vibrations of embedded nanowires under axial compression with high-order surface stress effects
Physica E 66 (2015) 238–244
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Physica E journal homepage: www.elsevier.com/locate/physe
Transverse vibrations of embedded nanowires under axial compression with high-order surface stress effects Y.Q. Zhang a, M. Pang a,n, W.Q. Chen b a b
College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, China
H I G H L I G H T S
G R A P H I C A L
Transverse vibration of an embedded nanowire is investigated based on the high-order surface stress model. Closed-form expressions for the natural frequency and axial buckling load are derived. Influences of axial load, high-order surface stress and surrounding medium on the natural frequency are analyzed. Effects of high-order surface stress and surrounding medium on the buckling load are discussed.
Effects of compressive axial load, high-order surface stress and surrounding elastic medium on the natural frequency of nanowire in the state of transverse vibration.
art ic l e i nf o
a b s t r a c t
Article history: Received 22 August 2014 Received in revised form 15 October 2014 Accepted 21 October 2014 Available online 23 October 2014
Implementing the high-order surface stress model into the Bernoulli–Euler beam theory, the transverse vibration of an axially compressed nanowire embedded in elastic medium is investigated. Closed-form expression is obtained for the natural frequency of a simply supported nanowire. The influences of compressive axial load, high-order surface stress and surrounding elastic medium on the natural frequency are discussed. Additionally, the analytical solution of axial buckling load for the simply supported nanowire is derived, which takes into account the effects of high-order surface stress and surrounding elastic medium. It is concluded from numerical results that the natural frequency of transverse vibration of the nanowire is dependent upon axial load, surrounding elastic medium, and high-order surface stress. Similarly, the dependences of the buckling load on surrounding elastic medium and high-order surface stress are significant. & Elsevier B.V. All rights reserved.
Keywords: Transverse vibration Embedded nanowire Axial compression High-order surface stress effect
A B S T R A C T
1. Introduction It is known that nanostructures have received tremendous attention from various branches of science in the last two decades. By the use of varieties of experimental, theoretical and computer simulation approaches, extensive research studies of properties of nanostructures have been carried out [1–12]. As a kind of n
Corresponding author. E-mail addresses: [email protected], [email protected] (M. Pang).
http://dx.doi.org/10.1016/j.physe.2014.10.027 1386-9477/& Elsevier B.V. All rights reserved.
nanostructure, nanowires can be defined as structures that have a thickness or diameter constrained to tens of nanometers or less and an unconstrained length. Nanowires have a wide range of potential applications, such as actuators, sensors, resonators, transistors and probes in nanoelectromechnical systems (NEMS) and biotechnology [13–17]. As a Consequence, a thorough understanding of the mechanical responses of individual nanowires is of great necessity for their potential applications. Due to the high ratio of surface area to volume at the nanoscale, the significant surface effects on the mechanical behavior of
Y.Q. Zhang et al. / Physica E 66 (2015) 238–244
nanowires have been observed both theoretically and experimentally [18–20]. A large number of numerical approaches including the molecular simulations and the finite element-based methods have been developed to account for surface effects of nanowires [21–28]. For simplicity and efficiency, analytical continuum models including surface effects have been widely employed to study the mechanical behavior of nanowires. Based on the Euler-Bernoulli beam theory and Young–Laplace equation, He and Lilley [29] studied the dependence of the surface effect on the overall Young's modulus of nanowires for different boundary conditions. Using a continuum surface model, Zhang et al. [30] investigated the surface stress effects on size-dependent elastic moduli and yield strength of nanowires. Wang and Feng [31,32] analyzed the influence of surface energy on the buckling and vibration of nanowires and obtained analytical solutions for the critical buckling load and the natural frequency with surface effects. Using a comprehensive Timoshenko beam model, Jiang and Yan [33] studied the surface effects on the elastic behavior of static bending nanowires. Liu et al. [34] conducted an investigation of large displacement of a static bending nanowire considering surface effects. Taking account of the surface effects, Juntarasaid et al. [35] derived the analytical solutions for static displacement and buckling load of nanowires. It is noted that all the analytical continuum models developed in the studies mentioned above are mainly on the basis of the Gurtin–Murdoch framework [36,37] of membrane theory. In this framework, the thin interface is thought of as a curved membrane sustaining only in-plane stresses. This assumption yields the generalized Young–Laplace (YL) equation which characterizes the field jumps across the interface [38]. The generalized YL equation can be regarded as a first-order approximation of the mechanical properties of a surface or interface. However, via experimental study, Liu et al. [39] found that there existed a local bending phenomenon in addition to a uniform bending mode in a nanoscaled thin film, which could be owing to the effect of surface moment. As a result, it can be concluded that the thin layer is capable to sustain both the surface stress and the surface moment. From the point of view of mechanics, the presence of surface moment is attributed to the non-uniform in-plane stress across the thin layer thickness, and thus the incorporation of surface moment is more relevant to the real deformation mechanism [40]. In the framework of theory of elasticity, the surface moment can be considered as a high-order term of surface stress. It has been demonstrated that high-order surface stress effects on mechanical properties of nanowires are significant and should not be overlooked [40,41]. Recently, Chiu and Chen [42] studied the high-order surface stress effects on vibration behaviors of nanowires. The high-order surface stress effect includes the influences of the in-plane membrane surface stress and the surface moment. In their work, the vibrational properties of nanowires were discussed only for the case of no axial stress. As a matter of fact, the nanowires in such applications as sensors and transistors often need to sustain axial compression [31]. Hence, the effect of axial load on the mechanical properties of nanowires is of practical interest. In addition, nanowires are often embedded in substrate media in NEMS, and it is necessary to investigate the influences of surrounding media on the mechanical properties of nanowires [43]. Consequently, in this present study, the transverse vibration of an axially compressed nanowire embedded in elastic medium is analyzed. The effects of axial load, surrounding elastic medium and highorder surface stress on the vibrational properties of the nanowire with simply supported ends are discussed. Also, the critical axial buckling load for the nanowire is derived.
239
2. Formulation Surface stresses can be delineated by a thin layer which is firmly bonded to a bulk volume and subjected to in-plane stress along the curved plane. The surface stress tensor is related to the deformation-dependent surface energy and can be expressed as the derivatives of surface energy. A mathematical framework for surfaces or interfaces with surface stress was developed by Gurtin and Murdoch [36]. This mathematical relation is generally referred to as generalized YL equation, which links the fields between two neighboring regions. It was shown by Chen et al. [38] that the generalized YL equation can be directly constructed by considering the force balance condition along the curved thin interface, in which only in-plane membrane stress is taken into account. A more rigorous approach in the modeling of surface stress is to allow that the in-plane membrane stress could be varying across the layer thickness. In this context, the membrane can sustain both in-plane stress and surface moment. In contrast to the conventional surface stress, it is referred as high-order surface stress which includes surface stress and surface moment [42]. In what follows, the influences of high-order surface stress on the vibrational properties of nanowires are investigated in the framework of Euler–Bernoulli beam theory. Taking account of the high-order surface stress effects, the equivalent flexural rigidity (EI)⁎ for rectangular and circular cross sections has been derived, which is expressed as [40]
(EI)⁎ =
1 Ewh3 12
1
1
+ 2 Eswh2 + 6 Esh3 + Eswhh1 + 8Ds w ,
(rectangular)
(EI)⁎ =
1 πEd 4 64
1
(1a)
+ 8 πEsd3,
(circular)
(1b)
where E is the Young's modulus, Es is the surface Young's modulus, Ds is the surface bending stiffness, h1 is the surface layer thickness, w and h are the width and height of a rectangular cross section, and d is the diameter of a circular nanowire. It should be pointed out that the last two terms in Eq. (1a) can be ignored as their values are infinitesimal compared with other terms, and the equivalent flexural rigidity given by Eq. (1b) for a circular nanowire is the same as that with pure surface stress effect [40]. Hence, the equivalent flexural rigidity (EI)⁎ in the present context has the same expression as that given by Wang and Feng [31]. Additionally, Chiu and Chen [40] showed that the effect of high-order surface stress will generate a distributed transverse force acting on the nanowire. Assuming that one end of the nanowire is the origin of the coordinate system and the x coordinate axis is along the axial direction of the wire, the distributed transverse force per unit axial length can be derived by [40]
qs = H
∂ 2v ∂ 4v −K 4 2 ∂x ∂x
(2)
where v denotes the transverse displacement of the wire, and
H = 2σsw ,
K = 2Ds w
H = 2σsd,
K = 2Ds d
(rectangular) (circular)
(3a) (3b)
where Ds is the surface bending stiffness and σs is the surface stress which is given by
σs = τ0 + Esεs
(4)
where τ0 is the residual surface tension when the bulk is unstrained, and εs is the longitudinal surface strain of the nanowire. For an embedded nanowire, the pressure per unit axial length, acting on the wire due to the surrounding elastic medium, can be
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described by a Winkler-like model [44]
qm = − ηv
It follows from Eq. (14) that
(5)
where the negative sign indicates that the pressure qm is opposite to the deflection of the wire and η is the stiffness modulus reflecting the interaction of the wire and the surrounding elastic medium. As known, the Bernoulli–Euler beam theory is based upon the assumption that plane cross-sections of a beam remain plane during flexure and that the radius of curvature of a bent beam is large compared with the beam's depth. Considering the effects of high-order surface stress, the general equation for transverse vibrations of an embedded nanowire under compressive axial load and distributed transverse pressure is expressed by
qs + qm = (EI)⁎ v IV + F v″ + ρAv¨
(6)
where F is the compressive axial load, A is the cross-sectional area of the wire, and ρ is the mass density of the nanowire. In addition, we define
v′ =
∂v , ∂x
v˙ =
∂v ∂t
(7)
where t is time. Substituting Eqs. (2) and (5) into Eq. (6), we have
[(EI)⁎ + K]v IV + (F − H) v″ + ηv + ρAv¨ = 0
(8)
Introducing Eqs. (3) and (4) into Eq. (8) and using small deformation approximations εs ≈ − (h/2)v″ and εs ≈ − (d/2)v″ for rectangular and circular nanowires respectively [29], Eq. (8) becomes
[(EI)⁎ + K]v IV + (F − H0)v″ + ηv + ρAv¨ = 0
(9)
where the component of (v″)2 is ignored for small deformation, and
H0 = 2τ0w
(rectangular)
(10a)
H0 = 2τ0d
(circular)
(10b)
Letting η = F = 0, Eq. (9) can be simplified to the governing equation derived by Chiu and Chen [42].
ρAT¨n + [(EI)⁎k n4 + Kk n4 + η − (F − H0)k n2]Tn = 0 The solution of Eq. (15) can be given by
Tn(t) = Cne jωnt ,
Suppose that the length of the nanowire is l and the ends are simply supported, the boundary conditions are given by
v(0, t) = v(l, t) = 0,
v″(0, t) = v″(l, t) = 0
j=
−1
(16)
where ωn denotes the natural frequency of the nanowire and Cn represents the amplitude coefficient. Substituting Eq. (16) into Eq. (15), we have
[(EI)⁎k n4 + Kk n4 + η − (F − H0)k n2 − ρAωn2]Cn = 0
(17)
When the nanowire is in the state of vibration, the amplitude coefficient Cn cannot be equal to zero, which yields the following frequency (characteristic) equation
(EI)⁎k n4 + Kk n4 + η − (F − H0)k n2 − ρAωn2 = 0
(18)
Then from the characteristic Eq. (18), we obtain
ωn =
(EI)⁎k n4 + Kk n4 + η − (F − H0)k n2 ρA
(19)
When the influences of surrounding elastic medium and axial compressive load are ignored (η = F = 0), Eq. (19) can be reduced to the solution for the natural frequency given by Chiu and Chen [42], which includes the effects of surface stress and surface moment and is expressed as
ωnc =
nπ l2
[(EI)⁎ + K]n2π 2 + H0l2 ρA
(20)
Assuming η = Κ = F = 0, Eq. (19) can be reduced to the solution with the effect of pure surface stress, which is obtained by Wang and Feng [45] and expressed as
ωnw =
nπ l2
(EI)⁎n2π 2 + H0l2 ρA
(21)
Neglecting surface effect and influences of surrounding elastic medium and axial compressive load (η = Κ = Η0 = F = 0 and (EI)⁎ = EI ), the classical solution of natural frequency can be recovered from Eq. (19), which is given by [46]
ωn0 = 3. Solution of the problem
(15)
n2π 2 l2
EI ρA
(22)
4. Application and discussion
(11)
The homogeneous partial differential Eq. (9) with the governing boundary conditions can be solved by the Bernoulli–Fourier method assuming the solution in the form
For numerical illustrations of mechanical properties of nanowires, a circular silver nanowire with simply supported ends is herein considered. The values of some material parameters for the silver nanowire are listed as follows [29,31,42]:
∞
v(x, t) =
E = 76 GPa,
∑ X n(x)Tn(t)
where Tn(t) is the unknown time function and Xn(x) is the known mode shape function for the simply supported wire, which is expressed as
X n(x) = sin(k nx),
kn =
nπ , l
n = 1, 2, 3, …
Es = 1.22 N /m
At a sufficiently large compressive axial force the natural frequency becomes zero and the nanowire with ends simply supported transversely buckles. Putting ωn = 0 in Eq. (18), we have
(14)
(EI)⁎k n4 + Kk n4 + η − (F − H0)k n2 = 0
∞
∑ {ρAT¨n + [(EI)⁎kn4 + Kkn4 + η − (F − H0)kn2]Tn}X n = 0
4.1. Axial buckling of nanowires
(13)
Introduction of Eq. (12) into Eq. (9) yields
n= 1
τ0 = 0.89 N /m,
(12)
n= 1
(23)
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241
It follows from Eq. (23) that the value of the buckling force corresponding to vibration mode n can be obtained by
Fb =
n2π 2 l2η [(EI)⁎ + K] + 2 2 + H0 2 l nπ
(24)
When η = 0 and n = 1, the critical buckling load derived by Chiu and Chen [42] can be recovered from Eq. (24), which is expressed as
Fcrc =
π2 [(EI)⁎ + K] + H0 l2
(25)
When η = 0 and ignoring the effects of surface stress and surface moment, the classical buckling load can be obtained from Eq. (24), which is known as [47]
Fbc =
n2π 2EI l2
(26)
Furthermore, with n = 1, the classical critical buckling load can be given from Eq. (26), which is known as
Fcr0 =
π EI l2
(27)
To examine the influence of surrounding elastic medium on the buckling load, we define
χI =
Fig. 2. The ratio χI versus the aspect ratio l/d for various diameters of the wire.
2
Fb Fb
(28)
η= 0
Combination of Eqs. (1b), (3b), (10b), (24) and (28) gives
χI = 1 +
64η′Εl
χII =
4
n4 π 4d(πEd3 + 8πEsd2 + 128Ds ) + 128n2π 2l2τ0d
(29)
where η′ is a non-dimensional parameter, which is defined by η′ = η/E . Assuming d = 20 nm , l/d = 20, and Ds = 0, the effect of surrounding elastic medium on the buckling load is shown in Fig. 1. It is seen that the buckling load increases with the increase of the stiffness modulus of surrounding medium, and the effect of surrounding elastic medium is related to the mode number n. The larger the mode number n is, the less significant the effect of surrounding medium on the buckling load becomes. In addition, the effect of surrounding medium on the buckling load is also dependent on the aspect ratio and diameter of the nanowire. With n = 2, Ds = 0, and η′ = 0. 0001, the influences of the aspect ratio
Fig. 1. Effect of surrounding elastic medium on the buckling load.
and diameter on the effect of surrounding medium are shown in Fig. 2. It is indicated that the effect of surrounding elastic medium on the buckling load increases with the increase of the aspect ratio and diameter of the nanowire. To illustrate the effect of high-order surface stress on the buckling load, it is defined that
Fb Fbc
(30)
Substituting Eqs. (24) and (26) into Eq. (30) and combining Eqs. (1b), (3b) and (10b), we have
χII = 1 +
8Es 128Ds 128τ l2 64η′l 4 + + 4 5 4 + 2 30 3 3 Ed πEd n π d n π Ed
(31)
With d = 30 nm , η = 0, and n = 2, the variation of the ratio χII with the aspect ratio l/d is obtained for various values of Ds , as shown in Fig. 3. The values of Ds used for numerical calculations are the same as those used by Chiu and Chen [42]. It is observed from Fig. 3 that there exist discrepancies between the solutions based on the present model considering the effects of surface stress and surface moment and those by the pure surface stress model (generalized YL equation). The buckling load becomes larger with
Fig. 3. Effect of high-order surface stress on the buckling load.
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Fig. 4. The ratio χII versus the diameter d of the nanowire for various mode numbers (η = 0, l/d = 20, Ds = 1 × 10−14 N m)
Fig. 5. Effect of axial load on the natural frequency (η = Ds = 0, d = 30 nm, l/d = 20)
increasing the value of Ds , and the effect of high-order surface stress is related to the aspect ratio of the nanowire. The influence of high-order surface stress on the buckling load becomes more significant with the increase of the aspect ratio. Additionally, the effect of high-order surface stress on the buckling load is also sensitive to the mode number n and the diameter of the nanowire, as illustrated in Fig. 4. It is shown that the effect of high-order surface stress on the buckling load decreases with the increase of the mode number n and the diameter of the wire. 4.2. Influences on natural frequency In the following, let us discuss the effects of compressive axial load, surrounding elastic medium and high-order surface stress on the transverse vibration of the circular silver nanowire with simply supported ends. To investigate the influences of compressive axial load and surrounding elastic medium on the natural frequency of the nanowire, it is introduced that
ψI =
ωn ωnc
Fig. 6. The ratio ψI versus the aspect ratio l/d for various diameters of the wire (η = Ds = 0, n = 2, F ′ = 0.2) .
(32)
Inserting Eqs. (19) and (20) into Eq. (32) and combining Eqs. (1b), (3b) and (10b), we obtain
ψI =
1+
E(64η′l 4 − F ′n2π 5d 4) 3
(πEd + 8πEsd2 + 128Ds )n4 π 4d + 128n2π 2τ0dl2
(33)
with
F′ =
F Fcr0
(34)
Figs. 5 and 6 show the effects of axial load on the natural frequency of transverse vibration of the nanowire. It is found from Fig. 5 that the ratios ψI diminishes with increasing the axial load, which implies that the natural frequency becomes smaller when the axial load gets larger. It can also be found from Figs. 5 and 6 that the effect of the axial load is related to the vibration mode, aspect ratio and diameter of the nanowire. The effect of axial load on the natural frequency gets less significant with the increase of mode number n and aspect ratio l/d , while becoming more significant with the increase of diameter.
Fig. 7. Effect of surrounding elastic (F = Ds = 0, d = 30 nm, l/d = 20) .
medium on
the natural frequency
Figs. 7 and 8 illustrate the influences of surrounding elastic medium on the natural frequency of transverse vibration of the nanowire. From Fig. 7, it can be observed that the augmentation of
Y.Q. Zhang et al. / Physica E 66 (2015) 238–244
Fig. 8. The ratio ψI versus the aspect ratio l/d for various diameters of the wire (F = Ds = 0, n = 2, η′ = 0.0001)
243
Fig. 9. Effect of high-order surface stress on the natural frequency.
the non-dimensional parameterη′ brings about an increase of ratio ψI . In other words, the natural frequency gets larger with the increase of the stiffness modulus of surrounding medium. From Figs. 7 and 8, we note that the effect of surrounding elastic medium on the natural frequency is dependent upon the vibration mode, aspect ratio and diameter of the nanowire. It becomes more pronounced with the decrease of mode number n and the increase of aspect ratio and diameter of the wire. To examine the effect of high-order surface stress on the natural frequency, we define
ψII =
ωn ωne
(35)
ωne
where is the natural frequency excluding the surface effect, which is expressed as
ωne =
EIk n4 + η − Fk n2 ρA
(36)
Introduction of Eqs. (19) and (36) into Eq. (35) and combination of Eqs. (1b), (3b) and (10b) yields
ψII =
1+
Fig. 10. The ratio ψII versus the diameter d for various mode numbers. (F = η = 0, l/d = 20, Ds = 1 × 10−14 N m) .
8n2π 2d(n2π 3Esd2 + 16n2π 2Ds + 16τ0l2) E(n4 π 5d 4 + 64η′l 4 − n2π 5F ′d 4)
(37)
With d = 30 nm , η = F = 0, and n = 2, the relationship curves between the ratio ψII and the aspect ratio l/d are obtained for different values of Ds , as indicated in Fig. 9. As can be seen, Fig. 9 displays the discrepancies between the results on the basis of the present model including the effects of surface stress and surface moment and those based on the pure surface stress model (generalized YL equation). The natural frequency increases with the increase of the magnitude of Ds , and the effect of high-order surface stress is sensitive to the aspect ratio of the nanowire. The influence of high-order surface stress on the natural frequency becomes more significant with the increase of the aspect ratio. It is shown from Fig. 10 that the effect of high-order surface stress on the natural frequency is also dependent upon the mode number n and the diameter of the nanowire. It is indicated that the increase of the mode number n and the diameter of the wire will make the effect of high-order surface stress less significant. 5. Conclusions Taking into account the effects of surface stress and surface moment, the transverse vibration of an axially compressed
nanowire embedded in elastic medium is investigated on the basis of the high-order surface stress model and Bernoulli–Euler beam theory. Explicit expression is derived for natural frequency for the case of a simply supported nanowire, and the influences of compressive axial load, high-order surface stress and surrounding elastic medium on the natural frequency are analyzed. In addition, the analytical solution of axial buckling load for the simply supported nanowire is obtained, which includes the effects of highorder surface stress and surrounding elastic medium. Through numerical results and discussions for a circular silver nanowire, it is concluded that the natural frequency of the nanowire is dependent upon axial load, surrounding elastic medium, and high-order surface stress. The effects of axial load, surrounding elastic medium, and high-order surface stress on the natural frequency are sensitive to the vibration modes, diameter, and aspect ratio of the nanowire. Similarly, the dependences of the buckling load on surrounding elastic medium and high-order surface stress are significant, and their influences on the buckling load are related to the mode number, diameter, and aspect ratio of the nanowire.
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Acknowledgments Authors acknowledge support from National Natural Science Foundation of China (NSFC) (No. 11172265).
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Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Transverse vibrations of embedded nanowires under axial compression with high-order surface stress effects Y.Q. Zhang a, M. Pang a,n, W.Q. Chen b a b
College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, China
H I G H L I G H T S
G R A P H I C A L
Transverse vibration of an embedded nanowire is investigated based on the high-order surface stress model. Closed-form expressions for the natural frequency and axial buckling load are derived. Influences of axial load, high-order surface stress and surrounding medium on the natural frequency are analyzed. Effects of high-order surface stress and surrounding medium on the buckling load are discussed.
Effects of compressive axial load, high-order surface stress and surrounding elastic medium on the natural frequency of nanowire in the state of transverse vibration.
art ic l e i nf o
a b s t r a c t
Article history: Received 22 August 2014 Received in revised form 15 October 2014 Accepted 21 October 2014 Available online 23 October 2014
Implementing the high-order surface stress model into the Bernoulli–Euler beam theory, the transverse vibration of an axially compressed nanowire embedded in elastic medium is investigated. Closed-form expression is obtained for the natural frequency of a simply supported nanowire. The influences of compressive axial load, high-order surface stress and surrounding elastic medium on the natural frequency are discussed. Additionally, the analytical solution of axial buckling load for the simply supported nanowire is derived, which takes into account the effects of high-order surface stress and surrounding elastic medium. It is concluded from numerical results that the natural frequency of transverse vibration of the nanowire is dependent upon axial load, surrounding elastic medium, and high-order surface stress. Similarly, the dependences of the buckling load on surrounding elastic medium and high-order surface stress are significant. & Elsevier B.V. All rights reserved.
Keywords: Transverse vibration Embedded nanowire Axial compression High-order surface stress effect
A B S T R A C T
1. Introduction It is known that nanostructures have received tremendous attention from various branches of science in the last two decades. By the use of varieties of experimental, theoretical and computer simulation approaches, extensive research studies of properties of nanostructures have been carried out [1–12]. As a kind of n
Corresponding author. E-mail addresses: [email protected], [email protected] (M. Pang).
http://dx.doi.org/10.1016/j.physe.2014.10.027 1386-9477/& Elsevier B.V. All rights reserved.
nanostructure, nanowires can be defined as structures that have a thickness or diameter constrained to tens of nanometers or less and an unconstrained length. Nanowires have a wide range of potential applications, such as actuators, sensors, resonators, transistors and probes in nanoelectromechnical systems (NEMS) and biotechnology [13–17]. As a Consequence, a thorough understanding of the mechanical responses of individual nanowires is of great necessity for their potential applications. Due to the high ratio of surface area to volume at the nanoscale, the significant surface effects on the mechanical behavior of
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nanowires have been observed both theoretically and experimentally [18–20]. A large number of numerical approaches including the molecular simulations and the finite element-based methods have been developed to account for surface effects of nanowires [21–28]. For simplicity and efficiency, analytical continuum models including surface effects have been widely employed to study the mechanical behavior of nanowires. Based on the Euler-Bernoulli beam theory and Young–Laplace equation, He and Lilley [29] studied the dependence of the surface effect on the overall Young's modulus of nanowires for different boundary conditions. Using a continuum surface model, Zhang et al. [30] investigated the surface stress effects on size-dependent elastic moduli and yield strength of nanowires. Wang and Feng [31,32] analyzed the influence of surface energy on the buckling and vibration of nanowires and obtained analytical solutions for the critical buckling load and the natural frequency with surface effects. Using a comprehensive Timoshenko beam model, Jiang and Yan [33] studied the surface effects on the elastic behavior of static bending nanowires. Liu et al. [34] conducted an investigation of large displacement of a static bending nanowire considering surface effects. Taking account of the surface effects, Juntarasaid et al. [35] derived the analytical solutions for static displacement and buckling load of nanowires. It is noted that all the analytical continuum models developed in the studies mentioned above are mainly on the basis of the Gurtin–Murdoch framework [36,37] of membrane theory. In this framework, the thin interface is thought of as a curved membrane sustaining only in-plane stresses. This assumption yields the generalized Young–Laplace (YL) equation which characterizes the field jumps across the interface [38]. The generalized YL equation can be regarded as a first-order approximation of the mechanical properties of a surface or interface. However, via experimental study, Liu et al. [39] found that there existed a local bending phenomenon in addition to a uniform bending mode in a nanoscaled thin film, which could be owing to the effect of surface moment. As a result, it can be concluded that the thin layer is capable to sustain both the surface stress and the surface moment. From the point of view of mechanics, the presence of surface moment is attributed to the non-uniform in-plane stress across the thin layer thickness, and thus the incorporation of surface moment is more relevant to the real deformation mechanism [40]. In the framework of theory of elasticity, the surface moment can be considered as a high-order term of surface stress. It has been demonstrated that high-order surface stress effects on mechanical properties of nanowires are significant and should not be overlooked [40,41]. Recently, Chiu and Chen [42] studied the high-order surface stress effects on vibration behaviors of nanowires. The high-order surface stress effect includes the influences of the in-plane membrane surface stress and the surface moment. In their work, the vibrational properties of nanowires were discussed only for the case of no axial stress. As a matter of fact, the nanowires in such applications as sensors and transistors often need to sustain axial compression [31]. Hence, the effect of axial load on the mechanical properties of nanowires is of practical interest. In addition, nanowires are often embedded in substrate media in NEMS, and it is necessary to investigate the influences of surrounding media on the mechanical properties of nanowires [43]. Consequently, in this present study, the transverse vibration of an axially compressed nanowire embedded in elastic medium is analyzed. The effects of axial load, surrounding elastic medium and highorder surface stress on the vibrational properties of the nanowire with simply supported ends are discussed. Also, the critical axial buckling load for the nanowire is derived.
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2. Formulation Surface stresses can be delineated by a thin layer which is firmly bonded to a bulk volume and subjected to in-plane stress along the curved plane. The surface stress tensor is related to the deformation-dependent surface energy and can be expressed as the derivatives of surface energy. A mathematical framework for surfaces or interfaces with surface stress was developed by Gurtin and Murdoch [36]. This mathematical relation is generally referred to as generalized YL equation, which links the fields between two neighboring regions. It was shown by Chen et al. [38] that the generalized YL equation can be directly constructed by considering the force balance condition along the curved thin interface, in which only in-plane membrane stress is taken into account. A more rigorous approach in the modeling of surface stress is to allow that the in-plane membrane stress could be varying across the layer thickness. In this context, the membrane can sustain both in-plane stress and surface moment. In contrast to the conventional surface stress, it is referred as high-order surface stress which includes surface stress and surface moment [42]. In what follows, the influences of high-order surface stress on the vibrational properties of nanowires are investigated in the framework of Euler–Bernoulli beam theory. Taking account of the high-order surface stress effects, the equivalent flexural rigidity (EI)⁎ for rectangular and circular cross sections has been derived, which is expressed as [40]
(EI)⁎ =
1 Ewh3 12
1
1
+ 2 Eswh2 + 6 Esh3 + Eswhh1 + 8Ds w ,
(rectangular)
(EI)⁎ =
1 πEd 4 64
1
(1a)
+ 8 πEsd3,
(circular)
(1b)
where E is the Young's modulus, Es is the surface Young's modulus, Ds is the surface bending stiffness, h1 is the surface layer thickness, w and h are the width and height of a rectangular cross section, and d is the diameter of a circular nanowire. It should be pointed out that the last two terms in Eq. (1a) can be ignored as their values are infinitesimal compared with other terms, and the equivalent flexural rigidity given by Eq. (1b) for a circular nanowire is the same as that with pure surface stress effect [40]. Hence, the equivalent flexural rigidity (EI)⁎ in the present context has the same expression as that given by Wang and Feng [31]. Additionally, Chiu and Chen [40] showed that the effect of high-order surface stress will generate a distributed transverse force acting on the nanowire. Assuming that one end of the nanowire is the origin of the coordinate system and the x coordinate axis is along the axial direction of the wire, the distributed transverse force per unit axial length can be derived by [40]
qs = H
∂ 2v ∂ 4v −K 4 2 ∂x ∂x
(2)
where v denotes the transverse displacement of the wire, and
H = 2σsw ,
K = 2Ds w
H = 2σsd,
K = 2Ds d
(rectangular) (circular)
(3a) (3b)
where Ds is the surface bending stiffness and σs is the surface stress which is given by
σs = τ0 + Esεs
(4)
where τ0 is the residual surface tension when the bulk is unstrained, and εs is the longitudinal surface strain of the nanowire. For an embedded nanowire, the pressure per unit axial length, acting on the wire due to the surrounding elastic medium, can be
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described by a Winkler-like model [44]
qm = − ηv
It follows from Eq. (14) that
(5)
where the negative sign indicates that the pressure qm is opposite to the deflection of the wire and η is the stiffness modulus reflecting the interaction of the wire and the surrounding elastic medium. As known, the Bernoulli–Euler beam theory is based upon the assumption that plane cross-sections of a beam remain plane during flexure and that the radius of curvature of a bent beam is large compared with the beam's depth. Considering the effects of high-order surface stress, the general equation for transverse vibrations of an embedded nanowire under compressive axial load and distributed transverse pressure is expressed by
qs + qm = (EI)⁎ v IV + F v″ + ρAv¨
(6)
where F is the compressive axial load, A is the cross-sectional area of the wire, and ρ is the mass density of the nanowire. In addition, we define
v′ =
∂v , ∂x
v˙ =
∂v ∂t
(7)
where t is time. Substituting Eqs. (2) and (5) into Eq. (6), we have
[(EI)⁎ + K]v IV + (F − H) v″ + ηv + ρAv¨ = 0
(8)
Introducing Eqs. (3) and (4) into Eq. (8) and using small deformation approximations εs ≈ − (h/2)v″ and εs ≈ − (d/2)v″ for rectangular and circular nanowires respectively [29], Eq. (8) becomes
[(EI)⁎ + K]v IV + (F − H0)v″ + ηv + ρAv¨ = 0
(9)
where the component of (v″)2 is ignored for small deformation, and
H0 = 2τ0w
(rectangular)
(10a)
H0 = 2τ0d
(circular)
(10b)
Letting η = F = 0, Eq. (9) can be simplified to the governing equation derived by Chiu and Chen [42].
ρAT¨n + [(EI)⁎k n4 + Kk n4 + η − (F − H0)k n2]Tn = 0 The solution of Eq. (15) can be given by
Tn(t) = Cne jωnt ,
Suppose that the length of the nanowire is l and the ends are simply supported, the boundary conditions are given by
v(0, t) = v(l, t) = 0,
v″(0, t) = v″(l, t) = 0
j=
−1
(16)
where ωn denotes the natural frequency of the nanowire and Cn represents the amplitude coefficient. Substituting Eq. (16) into Eq. (15), we have
[(EI)⁎k n4 + Kk n4 + η − (F − H0)k n2 − ρAωn2]Cn = 0
(17)
When the nanowire is in the state of vibration, the amplitude coefficient Cn cannot be equal to zero, which yields the following frequency (characteristic) equation
(EI)⁎k n4 + Kk n4 + η − (F − H0)k n2 − ρAωn2 = 0
(18)
Then from the characteristic Eq. (18), we obtain
ωn =
(EI)⁎k n4 + Kk n4 + η − (F − H0)k n2 ρA
(19)
When the influences of surrounding elastic medium and axial compressive load are ignored (η = F = 0), Eq. (19) can be reduced to the solution for the natural frequency given by Chiu and Chen [42], which includes the effects of surface stress and surface moment and is expressed as
ωnc =
nπ l2
[(EI)⁎ + K]n2π 2 + H0l2 ρA
(20)
Assuming η = Κ = F = 0, Eq. (19) can be reduced to the solution with the effect of pure surface stress, which is obtained by Wang and Feng [45] and expressed as
ωnw =
nπ l2
(EI)⁎n2π 2 + H0l2 ρA
(21)
Neglecting surface effect and influences of surrounding elastic medium and axial compressive load (η = Κ = Η0 = F = 0 and (EI)⁎ = EI ), the classical solution of natural frequency can be recovered from Eq. (19), which is given by [46]
ωn0 = 3. Solution of the problem
(15)
n2π 2 l2
EI ρA
(22)
4. Application and discussion
(11)
The homogeneous partial differential Eq. (9) with the governing boundary conditions can be solved by the Bernoulli–Fourier method assuming the solution in the form
For numerical illustrations of mechanical properties of nanowires, a circular silver nanowire with simply supported ends is herein considered. The values of some material parameters for the silver nanowire are listed as follows [29,31,42]:
∞
v(x, t) =
E = 76 GPa,
∑ X n(x)Tn(t)
where Tn(t) is the unknown time function and Xn(x) is the known mode shape function for the simply supported wire, which is expressed as
X n(x) = sin(k nx),
kn =
nπ , l
n = 1, 2, 3, …
Es = 1.22 N /m
At a sufficiently large compressive axial force the natural frequency becomes zero and the nanowire with ends simply supported transversely buckles. Putting ωn = 0 in Eq. (18), we have
(14)
(EI)⁎k n4 + Kk n4 + η − (F − H0)k n2 = 0
∞
∑ {ρAT¨n + [(EI)⁎kn4 + Kkn4 + η − (F − H0)kn2]Tn}X n = 0
4.1. Axial buckling of nanowires
(13)
Introduction of Eq. (12) into Eq. (9) yields
n= 1
τ0 = 0.89 N /m,
(12)
n= 1
(23)
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It follows from Eq. (23) that the value of the buckling force corresponding to vibration mode n can be obtained by
Fb =
n2π 2 l2η [(EI)⁎ + K] + 2 2 + H0 2 l nπ
(24)
When η = 0 and n = 1, the critical buckling load derived by Chiu and Chen [42] can be recovered from Eq. (24), which is expressed as
Fcrc =
π2 [(EI)⁎ + K] + H0 l2
(25)
When η = 0 and ignoring the effects of surface stress and surface moment, the classical buckling load can be obtained from Eq. (24), which is known as [47]
Fbc =
n2π 2EI l2
(26)
Furthermore, with n = 1, the classical critical buckling load can be given from Eq. (26), which is known as
Fcr0 =
π EI l2
(27)
To examine the influence of surrounding elastic medium on the buckling load, we define
χI =
Fig. 2. The ratio χI versus the aspect ratio l/d for various diameters of the wire.
2
Fb Fb
(28)
η= 0
Combination of Eqs. (1b), (3b), (10b), (24) and (28) gives
χI = 1 +
64η′Εl
χII =
4
n4 π 4d(πEd3 + 8πEsd2 + 128Ds ) + 128n2π 2l2τ0d
(29)
where η′ is a non-dimensional parameter, which is defined by η′ = η/E . Assuming d = 20 nm , l/d = 20, and Ds = 0, the effect of surrounding elastic medium on the buckling load is shown in Fig. 1. It is seen that the buckling load increases with the increase of the stiffness modulus of surrounding medium, and the effect of surrounding elastic medium is related to the mode number n. The larger the mode number n is, the less significant the effect of surrounding medium on the buckling load becomes. In addition, the effect of surrounding medium on the buckling load is also dependent on the aspect ratio and diameter of the nanowire. With n = 2, Ds = 0, and η′ = 0. 0001, the influences of the aspect ratio
Fig. 1. Effect of surrounding elastic medium on the buckling load.
and diameter on the effect of surrounding medium are shown in Fig. 2. It is indicated that the effect of surrounding elastic medium on the buckling load increases with the increase of the aspect ratio and diameter of the nanowire. To illustrate the effect of high-order surface stress on the buckling load, it is defined that
Fb Fbc
(30)
Substituting Eqs. (24) and (26) into Eq. (30) and combining Eqs. (1b), (3b) and (10b), we have
χII = 1 +
8Es 128Ds 128τ l2 64η′l 4 + + 4 5 4 + 2 30 3 3 Ed πEd n π d n π Ed
(31)
With d = 30 nm , η = 0, and n = 2, the variation of the ratio χII with the aspect ratio l/d is obtained for various values of Ds , as shown in Fig. 3. The values of Ds used for numerical calculations are the same as those used by Chiu and Chen [42]. It is observed from Fig. 3 that there exist discrepancies between the solutions based on the present model considering the effects of surface stress and surface moment and those by the pure surface stress model (generalized YL equation). The buckling load becomes larger with
Fig. 3. Effect of high-order surface stress on the buckling load.
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Fig. 4. The ratio χII versus the diameter d of the nanowire for various mode numbers (η = 0, l/d = 20, Ds = 1 × 10−14 N m)
Fig. 5. Effect of axial load on the natural frequency (η = Ds = 0, d = 30 nm, l/d = 20)
increasing the value of Ds , and the effect of high-order surface stress is related to the aspect ratio of the nanowire. The influence of high-order surface stress on the buckling load becomes more significant with the increase of the aspect ratio. Additionally, the effect of high-order surface stress on the buckling load is also sensitive to the mode number n and the diameter of the nanowire, as illustrated in Fig. 4. It is shown that the effect of high-order surface stress on the buckling load decreases with the increase of the mode number n and the diameter of the wire. 4.2. Influences on natural frequency In the following, let us discuss the effects of compressive axial load, surrounding elastic medium and high-order surface stress on the transverse vibration of the circular silver nanowire with simply supported ends. To investigate the influences of compressive axial load and surrounding elastic medium on the natural frequency of the nanowire, it is introduced that
ψI =
ωn ωnc
Fig. 6. The ratio ψI versus the aspect ratio l/d for various diameters of the wire (η = Ds = 0, n = 2, F ′ = 0.2) .
(32)
Inserting Eqs. (19) and (20) into Eq. (32) and combining Eqs. (1b), (3b) and (10b), we obtain
ψI =
1+
E(64η′l 4 − F ′n2π 5d 4) 3
(πEd + 8πEsd2 + 128Ds )n4 π 4d + 128n2π 2τ0dl2
(33)
with
F′ =
F Fcr0
(34)
Figs. 5 and 6 show the effects of axial load on the natural frequency of transverse vibration of the nanowire. It is found from Fig. 5 that the ratios ψI diminishes with increasing the axial load, which implies that the natural frequency becomes smaller when the axial load gets larger. It can also be found from Figs. 5 and 6 that the effect of the axial load is related to the vibration mode, aspect ratio and diameter of the nanowire. The effect of axial load on the natural frequency gets less significant with the increase of mode number n and aspect ratio l/d , while becoming more significant with the increase of diameter.
Fig. 7. Effect of surrounding elastic (F = Ds = 0, d = 30 nm, l/d = 20) .
medium on
the natural frequency
Figs. 7 and 8 illustrate the influences of surrounding elastic medium on the natural frequency of transverse vibration of the nanowire. From Fig. 7, it can be observed that the augmentation of
Y.Q. Zhang et al. / Physica E 66 (2015) 238–244
Fig. 8. The ratio ψI versus the aspect ratio l/d for various diameters of the wire (F = Ds = 0, n = 2, η′ = 0.0001)
243
Fig. 9. Effect of high-order surface stress on the natural frequency.
the non-dimensional parameterη′ brings about an increase of ratio ψI . In other words, the natural frequency gets larger with the increase of the stiffness modulus of surrounding medium. From Figs. 7 and 8, we note that the effect of surrounding elastic medium on the natural frequency is dependent upon the vibration mode, aspect ratio and diameter of the nanowire. It becomes more pronounced with the decrease of mode number n and the increase of aspect ratio and diameter of the wire. To examine the effect of high-order surface stress on the natural frequency, we define
ψII =
ωn ωne
(35)
ωne
where is the natural frequency excluding the surface effect, which is expressed as
ωne =
EIk n4 + η − Fk n2 ρA
(36)
Introduction of Eqs. (19) and (36) into Eq. (35) and combination of Eqs. (1b), (3b) and (10b) yields
ψII =
1+
Fig. 10. The ratio ψII versus the diameter d for various mode numbers. (F = η = 0, l/d = 20, Ds = 1 × 10−14 N m) .
8n2π 2d(n2π 3Esd2 + 16n2π 2Ds + 16τ0l2) E(n4 π 5d 4 + 64η′l 4 − n2π 5F ′d 4)
(37)
With d = 30 nm , η = F = 0, and n = 2, the relationship curves between the ratio ψII and the aspect ratio l/d are obtained for different values of Ds , as indicated in Fig. 9. As can be seen, Fig. 9 displays the discrepancies between the results on the basis of the present model including the effects of surface stress and surface moment and those based on the pure surface stress model (generalized YL equation). The natural frequency increases with the increase of the magnitude of Ds , and the effect of high-order surface stress is sensitive to the aspect ratio of the nanowire. The influence of high-order surface stress on the natural frequency becomes more significant with the increase of the aspect ratio. It is shown from Fig. 10 that the effect of high-order surface stress on the natural frequency is also dependent upon the mode number n and the diameter of the nanowire. It is indicated that the increase of the mode number n and the diameter of the wire will make the effect of high-order surface stress less significant. 5. Conclusions Taking into account the effects of surface stress and surface moment, the transverse vibration of an axially compressed
nanowire embedded in elastic medium is investigated on the basis of the high-order surface stress model and Bernoulli–Euler beam theory. Explicit expression is derived for natural frequency for the case of a simply supported nanowire, and the influences of compressive axial load, high-order surface stress and surrounding elastic medium on the natural frequency are analyzed. In addition, the analytical solution of axial buckling load for the simply supported nanowire is obtained, which includes the effects of highorder surface stress and surrounding elastic medium. Through numerical results and discussions for a circular silver nanowire, it is concluded that the natural frequency of the nanowire is dependent upon axial load, surrounding elastic medium, and high-order surface stress. The effects of axial load, surrounding elastic medium, and high-order surface stress on the natural frequency are sensitive to the vibration modes, diameter, and aspect ratio of the nanowire. Similarly, the dependences of the buckling load on surrounding elastic medium and high-order surface stress are significant, and their influences on the buckling load are related to the mode number, diameter, and aspect ratio of the nanowire.
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Acknowledgments Authors acknowledge support from National Natural Science Foundation of China (NSFC) (No. 11172265).
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