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The effect of thickness on the mechanics of nanobeams
International Journal of Engineering Science 123 (2018) 81–91
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International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
The effect of thickness on the mechanics of nanobeams Li Li∗, Haishan Tang, Yujin Hu State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
a r t i c l e
i n f o
Article history: Received 22 October 2017 Revised 19 November 2017 Accepted 19 November 2017
Keywords: Nonlocal strain gradient model Size-dependent effect Nanobeam Buckling Nanomechanics
a b s t r a c t Nonlocal strain gradient models are being used more and more extensively in examining the size-dependent effects on the statical and dynamical behaviors of micro/nanostructures. However, for the fake of simplification, the size-dependent effects are often assumed to be neglected in the thickness direction of beams and plates. This simplification is originally aiming at reducing the complexity of the studied beam and plate problems while retaining all the most important mechanical features. Based on the nonlocal strain gradient model without the thickness effect, the size-dependent effect may reveal stiffness-softening or stiffness-hardening effect, which only depends on the values of strain-gradient and stress-gradient parameters but is independent on the geometric feature. In this paper, a nonlocal strain gradient beam model incorporating the thickness effect is developed for the size-dependent buckling analysis of nanobeams, and closed-form solutions are derived for post-buckling configuration and critical buckling force. When incorporating the size-dependent effect of thickness, it is found that the stiffness-softening and stiffness-hardening effects depend not only on the ratio of stress-gradient parameter to strain-gradient parameter, but also on the geometric feature (slenderness ratio). Interestingly, the stiffness-softening effect can be only found for the longitudinal dispersion relation of Silicon, while the stiffness-hardening effect may be observed for the buckling behaviors of nanobeams made of Silicon due to the size-dependent effect of thickness. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction The rapid development towards nanoscience and nanotechnology has begun a new era for many areas, including engineering, chemical, medicine, electronics. Many nano/micro scale devices have been developed and being tested widely for applications, such as atomic force microscopy (AFM) (Eaton & West, 2010), nanosensors (Cui, Wei, Park, & Lieber, 2001; Duan, Li, Hu, & Wang, 2017), nanoactuators (Moya, Azzaroni, Farhan, Osborne, & Huck, 2005; Shi, Cheng, Pugno, & Gao, 2010), nano/micro-electro-mechanical systems (NEMS/MEMS), and frequency synthesis. Some nanobeams in engineering structures have been developed for recent applications (Demir & Civalek, 2017), including tunable oscillator, nanorelays, nonvolatile random access memory, rotational motors, nanotweezers, and feedback-controlled nanocantilevers. It has been pointed out by many experimental researches that many beam-type and plate-type materials at nano/micro scale under bending and buckling loads have size dependent effects on their mechanical and physical properties. For
∗
Corresponding author. E-mail address: [email protected] (L. Li).
https://doi.org/10.1016/j.ijengsci.2017.11.021 0020-7225/© 2017 Elsevier Ltd. All rights reserved.
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Fig. 1. Longitudinal dispersion relation in Silicon (Si) obtained by utilizing classical elasticity, nonlocal strain gradient elasticity and experimental data (Cochran, 1973). The mass density and Young’s modulus of Si used in numerical studies is considered as ρ = 2330 kg/m3 and E = 201.92 GPa, respectively. Here μ = 0.16544 nm and l = 0.0535 nm for nonlocal strain gradient elasticity.
example, Young’s modulus is length-dependent or diameter-dependent, and may be stiffness-hardening (Chen, Shi, Zhang, Zhu, & Yan, 2006; Cuenot, Frétigny, Demoustier-Champagne, & Nysten, 20 04; Jing et al., 20 06) or stiffness-softening (Zhao, Min, & Aluru, 2009). However, experimental researches are very difficult to control and are often very expensive. Some simulation methods including first-principle method, mixed atomistic-continuum mechanics, molecular dynamics, the density functional theory (DFT) can take the size-dependent effect into account. But the solutions of these atomistic-based methods are very time-consuming, even for supercomputers. For the sake of the ease of implementation, continuum mechanics can be alternatively chosen as a tool for theoretical analysis. To develop reliable continuum models for predicting the mechanical behaviors of structures at nano-scale, some nonclassical continuum theories have been proposed. The nonlocal strain gradient theory (Askes & Aifantis, 2009; Lim, Zhang, & Reddy, 2015) is one of the popular theories for studying size-dependent mechanical behaviors. Unlike the classical elasticity, the nonlocal strain gradient theory can capture both stiffness-hardening and stiffness-softening effects by accounting for both stain and stress gradient effects. The total stress tkl can be expressed as (Askes & Aifantis, 2009)
1 − μ2 ∇ 2 tkl = 1 − l 2 ∇ 2 Ckli j εi j
where Cklij denotes elastic coefficient; ε ij denotes strain tensor; and ∇ 2 is Laplacian operator. Furthermore, a stress-gradient parameter μ is introduced to incorporate the stress gradient effect, and a strain-gradient parameter l is extra introduced to incorporate the strain gradient effect. The gradient model (1) can be simplified to the nonlocal model (Eringen, 1983) if l = 0, and strain gradient model (Aifantis, 1992) if μ = 0. Therefore, the stain and stress gradient models are also known as nonlocal strain gradient models, which are being applied more and more extensively for examining the size-dependent effects on the statical and dynamical behaviors of micro/nano-structures (Barati & Zenkour, 2017; Ebrahimi & Barati, 2016b; Ebrahimi & Dabbagh, 2017; Farajpour & Rastgoo, 2017; Guo et al., 2016; Karami, Shahsavari, & Li, 2017; Li & Hu, 2015; Li, Hu, & Li, 2016; Li, Tang, & Hu, 2018; Li, Li, Hu, Ding, & Deng, 2017; Lim et al., 2015; Lu, Guo, & Zhao, 2017a; Mehralian, Beni, & Zeverdejani, 2017; Rajabi & Hosseini-Hashemi, 2017; Sahmani & Aghdam, 2017a; 2017b; Shahsavari, Karami, & Mansouri, 2018; Zeighampour, Beni, & Karimipour, 2017; Zhen & Zhou, 2017; Zhu & Li, 2017a). These works were recently motivated by the good match between the results of nonlocal strain gradient theory and the data of experimental studies and molecular dynamics simulations (Karami, Shahsavari, Janghorban, & Li, 2018; Li & Hu, 2016b; Li, Hu, & Ling, 2016; Li et al., 2018; Lim et al., 2015; Zeigampour & Beni, 2017; Zhu & Li, 2017a; 2017c). For example, based on the closed-form solution of longitudinal dispersion in Zhu and Li (2017c), Fig. 1 shows the longitudinal dispersion relations in the [100] direction of Si. Unlike the classical model, the nonlocal strain gradient elasticity model can show a good agreement with the experimental data. For one-dimensional (1D) problems, the constitutive equations can be simplified to (Li & Hu, 2017; Li, Li, & Hu, 2016; Lim et al., 2015)
1 − μ2 ∇ 2 txx = 1 − l 2 ∇ 2 E εxx
(1)
where E is the Young’s modulus. For longitudinally static and dynamic problems of rods, the longitudinal strain ε xx is independent of the width and thickness of rods. That is to say, the Laplacian operator can be simplified as ∇ 2 = ∂ 2 /∂ x2 , and this simplification is good in some studies (Li, Hu, & Li, 2016; Zhu & Li, 2017c; Fernandes, El-Borgi, Mousavi, Reddy, & Mechmoum, 2017).
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In the case of beam-type structure, the longitudinal strain ε xx depends on the position at the thickness direction. Considering a nanobeam with the length (in x− direction), width (in y− direction) and thickness (in z− direction), and according to the von–Kármán’s nonlinear strain-displacement relationship, the non-zero strain for the Euler–Bernoulli nanobeam can be yielded as (Li & Hu, 2017)
εxx =
2 ∂u 1 ∂w ∂ 2w + −z 2 ∂x 2 ∂x ∂x
(2)
2
where u denotes longitudinal displacement, w is transversal displacement, the nonlinearity term 21 ∂∂wx is due to the stretching effect of the mid-plane of nanobeam. It is clear that the longitudinal strain ε xx is dependent on z. However, for the fake of simplification, many researchers (Ebrahimi & Barati, 2016a; Li & Hu, 2015; 2016a; 2017; Lim et al., 2015; Lu, Guo, & Zhao, 2017b; S¸ ims¸ ek, 2016; Tang, Liu, & Zhao, 2018; Xu, Wang, Zheng, & Ma, 2017) assumed that the size-dependent behaviors of nanobeam can be neglected in the thickness direction, and also let the Laplacian operator be ∇ 2 = ∂ 2 /∂ x2 for various beam models. This simplification is originally aiming at reducing the complexity of the studied beam and plate problems while retaining all the most important mechanical features. In this study, we will show that the size-dependent effect of thickness due to ε xx, z is important, and the size-dependent model without the thickness effect may obtain misleading results. Therefore, the Laplacian operator should be ∇ 2 = ∂ 2 /∂ x2 + ∂ 2 /∂ z2 , and the size-dependent effect of thickness has to be reconsidered for analyzing the mechanical behaviors of nanobeams. Based on the nonlocal strain gradient model without the thickness effect, it has recently found by many researchers (Li & Hu, 2015; 2016a; 2017; Lu et al., 2017b; Xu et al., 2017) that the size-dependent effect may reveal “stiffness-softening” or “stiffness-hardening” effect, which is independent on the geometric feature but depends on the relative magnitude of the strain-gradient and stress-gradient parameters. However, it is reported by Li, Wei, Lu, Lu, and Gao (2010) that the “stiffnesssoftening” or ”stiffness-hardening” effect on the yield stress of nano-twinned Cu depends on the geometric feature of its microstructures at nano-scale. In this paper, based on the nonlocal strain gradient model incorporating the thickness effect, we will show that the “stiffness-softening” and “stiffness-hardening” effects depend not only on the ratio of stress-gradient parameter to strain-gradient parameter, but also on the geometric feature (slenderness ratio), which will be explored for the buckling of nanobeams. 2. Governing equations for post-buckling of nanobeam incorporating the size-dependent effect of thickness In this section, based on the von–Kármán’s nonlinear strain-displacement relationship (2), a nonlocal strain gradient beam model incorporating the size-dependent effect of thickness will be deduced for analyzing the size-dependent postbuckling behaviors of nanobeams. 2.1. Equilibrium equations in terms of the principle of minimum potential energy By considering the thickness effect in the Euler–Bernoulli nanobeam model, the total stress can be expressed as (0 ) (1 ) (1 ) txx = σxx + σxxx + σxxz
(3)
(1 ) (1 ) where σ xx is nonlocal stress, and σxxx , σxxz are the high order nonlocal stresses. The virtual strain energy δ U can be ex(1 ) (1 ) pressed as a function of nonlocal stress σ xx and strain ε xx , high order nonlocal stresses σxxx , σxxz , and strain gradients ε xx, x , εxx, z , that is,
(1 ) (1 ) δεxx,x + σxxz δεxx,z dV σxx δεxx + σxxx V 2 2 2 ∂u 1 ∂w ∂ 2w ∂ 3w ∂ w ∂ u ∂ w ∂ 2w (1 ) (1 ) = σxx δ + − z 2 + σxxx δ + − z 3 + σxxz δ − 2 dV ∂x 2 ∂x ∂x ∂ x2 ∂ x ∂ x2 ∂x ∂x V L 3 ∂ ∂w ∂ ∂ 2w ∂2 ∂w ∂ 2 (M + Q ) ∂ Mh = − N − N + N − δ wdx ∂x ∂x ∂ x h ∂ x2 ∂ x3 ∂ x2 h ∂ x ∂ x2 0 L L 2 ∂w ∂ 2 w ∂ 2 Mh ∂ ( M + Q ) ∂ ∂w ∂ Nh ∂ N + − + Nh 2 − + − N δ udx + N δw ∂x ∂x ∂x ∂x h ∂x ∂ x2 ∂x ∂ x2 0 0 L
L
L
L 2 ∂ Nh ∂ w ∂ Mh ∂w ∂u ∂ w δ u + Nh δ + N− + Nh + −M−Q δ − Mh δ 2 . ∂x ∂ x ∂ x ∂ x ∂ x ∂x 0 0 0 0
δU =
(4)
Here N and M are the resultant force and moment for longitudinal and transverse directions, respectively. Nh and Q are the high order resultant forces for longitudinal and transverse directions, respectively. Mh is the high order resultant moment.
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These resultants can be given by
2 ∂ u 1 ∂w (1 − μ2 ∇ 2 )N = EA + , ∂x 2 ∂x (1 − μ2 ∇ 2 )M = −EI
∂ 2w , ∂ x2
(5)
(6)
(1 − μ2 ∇ 2 )Nh = l 2 EA
∂ 2u ∂ w ∂ 2w + , ∂ x2 ∂ x ∂ x2
(1 − μ2 ∇ 2 )Mh = −l 2 EI
∂ 3w , ∂ x3
(8)
∂ 2w , ∂ x2
(9)
(1 − μ2 ∇ 2 )Q = −l 2 EA
(7)
where A is the area of cross section, and I signifies the inertia moment. With the aid of Eqs. (5)–(9), it can be made out that the resultants stresses and moment are the functions only concerning x due to the integrations on the cross section of beam. Thus, ∂ 2 /∂ z2 in the operator ∇ 2 cannot work and therefore can be omitted in the resultant forces and moment. (1 ) Remark 1. The work done by the high order stress term along the thickness direction σxxz and the corresponding strain gradient term ε xx, z , which was neglected in Li and Hu (2015); 2017), is considered in strain energy U. Since the actual distribution of strain ε xx in any given section is not invariable, the strain gradient in the thickness direction is not zero. Thus, the work due to (1 ) σxxz and ε xx, z will play a key role in the stiffness of nanobeams. This will be elaborated later.
With regard to buckling and post-buckling research, the virtual work done by the axial compressing force p at one end of the nanobeam can be taken as the following expression:
δW =
L
p 0
∂w ∂w δ dx. ∂x ∂x
(10)
According to the principle of minimum potential energy, we have
δ U − δ W = 0.
(11)
Substituting Eqs. (4) and (10) into Eq. (11) and using integration by parts, one can obtain
δu : −
∂ N ∂ 2 Nh + = 0, ∂x ∂ x2
δ w : −N
(12)
∂ 2 w ∂ N ∂ w ∂ 2 (M + Q ) ∂ 2 Nh ∂ w ∂ Nh ∂ 2 w ∂ 3 Mh ∂ 2w − − + + + + p 2 = 0, 2 2 2 2 3 ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x
with the classical boundary condition:
u=0
or
N = 0;
∂w ∂ 2 w ∂ (M + Q ) ∂ ∂w ∂ 2 Mh ∂w w = 0 or N +Nh 2 + − Nh − −p = 0; ∂x ∂x ∂x ∂x ∂x ∂x ∂ x2 ∂w = 0 or ∂x
− (M + Q ) + Nh
∂ w ∂ Mh + = 0; ∂x ∂x
and the non-classical boundary conditions:
∂u = 0 or Nh = 0, ∂x ∂ 2w = 0 or Mh = 0. ∂ x2
(13)
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2.2. Governing equations in terms of the displacement In this subsection, the governing equations in terms of the displacement will be deduced for analyzing the sizedependent post-buckling behaviors of hinged-hinged nanobeams. In the case of hinged-hinged beam, the longitudinal displacements at beam’s two ends are equal to zero based on the classical boundary condition, that is,
u ( 0 ) = u ( L ) = 0.
(14)
Furthermore, we may consider a pair of the non-classical boundary conditions, defined as (Li & Hu, 2017)
Nh (0 ) = Nh (L ) = 0.
(15)
With the help of Eqs. (5) and (7), we can easily gain the following relationship:
Nh = l 2
∂N . ∂x
(16)
Considering Eqs. (5), (7), (14), (12) and (16), and using some mathematical manipulations, the longitudinal resultant force N can be obtained as
A N= 2L
L
0
∂w ∂x
2
dx.
(17)
As can be seen, the extension exerted by the transverse deflection w can change the longitudinal resultant force N in the nanobeam. This is known as mid-plane stretching, and the geometric nonlinearity in the considered nanobeam is attributed to it. With the help of Eqs. (17) and (16), Eq. (13) can be simplified to
EA 2L
L
0
∂w ∂x
2
1 − μ2
dx − p
4 2 ∂ 2 ∂ 2w ∂ w ∂ 4w 2 ∂ − 1 − l E I 4 − l 2 E A 4 = 0. 2 2 2 ∂x ∂x ∂x ∂x ∂x
(18)
This is the governing equations for post-buckling of nanobeam incorporating the size-dependent effect of thickness. When neglecting the effect of mid-plane stretching, the linear buckling equation of motion can be obtained as
p 1 − μ2
4 2 ∂ 2 ∂ 2w ∂ w ∂ 4w 2 ∂ + 1 − l E I 4 + l 2 E A 4 = 0. 2 2 2 ∂x ∂x ∂x ∂x ∂x
If further neglecting the size-dependent effect of thickness, we obtain
p 1 − μ2
4 2 ∂ 2 ∂ 2w ∂ w 2 ∂ + 1 − l EI 4 = 0, ∂ x2 ∂ x2 ∂ x2 ∂x
which is the same as that developed by Lu et al. (2017b). When neglecting the effect of strain-gradient and stress-gradient parameters, the classical equation of motion can be recovered from Eq. (18) as
EA 2L
L
0
∂w ∂x
2
dx − p
∂ 2w ∂ 4w − EI 4 = 0. 2 ∂x ∂x
For the sake of generality, Eq. (18) can be also expressed in a dimensionless form by using the following non-dimensional quantities:
x , L
X=
W =
w , L
τ=
ea , L
l L
ζ= , γ=
I , AL2
P=
p . EA
Thus, Eq. (18) can be rewritten as
1 2
1 0
∂W ∂X
2
4 2 4 ∂ 2 ∂ 2W ∂ W 2 ∂ 2∂ W dX − P ( 1 − τ ) − 1 − ζ − ζ = 0. γ ∂X2 ∂X2 ∂X2 ∂X4 ∂X4 2
(19)
3. Closed-form solutions for the post-buckling of hinged-hinged nanobeam In this section, we will obtain the analytical solutions for the post-buckling of nanobeam based nonlocal strain gradient theory. As mentioned previously, the nonlocal strain gradient theory brings about the non-classical boundary conditions. The closed-form solution must satisfy both the equilibrium equations and boundary conditions. For the hinged-hinged nanobeam, the classical and non-classical boundary conditions can be satisfied provided that
w=
∂ 2w = 0, ∂ x2
(20)
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− (M + Q ) + Nh
∂ w ∂ Mh + = 0. ∂x ∂x
(21)
With the help of Eqs. (6), (8) and (9), Eq. (21) can be expanded as
3 2 L ∂ 2w 2 ∂ 4w ∂ 2w A ∂ u ∂ 2w ∂ 2w ∂ w ∂ 3w ∂w 2 2 l EA + EI − l EI 4 − μ l EA + + − dx + p = 0, 2 3 2 2 3 ∂x ∂x 2L 0 ∂x ∂x ∂x ∂x ∂x ∂x ∂ x2
2
which reveals that a simplified condition derived from Eq. (21) can be obtained as
∂ 4w = 0. ∂ x4
(22)
By using Eqs. (20), (22) and non-dimensional quantities, the hinged-hinged boundary conditions can be transformed into
∂ 2W ∂ 4W = = 0. ∂X2 ∂X4
W =
(23)
3.1. Post-buckling configuration For the sake of simplicity, we can define η= 12
1 ∂ W 2 0
∂X
dX . Afterwards, Eq. (19) can be concisely rewritten as
∂ 4W ∂ 2W ∂ W ζ 2γ − γ + ζ 2 + ητ 2 − P τ 2 + (η − P ) = 0. 6 4 ∂X ∂X ∂X2 6
(24)
According to the basic method of ordinary differential equations, the general solution can be assumed as
W = C1 cos λ1 X + C2 sin λ1 X + C3 cosh λ2 X + C4 sinh λ2 X + C5 X + C6 , where
λ = 2 1
β 2 − 4α β + β 2 − 4α 2 , λ2 = , 2ζ 2 γ 2ζ 2 γ
−β +
(25)
(26)
with
α = ζ 2 γ (η − P ), β = (η − P )τ 2 + γ + ζ 2 . Substituting the general solution (25) into the boundary conditions (23), one obtains
C1 = C3 = C4 = C5 = C6 = 0,
C2 sin λ1 = 0.
(27)
Solving this equation, we can obtain
λ1 = nπ for n = 1, 2, 3 · · ·
(28)
Substituting into Eq. (26), the axial force P can be expressed as
P = n2 π 2 γ
1 + n2 π 2 ζ 2 n2 π 2 ζ 2 + 2 2 2 + η. 2 2 2 1+n π τ n π τ +1
(29)
Besides, substituting Eq. (28) into the general solution (25), Eq. (25) can be given by
W = C2 sin nπ X,
(30)
According to Eq. (30), η can be also rewritten as
η=
n2 π 2 2 C2 . 4
(31)
Combining Eqs. (29) and (31), C2 can be yielded as
C2 = ±2
P 1 + n2 π 2 ζ 2 ζ2 −γ − . n2 π 2 1 + n2 π 2 τ 2 1 + n2 π 2 τ 2
(32)
Substituting Eq. (32) into Eq. (30) yields
W = ±2
P n2
π
2
−γ
1 + n2 π 2 ζ 2 ζ2 − sin nπ X. 2 2 2 1+n π τ 1 + n2 π 2 τ 2
(33)
Eq. (33) reflects the buckling configuration of the nanobeam undergoing compressing forces in the dimensionless form, and the dimension form can be easily obtained by using w = LW . It can be observed that the nanobeam will remain its original equilibrium position, as long as P does not exceed a certain value Pcr , called the critical buckling force. However, when considering P > Pcr , the nanobeam will move away from the original position and look for a new equilibrium position, known as post-buckling behavior. Under such case, it is said to be unstable.
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Fig. 2. Comparison of normalized critical buckling force of nanobeam with and without the thickness effect as a function of the slenderness ratio L/h.
3.2. Critical buckling force The critical buckling force (CBF) Pcr can be determined by setting W = 0 and n = 1 in Eq. (33), that is
Pcr = π 2
1 + π 2ζ 2 ζ2 + . γ 1 + π 2τ 2 1 + π 2τ 2
(34)
By utilizing the dimensionless transformation variables and the dimensionless form of critical buckling force (34), the CBF in the physical form pcr can be expressed as:
EIπ 2 pcr = L2
L2 + π 2 l 2 AL2 l2 + I L 2 + π 2 μ2 L 2 + π 2 μ2
.
(35)
If the cross section of beam is considered as a rectangle with wide b and thickness h, we have A = bh and I = bh3 /12, and the critical buckling force can be then simplified as
2 2 2 2 EIπ 2 L + π + 12λ l pcr = L2 L 2 + π 2 μ2
(36)
where the slenderness ratio (length-to-thickness ratio) λ = L/h. If the considered length is much larger than the values of small-scale parameters (strain-gradient and stress-gradient parameters), the size-dependent effect will be not significant and the classical model can be alternatively used in this case. Furthermore, when neglecting the size-dependent effect of thickness, the CBF can be reduced to
pcr =
EIπ 2 L2 + π 2 l 2 , L 2 L 2 + π 2 μ2
which is the same as that given in Li and Hu (2015); 2017). Clearly, when neglecting the size-dependent effect of thickness, the CBF is underestimated. 3.3. Remarks on size-dependent effect To clearly show the size-dependent effect, the CBF is normalized by the classical CBF, EIπ 2 /L2 . The plot of the normalized CBF versus the slenderness ratio λ is shown in Fig. 2 for nanobeam with and without the thickness effect. In the case of nanobeam without the thickness effect, it has been shown by Li and Hu (2015); 2017) and Lu et al. (2017b) that the nanobeams will always reveal a stiffness-softening effect if μ/l > 1, and reveal a stiffness-hardening effect if μ/l < 1. That is, the stiffness-softening and stiffness-hardening effects are only dependent on the ratio of stress-gradient parameter μ to strain-gradient parameter l. In this study, it is found that, when taking the thickness effect into account, the stiffnesssoftening and stiffness-hardening effects depend not only on the ratio of stress-gradient parameter to strain-gradient parameter, but also on the slenderness ratio λ. In fact, when
λ < 2
π 2 μ2 12
l2
−1 ,
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Fig. 3. The zone of the stiffness-softening and stiffness-hardening effects (L = 100 nm).
Fig. 4. Comparison of normalized critical buckling forces in nanobeam made of Silicon obtained by utilizing nonlocal strain gradient elasticity models without and with the size-dependent effect of thickness. Here μ = 0.16544 nm and l = 0.0535 nm for nonlocal strain gradient elasticity. The lattice parameter ˚ a = 5.43 A.
the nanobeam reveals a stiffness-softening effect. And when
λ2 >
π 2 μ2 12
l2
−1 ,
the nanobeam reveals a stiffness-hardening effect. These two critical size criteria can be easily used to distinguish stiffnesshardening or stiffness-hardening effect. It should be noted that the nanobeams will always reveal a stiffness-softening effect if μ/l < 1. With the critical size criteria, Fig. 3 shows clearly the zone of the stiffness-softening and stiffness-hardening effects when considering L = 100 nm. As can be observed, the stiffness-softening effect increases with increasing the ratio of stressgradient parameter μ to strain-gradient parameter l (μ/l), and the stiffness-hardening effect increases with increasing the slenderness ratio λ. Based on the values of small-scale parameters in Fig. 1, comparison of the normalized critical buckling forces in nanobeam made of Silicon obtained by utilizing nonlocal strain gradient elasticity without and with the thickness effect is plotted in Fig. 4. As can be seen, the size-dependent effect of thickness due to ε xx, z is important, and the size-dependent model without the thickness effect may obtain misleading results (the nonlocal strain gradient beam model with the thickness effect reveals stiffness-stiffening results, while the nonlocal strain gradient beam model without the thickness effect
L. Li et al. / International Journal of Engineering Science 123 (2018) 81–91
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reveals stiffness-softening results). Interestingly, the stiffness-softening effect can be found for the longitudinal dispersion relation plotted in Fig. 1, however, the stiffness-hardening effect is only observed for the buckling behaviors of nanobeams due to the size-dependent effect of thickness. This may be used to explain that, micro-scale beam is often observed stiffnesshardening effect in experiment researches (Chen & Ngan, 2012; Lam, Yang, Chong, Wang, & Tong, 2003), while the dispersion relation of nanostructures usually show stiffness-softening phenomena (Zhu & Li, 2017b; 2017c). 3.4. Generalized Young’s modulus In the view of Eq. (36), the generalized flexural rigidity Df may be expressed as
D f = EI
L2 +
π 2 + 12λ2 l 2 L 2 + π 2 μ2
By using the generalized flexural rigidity, the CBF of the nonlocal strain gradient beam model can be obtained via a traditional buckling analysis in a familiar manner applied for classical beam model. Notice that the classical flexural rigidity (D f = EI) can be recovered if we consider μ = l = 0. Under such case, the generalized Young’s modulus Egh of the nonlocal strain gradient model with the thickness effect can be defined as
Egh
=E
L2 +
π 2 + 12λ2 l 2 , L 2 + π 2 μ2
(37)
which is dependent on both length and thickness. However, the generalized Young’s modulus Egnh of the nonlocal strain gradient model without the thickness effect can be defined as
Egnh = E
L2 + π 2 l 2 , L 2 + π 2 μ2
(38)
which is only length-dependent. This is unusual since length-dependent or diameter-dependent Young’s modulus has been observed for experimental or atom dynamics data (Chen et al., 2006; Zhao et al., 2009). Furthermore, as can be observed from Eq. (37), the stiffness-softening and stiffness-hardening effects on Young’s modulus depend not only on the ratio of stress-gradient parameter to strain-gradient parameter, but also on the slenderness ratio. This may be used to explain that Young’s modulus is length-dependent or diameter-dependent, and may be stiffness-softening (Zhao et al., 2009) or stiffnesshardening (Chen et al., 2006). 4. Conclusion A nonlocal strain gradient beam model incorporating the thickness effect is developed for the size-dependent buckling analysis of nanobeams, and closed-form solutions are derived for post-buckling configuration and critical buckling force. When incorporating the size-dependent effect of thickness, the stiffness-softening and stiffness-hardening effects depend not only on the ratio of stress-gradient parameter to strain-gradient parameter, but also on the geometric feature (slenderness ratio). It is found that the size-dependent effect of thickness is important, and the size-dependent model without the thickness effect may obtain misleading results. It is shown that, the stiffness-softening effect can be only found for the longitudinal dispersion relation of Silicon, however, the stiffness-hardening effect may be observed for the buckling behaviors of nanobeams made of Silicon due to the size-dependent effect of thickness. Acknowledgments This work is supported by the National Natural Science Foundation of China (Grant Nos. 51605172 and 51775201), the Natural Science Foundation of Hubei Province (Grant No. 2016CFB191) and the Fundamental Research Funds for the Central Universities (Grant No. 2015MS014). References Aifantis, E. C. (1992). On the role of gradients in the localization of deformation and fracture. International Journal of Engineering Science, 30(10), 1279–1299. Askes, H., & Aifantis, E. C. (2009). Gradient elasticity and flexural wave dispersion in carbon nanotubes. Physical Review B, 80(19), 195412. Barati, M. R., & Zenkour, A. (2017). A general bi-helmholtz nonlocal strain-gradient elasticity for wave propagation in nanoporous graded double-nanobeam systems on elastic substrate. Composite Structures, 168, 885–892. Chen, C., Shi, Y., Zhang, Y. S., Zhu, J., & Yan, Y. (2006). Size dependence of youngs modulus in zno nanowires. Physical Review Letters, 96(7), 075505. Chen, X., & Ngan, A. (2012). Tensile deformation of silver micro-wires of small thickness-to-grain-size ratios. Materials Science and Engineering: A, 539, 74–84. Cochran, W. (1973). The dynamics of atoms in crystals. Edward Arnold London. Cuenot, S., Frétigny, C., Demoustier-Champagne, S., & Nysten, B. (2004). Surface tension effect on the mechanical properties of nanomaterials measured by atomic force microscopy. Physical Review B, 69(16), 165410. Cui, Y., Wei, Q., Park, H., & Lieber, C. M. (2001). Nanowire nanosensors for highly sensitive and selective detection of biological and chemical species. Science, 293(5533), 1289–1292. Demir, C., & Civalek, O. (2017). On the analysis of microbeams. International Journal of Engineering Science, 121(Supplement C), 14–33. doi:10.1016/j.ijengsci. 2017.08.016.
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Wave propagation in fluid-conveying viscoelastic carbon nanotubes based on nonlocal strain gradient theory. Computational Materials Science, 112, 282–288. Li, L., & Hu, Y. (2017). Post-buckling analysis of functionally graded nanobeams incorporating nonlocal stress and microstructure-dependent strain gradient effects. International Journal of Mechanical Sciences, 120, 159–170. Li, L., Hu, Y., & Li, X. (2016). Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory. International Journal of Mechanical Sciences, 115, 135–144. Li, L., Hu, Y., & Ling, L. (2016). Wave propagation in viscoelastic single-walled carbon nanotubes with surface effect under magnetic field based on nonlocal strain gradient theory. Physica E: Low-dimensional Systems and Nanostructures, 75, 118–124. Li, L., Li, X., & Hu, Y. (2016). Free vibration analysis of nonlocal strain gradient beams made of functionally graded material. International Journal of Engineering Science, 102, 77–92. Li, L., Tang, H., & Hu, Y. (2018). Size-dependent nonlinear vibration of beam-type porous materials with an initial geometrical curvature. Composite Structures, 184, 1177–1188. Li, X., Li, L., Hu, Y., Ding, Z., & Deng, W. (2017). Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory. Composite Structures, 165, 250–265. Li, X., Wei, Y., Lu, L., Lu, K., & Gao, H. (2010). Dislocation nucleation governed softening and maximum strength in nano-twinned metals. Nature, 464(7290), 877–880. Lim, C. W., Zhang, G., & Reddy, J. N. (2015). A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. Journal of the Mechanics and Physics of Solids, 78, 298–313. Lu, L., Guo, X., & Zhao, J. (2017a). Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory. International Journal of Engineering Science, 116, 12–24. Lu, L., Guo, X., & Zhao, J. (2017b). A unified nonlocal strain gradient model for nanobeams and the importance of higher order terms. International Journal of Engineering Science, 119(Supplement C), 265–277. doi:10.1016/j.ijengsci.2017.06.024. Mehralian, F., Beni, Y. T., & Zeverdejani, M. K. (2017). Nonlocal strain gradient theory calibration using molecular dynamics simulation based on small scale vibration of nanotubes. Physica B: Condensed Matter, 514, 61–69. Moya, S., Azzaroni, O., Farhan, T., Osborne, V. L., & Huck, W. T. (2005). Locking and unlocking of polyelectrolyte brushes: toward the fabrication of chemically controlled nanoactuators. Angewandte Chemie International Edition, 44(29), 4578–4581. Rajabi, K., & Hosseini-Hashemi, S. (2017). Size-dependent free vibration analysis of first-order shear-deformable orthotropic nanoplates via the nonlocal strain gradient theory. Materials Research Express, 4(7), 075054. Sahmani, S., & Aghdam, M. (2017a). 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Zhu, X., & Li, L. (2017a). Closed form solution for a nonlocal strain gradient rod in tension. International Journal of Engineering Science, 119, 16–28. Zhu, X., & Li, L. (2017b). Longitudinal and torsional vibrations of size-dependent rods via nonlocal integral elasticity. International Journal of Mechanical Sciences, 133, 639–650. Zhu, X., & Li, L. (2017c). On longitudinal dynamics of nanorods. International Journal of Engineering Science, 120, 129–145.
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International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
The effect of thickness on the mechanics of nanobeams Li Li∗, Haishan Tang, Yujin Hu State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
a r t i c l e
i n f o
Article history: Received 22 October 2017 Revised 19 November 2017 Accepted 19 November 2017
Keywords: Nonlocal strain gradient model Size-dependent effect Nanobeam Buckling Nanomechanics
a b s t r a c t Nonlocal strain gradient models are being used more and more extensively in examining the size-dependent effects on the statical and dynamical behaviors of micro/nanostructures. However, for the fake of simplification, the size-dependent effects are often assumed to be neglected in the thickness direction of beams and plates. This simplification is originally aiming at reducing the complexity of the studied beam and plate problems while retaining all the most important mechanical features. Based on the nonlocal strain gradient model without the thickness effect, the size-dependent effect may reveal stiffness-softening or stiffness-hardening effect, which only depends on the values of strain-gradient and stress-gradient parameters but is independent on the geometric feature. In this paper, a nonlocal strain gradient beam model incorporating the thickness effect is developed for the size-dependent buckling analysis of nanobeams, and closed-form solutions are derived for post-buckling configuration and critical buckling force. When incorporating the size-dependent effect of thickness, it is found that the stiffness-softening and stiffness-hardening effects depend not only on the ratio of stress-gradient parameter to strain-gradient parameter, but also on the geometric feature (slenderness ratio). Interestingly, the stiffness-softening effect can be only found for the longitudinal dispersion relation of Silicon, while the stiffness-hardening effect may be observed for the buckling behaviors of nanobeams made of Silicon due to the size-dependent effect of thickness. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction The rapid development towards nanoscience and nanotechnology has begun a new era for many areas, including engineering, chemical, medicine, electronics. Many nano/micro scale devices have been developed and being tested widely for applications, such as atomic force microscopy (AFM) (Eaton & West, 2010), nanosensors (Cui, Wei, Park, & Lieber, 2001; Duan, Li, Hu, & Wang, 2017), nanoactuators (Moya, Azzaroni, Farhan, Osborne, & Huck, 2005; Shi, Cheng, Pugno, & Gao, 2010), nano/micro-electro-mechanical systems (NEMS/MEMS), and frequency synthesis. Some nanobeams in engineering structures have been developed for recent applications (Demir & Civalek, 2017), including tunable oscillator, nanorelays, nonvolatile random access memory, rotational motors, nanotweezers, and feedback-controlled nanocantilevers. It has been pointed out by many experimental researches that many beam-type and plate-type materials at nano/micro scale under bending and buckling loads have size dependent effects on their mechanical and physical properties. For
∗
Corresponding author. E-mail address: [email protected] (L. Li).
https://doi.org/10.1016/j.ijengsci.2017.11.021 0020-7225/© 2017 Elsevier Ltd. All rights reserved.
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Fig. 1. Longitudinal dispersion relation in Silicon (Si) obtained by utilizing classical elasticity, nonlocal strain gradient elasticity and experimental data (Cochran, 1973). The mass density and Young’s modulus of Si used in numerical studies is considered as ρ = 2330 kg/m3 and E = 201.92 GPa, respectively. Here μ = 0.16544 nm and l = 0.0535 nm for nonlocal strain gradient elasticity.
example, Young’s modulus is length-dependent or diameter-dependent, and may be stiffness-hardening (Chen, Shi, Zhang, Zhu, & Yan, 2006; Cuenot, Frétigny, Demoustier-Champagne, & Nysten, 20 04; Jing et al., 20 06) or stiffness-softening (Zhao, Min, & Aluru, 2009). However, experimental researches are very difficult to control and are often very expensive. Some simulation methods including first-principle method, mixed atomistic-continuum mechanics, molecular dynamics, the density functional theory (DFT) can take the size-dependent effect into account. But the solutions of these atomistic-based methods are very time-consuming, even for supercomputers. For the sake of the ease of implementation, continuum mechanics can be alternatively chosen as a tool for theoretical analysis. To develop reliable continuum models for predicting the mechanical behaviors of structures at nano-scale, some nonclassical continuum theories have been proposed. The nonlocal strain gradient theory (Askes & Aifantis, 2009; Lim, Zhang, & Reddy, 2015) is one of the popular theories for studying size-dependent mechanical behaviors. Unlike the classical elasticity, the nonlocal strain gradient theory can capture both stiffness-hardening and stiffness-softening effects by accounting for both stain and stress gradient effects. The total stress tkl can be expressed as (Askes & Aifantis, 2009)
1 − μ2 ∇ 2 tkl = 1 − l 2 ∇ 2 Ckli j εi j
where Cklij denotes elastic coefficient; ε ij denotes strain tensor; and ∇ 2 is Laplacian operator. Furthermore, a stress-gradient parameter μ is introduced to incorporate the stress gradient effect, and a strain-gradient parameter l is extra introduced to incorporate the strain gradient effect. The gradient model (1) can be simplified to the nonlocal model (Eringen, 1983) if l = 0, and strain gradient model (Aifantis, 1992) if μ = 0. Therefore, the stain and stress gradient models are also known as nonlocal strain gradient models, which are being applied more and more extensively for examining the size-dependent effects on the statical and dynamical behaviors of micro/nano-structures (Barati & Zenkour, 2017; Ebrahimi & Barati, 2016b; Ebrahimi & Dabbagh, 2017; Farajpour & Rastgoo, 2017; Guo et al., 2016; Karami, Shahsavari, & Li, 2017; Li & Hu, 2015; Li, Hu, & Li, 2016; Li, Tang, & Hu, 2018; Li, Li, Hu, Ding, & Deng, 2017; Lim et al., 2015; Lu, Guo, & Zhao, 2017a; Mehralian, Beni, & Zeverdejani, 2017; Rajabi & Hosseini-Hashemi, 2017; Sahmani & Aghdam, 2017a; 2017b; Shahsavari, Karami, & Mansouri, 2018; Zeighampour, Beni, & Karimipour, 2017; Zhen & Zhou, 2017; Zhu & Li, 2017a). These works were recently motivated by the good match between the results of nonlocal strain gradient theory and the data of experimental studies and molecular dynamics simulations (Karami, Shahsavari, Janghorban, & Li, 2018; Li & Hu, 2016b; Li, Hu, & Ling, 2016; Li et al., 2018; Lim et al., 2015; Zeigampour & Beni, 2017; Zhu & Li, 2017a; 2017c). For example, based on the closed-form solution of longitudinal dispersion in Zhu and Li (2017c), Fig. 1 shows the longitudinal dispersion relations in the [100] direction of Si. Unlike the classical model, the nonlocal strain gradient elasticity model can show a good agreement with the experimental data. For one-dimensional (1D) problems, the constitutive equations can be simplified to (Li & Hu, 2017; Li, Li, & Hu, 2016; Lim et al., 2015)
1 − μ2 ∇ 2 txx = 1 − l 2 ∇ 2 E εxx
(1)
where E is the Young’s modulus. For longitudinally static and dynamic problems of rods, the longitudinal strain ε xx is independent of the width and thickness of rods. That is to say, the Laplacian operator can be simplified as ∇ 2 = ∂ 2 /∂ x2 , and this simplification is good in some studies (Li, Hu, & Li, 2016; Zhu & Li, 2017c; Fernandes, El-Borgi, Mousavi, Reddy, & Mechmoum, 2017).
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In the case of beam-type structure, the longitudinal strain ε xx depends on the position at the thickness direction. Considering a nanobeam with the length (in x− direction), width (in y− direction) and thickness (in z− direction), and according to the von–Kármán’s nonlinear strain-displacement relationship, the non-zero strain for the Euler–Bernoulli nanobeam can be yielded as (Li & Hu, 2017)
εxx =
2 ∂u 1 ∂w ∂ 2w + −z 2 ∂x 2 ∂x ∂x
(2)
2
where u denotes longitudinal displacement, w is transversal displacement, the nonlinearity term 21 ∂∂wx is due to the stretching effect of the mid-plane of nanobeam. It is clear that the longitudinal strain ε xx is dependent on z. However, for the fake of simplification, many researchers (Ebrahimi & Barati, 2016a; Li & Hu, 2015; 2016a; 2017; Lim et al., 2015; Lu, Guo, & Zhao, 2017b; S¸ ims¸ ek, 2016; Tang, Liu, & Zhao, 2018; Xu, Wang, Zheng, & Ma, 2017) assumed that the size-dependent behaviors of nanobeam can be neglected in the thickness direction, and also let the Laplacian operator be ∇ 2 = ∂ 2 /∂ x2 for various beam models. This simplification is originally aiming at reducing the complexity of the studied beam and plate problems while retaining all the most important mechanical features. In this study, we will show that the size-dependent effect of thickness due to ε xx, z is important, and the size-dependent model without the thickness effect may obtain misleading results. Therefore, the Laplacian operator should be ∇ 2 = ∂ 2 /∂ x2 + ∂ 2 /∂ z2 , and the size-dependent effect of thickness has to be reconsidered for analyzing the mechanical behaviors of nanobeams. Based on the nonlocal strain gradient model without the thickness effect, it has recently found by many researchers (Li & Hu, 2015; 2016a; 2017; Lu et al., 2017b; Xu et al., 2017) that the size-dependent effect may reveal “stiffness-softening” or “stiffness-hardening” effect, which is independent on the geometric feature but depends on the relative magnitude of the strain-gradient and stress-gradient parameters. However, it is reported by Li, Wei, Lu, Lu, and Gao (2010) that the “stiffnesssoftening” or ”stiffness-hardening” effect on the yield stress of nano-twinned Cu depends on the geometric feature of its microstructures at nano-scale. In this paper, based on the nonlocal strain gradient model incorporating the thickness effect, we will show that the “stiffness-softening” and “stiffness-hardening” effects depend not only on the ratio of stress-gradient parameter to strain-gradient parameter, but also on the geometric feature (slenderness ratio), which will be explored for the buckling of nanobeams. 2. Governing equations for post-buckling of nanobeam incorporating the size-dependent effect of thickness In this section, based on the von–Kármán’s nonlinear strain-displacement relationship (2), a nonlocal strain gradient beam model incorporating the size-dependent effect of thickness will be deduced for analyzing the size-dependent postbuckling behaviors of nanobeams. 2.1. Equilibrium equations in terms of the principle of minimum potential energy By considering the thickness effect in the Euler–Bernoulli nanobeam model, the total stress can be expressed as (0 ) (1 ) (1 ) txx = σxx + σxxx + σxxz
(3)
(1 ) (1 ) where σ xx is nonlocal stress, and σxxx , σxxz are the high order nonlocal stresses. The virtual strain energy δ U can be ex(1 ) (1 ) pressed as a function of nonlocal stress σ xx and strain ε xx , high order nonlocal stresses σxxx , σxxz , and strain gradients ε xx, x , εxx, z , that is,
(1 ) (1 ) δεxx,x + σxxz δεxx,z dV σxx δεxx + σxxx V 2 2 2 ∂u 1 ∂w ∂ 2w ∂ 3w ∂ w ∂ u ∂ w ∂ 2w (1 ) (1 ) = σxx δ + − z 2 + σxxx δ + − z 3 + σxxz δ − 2 dV ∂x 2 ∂x ∂x ∂ x2 ∂ x ∂ x2 ∂x ∂x V L 3 ∂ ∂w ∂ ∂ 2w ∂2 ∂w ∂ 2 (M + Q ) ∂ Mh = − N − N + N − δ wdx ∂x ∂x ∂ x h ∂ x2 ∂ x3 ∂ x2 h ∂ x ∂ x2 0 L L 2 ∂w ∂ 2 w ∂ 2 Mh ∂ ( M + Q ) ∂ ∂w ∂ Nh ∂ N + − + Nh 2 − + − N δ udx + N δw ∂x ∂x ∂x ∂x h ∂x ∂ x2 ∂x ∂ x2 0 0 L
L
L
L 2 ∂ Nh ∂ w ∂ Mh ∂w ∂u ∂ w δ u + Nh δ + N− + Nh + −M−Q δ − Mh δ 2 . ∂x ∂ x ∂ x ∂ x ∂ x ∂x 0 0 0 0
δU =
(4)
Here N and M are the resultant force and moment for longitudinal and transverse directions, respectively. Nh and Q are the high order resultant forces for longitudinal and transverse directions, respectively. Mh is the high order resultant moment.
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These resultants can be given by
2 ∂ u 1 ∂w (1 − μ2 ∇ 2 )N = EA + , ∂x 2 ∂x (1 − μ2 ∇ 2 )M = −EI
∂ 2w , ∂ x2
(5)
(6)
(1 − μ2 ∇ 2 )Nh = l 2 EA
∂ 2u ∂ w ∂ 2w + , ∂ x2 ∂ x ∂ x2
(1 − μ2 ∇ 2 )Mh = −l 2 EI
∂ 3w , ∂ x3
(8)
∂ 2w , ∂ x2
(9)
(1 − μ2 ∇ 2 )Q = −l 2 EA
(7)
where A is the area of cross section, and I signifies the inertia moment. With the aid of Eqs. (5)–(9), it can be made out that the resultants stresses and moment are the functions only concerning x due to the integrations on the cross section of beam. Thus, ∂ 2 /∂ z2 in the operator ∇ 2 cannot work and therefore can be omitted in the resultant forces and moment. (1 ) Remark 1. The work done by the high order stress term along the thickness direction σxxz and the corresponding strain gradient term ε xx, z , which was neglected in Li and Hu (2015); 2017), is considered in strain energy U. Since the actual distribution of strain ε xx in any given section is not invariable, the strain gradient in the thickness direction is not zero. Thus, the work due to (1 ) σxxz and ε xx, z will play a key role in the stiffness of nanobeams. This will be elaborated later.
With regard to buckling and post-buckling research, the virtual work done by the axial compressing force p at one end of the nanobeam can be taken as the following expression:
δW =
L
p 0
∂w ∂w δ dx. ∂x ∂x
(10)
According to the principle of minimum potential energy, we have
δ U − δ W = 0.
(11)
Substituting Eqs. (4) and (10) into Eq. (11) and using integration by parts, one can obtain
δu : −
∂ N ∂ 2 Nh + = 0, ∂x ∂ x2
δ w : −N
(12)
∂ 2 w ∂ N ∂ w ∂ 2 (M + Q ) ∂ 2 Nh ∂ w ∂ Nh ∂ 2 w ∂ 3 Mh ∂ 2w − − + + + + p 2 = 0, 2 2 2 2 3 ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x
with the classical boundary condition:
u=0
or
N = 0;
∂w ∂ 2 w ∂ (M + Q ) ∂ ∂w ∂ 2 Mh ∂w w = 0 or N +Nh 2 + − Nh − −p = 0; ∂x ∂x ∂x ∂x ∂x ∂x ∂ x2 ∂w = 0 or ∂x
− (M + Q ) + Nh
∂ w ∂ Mh + = 0; ∂x ∂x
and the non-classical boundary conditions:
∂u = 0 or Nh = 0, ∂x ∂ 2w = 0 or Mh = 0. ∂ x2
(13)
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85
2.2. Governing equations in terms of the displacement In this subsection, the governing equations in terms of the displacement will be deduced for analyzing the sizedependent post-buckling behaviors of hinged-hinged nanobeams. In the case of hinged-hinged beam, the longitudinal displacements at beam’s two ends are equal to zero based on the classical boundary condition, that is,
u ( 0 ) = u ( L ) = 0.
(14)
Furthermore, we may consider a pair of the non-classical boundary conditions, defined as (Li & Hu, 2017)
Nh (0 ) = Nh (L ) = 0.
(15)
With the help of Eqs. (5) and (7), we can easily gain the following relationship:
Nh = l 2
∂N . ∂x
(16)
Considering Eqs. (5), (7), (14), (12) and (16), and using some mathematical manipulations, the longitudinal resultant force N can be obtained as
A N= 2L
L
0
∂w ∂x
2
dx.
(17)
As can be seen, the extension exerted by the transverse deflection w can change the longitudinal resultant force N in the nanobeam. This is known as mid-plane stretching, and the geometric nonlinearity in the considered nanobeam is attributed to it. With the help of Eqs. (17) and (16), Eq. (13) can be simplified to
EA 2L
L
0
∂w ∂x
2
1 − μ2
dx − p
4 2 ∂ 2 ∂ 2w ∂ w ∂ 4w 2 ∂ − 1 − l E I 4 − l 2 E A 4 = 0. 2 2 2 ∂x ∂x ∂x ∂x ∂x
(18)
This is the governing equations for post-buckling of nanobeam incorporating the size-dependent effect of thickness. When neglecting the effect of mid-plane stretching, the linear buckling equation of motion can be obtained as
p 1 − μ2
4 2 ∂ 2 ∂ 2w ∂ w ∂ 4w 2 ∂ + 1 − l E I 4 + l 2 E A 4 = 0. 2 2 2 ∂x ∂x ∂x ∂x ∂x
If further neglecting the size-dependent effect of thickness, we obtain
p 1 − μ2
4 2 ∂ 2 ∂ 2w ∂ w 2 ∂ + 1 − l EI 4 = 0, ∂ x2 ∂ x2 ∂ x2 ∂x
which is the same as that developed by Lu et al. (2017b). When neglecting the effect of strain-gradient and stress-gradient parameters, the classical equation of motion can be recovered from Eq. (18) as
EA 2L
L
0
∂w ∂x
2
dx − p
∂ 2w ∂ 4w − EI 4 = 0. 2 ∂x ∂x
For the sake of generality, Eq. (18) can be also expressed in a dimensionless form by using the following non-dimensional quantities:
x , L
X=
W =
w , L
τ=
ea , L
l L
ζ= , γ=
I , AL2
P=
p . EA
Thus, Eq. (18) can be rewritten as
1 2
1 0
∂W ∂X
2
4 2 4 ∂ 2 ∂ 2W ∂ W 2 ∂ 2∂ W dX − P ( 1 − τ ) − 1 − ζ − ζ = 0. γ ∂X2 ∂X2 ∂X2 ∂X4 ∂X4 2
(19)
3. Closed-form solutions for the post-buckling of hinged-hinged nanobeam In this section, we will obtain the analytical solutions for the post-buckling of nanobeam based nonlocal strain gradient theory. As mentioned previously, the nonlocal strain gradient theory brings about the non-classical boundary conditions. The closed-form solution must satisfy both the equilibrium equations and boundary conditions. For the hinged-hinged nanobeam, the classical and non-classical boundary conditions can be satisfied provided that
w=
∂ 2w = 0, ∂ x2
(20)
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L. Li et al. / International Journal of Engineering Science 123 (2018) 81–91
− (M + Q ) + Nh
∂ w ∂ Mh + = 0. ∂x ∂x
(21)
With the help of Eqs. (6), (8) and (9), Eq. (21) can be expanded as
3 2 L ∂ 2w 2 ∂ 4w ∂ 2w A ∂ u ∂ 2w ∂ 2w ∂ w ∂ 3w ∂w 2 2 l EA + EI − l EI 4 − μ l EA + + − dx + p = 0, 2 3 2 2 3 ∂x ∂x 2L 0 ∂x ∂x ∂x ∂x ∂x ∂x ∂ x2
2
which reveals that a simplified condition derived from Eq. (21) can be obtained as
∂ 4w = 0. ∂ x4
(22)
By using Eqs. (20), (22) and non-dimensional quantities, the hinged-hinged boundary conditions can be transformed into
∂ 2W ∂ 4W = = 0. ∂X2 ∂X4
W =
(23)
3.1. Post-buckling configuration For the sake of simplicity, we can define η= 12
1 ∂ W 2 0
∂X
dX . Afterwards, Eq. (19) can be concisely rewritten as
∂ 4W ∂ 2W ∂ W ζ 2γ − γ + ζ 2 + ητ 2 − P τ 2 + (η − P ) = 0. 6 4 ∂X ∂X ∂X2 6
(24)
According to the basic method of ordinary differential equations, the general solution can be assumed as
W = C1 cos λ1 X + C2 sin λ1 X + C3 cosh λ2 X + C4 sinh λ2 X + C5 X + C6 , where
λ = 2 1
β 2 − 4α β + β 2 − 4α 2 , λ2 = , 2ζ 2 γ 2ζ 2 γ
−β +
(25)
(26)
with
α = ζ 2 γ (η − P ), β = (η − P )τ 2 + γ + ζ 2 . Substituting the general solution (25) into the boundary conditions (23), one obtains
C1 = C3 = C4 = C5 = C6 = 0,
C2 sin λ1 = 0.
(27)
Solving this equation, we can obtain
λ1 = nπ for n = 1, 2, 3 · · ·
(28)
Substituting into Eq. (26), the axial force P can be expressed as
P = n2 π 2 γ
1 + n2 π 2 ζ 2 n2 π 2 ζ 2 + 2 2 2 + η. 2 2 2 1+n π τ n π τ +1
(29)
Besides, substituting Eq. (28) into the general solution (25), Eq. (25) can be given by
W = C2 sin nπ X,
(30)
According to Eq. (30), η can be also rewritten as
η=
n2 π 2 2 C2 . 4
(31)
Combining Eqs. (29) and (31), C2 can be yielded as
C2 = ±2
P 1 + n2 π 2 ζ 2 ζ2 −γ − . n2 π 2 1 + n2 π 2 τ 2 1 + n2 π 2 τ 2
(32)
Substituting Eq. (32) into Eq. (30) yields
W = ±2
P n2
π
2
−γ
1 + n2 π 2 ζ 2 ζ2 − sin nπ X. 2 2 2 1+n π τ 1 + n2 π 2 τ 2
(33)
Eq. (33) reflects the buckling configuration of the nanobeam undergoing compressing forces in the dimensionless form, and the dimension form can be easily obtained by using w = LW . It can be observed that the nanobeam will remain its original equilibrium position, as long as P does not exceed a certain value Pcr , called the critical buckling force. However, when considering P > Pcr , the nanobeam will move away from the original position and look for a new equilibrium position, known as post-buckling behavior. Under such case, it is said to be unstable.
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87
Fig. 2. Comparison of normalized critical buckling force of nanobeam with and without the thickness effect as a function of the slenderness ratio L/h.
3.2. Critical buckling force The critical buckling force (CBF) Pcr can be determined by setting W = 0 and n = 1 in Eq. (33), that is
Pcr = π 2
1 + π 2ζ 2 ζ2 + . γ 1 + π 2τ 2 1 + π 2τ 2
(34)
By utilizing the dimensionless transformation variables and the dimensionless form of critical buckling force (34), the CBF in the physical form pcr can be expressed as:
EIπ 2 pcr = L2
L2 + π 2 l 2 AL2 l2 + I L 2 + π 2 μ2 L 2 + π 2 μ2
.
(35)
If the cross section of beam is considered as a rectangle with wide b and thickness h, we have A = bh and I = bh3 /12, and the critical buckling force can be then simplified as
2 2 2 2 EIπ 2 L + π + 12λ l pcr = L2 L 2 + π 2 μ2
(36)
where the slenderness ratio (length-to-thickness ratio) λ = L/h. If the considered length is much larger than the values of small-scale parameters (strain-gradient and stress-gradient parameters), the size-dependent effect will be not significant and the classical model can be alternatively used in this case. Furthermore, when neglecting the size-dependent effect of thickness, the CBF can be reduced to
pcr =
EIπ 2 L2 + π 2 l 2 , L 2 L 2 + π 2 μ2
which is the same as that given in Li and Hu (2015); 2017). Clearly, when neglecting the size-dependent effect of thickness, the CBF is underestimated. 3.3. Remarks on size-dependent effect To clearly show the size-dependent effect, the CBF is normalized by the classical CBF, EIπ 2 /L2 . The plot of the normalized CBF versus the slenderness ratio λ is shown in Fig. 2 for nanobeam with and without the thickness effect. In the case of nanobeam without the thickness effect, it has been shown by Li and Hu (2015); 2017) and Lu et al. (2017b) that the nanobeams will always reveal a stiffness-softening effect if μ/l > 1, and reveal a stiffness-hardening effect if μ/l < 1. That is, the stiffness-softening and stiffness-hardening effects are only dependent on the ratio of stress-gradient parameter μ to strain-gradient parameter l. In this study, it is found that, when taking the thickness effect into account, the stiffnesssoftening and stiffness-hardening effects depend not only on the ratio of stress-gradient parameter to strain-gradient parameter, but also on the slenderness ratio λ. In fact, when
λ < 2
π 2 μ2 12
l2
−1 ,
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L. Li et al. / International Journal of Engineering Science 123 (2018) 81–91
Fig. 3. The zone of the stiffness-softening and stiffness-hardening effects (L = 100 nm).
Fig. 4. Comparison of normalized critical buckling forces in nanobeam made of Silicon obtained by utilizing nonlocal strain gradient elasticity models without and with the size-dependent effect of thickness. Here μ = 0.16544 nm and l = 0.0535 nm for nonlocal strain gradient elasticity. The lattice parameter ˚ a = 5.43 A.
the nanobeam reveals a stiffness-softening effect. And when
λ2 >
π 2 μ2 12
l2
−1 ,
the nanobeam reveals a stiffness-hardening effect. These two critical size criteria can be easily used to distinguish stiffnesshardening or stiffness-hardening effect. It should be noted that the nanobeams will always reveal a stiffness-softening effect if μ/l < 1. With the critical size criteria, Fig. 3 shows clearly the zone of the stiffness-softening and stiffness-hardening effects when considering L = 100 nm. As can be observed, the stiffness-softening effect increases with increasing the ratio of stressgradient parameter μ to strain-gradient parameter l (μ/l), and the stiffness-hardening effect increases with increasing the slenderness ratio λ. Based on the values of small-scale parameters in Fig. 1, comparison of the normalized critical buckling forces in nanobeam made of Silicon obtained by utilizing nonlocal strain gradient elasticity without and with the thickness effect is plotted in Fig. 4. As can be seen, the size-dependent effect of thickness due to ε xx, z is important, and the size-dependent model without the thickness effect may obtain misleading results (the nonlocal strain gradient beam model with the thickness effect reveals stiffness-stiffening results, while the nonlocal strain gradient beam model without the thickness effect
L. Li et al. / International Journal of Engineering Science 123 (2018) 81–91
89
reveals stiffness-softening results). Interestingly, the stiffness-softening effect can be found for the longitudinal dispersion relation plotted in Fig. 1, however, the stiffness-hardening effect is only observed for the buckling behaviors of nanobeams due to the size-dependent effect of thickness. This may be used to explain that, micro-scale beam is often observed stiffnesshardening effect in experiment researches (Chen & Ngan, 2012; Lam, Yang, Chong, Wang, & Tong, 2003), while the dispersion relation of nanostructures usually show stiffness-softening phenomena (Zhu & Li, 2017b; 2017c). 3.4. Generalized Young’s modulus In the view of Eq. (36), the generalized flexural rigidity Df may be expressed as
D f = EI
L2 +
π 2 + 12λ2 l 2 L 2 + π 2 μ2
By using the generalized flexural rigidity, the CBF of the nonlocal strain gradient beam model can be obtained via a traditional buckling analysis in a familiar manner applied for classical beam model. Notice that the classical flexural rigidity (D f = EI) can be recovered if we consider μ = l = 0. Under such case, the generalized Young’s modulus Egh of the nonlocal strain gradient model with the thickness effect can be defined as
Egh
=E
L2 +
π 2 + 12λ2 l 2 , L 2 + π 2 μ2
(37)
which is dependent on both length and thickness. However, the generalized Young’s modulus Egnh of the nonlocal strain gradient model without the thickness effect can be defined as
Egnh = E
L2 + π 2 l 2 , L 2 + π 2 μ2
(38)
which is only length-dependent. This is unusual since length-dependent or diameter-dependent Young’s modulus has been observed for experimental or atom dynamics data (Chen et al., 2006; Zhao et al., 2009). Furthermore, as can be observed from Eq. (37), the stiffness-softening and stiffness-hardening effects on Young’s modulus depend not only on the ratio of stress-gradient parameter to strain-gradient parameter, but also on the slenderness ratio. This may be used to explain that Young’s modulus is length-dependent or diameter-dependent, and may be stiffness-softening (Zhao et al., 2009) or stiffnesshardening (Chen et al., 2006). 4. Conclusion A nonlocal strain gradient beam model incorporating the thickness effect is developed for the size-dependent buckling analysis of nanobeams, and closed-form solutions are derived for post-buckling configuration and critical buckling force. When incorporating the size-dependent effect of thickness, the stiffness-softening and stiffness-hardening effects depend not only on the ratio of stress-gradient parameter to strain-gradient parameter, but also on the geometric feature (slenderness ratio). It is found that the size-dependent effect of thickness is important, and the size-dependent model without the thickness effect may obtain misleading results. It is shown that, the stiffness-softening effect can be only found for the longitudinal dispersion relation of Silicon, however, the stiffness-hardening effect may be observed for the buckling behaviors of nanobeams made of Silicon due to the size-dependent effect of thickness. Acknowledgments This work is supported by the National Natural Science Foundation of China (Grant Nos. 51605172 and 51775201), the Natural Science Foundation of Hubei Province (Grant No. 2016CFB191) and the Fundamental Research Funds for the Central Universities (Grant No. 2015MS014). References Aifantis, E. C. (1992). On the role of gradients in the localization of deformation and fracture. International Journal of Engineering Science, 30(10), 1279–1299. Askes, H., & Aifantis, E. C. (2009). Gradient elasticity and flexural wave dispersion in carbon nanotubes. Physical Review B, 80(19), 195412. Barati, M. R., & Zenkour, A. (2017). A general bi-helmholtz nonlocal strain-gradient elasticity for wave propagation in nanoporous graded double-nanobeam systems on elastic substrate. Composite Structures, 168, 885–892. Chen, C., Shi, Y., Zhang, Y. S., Zhu, J., & Yan, Y. (2006). Size dependence of youngs modulus in zno nanowires. Physical Review Letters, 96(7), 075505. Chen, X., & Ngan, A. (2012). Tensile deformation of silver micro-wires of small thickness-to-grain-size ratios. Materials Science and Engineering: A, 539, 74–84. Cochran, W. (1973). The dynamics of atoms in crystals. Edward Arnold London. Cuenot, S., Frétigny, C., Demoustier-Champagne, S., & Nysten, B. (2004). Surface tension effect on the mechanical properties of nanomaterials measured by atomic force microscopy. Physical Review B, 69(16), 165410. Cui, Y., Wei, Q., Park, H., & Lieber, C. M. (2001). Nanowire nanosensors for highly sensitive and selective detection of biological and chemical species. Science, 293(5533), 1289–1292. Demir, C., & Civalek, O. (2017). On the analysis of microbeams. International Journal of Engineering Science, 121(Supplement C), 14–33. doi:10.1016/j.ijengsci. 2017.08.016.
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