Analytical solutions for thermal vibration of nanobeams with elastic boundary conditions

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Analytical solutions for thermal vibration of nanobeams with elastic boundary conditions Jingnong Jiang, Lifeng Wang∗ State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, 210016 Nanjing, China

a r t i c l e

i n f o

a b s t r a c t

Article history:

A nonlocal Euler beam model with second-order gradient of stress taken into consideration

Received 30 November 2016

is used to study the thermal vibration of nanobeams with elastic boundary. An analytical

Revised 28 July 2017

solution is proposed to investigate the free vibration of nonlocal Euler beams subjected to

Accepted 7 August 2017

axial thermal stress. The effects of the nonlocal parameter, thermal stress and stiffness

Available online xxx

of boundary constraint on the vibration behaviors of nanobeams are revealed. The results show that natural frequencies including the thermal stress are lower than those without the

Keywords:

thermal stress when temperature rises. The boundary-constrained springs have significant

Nonlocal Euler beam

effects on the vibration of nanobeams. In addition, numerical simulations also indicate the

Thermal stress

importance of small-scale effect on the vibration of nanobeams.

Vibration

© 2017 Published by Elsevier Ltd on behalf of Chinese Society of Theoretical and Applied Mechanics.

Elastic boundary conditions Nanobeam

1.

Introduction

Nanostructures have attracted considerable attention for their outstanding mechanical, chemical and thermal properties [1–4]. Hence, nanobeams hold exciting promise as transistors, probes, sensors, actuators, and resonators in nanoelectromechanical systems. Since the experiments are very difficult, and molecular dynamics simulations remain expensive at the nano-scale, continuum elastic models have been widely used to study wave propagation and vibration of nanobeams [5–9]. Experimental results and molecular dynamics simulation show that the size effects play a major role in the mechanical properties of microstructures [10–13] and nanostructures [5,6,14]. Because of the lacking of size-dependent material length-scale parameter, the classical continuum elasticity theory fails to describe the structural behavior at micron-

∗

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

and nanometer-scale accurately. To overcome this weakness, several nonlocal elasticity theories which incorporate the internal material length-scale parameters, such as the couple stress theory [15–17], the strain gradient theory [18–20] and the stress gradient theory [21–26], have been employed to describe the behaviors of material with microstructure and nanostructure. The stress gradient elasticity theory proposed by Eringen is a nonlocal model of the gradient type, which introduces high-order gradient of stress into the constitutive relation [21]. In recent years, many researchers have applied the stress gradient elasticity theory to the bending, buckling, and vibration analysis of nanostructures [22–26]. Thermal vibration analysis is needed for the nanobeams often subjected to thermal loading. Zhang et al. [27] developed a double-elastic beam model for studying transverse vibrations of double-walled carbon nanotubes. Benzair et al. [28] used the nonlocal Timoshenko beam model for free vibration analysis of single-walled carbon nanotubes including the thermal effect. Murmu and Pradhan [29] developed a singleelastic beam model to analyze the thermal vibration of singlewalled carbon nanotubes based on nonlocal elasticity theory.

http://dx.doi.org/10.1016/j.camss.2017.08.001 0894-9166/© 2017 Published by Elsevier Ltd on behalf of Chinese Society of Theoretical and Applied Mechanics. Please cite this article as: J. Jiang, L. Wang, Analytical solutions for thermal vibration of nanobeams with elastic boundary conditions, Acta Mechanica Solida Sinica (2017), http://dx.doi.org/10.1016/j.camss.2017.08.001

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Ebrahimi and Salari [30] investigated the thermal effect on the buckling and free vibration characteristics of the functionally graded size-dependent Timoshenko nanobeams subjected to an in-plane thermal loading. In the above-mentioned studies, the boundary conditions of nanobeams are all classical cases, such as free, hinged or clamped ones. In practice, the interaction between nanobeams and substrates is van der Waals force. Thus the boundary condition of the nanobeam is commonly elastically constrained. It has been widely accepted that it is very hard to obtain an analytical solution for beams or plates except for very few simple boundary cases. Thus, some efficient numerical solution techniques, such as the modified Fourier series method, the meshless method and the differential quadrature method have been employed to solve the vibration problems of beams with arbitrary boundary conditions. Li and coworkers [31,32] and Jin et al. [33] presented a modified Fourier series method for the vibration of beams with general boundary conditions based on the classical Euler beam model. Kiani [34] used the reproducing kernel particle method to study the free transverse vibration of embedded single-walled nanotubes with arbitrary boundary conditions by the nonlocal Euler beam, Timoshenko beam, and higher-order beam models. Rosa and Lippiello [35] adopted the differential quadrature method to investigate free vibrations of embedded singlewalled carbon nanotubes based on the Euler beam theory. The attempt of this work is to propose an analytical solution for studying the vibration of elastically-supported nanobeams with second-order stress gradient elastic theory subjected to thermal stress. For this purpose, the nonlocal Euler beam model with elastic boundary is presented in Section 2. Then, an analytical solution for boundary value problems for the free vibrations of a nanobeam is derived in Section 3. Vibration analysis of the nanobeams with elastic boundary conditions are presented and discussed in Section 4. Finally, some concluding remarks are made in Section 5.

(1)

∂w(x, y, t) , uy = 0, uz = w(x, y, t) ∂x

(3)

The relation between the shear force Q and the bending moment M is expressed as Q−

∂M = 0, ∂x

(4)

with axial force considered, ∂Q ∂ 2w ∂ 2w = ρA 2 − N 2 , ∂x ∂t ∂x

(5)

where ρ is the mass density of the material, A is the area of the cross section, and N denotes an additional axial force dependent on temperature change T and thermal coefficient α T of the nanobeam, which can be expressed as N = −EAαT T.

(6)

The bending moment  M=

A

zσx dA.

(7)

A combination of Eqs. (1), (3) and (7) leads to M − μ2

∂ 2M ∂ 2w = −EI 2 , ∂ x2 ∂x

(8)

where I is the moment of inertia of the cross section. Eliminating Q from Eqs. (4) and (5) yields ∂ 2M ∂ 2w ∂ 2w = ρA 2 − N 2 . 2 ∂x ∂t ∂x

M = −μ2 N

(9)

(2)

where t denotes time, and u and w are displacements of the middle line. The strain field can be expressed as

∂ 2w ∂ 2w − ρA 2 2 ∂x ∂t

 − EI

∂ 2w . ∂ x2

(10)

From Eqs. (4) and (10), the shear force Q for the nonlocal Euler beam model can be expressed as  Q = −μ2 N

where E represents Young’s modulus and ε x is the axial strain; μ = e0 a is the nonlocal parameter reflects the influence of the microstructure on the strain in the nonlocal elastic material [21], with e0 a constant appropriate to each material, and a an internal characteristic length of the material. For a thin beam, the displacement components can be written as ux = −z

∂ 2w . ∂ x2



According to Eringen’s nonlocal elasticity theory, the constitutive law between stress and strain in the one-dimensional case can be expressed as ∂ 2 σx = E εx , ∂ x2

εx = −z

Then, substituting Eq. (9) into Eq. (8), the bending moment M for the nonlocal Euler beam model with axial force can be expressed as

2. Nonlocal Euler beam model with elastic boundary

σx − μ2

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∂ 3w ∂ 3w − ρA 2 3 ∂x ∂ t ∂x

 − EI

∂ 3w . ∂ x3

(11)

Substituting Eq. (11) into Eq. (5), the governing equation of the nonlocal elastic beam can be derived as [EI + μ2 N]

∂ 4w ∂ 2w ∂ 2w ∂ 4w − N 2 + ρA 2 − μ2 ρA 2 2 = 0. 4 ∂x ∂x ∂t ∂ x ∂t

(12)

In particular, the new beam model can be degenerated to the classical Euler beam model with axial force, if the material length-scale parameter is set to be zero. The equivalent continuum model of a nanobeam supported by an elastic medium at both ends is shown in Fig. 1.

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∂W ∂ 2W = −[μ2 N + EI] 2 − μ2 ρAω2W, (17d) ∂x ∂x at x = L. The dimensionless parameters for the nonlocal Euler beam that is elastically restrained are defined as KL

Fig. 1 – An elastically constrained nanobeam.

ξ=

The boundary conditions for the elastically restrained nonlocal beam are given as follows: k0 w = Q, K0

∂w = −M, at x = 0, ∂x

(13a)

∂w = M, at x = L, ∂x

(13b)

kL w = −Q, KL

where k0 and kL are the translational spring constants, K0 and KL are the rotational spring constants at x = 0 and x = L, respectively. Actually, Eq. (13) represents a set of general boundary conditions. Particularly, all the classical boundary conditions can be obtained by setting the spring constants to be extremely large or small. Substituting Eqs. (10) and (11) into Eq. (13) yields  ∂ 3w ∂ 3w ∂ 3w k0 w = −μ N 3 − ρA 2 − EI 3 , ∂x ∂ t ∂x ∂x   ∂w ∂ 2w ∂ 2w ∂ 2w K0 = μ2 N 2 − ρA 2 + EI 2 , ∂x ∂x ∂t ∂x

x W (x ) μ ρAL4 ˜ k0 L3 , W (ξ ) = , g= , 2 = ω2 , k0 = , L L L EI EI

kL L3 K0 L KL L NL2 k˜ 1 = , K˜ 0 = , K˜ 1 = ,δ = − EI EI EI EI

(18)

where g is the dimensionless nonlocal effect parameter which can reflect a small-scale effect on the response of structures in nanosize, and δ is named thermal effect parameter which can reflect the effect of thermal stress. Using these parameters, the dimensionless form of Eq. (16) can be written as [1 − g2 δ]

∂ 4W ∂ 2W + [g2 2 + δ] 2 − 2W = 0. ∂ξ 4 ∂ξ

(19)

The dimensionless elastic boundary conditions are as follows ∂ W ∂W , k˜ 0W = − [1 − g2 δ] 3 − g2 2 ∂ξ ∂ξ

(20a)

∂W ∂ 2W = [1 − g2 δ] 2 + g2 2W, ∂ξ ∂ξ

(20b)

3



2

(14a) (14b)

at ξ = 0, and 3

∂ W ∂W k˜ 1W = [1 − g2 δ] 3 + g2 2 , ∂ξ ∂ξ

at x = 0, and  ∂ 3w ∂ 3w ∂ 3w − ρA + EI 3 , 3 2 ∂x ∂ t ∂x ∂x   2 2 ∂w ∂ w ∂ w ∂ 2w KL = −μ2 N 2 − ρA 2 − EI 2 , ∂x ∂x ∂t ∂x

K˜ 0

(20c)



kL w = μ2 N

(14c) (14d)

at x = L. In accordance with the classical solution, for free vibration of beam, the solution w(x, t) is assumed as follows:

∂W ∂ 2W = −[1 − g2 δ] 2 − g2 2W, ∂ξ ∂ξ at ξ = 1. K˜ 1

3. Solution of boundary value problem for free vibration of a nanobeam For convenience, the following parameters are introduced:

w(x, t) = W(x )eiωt ,

(15) e = 1 − g2 δ, p = g2 c, c = 2 , b = p + δ,

where W(x) is the displacement of the beam, ω is the natural √ frequency of the beam and i ≡ −1. Substituting Eq. (15) into Eq. (12) gives ∂ 4W ∂ 2W [EI + μ N] 4 + [μ2 ω2 ρA − N] 2 − ρAω2W = 0. ∂x ∂x 2

(16)

∂ 3W ∂ x3

− μ2 ρAω2

∂W , ∂x

∂W ∂ 2W K0 = [μ2 N + EI] 2 + μ2 ρAω2W, ∂x ∂x

∂ 3W ∂W + μ2 ρAω2 , ∂x ∂ x3

∂ 4W ∂ 2W + b 2 − cW = 0, 4 ∂ξ ∂ξ

(22)

and ∂ W ∂W , k˜ 0W = − e 3 − p ∂ξ ∂ξ

(23a)

∂W ∂ 2W = e 2 + pW, ∂ξ ∂ξ

(23b)

∂ W ∂W , k˜ 1W = e 3 + p ∂ξ ∂ξ

(23c)

3

(17a) K˜ 0 (17b)

3

at x = 0, and kLW = [μ2 N + EI]

(21)

Eqs. (19) and (20) can be expressed as e

Substituting Eq. (15) into Eq. (14) gives k0W = −[μ2 N + EI]

(20d)

(17c)

K˜ 1

∂W ∂ 2W = −e 2 − pW. ∂ξ ∂ξ

(23d)

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The solution for Eq. (22) is in the form of

Using a series of elementary transformation, Eq. (26) can be written as

W(ξ ) = a1 sin(αξ ) + a2 cos(αξ ) + a3 sinh(βξ ) + a4 cosh(βξ ), (24)

⎛ 1 ⎜0 ⎜ ⎜ ⎝0 0

where a1 , a2 , a3 and a4 are constants of integration to be determined from the boundary conditions of the problem, and  α=

b+

   b2 + 4ec −b + b2 + 4ec ,β= , 2e 2e

c11 ⎜c ⎜ 21 ⎜ ⎝c31 c41

c12 c22 c32 c42

c13 c23 c33 c43

⎞⎛ ⎞ c14 a1 ⎜ ⎟ c24 ⎟ ⎟⎜a2 ⎟ ⎟⎜ ⎟ = 0. c34 ⎠⎝a3 ⎠ c44 a4

s

33

s43

c12 c22 c32 c42

c13 c23 c33 c43

⎞⎛ ⎞ s14 a1 ⎜ ⎟ s24 ⎟ ⎟⎜a2 ⎟ ⎟⎜ ⎟ = 0 , s34 ⎠⎝a3 ⎠ s44 a4

(29)

s34

= 0. s44

(30)

By Eqs. (29) and (30), it is obtained as follows: (26)

rb =

In order to have a non-trivial solution, the determinant of the coefficient matrix of the unknowns a1 , a2 , a3 and a4 should satisfy the following condition:

c11

c

21

c31

c 41

s13 s23 s33 s43

in which new parameters are shown in Appendix B. The existence of a non-trivial solution of Eq. (26) implies

(25)

Substituting Eq. (24) into Eq. (23), one obtains the following system of equations in matrix form to be solved for the constants ai (i = 1, 2, 3, 4). ⎛

0 1 0 0

c14

c24

= 0, c34

c

(27)

a4 s43 =− , a3 s44

(31a)

a4 = rba3 ,

(31b)

a1 = −s13 a3 − s14 a4 = −s13 a3 − s14 rba3 ,

(31c)

a2 = −s23 a3 − s24 a4 = −s23 a3 − s24 rba3 .

(31d)

Then the mode shape of vibration can be obtained as

44

W(ξ ) = a3 [(−s13 − s14 rb) sin(αξ ) + (−s23 − s24 rb) cos(αξ ) in which the new parameters are shown in Appendix A. Eq. (27) is the frequency equation which provides all the natural frequencies of the Euler beam with elastic boundary supports. All the natural frequencies can be obtained numerically through a direct iterative process. Once the natural frequencies have been obtained from Eq. (27), the mode shapes corresponding to the natural frequencies can be readily determined as follows. It is easy to prove that

c

11

c21

c12

= α[(eα 2 − p)2 + K˜ 0 k˜ 0 ] = 0. c22

+ sinh(βξ ) + rb cosh(βξ )].

4.

(32)

Numerical results and discussions

In this section, the vibration of nanobeam with elastic boundary condition is investigated. First, the classical boundary conditions can be viewed as special cases of elastic boundary conditions. For a clamped-supported edge (C), the dimensionless stiffnesses of the translational and rotational boundary springs are infinitely large (1.0 × 1010 ). For a hinged edge (H), the dimensionless stiffness of the translational spring

(28)

Table 1 – Comparisons of the dimensionless frequencies  of nanobeams with the dimensionless nonlocal effect parameter g = 0.05 and the thermal effect parameter δ = 1(k˜ 0 = 1010 , k˜ 1 = 1010 , K˜ 0 = 1010 , K˜ 1 = 1010 for C–C case; k˜ 0 = 1010 , k˜ 1 = 0, K˜ 0 = 1010 , K˜ 1 = 0for C–F case; k˜ 0 = 1010 , k˜ 1 = 1010 , K˜ 0 = 1010 , K˜ 1 = 0 for C–H case; k˜ 0 = 1010 , k˜ 1 = 1010 , K˜ 0 = 0, K˜ 1 = 0 for H–H case). Boundary conditions

C–C C–F C–H H–H

Source

Present DQ Present DQ Present DQ Present DQ Exact

Mode 1

2

3

4

5

6

7

8

21.736 21.736 3.6356 3.6366 14.808 14.808 9.2301 9.2301 9.2301

57.967 57.967 21.346 21.347 47.013 47.013 37.136 37.136 37.136

107.74 107.74 58.005 58.005 93.277 93.277 79.797 79.797 79.797

166.61 166.61 107.74 107.74 149.43 149.43 133.12 133.12 133.12

231.10 231.10 166.61 166.61 211.88 211.88 193.41 193.41 193.41

298.71 298.72 231.10 231.13 277.97 277.99 257.88 257.90 257.88

367.85 367.91 298.71 297.72 345.96 346.03 324.64 324.81 324.64

437.55 437.03 367.85 362.08 414.78 412.70 392.51 390.85 392.51

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Table 2 – The dimensionless frequencies  of nanobeams with the dimensionless nonlocal effect parameter g = 0.05 and the thermal effect parameter δ = 1. k˜ 0 = k˜ 1 ,

10

2

104

106

108

K˜ 0 = K˜ 1

2

10 104 106 102 104 106 102 104 106 102 104 106

Mode 1

2

3

4

5

6

7

8

12.161 12.316 12.318 20.774 21.556 21.565 20.928 21.725 21.734 20.930 21.727 21.736

21.253 21.254 21.254 54.861 56.750 56.771 55.953 57.934 57.955 55.964 57.945 57.967

41.847 42.172 42.176 100.70 103.84 103.88 104.26 107.67 107.71 104.29 107.71 107.74

80.903 81.912 81.922 153.39 157.69 157.74 161.59 166.48 166.53 161.66 166.55 166.61

132.59 134.20 134.22 209.03 214.09 214.15 224.60 230.89 230.97 224.72 231.02 231.09

191.96 194.04 194.06 264.78 269.71 269.77 290.82 298.43 298.52 291.01 298.62 298.71

255.89 258.27 258.29 319.41 322.86 322.91 358.66 367.49 367.59 358.93 367.74 367.84

322.33 324.89 324.92 373.69 374.69 374.71 427.15 437.11 437.23 427.50 437.42 437.54

is infinitely large (1.0 × 1010 ), but the dimensionless stiffness of the rotational spring is set to be 0. For a free edge (F), the dimensionless stiffnesses of both the translational springs and the rotational boundary springs are set to be 0. To validate accuracy and reliability of the analytical solutions, the comparisons of the first eight dimensionless frequencies of nanobeams are made between the present results and the available results obtained by the differential

quadrature method (DQ) [29,36] with four kinds of classical boundary conditions, the dimensionless nonlocal effect parameter g = 0.05 and thermal effect parameter δ = 1, as shown in Table 1. The exact solution of the frequency of hingedsupported nanobeams can be easily obtained by Eq. (19) as  2 2

= mπ [(1 − g2 δ)(mπ ) − δ]/[1 + g2 (mπ ) ],

(33)

Fig. 2 – Effect of the boundary spring stiffness on the dimensionless frequencies of nanobeams with the nonlocal effect parameter g = 0.05. Please cite this article as: J. Jiang, L. Wang, Analytical solutions for thermal vibration of nanobeams with elastic boundary conditions, Acta Mechanica Solida Sinica (2017), http://dx.doi.org/10.1016/j.camss.2017.08.001

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Fig. 3 – Effect of the boundary spring stiffness on the temperature frequency ratios of nanobeams with the nonlocal effect parameter g = 0.05.

Fig. 4 – Effect of the boundary spring stiffness on the temperature frequency ratios with different nonlocal effect parameters for nanobeams of the thermal effect parameter δ = 1.

Fig. 5 – Effect of the thermal effect parameter δ on the temperature frequency ratios for nanobeams with the nonlocal effect parameter g = 0.05.

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Fig. 6 – Effect of nonlocal effect parameter on the dimensionless frequencies of nanobeams with the thermal effect parameter δ = 1. where m is the half-wave number. It is observed that the present results which set k˜ 0 = 1010 , k˜ 1 = 1010 , K˜ 0 = 1010 , K˜ 1 = 1010 for C–C case, k˜ 0 = 1010 , k˜ 1 = 0, K˜ 0 = 1010 , K˜ 1 = 0 for C–F case, k˜ 0 = 1010 , k˜ 1 = 1010 , K˜ 0 = 1010 , K˜ 1 = 0 for C–H case and k˜ 0 = 1010 , k˜ 1 = 1010 , K˜ 0 = 0, K˜ 1 = 0 for H–H case are in good agreement with those obtained by the DQ. In particular,

Fig. 8 – The first- and fourth-order mode shapes for nanobeams with the rotational spring stiffness K˜ = 0 and the nonlocal effect parameter g = 0.2.

Fig. 7 – The first- and fourth-order mode shapes for nanobeams with the rotational spring stiffness K˜ = 0 and the nonlocal effect parameter g = 0.2.

the present results are in very excellent agreement with the exact solution for hinged-supported case. It also indicates that the spring stiffness value 1.0 × 1010 is sufficiently large to represent clamped-supported edge and hinged edge. Furthermore, Table 2 shows the first eight dimensionless frequencies of the nanobeams with 12 different elastic stiffness constants at the boundary, the dimensionless nonlocal effect parameter g = 0.05 and the thermal effect parameter δ = 1. In order to study the influence of thermal effect on vibrations, we define temperature frequency ratio which reflects the relation between the natural frequency of free vibration beam including the thermal stress and that without the ther¯ Here, and

¯ are the dimensionless fremal stress by / . quencies of free vibrational beam including and without the thermal stress, respectively. For simplicity, in the following discussion, it is assumed that translational spring stiffness and rotational spring stiffness at both ends keep the same, i.e., k˜ 0 = k˜ 1 = k˜ and K˜ 0 = K˜ 1 = K˜ . Next, the influences of spring stiffness values at the boundaries on the thermal effect of nanobeams are investigated. The first-order vibrational frequencies of nanobeams with the nonlocal effect parameter g = 0.05 as functions of the dimensionless boundary spring stiffness k˜ or K˜ can be seen in Fig. 2. From Fig. 2(a) and (b), the dimensionless frequencies increase rapidly as the

Please cite this article as: J. Jiang, L. Wang, Analytical solutions for thermal vibration of nanobeams with elastic boundary conditions, Acta Mechanica Solida Sinica (2017), http://dx.doi.org/10.1016/j.camss.2017.08.001

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Fig. 9 – The first- and fourth-order mode shapes for nanobeams with the translational spring stiffness k˜ = 1010 and the nonlocal effect parameter g = 0.2.

dimensionless translational spring stiffness k˜ is in the range of 10–103 . However, the dimensionless translational spring stiffness k˜ has little effect on dimensionless frequencies when k˜ > 103 . In Fig. 2(c) and (d), the variation tendencies of the dimensionless frequencies versus the dimensionless rotational spring stiffness K˜ are similar to those in Fig. 2(a) and (b). To show the influence of thermal effect, the temperature frequency ratios of the first-order vibration mode of nanobeams with the nonlocal effect parameter g = 0.05 versus the dimensionless boundary spring stiffness are presented in Fig. 3. According to Fig. 3(a) and (b), the temperature frequency ratios decrease monotonically with the increase of dimensionless translational spring stiffness k˜ . However, the temperature frequency ratios increase monotonically with the increase of dimensionless rotational spring stiffness K˜ , as shown in Fig. 3(c) and (d). Remarkable changes of the temperature frequency ratios are found as the dimensionless boundary spring stiffness k˜ or K˜ ranges from 10 to 103 . Especially, the temperature frequency ratios change more considerable for the larger thermal effect parameters δ. To study the influence of the nonlocal effect parameter g on the vibration of nanobeams, the temperature frequency ratios of the first-order vibration mode for nanobeams with four different values of nonlocal effect parameter g = 0, 0.05, 0.1 and 0.15 when the thermal effect parameter δ = 1 are displayed in

Fig. 10 – The first- and fourth-order mode shapes for nanobeams with the translational spring stiffness k˜ = 1010 and the nonlocal effect parameter g = 0.2.

Fig. 4. They reveal that the thermal effect becomes more significant for the larger nonlocal effect parameter. In addition, the thermal effect is sensitive as the dimensionless stiffness values are in the range of 10–103 . Fig. 5 shows the influences of the thermal effect parameter δ on the vibration of nanobeams with different spring stiffness. The temperature frequency ratios of the first-order vibration mode decrease as δ increases for nonlocal effect parameter g = 0.05. Fig. 5(a) and (b) indicates that the translational spring stiffness k˜ and the rotational spring stiffness K˜ almost have no effect on the temperature frequency ratios for k˜ >102 or K˜ >102 , respectively. Fig. 6 shows the effects of nonlocal effect parameter g on the first-order vibrational frequencies of nanobeams by using nonlocal Euler models with classical elastic and nonlocal elastic boundary conditions for the thermal effect parameter δ = 1. As the nonlocal effect parameter g = 0 in Eq. (20) for boundary condition, the nonlocal elastic boundaries (NEB) can be reduced to classical elastic cases (CEB). Here, the results of CEB are obtained by using equation of motion (19) including the nonlocal effect parameter and CEB boundary equation without the nonlocal effect parameter g. The results of NEB can be directly obtained using Eqs. (19) and (20). From Fig. 6, it is apparent that the dimensionless frequencies of nanobeams are

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almost the same for NEB and CEB. That is, the nonlocal effect parameter g at the boundaries has almost no effect on the results. Thus the classical elastic boundary conditions, instead of the nonlocal elastic boundary conditions, can describe the thermal vibration of elastically-supported nanobeams with second-order gradient of stress elastic theory. The shapes of first and fourth vibration modes of the nonlocal Euler beams with the nonlocal effect parameter g = 0.2, thermal effect parameters in the cases of δ = 0 or δ = 2.5 with different elastic boundary conditions are plotted in Figs. 7–10. The elastic boundary conditions of Figs. 8 and 10 correspond to hinged-supported and clamped-supported boundary conditions, respectively. It is very hard to see the difference of the mode shapes between the nanobeams including the thermal stress and without the thermal stress. The thermal stress has little effect on the mode shape of nanobeams.

5.

c23 = eβ 3 + pβ,

(A.7)

c24 = k˜ 0 ,

(A.8)

c31 = −eα 2 sin(α) + K˜ 1 α cos(α) + p sin(α),

(A.9)

c32 = −eα 2 cos(α) − K˜ 1 α sin(α) + p cos(α),

(A.10)

c33 = eβ 2 sinh(β ) + K˜ 1 β cosh(β ) + p sinh(β ),

(A.11)

c34 = eβ 2 cosh(β ) + K˜ 1 β sinh(β ) + p cosh(β ),

(A.12)

c41 = eα 3 cos(α) − pα cos(α) + k˜ 1 sin(α),

(A.13)

c42 = −eα 3 sin(α) + pα sin(α) + k˜ 1 cos(α),

(A.14)

c43 = −eβ 3 cosh(β ) − pβ cosh(β ) + k˜ 1 sinh(β ),

(A.15)

c44 = −eβ 3 sinh(β ) − pβ sinh(β ) + k˜ 1 cosh(β ).

(A.16)

Concluding remarks Appendix B

A nonlocal Euler beam model with second-order stress gradient taken into consideration is used to study the vibration of nanobeams with elastic boundary conditions. An analytical solution is proposed to investigate the free vibration of nonlocal Euler beams subjected to thermal stress. Some numerical examples, which include rigidly and elastically constrained boundary conditions, are studied. The results show that the natural frequencies including the thermal stress are lower than those without the thermal stress when temperature rises. It is found that the constrained stiffness has drastic effects on the temperature frequency ratios when the dimensionless stiffness ranges from 10 to 103 . In addition, numerical simulations also indicate the importance of small-scale effect on the vibration of nanobeams.

The new parameters in Eq. (29) are expressed as s13 =

β (eβ 2 + p)(p − eα 2 ) + K˜ 0 k˜ 0 , 2 α (eα 2 − p) + K˜ 0 k˜ 0

s14 = −

ek˜ 0 (α 2 + β 2 ) 2 α[(eα 2 − p) + K˜ 0 k˜ 0 ]

,

(B.1)

(B.2)

s23 =

βeK˜ 0 (α 2 + β 2 ) , 2 (eα 2 − p) + K˜ 0 k˜ 0

(B.3)

s24 =

(eβ 2 + p)(p − eα 2 ) + K˜ 0 k˜ 0 , 2 (eα 2 − p) + K˜ 0 k˜ 0

(B.4)

s33 = αβ(eβ 2 + p)[(eα 2 − p)2 + K˜ 0 k˜ 0 ] cosh(β ) − α k˜ 1 [(eα 2 − p)2

Acknowledgments

+ K˜ 0 k˜ 0 ] sinh(β ) + αβ{−(eβ 2 + p)[(eα 2 − p)2 + K˜ 0 k˜ 0 ]

This work was supported in part by the National Natural Science Foundation of China (Nos. 11522217, 11632003), in part by The 333 Talent Program in Jiangsu Province (No. BRA2017374), Funding of Jiangsu Province Innovation Program for Graduate Education (KYLX15-0234), and in part by the Fundamental Research Funds for the Central Universities of China (No. NE2012001).

+ K˜ 0 e(α 2 + β 2 )(k˜ 0 + k˜ 1 )} cos(α) − β{(eα 2 − p) × [K˜ 0 eα 2 (α 2 + β 2 ) + k˜ 1 (eβ 2 + p)] − K˜ 0 k˜ 0 k˜ 1 } sin(α),

(B.5)

s34 = −α k˜ 1 [(eα 2 − p)2 + K˜ 0 k˜ 0 ] cosh(β ) + αβ(eβ 2 + p)[(eα 2 − p)2 + K˜ 0 k˜ 0 ] sinh(β ) − α{(eα 2 − p)[k˜ 0 e(α 2 + β 2 ) + k˜ 1 (eβ 2 + p)] − K˜ 0 k˜ 0 k˜ 1 } cos(α) + {α 2 (eβ 2 + p) × [(eα 2 − p)2 + K˜ 0 k˜ 0 ] − k˜ 0 e(α 2 + β 2 )(K˜ 0 α 2 + k˜ 1 )} sin(α), (B.6)

Appendix A

s43 = αβ K˜ 1 [(eα 2 − p)2 + K˜ 0 k˜ 0 ] cosh(β ) + α(eβ 2 + p)[(eα 2 − p)2

The new parameters in Eq. (27) are expressed as c11 = K˜ 0 α,

(A.1)

c12 = eα 2 − p,

(A.2)

c13 = K˜ 0 β,

(A.3)

2

c14 = −eβ − p, 3

c21 = −eα + pα, c22

= k˜ 0 ,

(A.4)

+ K˜ 0 k˜ 0 ] sinh(β ) + αβ{(eα 2 − p)[K˜ 0 e(α 2 + β 2 ) + K˜ 1 (eβ 2 + p)] − K˜ 0 K˜ 1 k˜ 0 } cos(α) − β{(eβ 2 + p)[(eα 2 − p)2 + K˜ 0 k˜ 0 ] − K˜ 0 e(α 2 + β 2 )(K˜ 1 α 2 + k˜ 0 )} sin(α),

(B.7)

s44 = α(eβ 2 + p)[(eα 2 − p)2 + K˜ 0 k˜ 0 ] cosh(β ) + αβ K˜ 1 [(eα 2 − p)2 + K˜ 0 k˜ 0 ] sinh(β ) − α{(eβ 2 + p)[(eα 2 − p)2

(A.5)

+ K˜ 0 k˜ 0 ] − k˜ 0 e(α 2 + β 2 )(K˜ 1 + K˜ 0 )} cos(α) − {(eα 2 − p)

(A.6)

× [K˜ 1 α 2 (eβ 2 + p) + k˜ 0 e(α 2 + β 2 )] − α 2 K˜ 0 K˜ 1 k˜ 0 } sin(α).

(B.8)

Please cite this article as: J. Jiang, L. Wang, Analytical solutions for thermal vibration of nanobeams with elastic boundary conditions, Acta Mechanica Solida Sinica (2017), http://dx.doi.org/10.1016/j.camss.2017.08.001

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Please cite this article as: J. Jiang, L. Wang, Analytical solutions for thermal vibration of nanobeams with elastic boundary conditions, Acta Mechanica Solida Sinica (2017), http://dx.doi.org/10.1016/j.camss.2017.08.001