The nonlocal frequency behavior of nanomechanical mass sensors based on the multi-directional vibrations of a buckled nanoribbon

Applied Mathematical Modelling 77 (2020) 1780–1796

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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

The nonlocal frequency behavior of nanomechanical mass sensors based on the multi-directional vibrations of a buckled nanoribbon Haibo Li a,b,c, Xi Wang a,∗, Heling Wang b, Jubing Chen a a

School of Naval Architecture, Ocean and Civil Engineering (State Key Laboratory of Ocean Engineering), Shanghai Jiaotong University, Shanghai 200240, China b Departments of Civil and Environmental Engineering, Mechanical Engineering, and Materials Science and Engineering, Northwestern University, Evanston, IL 60208, USA c AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China

a r t i c l e

i n f o

Article history: Received 3 March 2019 Revised 13 August 2019 Accepted 12 September 2019 Available online 21 September 2019 Keywords: Buckled nanostructures Nanomechanical mass sensor Compressive strain Nonlocal effect Analytical method

a b s t r a c t Considering the potential applications of the buckled structures as nanomechanical mass sensors, this paper presents an analytical method to solve the frequency shifts of the firstorder transverse and longitudinal vibration modes when a mass attaches on the surface of a buckled nanoribbon based on the nonlocal elastic theory and the Lagrange’s motion equation. The first-order transverse and longitudinal vibration modes of the buckled nanoribbon are introduced. A comparison between the analytic solution and the finite element analysis (FEA) is presented. Then, the effects of the compressive strain, the magnitude and location of attached mass, the nonlocal parameter, and attached mass numbers on the frequency shifts are presented. From example calculations, it is seen that the magnitude of attached mass increases the frequency shifts of the first-order modes, except the first-order transverse vibration mode at the location Z1=0. The frequency shifts for the first-order transverse and longitudinal vibration modes are different, and could be used as important principles in mass sensing. What’s more, the compressive strain and the nonlocal parameter play significant roles on the sensing process of the buckled nanoribbon. The results could serve as useful references for the fabrications and applications of buckled structures based nanomechanical mass sensors. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Two-dimensional (2D) microfilms or nanofilms, such as graphene [1,2], boron nitride (BN) [3], molybdenum disulfide (MoS2 ) [4], molybdenum diselenide (MoSe2 ) and tungsten diselenide (WSe2 ) [5], have attracted huge attentions due to their excellent material properties. However, one of the key factors restricted their further applications is the phenomenon of wrinkling and buckling in these films due to compressive forces or strains originating from mechanical, thermal, or osmotic [6–8]. This phenomenon might induce changes of conductivity, magnetic states and electricity of these materials [9–11]. At the same time, wrinkling and buckling, the status after postbuckling [12,13], can also be used to extend 2D structures ∗

Corresponding author. E-mail address: [email protected] (X. Wang).

https://doi.org/10.1016/j.apm.2019.09.023 0307-904X/© 2019 Elsevier Inc. All rights reserved.

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into three-dimensional (3D) buckled structures. These wrinkled and buckled structures formed from planar films show great potentials in the applications of some frontier science, such as biology [14,15], free-electron laser [16], sensors [17], modern metrology methods [18,19], soft and stretchable electronics [20–22]. The wrinkled and buckled structures could not only maintain some original excellent material properties of the original 2D films, but also perform innovative material and structural properties. For example, Fedorchenko et al. [16] demonstrated that the wrinkled Si1– x Gex /Si1– y Gey films could serve as radiation sources, which emit electromagnetic waves in a very wide range of the frequencies including the terahertz band from 0.3 to 3 THz. Based on the first-principles calculations, Guo and Guo [23] studied the electronic and field emission properties of wrinkled graphene under charge injecting and external electric field and showed its potentials in the applications of field emitters. By depositing Au nanoparticles on wrinkled graphene, Chen et al. [24] showed a flexible and stretchable surface-enhanced Raman scattering platform with ultrahigh sensitivity and outstanding stability for chemical detection and analysis. Zang et al. [25] reported an approach to reversibly control the crumpling and unfolding of large-area graphene sheets with characteristics of super hydrophobicity, high transparency, and tunable wettability and transmittance, and demonstrated the potentials of crumpled graphene–polymer laminates as artificial-muscle actuators. Wang et al. [26] presented a buckling approach for graphene and graphene ribbons on stretchable polydimethylsiloxane (PDMS) films, and the composited device could be used as strain sensors. Cheng et al. [27] fabricated a wrinkle structure based high-performance triboelectric nanogenerator, and the wrinkled structure made the current and surface charge density increase by 810% and 528%, respectively. What is more, the innovative 3D mechanical self-assembly technologies [28–30] allow various of 3D buckled structures to find their potential applications in multidirectional energy harvesting, simultaneous evaluation of multiple mechanical properties, anisotropic mechanical property measurement, multi-vibration mechanical resonators and other applications based on 3D space. In view of its multi-directional vibration characteristics, one of the most promising applications for buckled structures is the use as nanomechanical mass sensors. The existing theoretical, numerical and experimental studies on nanomechanical mass-sensors are mainly based on the vibrations of the one-dimensional beams or 2D films [31,32]. For example, Eltaher et al. [31] studied the potential application of CNTs as a nanomechanical mass sensor in measuring a zeptogram mass with a simplified nonlocal finite element model. Jensen et al. [33] demonstrated a room temperature, carbon-nanotube-based nanomechanical mass sensor with extreme mass sensitivity as 1.3 × 10−25 kg Hz−1/2 , which is equivalent to 0.40 gold atoms Hz−1/2 . Cho et al. [34] described and discussed mass sensing characteristics of SWCNT-based nano-mechanical resonators using continuum mechanics based finite element analysis. Then, Lei et al. [35] theoretically studied an atomic-resolution nanomechanical mass sensor by using a monolayer graphene sheet with fixed boundary conditions and showed the relationships of the frequency shifts with absorbed masses under several factors. Farajpour et al. [36] reported nonlocal motion equations for different anostructures including nanorods, nanorings, nanobeams, nanoplates and nanoshells and reviewed the size-dependent mechanical behavior for the vibration, bending and buckling of nanostructures. Zhou et al. [37] analyzed and discussed Transverse vibration of circular graphene sheet-based mass sensor via nonlocal Kirchhoff plate theory. Li and Wang [38] studied the active voltage controlled nanomechanical mass sensor by using the graphene-elastic–piezoelectric laminated films. It is known from the above literatures that the mass sensors are mainly dependent on the structural vibrations in one direction, which might not precisely obtain the information of absorbed masses. Thereby, based on our former analytical models of the vibration behaviors of 3D buckled structures [39,40], this paper presents the nanomechanical mass sensors based on the multi-directional vibrations of a buckled nanoribbon. An analytical model considering the effect of the nonlocal parameter is established to solve the frequency shifts of the multi-directional vibrations when a mass attaches on the surface of a buckled nanoribbon. The governing equations of the multi-directional vibrations of the buckled nanoribbon attached with the mass are constructed with Lagrange’s motion equation by introducing the nonlocal elastic theory. The first-order transverse vibration mode and the first-order longitudinal vibration mode of the buckled nanoribbon are considered. To validate the present analytic model, the analytical solutions of the natural frequencies are compared with the finite element simulations. Then, the effects of some key factors on the influence of the frequency shifts are described, including the compressive strain, the magnitude of the attached mass, the nonlocal parameter and the attached location. The results could serve as useful references for the design of buckled structures based nanomechanical mass sensors. 2. Nonlocal governing equation and solving process The schematic of a buckled nanoribbon formed by compressing a straight nanoribbon and its multi-directional vibrations with an attached mass is shown in Fig. 1. The attached mass could be gas molecular, biomolecules, viruses, buckyballs and so on. As shown in Fig. 1a, the buckled nanoribbons with length L, width b and thickness h under different compressive strains are formed by releasing a pre-strained or thermal expanded substrate. The coordinate systems used in the construction of the governing equations are shown in Fig. 1(b). Fig. 1(c) and (d) gives the mode shapes of first-order transverse vibration and first-order longitudinal vibration with an attached mass, respectively. It should be noted that the transverse direction and longitudinal direction are respectively normal and parallel to the upper plane of the substrate. Phase I and II in Fig. 1(c) and (d) represent the two displacement limits during the vibrations with attached mass. It is assumed that the nanoribbon’s thickness is much smaller than its width and length, i.e. h << b and h << L. Thereby, the finite-deformation beam theory without transverse shear deformation could be introduced. As shown in Fig. 1(b), the deformations of the nanoribbon could be expressed by the transverse displacement u1 and longitudinal displacement u3 in the X–Z plane.

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Fig. 1. The schematic of the buckled nanoribbon formed by compressing a straight nanoribbon and its transverse and longitudinal vibrations with an attached mass, (a) the shapes of nanoribbon under different compressive strains, (b) the coordinate systems in the vibrations of the buckled nanoribbon, (c) the transverse vibration mode with an attached mass, (d) the longitudinal vibration mode with an attached mass, where phases I and II are corresponding to the largest amplitudes during vibration.

2.1. The nonlocal governing equations Based on the nonlocal elastic theory, the nonlocal physical relationship of a homogeneous material between stress component σ and strain component ε = (λ − 1) − Xκ could be written as



1 − μ2 ∇ 2



σ = E [ (λ − 1 ) − X κ ]

(1)

where μ = e0 a is the nonlocal parameter, e0 represents a constant of specified material, a is the internal characteristic length depending on the two bonds distance as atom diameters. Eis the elastic modulus of nanoribbon material. In addition, λ and κ are the stretch ratio and curvature of the buckled nanoribbon, respectively. The total displacements of buckled nanoribbon during the two-dimensional vibrations could be, respectively, described as

u1 (Z, t ) = u1(0 ) + u1 (Z, t ), u3 (Z, t ) = u3(0 ) + u3 (Z, t ).

(2)

where u1(0) and u3(0) are the initial transverse and longitudinal displacements of the buckled nanoribbon with two clamped ends, which can be expressed as



u1(0 ) = A0 cos

 2π 







π A20 4π Z + 1 , u3 ( 0 ) = sin Z − εcompre Z, L 4L L

(3)

and u1 (Z,t), u2 (Z,t) express the transverse and longitudinal dynamic displacements of the buckled nanoribbon during vibrations, as follows

u1 (Z, t ) =

n 

a(k) (t )ϕ(k) (Z ), u3 (Z, t ) =

k=1

n 

a(k) (t )φ(k) (Z ),

(4)

k=1

√ In Eqs. (3) and (4), A0 = (L/π ) εcompre − εcr is the static transverse deflection amplitude of the buckled nanoribbon, εcompre = (L − l )/L is the compressive strain between two clamped bonding sites, and is much larger than the critical strain of the nanoribbon ε cr = π 2 h2 /(3L2 ), and {a(1) ,a(2) ,..., a(n) }T =a is a n × 1 vector. The clamped boundary conditions in the forms of displacement components can be written as

u1 (±L/2 ) = 0, u3 (±L/2 ) = 0,

du1 (±L/2 ) = 0. dZ

(5)

To satisfy the clamped boundary conditions in Eq. (5), φ (k) (Z) and φ (k) (Z)in Eq. (4) should be

ϕ(k) (±L/2 ) = 0, φ(k) (±L/2 ) = 0,

dϕk (±L/2 ) = 0. dZ

(6)

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The total kinetic energy induced by the vibration process of buckled nanoribbon with attached mass is composed of two parts which are, respectively, the kinetic energy of buckled nanoribbon and the kinetic energy of attached mass, as follows; The kinetic energy of only buckled nanoribbon is written as

1 T1 (a˙ ) = 2



L/2 −L/2



γ

∂ u1 (Z, t ) ∂t

2



∂ u3 (Z, t ) + ∂t

2

dZ =

1 a˙ T Mn a˙ . 2

(7)

where the 3rd and higher order items of A in Eq. (7) is neglected, a˙ = {a˙ (1 ) , a˙ (2 ) , ..., a˙ (n ) }T is the time derivative of a, i.e. a˙ = d (a )/dt, γ = ρ bh, ρ is the density of the nanoribbon material, and Mn is an n × n mass matrix induced by the nanoribbon material. The attached mass could be recognized as a concentrated mass on the surface of the buckled nanoribbon. Thereby, the kinetic energy induced by the attached mass could be written as



N 1 T2 (a˙ ) = mk δ ( Z − Zk ) 2 k=1

∂ u1 (Z, t ) ∂t

2



∂ u3 (Z, t ) + ∂t

2

=

1 a˙ Mm a˙ . 2

(8)

where mk represents the kth mass attached at location Zk , Nrepresents the total attached mass number, Mm is an mass matrix induced by the attached mass, and δ is the Dirac delta function defined by



δ ( Z − Zk ) =

1

Z = Zk

0

Z = Zk

(9)

Thus, the total kinetic energy induced by the buckled nanoribbon with attached mass is written as

T (a˙ ) = T1 (a˙ ) + T2 (a˙ ) =

1 a˙ T M a˙ . 2

(10)

where M = Mm + Mn. From Eq. (1), the potential energy W during the vibrations process of the buckled nanoribbon could be written as

W =

1 2



1 2

σ [(λ − 1 ) − X κ ]dV = aT K a,

(11)

V

where K is the stiffness matrix with n × n dimensions. Lagrange’s equation of motion requires that

∂ (T − W ) d ∂ (T − W ) − = 0. ∂ a dt ∂ a˙

(12)

Substituting Eqs. (10) and (11) into Eq. (12), gives

L/2     ∂ 2 1 − μ2 ∇ 2 [T1 (a˙ ) + T2 (a˙ )] − α (λ − 1 ) + 2βκ 2 dZ ∂ a −L/2 , L/2     d ∂ 2 − 1 − μ2 ∇ 2 [T1 (a˙ ) + T2 (a˙ )] − α (λ − 1 ) + 2βκ 2 dZ = 0 dt ∂ a˙ −L/2

(13)

where

α = E bh, β =

E bh3 . 24

(14)

By simplifying Eq. (13), the nonlocal vibration governing equation of the buckled nanoribbon with attacted mass is given by

M a¨ + K a = 0.

(15)

By using the substation of a = Asin (ωt), Eq. (15) is derived as





K − ω 2 M A = 0

(16)

where A = {A1 ,A2 ,…, An }T . Then the natural frequencies ω of the buckled nanoribbon with attacted mass could be determined by solving the eigenvalue problems as

  K − ω2 M  = 0

where |K − ω2 M| represents the eigenvalue of K − ω2 M.

(17)

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2.2. The nonlocal frequency shifts of the first-order transverse mode with attached mass For the first-order transverse vibration mode, the transverse and longitudinal dynamic displacements in Eq. (13) can be well characterized by the superposition of the following base functions [39,40], with two terms (i.e. n = 2) in Eq. (4), as

ϕ(1) (Z ) = 1 + cos π A0

φ (1 ) ( Z ) =

 2π Z  L

sin

2L

,

 4π Z  L

ϕ(2) (Z ) = 1 − cos ,

 4π Z  L

π A0 

φ (2 ) ( Z ) =

3L

6 sin

,

(18)

 2π Z  L

− 2 sin

 6π Z  L

.

(19)

Considering the influence of the attached location on the natural frequencies of the buckled nanoribbon with an attached mass, three different attached locations as examples are mainly discussed, including Z1 = 0, Z1 = −L/6 and Z1 = −L/4. Then substituting Eqs. (18) and (19) into Eq. (13), gives the governing equations with an attached mass as



M11

M12

M12

M22

where

 M11 =

γL



  a¨ (1) K11 + a¨ (2) 0

π 2 A20 8L2

3 + 2





20π 2 A20 3 = γL + 2 9L2

M22

2π 4 β = 3L3

K11



0 K22

4 π 2 μ2 γ + L







   a ( 1 ) 0 = , 0 a ( 2 )

π 2 A20 L2

16π 2 μ2 γ + L



(20)

+ g11 γ L,

+1



2π 2 A20 +1 L2

+ g22 γ L, M12 = γ L + g12 γ L,

     2 2  4α 2 2π 4 β α 2 2 2 A0 2 A0 6A0 − h − 24 5π 2 − 1 , K22 = 384 2π 2 + 1 − h . β β L 3L3 L

(21)

where g11 , g22 and g12 are three non-dimensional parameters, and they can be written as

  ⎧  8π 2 μ2 π 2 A20 m1 ⎪ ⎪ 4 − + 1 ⎪ γL ⎪ L2 L2 ⎪ ⎪  ⎨  3π 2 A20 m1 9 4π 2 μ2 π 2 A20 g11 = + + γL 4 16L2 L2 L2 ⎪ ⎪   2 2  ⎪ ⎪ 2 2 ⎪ π A0 ⎪ ⎩ m1 1 − 8π 2μ +1 γL L L2

g22

Z1 = 0 Z1 = − 6L , g12 = Z1 = − 4L

⎧ 0 ⎪ ⎪ ⎪  ⎪  ⎪ ⎨ m1 9 3π 2 A20 48π 2 μ2 π 2 A20 = Y γ L 4 + L2 − L2 L2 ⎪    ⎪ ⎪ 2 2 2 2 ⎪ m1 64π A0 64π 2 A20 8π μ ⎪ ⎩ 4+ + + 24 γL 9L2 3L2 L2

⎧ m 64π 2 μ2 1 ⎪ γL L2 ⎪ ⎪ ⎪ ⎪ ⎨ m 27π 2 μ2  π 2 A2 1

γ L L2 ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎩ m1 16π μ 2 γL L

0

L2

+1

Z1 = 0 Z1 = − 6L , Z1 = − 4L

Z1 = 0 Z1 = − 6L .

(22)

Z1 = − 4L

Using Eq. (17), we have

 K11 − ω2 M11   −ω2 M12

  =0 2 K22 − ω M22  −ω2 M12

(23)

Leading to the first-order transverse vibration frequency of buckled nanoribbon with an attached mass could be solved as

fI =

h L2



E

ρ



· fˆI

A0 A0 , h L



(24)

where fˆI is the normalized natural frequency with an attached mass on selected location, whose explicit form is given in Appendix A. It can be seen from Eq. (24) that the normalized frequency of the first-order transverse vibration mode depends on two parameters, A0 /h and A0 /L, and it can reduce to a single-variable function when the static deflection amplitude of the buckled nanoribbon is much larger than the nanoribbon thickness, i.e. A0 > >h. In the limit of A0 /h→∞, the normalized linear natural frequency fˆI could be approximately written as

√ fˆI = 2 3π





288 μLπ 2 2

2





3 + 8π 2 A20 /L2

1 + 2π 2 A20 /L2









+ 27 + 40π 2 A20 /L2 + 18gˆ22

(25)

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It is found from Eq. (25) that the linear natural frequency is independent of A0 /h. Together with the simplified relationship between the compressive strain ε compre and A0 /L, i.e. εcompre ≈ (π 2 A20 )/L2 , as well as the non-dimensional small scale nonlocal χ = μ/L, the normalized nonlocal frequency fˆI of the first-order transverse vibration mode could be further written as

√ fˆI = 2 3π where



3 + 8εcompre 288φ 2 π 2 (1 + 2εcompre ) + 27 + 40εcompre + 18gˆ22

⎧ m1   ⎪ 64π 2 χ 2 ⎪ ⎪ γL ⎪ ⎪  ⎨ m1  9 2 2 + 3 ε − 48 π χ ε compre compre gˆ22 = γ L 4 ⎪   ⎪ ⎪ ⎪ m 64 64π 2 χ 2 ⎪ ⎩ 1 4 + εcompre + (8εcompre + 3) γL 9 3

(26)

Z1 = 0 Z1 = − 6L .

(27)

Z1 = − 4L

The absolute value of the nonlocal frequency shift of the first-order transverse vibration mode induced by the attached mass is then solved as

     ˆ  ˆ ˆ  fI  =  fI − fI 

    m1 =0

(28)

2.3. The nonlocal frequency shift of the first-order longitudinal mode with attached mass For the first-order longitudinal vibration mode, the transverse and longitudinal dynamic displacements in Eq. (13) can be well described by the following base functions [39], with only one term (i.e. n = =1) in Eq. (4), as follows:

ϕ(1) (Z ) = sin

 2π Z  L

−

12π A0 Z 2 φ (1 ) ( Z ) = + L3



πZ L

+

4π Z 3 , L3

(29)

24π A0 Z 2 12A0 6π A0 + − πL L L3

cos2

πZ  L

+



12A0 Z 2π Z sin L L2



−

4π A0 cos4 L

πZ  L

−

3π A0 . L (30)

Also, three different attached locations are discussed, including Zk = 0, Zk = −L/6 and Zk = −L/4. Substitution of Eqs. (29) and (30) into Eq. (13), gives a¨ M11 (1 ) + K11 a(1 ) = 0,

where

 M

11

= 2γ L +

π2 105

+

μ2 π 2  98 L2

15

3 1 − 2 + 4 π −



= K11

(31)

 24

π2

2π 4 48β − + 2π 2 + 3 15 L

 111 80

+ g



11

π4 3

π2 −



121 567 A20 + + 6 8π 2 L2

μ2 π 2  18 L2

5

π2 −



100 27 A20 + 2 , 3 π L2

(32)

γL

−

16π 2 +3 9

π 2 A20



L2

,

(33)

and the coefficient g 11 in Eq. (32) is detailly described in Appendix B. Thereby, the first-order longitudinal vibration frequency of the buckled nanoribbon with an attached mass could be solved as

h fII = 2 L



E ˆ fII



ρ

where

√ ˆfII = 2 21π 3



A0 L



,

(34)

 



  π 2 A20 /L2    , τ1 + τ2 π 2 A20 /L2 + μ2 π 2 /L2 0 + 840π 4 g 11

−6π 4 + 90π 2 + 15π 4 − 80π 2 + 135

(35)

where τ 1 = 16π 6 + 420π 4 − 5040π 2 , τ 2 = 2331π 4 − 33880π 2 + 119070, and

     0 = 6048π 4 − 5600π 2 + 45360 π 2 A20 /L2 + 4704π 4 − 40320π 2 .

(36)

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H. Li, X. Wang and H. Wang et al. / Applied Mathematical Modelling 77 (2020) 1780–1796 Table 1 Comparison of the non-dimensional frequency fˆI of the first-order transverse vibration mode via the precise solution in Eq. (24), the approximate solution in Eq. (26) and FEA.

ε compre

Precise solution

Approximate solution

FEA

0.05 0.10 0.15 0.20 0.25 0.30

3.8248 3.8643 3.9208 3.9755 4.0246 4.0705

3.7263 3.8102 3.8825 3.9453 4.0006 4.0495

3.7286 3.8154 3.8712 3.9218 3.9770 4.0116

Maximum relative error (%) 2.58% 1.28% 1.28% 1.37% 1.20% 1.47%

Table 2 Comparisons of the non-dimensional frequency fˆII of the first-order longitudinal vibration mode vibration via the precise solution in Eq. (37) and FEA.

ε compre

Precise Solution

FEA

Relative error (%)

0.05 0.10 0.15 0.20 0.25 0.30

2.10725 2.14096 2.16948 2.19394 2.21515 2.23372

2.11946 2.10781 2.09688 2.07666 2.0446 2.01236

0.58% 1.57% 3.46% 5.65% 8.34% 11.00%

When the static deflection amplitude of the buckled ribbon is much larger than its thickness, fˆII can also be expressed as a single-variable function of the compressive strain, i.e.

√ 2 21π fˆII = 3







−6π 4 + 90π 2 + 15π 4 − 80π 2 + 135

τ1 + τ2 εcompre + π χ 1 + 840π 2

2

εcompre

4 g

,

(37)

11

where 1 = (6048π 4 − 5600π 2 + 45360)ε compre + (4704π 4 − 40320π 2 ). Then the non-dimensional nonlocal frequency shift of the first-order longitudinal mode induced by the attached mass is derived as

     ˆ   ˆ   fII  =  fII − fˆII 

    m1 =0

(38)

3. Results and discussions In this section, the initial geometry sizes of the nanoribbon before buckling are taken as L = =50 nm, b = =5 nm, h = =0.34 nm, and the corresponding material properties are taken as elastic modulus E = =1.0 TPa, density ρ = =2250 Kg/m3 . 3.1. Validation To validate the present analytic model, Tables 1 and 2 give the simple comparison between the present analytical solution and the finite element analysis (FEA), in which the results from finite element simulation are performed by the commercial software ABAQUS, and the influences of the nonlocal parameter and the attached mass are neglected. In the finite element analysis, the material was assumed to be homogeneous and linear elasticity. Then the ribbon was discretized by Four-node finite-strain shell elements (S4) with at least 20 elements were along the width direction of the ribbon to guarantee the convergence. The boundary conditions of the buckled ribbon were set as clamped-clamped. It is clearly seen from Table 1 that the maximum relative error of the non-dimensional natural frequency fˆI among the precise solution in Eq. (24), the approximate solution in Eq. (26) and the FEA results gradually decreases with the compressive strain increases, and the maximum relative error shows 2.58% under ε compre = 0.05 and just 1.47% under ε compre = 0.30. However, the relative error of the non-dimensional natural frequency fˆII between the precise solution in Eq. (37) and the FEA results dramatically increases as the compressive strain increases, and the relative error shows 0.58% under ε compre = 0.05, but increases to 11.00% under ε compre = 0.30. It has been demonstrated in our former publication [39] that the analytic solution for in-plane vibration (defined as the first-order longitudinal vibration in Ref. [39]) agrees reasonably well with the FEA results when the compressive strain is smaller than 0.25, but shows increasing error as the compressive strain increases because of the inaccuracy of the base functions for in-plane vibration when the compressive strain is greater than 0.25. Thereby, when the compressive strain is greater than 0.25, the present analytical solution for first-order longitudinal vibration mode should

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Fig. 2. The effect of the compressive strain ε compre on the non-dimensional frequency shift | fˆI | of the first-order transverse vibration mode under different nonlocal parameters.

Fig. 3. The effect of the compressive strain ε compre on the non-dimensional frequency shift | fˆII | of the first-order longitudinal vibration mode under different nonlocal parameters.

be improved by amending the base functions Eqs. (29) and (30) in our future work. So that, it is indirectly validated from Tables 1 and 2 that when the compressive strain is less than less than or equal to 2.5, the nonlocal solution in Eqs. (24) and (37) could approximately describe the relationships between first-order transverse and longitudinal vibration frequencies and the compressive strain under the moving mass. Now Figs. 4 and 5 in the original manuscript have been adjusted to Fig. 8 and Fig. 9 in the revised manuscript, respectively. As Tables 1 and 2 presented, the maximum relative errors under ε compre = 0.05 and ε compre = 0.25 for the first-order transverse and longitudinal vibration modes, respectively, are smaller than the other compressive strains (when the compressive strain is less than 0.3), so that different compressive strains for these two vibration modes are used in Figs. 4, 8, 14 and Figs. 5, 9, 15, respectively. 3.2. Nonlocal effect To study the effect of the non-dimensional nonlocal parameter on influence of the natural frequency shifts of the buckled nanoribbon with an attached mass, Figs. 2 and 3 gives the changes of the non-dimensional natural frequency shifts | fˆI | and | fˆII | of the transverse and longitudinal vibration modes as the compressive strain ε compre increases by varying the nonlocal parameter (χ =0.0 0 0, 0.025, 0.050, 0.075, 0.100), where the non-dimensional attached mass is taken as m1 = 0.1γ L, and the

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Fig. 4. The effect of the non-dimensional nonlocal parameter on the non-dimensional frequency shift | fˆI | of the first-order transverse vibration mode when a mass attached at different locations.

Fig. 5. The effect of the non-dimensional nonlocal parameter on the non-dimensional frequency shift | fˆII | of the first-order longitudinal vibration mode when a mass attached at different locations.

attached location is Z1 = −L/4. It can be seen from Fig. 2 that | fˆI | increases with ε compre increases, and the nonlocal parameter could obviously decrease the frequency shift value | fˆI |. Results in Fig. 3 show that | fˆII | firstly increases with εcompre increases without considering the nonlocal parameter, then decreases as ε compre increases when χ =0.0 0 0, 0.025, 0.050, 0.075, 0.100. It is found that differences of the frequency shift between the first-order transverse and longitudinal vibration modes could be regarded as important principles in the sensing of attached mass. To further study the effect of the non-dimensional nonlocal parameter on influence of the natural frequency of the buckled nanoribbon with an attached mass, Figs. 4 and 5 plot the non-dimensional natural frequency shifts | fˆI | and | fˆII | of the transverse and longitudinal vibration modes when an mass attached on four different locations: Z1 = 0, Z1 = −L/6,Z1 = −0.19L and Z1 = −L/4, where the non-dimensional attached mass is taken as m1 = 0.1γ L, and the compressive strains are taken as ε compre = 0.25 in Fig. 4 and εcompre = 0.05 in Fig. 5. The results in Fig. 4 show that the frequency shift of the first-order transverse vibration mode under Z1 = −L/6, Z1 = −0.19L and Z1 = −L/4 gradually decreases with χ increases, and keeps zero for the case at Z1 = 0. Fig. 5 show that the frequency shift of the first-order longitudinal vibration mode under Z1 = −L/6, Z1 = −0.19L and Z1 = −L/4 gradually decreases with χ increases, however the results under Z1 = 0 increase and then decrease as χ increases.

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Fig. 6. The effect of the compressive strain ε compre on the non-dimensional frequency shift | fˆI | of the first-order transverse vibration mode when a mass attached at different locations. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 7. The effect of the compressive strain ε compre on the non-dimensional frequency shift | fˆII | of the first-order longitudinal vibration mode when a mass attached at different locations.

3.3. Effect of initial compressive strain To study the effect of the initial compressive strain ε compre on the influence of the natural frequency of buckled nanoribbon when attached with an mass, Figs. 6 and 7 plots the non-dimensional frequency shifts | fˆI | and | fˆII | of the firstorder transverse and longitudinal vibration modes when an mass attached on four different locations: Z1 = 0, Z1 = −L/6, Z1 = −0.19L and Z1 = −L/4, where the non-dimensional nonlocal parameter and the non-dimensional attached mass are taken as χ = 0.10 and m1 = 0.1γ L, respectively. It is found from Fig. 6 that the non-dimensional frequency shift of the firstorder transverse vibration mode is zero when the attached location is equal to zero, i.e. Z1 = 0, and while the result under Z1 = −L/6 decreases as the compressive strain increases and the result under Z1 = −L/4 experiences an increase trend with εcompre increases. It is also found that there exists a location Z1 ≈ −0.19L between −L/6 and −L/4 so that the frequency shift keeps 0.20 while the compressive strains varies, as the blue line shown in Fig. 6. At the same time, the results under condition Z1 = −L/4 is much larger than the other three cases. As shown in Fig. 7, the non-dimensional frequency shift | fˆII | for the first-order longitudinal vibration mode decreases as the compressive strain ε compre increases, and the frequency shifts are larger at Z1 = −0.19L than that at Z1 = −L/6 and Z1 = −L/4.

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Fig. 8. The effect of the magnitude of attached mass on the non-dimensional frequency shift | fˆI | of the first-order transverse vibration mode when a mass attached at different locations.

Fig. 9. The effect of the magnitude of attached mass on the non-dimensional frequency shift | fˆII | of the first-order longitudinal vibration mode when a mass attached at different locations.

3.4. Effect of attached mass The effects of the magnitude of the attached mass on the nonlocal frequency of buckled nanoribbon when attached with an mass are, respectively, shown in Figs. 8 and 9, where the non-dimensional frequency shifts | fˆI | and | fˆII | of the first-order transverse and longitudinal vibration modes are described when the mass attached on four different locations: Z1 = 0, Z1 = −L/6, Z1 = −0.19L and Z1 = −L/4, where the non-dimensional nonlocal parameter is taken as χ = 0.10, and the compressive strains are taken as ε compre = 0.25 in Fig. 8 and ε compre = 0.05 in Fig. 9, respectively. Results show that the non-dimensional frequency shifts | fˆI | and | fˆII | for the first-order transverse and longitudinal vibration modes linearly increase with the attached mass increases at Z1 = −L/6, Z1 = −0.19L and Z1 = −L/4 points, and the frequency shift of the first-order transverse vibration mode under Z1 = 0 keep zero, while the results of the first-order longitudinal vibration mode under Z1 = 0 also increases with attached mass increases. It is also seen that the results in Fig. 8 under the case Z1 = −L/4 is much larger than the other three cases, and results in Fig. 9 under the case Z1 = −0.19L is larger than the other three cases. Further, Figs. 10 and 11 demonstrates the variations of the frequency shifts  f I and  fII of the first-order transverse and longitudinal vibration modes under different non-dimensional nonlocal parameters (χ = 0.00, χ = 0.05, χ = 0.10) when

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Fig. 10. The frequency shift  f I of the first-order transverse vibration mode with different nonlocal parameters when a mass attached in the range [-L/2, L/2].

Fig. 11. The frequency shift  f II of the first-order longitudinal vibration mode with different nonlocal parameters when a mass attached in the range [-L/2, L/2].

an mass attached in the range [−L/2, L/2] of the buckle nanoribbon, where the non-dimensional attached mass is taken as m1 = 0.1γ L, and the compressive strains are taken as ε compre = 0.25 in Fig. 10 and ε compre = 0.05 in Fig. 11. It can be seen from Figs. 10 and 11 that the frequency shifts are symmetric along the central axis of Z1 = 0 in the range [−L/2, L/2], and the results in Fig. 10 are further symmetric along the central axis of Z1 = −L/4 and Z1 = L/4 from [−L/2, 0] and [0, L/2], respectively. The frequency shift  fI and  fII in the range [−L/4, 0] experiences an increase and then decreases from Z1 = −L/4 to Z1 = 0 under χ = 0.05 and χ = 0.10, while the frequency shifts  f I and  fII without the nonlocal effect decreases from Z1 = −L/4 to Z1 = 0. The most obvious difference between Figs. 10 and 11 is the frequency shift at location Z1 = 0, which exhibits the frequency shift  fI keeps zero while the frequency shift  fII increases with the non-dimensional nonlocal parameter increases. It is concluded that the nonlocal parameter plays a significant role in the dynamic sensing process of the attached mass on the surface of the buckled nanoribbon. 3.5. Multiple mass factor The above calculations are based on a single attached mass. To simulate the effect of multiple mass on the frequency shifts, Figs. 12 and 13 present the changes of the non-dimensional frequency shifts| fˆI | and | fˆII | for the first-order transverse and longitudinal vibration modes as the compressive strain increases, respectively, when the buckled nanoribbon is

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Fig. 12. The effect of the compressive strain ε compre on the non-dimensional frequency shift | fˆI | of the first-order transverse vibration mode when the buckled nanoribbon is attached with different mass numbers at different locations. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 13. The effect of the compressive strain ε compre on the non-dimensional frequency shift | fˆII | of the first-order longitudinal vibration mode when the buckled nanoribbon is attached with different mass numbers at different locations. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

attached with different mass numbers at different locations. For each attached mass, the magnitude of the mass is taken as m1 = 0.1γ L. The black lines in Figs. 12 and 13 represent the non-dimensional frequency shift | fˆI | and | fˆII | when a single mass attaches at location Z1 = −L/6, and red lines represent | fˆI | and | fˆII | when two mass attaches at location Z1 = −L/6 and Z1 = −0.19L, respectively, and blue lines represent | fˆI | and | fˆII | when three mass attaches at location Z1 = −L/6, Z1 = −0.19L and Z1 = −L/4, respectively. It can be seen in Fig. 12 that there is an obvious difference between these three lines, while the red line and blue line show a tiny difference in Fig. 13. What is more, Figs. 14 and 15 investigates the effect of the magnitude of attached mass on the non-dimensional frequency shift | fˆI | and | fˆII | of the first-order transverse and longitudinal vibration modes when the buckled nanoribbon is attached with different mass numbers at different locations. Results in Figs. 14 and 15 show that increase the attached mass could increase the frequency shifts of the two vibration modes, and the attached location plays a significant role on the detection of the attached mass.

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Fig. 14. The effect of the magnitude of attached mass on the non-dimensional frequency shift | fˆI | of the first-order transverse vibration mode when different mass numbers at different locations.

Fig. 15. The effect of the magnitude of attached mass on the non-dimensional frequency shift | fˆII | of the first-order longitudinal vibration mode when different mass numbers at different locations.

4. Conclusions Here, an analytical method is presented to solve the frequency shifts of the first-order transverse and longitudinal vibration modes when a mass attaches on the surface of a buckled nanoribbon based on the nonlocal elastic theory and Lagrange’s motion equation. A comparison between the analytical solutions and the result from FEA by using commercial software ABAQUS is demonstrated. Then, the effects of some key factors on the influence of the frequency shifts of the firstorder transverse and longitudinal vibration modes are calculated, including the compressive strain, the magnitude of the attached mass, the nonlocal parameter and the attached location. Results show that increase the magnitude of the attached mass could increase the frequency shifts of two different vibration directions, however the frequency shift of the first-order transverse vibration mode keeps zero at location Z1 = 0. The frequency shifts for the first-order transverse and longitudinal vibration modes are quite different, and could be used as important principles in mass sensing. The frequency shifts of the first-order transverse and longitudinal vibration modes could be controlled by adjusting the initial compressive strain, and the nonlocal parameter and attached position play significant roles in the dynamic sensing process of the attached

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mass on the surface of the buckled nanoribbon. The results could serve as useful theoretical references for the designs and applications of buckled structures based nanomechanical mass sensors. Acknowledgment This work was financially supported from National Science Foundation of China (NSFC) under Number 11732009. H.B. Li thanks the support from China Scholarship Council, and National Postdoctoral Program for Innovative Talents. Appendix A The normalized linear natural frequency with nonlocal effect in Eq. (24) could be written as





fˆI = π 40π 4

A0 L



4



A0 h

2 

 − 1 + 162 + 512 where

 1 = 1856π

4

A0 L

 2 = 18432π

4

A0 L

+ 507π 2

160π 2

×



A0 L

+ 180 +

2

 − 85π 4

μ2 π 2

 + 9344π

2

A0 L

4

 + 27648π

2

A0 L

μ2 π 2

3 +

L2

4

A0 L

2

L2

μ2 π 2 L2

4

μ4 π 4

1 +



A0 + 108 h

4 + 36864

2 +  3 +

L4

2

 + 504π 2

μ4 π 4 L4

μ2 π 2

1/2 1/2

A0 L

L2

− 12

4

·

2 + 216 +

μ2 π 2 L2

1 + 2 ,

(A1)

5

2 + 4320, A0 L

(A2)

2 + 9216,

(A3)

   A0 2 3 = 320π 2 g11 + 18π 2 g22 + 144g11 g22 − 144g212 + 216g11 − 288g12 + 216g22 ,

(A4)

   A0 2 4 = 4608π 2 g11 + 576π 2 g22 + 2304g11 + 576g22 ,

(A5)

L

L

 1 = −576π



2

2

A0 L

 + 2304π

2 = 384π g11 − 60π g22 2

2

 1 = 46656 + 32882π 8

 + 248064π 6

 + 106272π

2

A0 L

A0 L

2

  A0 2 L

A0 L



 2





6 

4 

A0 + 260928 L

A0 h

2

6

A0 h

4

A0 h

A0 L

4





2

 + 25600π 4

 + 34560π 2



A0 − 3204 h

2



+ 1440g22 π

4 

+ 1728π

A0 L

2 + 576,

(A6)

(A7)

2

A0 h

 2

,

A0 L

2

A0 L

A0 L

4 

2 

A0 h



A0 + 972 h

A0 h

+ 36096g11 − 11520g12 + 2

2

A0 + 1152 h

2

− (2832g22 + 7680g11 )π 4

225g222

  A0 4 h

A0 L

A0 L



A0 L

+ 2880g11 g22 + 5130g22 + + 972g22 +

− 384π

− 58027π 6

A0 h



+ 144g11 + 72g22

− 258084π 4

324g222

2 

6 2 

A0 L

8

2 = (8704g11 + 1360g22 )π 6



2 

A0 h

2 

A0 L

9216g211

A0 h

4

4

 + 454032π 4

4

4 ,

(A8)

4

−

A0 L

5760g212



 π

4

A0 L

 2 A0 + (5184g12 − 1944g11 ) h

4

H. Li, X. Wang and H. Wang et al. / Applied Mathematical Modelling 77 (2020) 1780–1796











+ 13824g12 − 6966g22 − 3456g11 g22 − 8064g11 + 6912g212 − 540g222

+ 1080g11 g22 + 22680g11 − 2160g212 + 6912g211 + 1620g22 − 4320g12



π2

A0 h

 π2



2 

A0 L

2

A0 L

1795

2

+ 1296g211 + 3888g11 ,

 3 = 2448π

8

A0 L



+ 1440π 4

 + 1692π 2

(A9)

8

 + 11410π

A0 h

A0 h

4 

4 

A0 L

A0 L

6

4

A0 L

6

 − 3792π



− 10350π 4

2

 + 4131π 2

 4 = (2654208g11 + 414720g22 )π

A0 L

6



+ (368640g22 + 3354624g11 )π

4

A0 h

A0 L

6

2 

2

A0 h A0 L

2 

A0 L

4

6

 + 11532π



− 6678π 2



− 1215

A0 h

2

A0 h

 + 486

2  4

A0 h

6 4

 + 165888g22

A0 h

A0 L

A0 L

4

2 ,

(A10)

+ 486

 − (774144g22 + 1769472g11 )π

A0 L

4

4

4

A0 h

2 

+ 331776g22 π

 2

A0 L

A0 h

4

4 

 2  2  2 A0 A0 A0 − (82944g22 + 331776g11 ) − (580608g22 + 1548288g11 )π 2 h h L  2 A0 + (1327104g11 + 69120g22 )π 2 + 165888g11

A0 L

2 ,

(A11)

L

 5 = 324π

8

A0 L

 − 600π

4

8 + 576π

A0 h

 + 144π 2



2 

A0 h

A0 L

4 

6

4

A0 L

A0 L

6

 − 432π



+ 144π

2

4

 + 36

A0 h

A0 h

6

4 

4

A0 h A0 L

2  4

+ 96π 2

A0 L

6

 + 364π



− 264π



A0 L

2

2

A0 h

 − 36

4

2 

A0 h

2

A0 L A0 L

4 2 .

(A12)

+9

Appendix B The coefficient in Eq. (32) could be written as

⎧     4π 2 A20 12 2 π 2 A20 2π 2 μ2 48 π 2 A20 m1 ⎪ ⎪ 1− 2 − − +1 Z1 = 0 ⎪ ⎪ γL π L2 L2 π 2 L2 L2 ⎪ ⎪  ⎪  2 2 2 √ ⎪ ⎪ √ 2 ⎪ π A 0 2 π 2 μ2 12 3 108 m1 1  1 ⎪ ⎪ −27 3 + 8 π + 11 − − − ⎪ 2 ⎪ γ L 2916 144 π π L2 27L2 ⎪ ⎪  √ √ 2 2  √  ⎨ √ 243 3 972 L 972 3 π A0 54 3 g 11 = · 141 − − 2 + + 8 3 π − 94 + Z1 = − . . 3 ⎪ π π 6 π π L2 ⎪ ⎪  ⎪    ⎪ ⎪ 3 6 2 π 2 A20 3 π 2 μ2 m1 3π 2 ⎪ ⎪ −1 + + 1− − 2 − ⎪ ⎪ γL 16 π π L2 8L2 ⎪ ⎪  ⎪    2 2 ⎪ 192 64 L 384 π A0 32 ⎪ ⎪ · 44 − − + + 4 π − 27 + Z1 = − ⎩ π π 4 π2 π3 L2

(A13)

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