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Thermal effects on the stability of embedded carbon nanotubes
Computational Materials Science 47 (2010) 721–726
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Thermal effects on the stability of embedded carbon nanotubes T. Murmu, S.C. Pradhan * Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur, West Bengal 721 302, India
a r t i c l e
i n f o
Article history: Received 16 May 2009 Received in revised form 12 September 2009 Accepted 20 October 2009 Available online 20 November 2009 PACS: 61.46.Fg 46.70.p 46.15.x
a b s t r a c t In this paper, nonlocal beam model is applied to the buckling analysis of single-walled carbon nanotubes (SWCNT) with effect of temperature change and surrounding elastic medium. The SWCNT is considered to be embedded in a Winkler-type elastic medium. The small scale and the thermal effects in SWCNT are incorporated through the nonlocal and thermal elasticity mechanics, respectively. Small-scale effects on buckling load are examined considering various parameters. These parameters include temperature change, aspect ratios, stiffness of Winkler-type elastic medium and mode numbers. The present study shows that at low temperature changes and large scale coefficient, the difference between local buckling load and nonlocal buckling load is comparatively large. Further it is found that the influence of temperature change on buckling load decreases in case of stiffer elastic medium. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Nonlocal elasticity Thermal elasticity Single-walled carbon nanotubes
1. Introduction For studying the mechanical behavior of CNTs such as bending, vibration and buckling, recently elastic continuum models are being widely used. This is because, though not impossible, at current state there are difficulties in experimental characterization of nanotubes. Moreover studying the behavior of CNTs via molecular dynamic simulation consumes much time and is computationally expensive. Thus because of the simplicity and accuracy, continuum mechanics are often applied for study of mechanical response of CNTs [1–3]. The study of vibration and buckling of carbon nanotubes is of practical interest for better understanding of mechanical responses of CNTs [4–6]. Similarly the investigation of thermal effect on the mechanical properties of CNTs is of great importance and necessity. Thermal effect can induce an axial force within CNTs and may lead to bending and buckling. However, only a limited portion of the literature is concerned with the vibration and buckling analysis of carbon nanotubes considering the thermal effects. Zhang et al. [7] studied the thermal effect on the vibration of doublewalled carbon nanotubes using thermal elasticity. Wang et al. [8] studied the thermal effects on the vibration and instability of conveying fluid CNTs based on thermal elasticity mechanics. Hsu et al. [9] analyzed the frequency of chiral single-walled CNT subjected to thermal vibration and using Timoshenko beam model. Ni et al. [10] * Corresponding author. Tel.: +91 3222 283008; fax: +91 3222 282242. E-mail address: [email protected] (S.C. Pradhan). 0927-0256/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2009.10.015
conducted an analysis of buckling behavior of single-walled CNTs subjected to axial compression under a thermal environment. Yao and Han [11] performed a buckling analysis of multi-walled CNTs subjected to torsional load under temperature field. They concluded that at room or lower temperature the critical load for infinitesimal buckling of a multi-walled carbon nanotubes (MWCNT) increases as the value of the temperature change increases, while at a temperature higher than room temperature theoretical load for infinitesimal buckling of a MWCNT decrease as the value of the temperature change increases. Tylikowski [12] studied the instability of thermally induced vibrations of carbon nanotubes by double-elastic shell model. Based on thermal elasticity mechanics, Zhang et al. [13] developed elastic multiple column model for column buckling of MWCNTs with large aspect ratios under axial compression coupling with temperature change. They concluded that at low or room temperature the buckling strain including thermal effect is larger than that excluding the thermal effect and increases with the increase of temperature change. In all the references discussed above, the size effects of the CNTs were not taken into consideration. While dealing with nano-scale structures such as CNTs, the effect of small size becomes prominent and cannot be neglected. Classical continuum mechanics however fails to capture the small-scale effects. Though some thermal based works with molecular dynamics (MD) solution have been reported [14–16]. One widely used theory which accounts for the effect of small scale is the nonlocal elasticity theory initiated by Eringen [17]. Unlike the local elasticity theory, in nonlocal elasticity theory the
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small-scale effects are captured by assuming that the stress at a point as a function not only of the strain at that point but also a function of the strains at all other points of the domain. Both experimental and atomistic simulation results have shown a significant ‘size-effect’ in mechanical properties when the dimensions of these structures become small. As the length scales are reduced, the influences of long-range inter-atomic and intermolecular cohesive forces on the static and dynamic properties tend to be significant and cannot be neglected. The classical theory of elasticity being the long wave limit of the atomic theory excludes these effects. Thus the traditional classical continuum mechanics would fail to capture the small-scale effects when dealing in nano structures. The small size analysis using local theory over predicts the results. Thus the consideration of small-scale effects is necessary for correct prediction of micro/nano structures. The importance of nonlocal elasticity theory motivated the scientific community to explore the behavior of the micro/nano structures [18–25] much accurately and easily. Prompted by the lack of study on the buckling of embedded SWCNT with allowance for small scale and thermal effects, this article presents a buckling analysis of SWCNT under axial compression coupling with temperature change. In this model, the small scale and thermal effects of SWCNT are incorporated through the nonlocal and thermal elasticity mechanics, respectively. A Winkler-type foundation model is assumed for simulating the interaction of the CNT and the elastic medium (polymer matrix). A differential quadrature (DQ) approach is being utilized and numerical solutions for the critical buckling load are obtained. Influence of nonlocal small-scale effects, Winkler modulus parameter, temperature change and buckling modes of the SWCNT on the buckling load are investigated and discussed. 2. Thermal nonlocal beam model for embedded SWCNT The most general form of the constitutive relation for nonlocal elasticity involves an integral over the whole body. The basic equations for a linear homogenous nonlocal elastic body neglecting the body force are given as
rij;j ¼ 0; rij ðxÞ ¼
Z
By combinations of Eqs. (1) and (2) one can obtain
ð1 s2 ‘2 r2 Þr ¼ t;
0
0
kðjx x j; sÞC ijkl ekl ðx ÞdVðx Þ; 8x 2 V;
ð1Þ
1 2
kðjxj; sÞ ¼ ð2p‘2 s2 Þ1 K 0
where K0 is the modified Bessel function.
ð2Þ
ð4Þ
It is noted that the parameter e0 is estimated such that the relations (3) of the model could provide satisfied approximation of atomic dispersion curves of plane waves with those of atomic lattice dynamics. The above nonlocal constitutive Eq. (3) has been recently employed widely for the study of micro and nanostructure elements due to its simplicity. The nonlocal constitutive relation (Eq. (3)) can be approximated to a one-dimensional form as
rðxÞ ðe0 aÞ2
@ 2 rðxÞ ¼ EeðxÞ @x2
ð5Þ
where E is the Young’s modulus. Thus, the scale coefficient (e0a) in the modeling will lead to small-scale effect on the response of structures in nano-size. Many studies showed that the classic Euler–Bernoulli beam theory offers a simple and reliable model for an overall mechanical deformation of CNTs, provided the characteristic wavelength is much larger than the diameter of CNTs. Therefore, the present work studies the thermal effect on the buckling of SWCNTs using Eringen’s nonlocal elasticity theory [17] and classic Euler–Bernoulli beam model. In the present theory the plane cross sections of the SWCNT remain plane during flexure and that the radius of curvature of a bent SWCNT is large compared to the beam’s depth. In addition, CNT is assumed to be simply supported at both ends, with or without being embedded into an elastic medium such as polymer. The displacement function for Euler–Bernoulli beam theory is given as
u1 ¼ uðx; tÞ z
@w ; @x
u2 ¼ 0;
u3 ¼ wðx; tÞ
ð6Þ
where (u, w) are the axial and transverse displacements of the point (x, 0) on the mid-plane (i.e., z = 0) of the beam. The nonzero strains of Euler–Bernoulli theory can be written as
@u @2w z 2 : @x @x
ð7Þ
The following stress resultants are introduced for the present analysis
N¼
pffiffiffiffiffiffiffiffiffi x x=‘s
ð3Þ
r2 ¼ ð@ 2 =@x2 þ @ 2 =@y2 Þ:
eij ¼ ðui;j þ uj;i Þ: The terms rij, ekl Cijkl are the stress, strain and fourth order elasticity tensors, respectively. kðjx x0 j; sÞ is the nonlocal modulus or attenuation function incorporating into constitutive equations the nonlocal effects at the reference point x produced by local strain at 0 the source x0 |x x | represents the distance in Euclidean form and s is a material constant that depends on the internal (e.g. lattice parameter, granular size, distance between C–C bonds) and external characteristics lengths (e.g. crack length, wave length). Material constant s is defined as e0a/‘. e0 is a constant for adjusting the model in matching with experimental results and by other models. The parameter e0 is estimated such that the relations of the nonlocal elasticity model could provide satisfied approximation of atomic dispersion curves of plane waves with those of atomic lattice dynamics. The terms a and l denotes the internal and external lengths, respectively. The kernel function kðjx x0 j; sÞ is given by Eringen [17] as
e0 a ‘
where t = C:e, and ‘:’ represents the double dot product. r2 is the Laplacian operator and is expressed as
exx ¼ 0
s¼
M¼
Z
ZA
rxx dA;
ð8aÞ
zrxx dA:
ð8bÞ
A
Using the nonlocal constitutive relations Eqs. (5) and (8a) we can obtain
N ðe0 aÞ2
@2N @u ¼ EA : @x2 @x
ð9Þ
Similarly using the nonlocal constitutive relations Eqs. (5) and (8b) we can obtain
M ðe0 aÞ2
@2M @2w ¼ EI 2 : 2 @x @x
ð10Þ
The Euler Lagrange equation are expressed as
@N þf ¼0 @x @2M @ @w ¼0 þ q þ kw N @x2 @x @x
ð11Þ ð12Þ
where f and q are the axial and transverse distributed force measured per unit length and N is the applied axial force.
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Substituting first order derivative from Eq. (11) to Eq. (9) we get
N ¼ EA
@u @f ðe0 aÞ2 @x @x
ð13Þ
4
Substituting second order derivative from Eq. (12) to Eq. (10) we get
R
ð14Þ
4
d w 4
dx
2
þ qðxÞ þ kw N
d w 2
dx
2
ðe0 aÞ2
d
2
dx
qðxÞ þ kw N
2
dx
¼0 ð15Þ
where x is an axial coordinate, w(x) is deflection of the CNT, I is the moment of inertia of the cross section of the CNT, A is the area of the innermost cross section of the CNT. The term k denotes the Winkler constant of the surrounding elastic medium described as a Winklertype elastic foundation [26]. The term (e0a) denotes the small-scale coefficient accounting the small size effects. It should be noted that the Eq. (15) reduces to the conventional Euler–Bernoulli equation when scale coefficient (e0a) is reduced to zero. Here N represents the axial force on the CNT and is expressed as
N ¼ Nm þ Nh
ð16Þ
where Nm the axial force due to the mechanical loading prior to buckling. While Nh is the axial force due to the influence of temperature change. Based on the theory of thermal elasticity mechanics, the thermal axial force Nh can be written as [13]
Nh ¼
EA ax h 1 2m
! @2w @2w Mð0; LÞ ¼ EI 2 þ ðe0 aÞ2 q kw þ N 2 ¼ 0: @x @x
2
2
dx
¼
2
dx
¼ 0:
2
þ Q þ KW ðNm þ Nh Þ
d W dX
2
d
dX
Q þ KW ðNm þ Nh Þ
2
w2 ! 2 d W 2
dX
2
¼ 0:
ð22Þ
If the influence of thermal effects and nonlocal parameter are not taken into account (i.e., Nh = 0, w = 0), the equation reduces to that of classical Euler–Bernoulli beam model. Though for the solution of Eq. (22), analytical solutions can be obtained, however we adopt a differential quadrature (DQ) method for the solution purpose. Various complex advanced nonlocal problems can be handled with DQ method much easily. 3. Solution approach by DQM The differential quadrature method is proved to be a good computational method for various engineering problems [28–30]. The essence of DQ method is that a derivative of a function F is approximated as a weighted linear sum of all functional values within the computational domain. The derivative of a function F is assumed as n NG X d F ðnÞ ¼ C ij Fðxj Þ n dx x¼xi j¼1
Rðxi Þ ; ðxi xj ÞRðxj Þ
ð1Þ
ð18Þ
C ij ¼
ð23Þ
Rðxi Þ ¼
NG Y
For the sake of simplicity the following nondimensional variables are introduced:
i–j;
ð24Þ
ðxi xj Þ;
i–j
ð25Þ
j¼1 ð1Þ
ð1Þ
C ij ¼ C ii ¼
NG X
ð1Þ
C ik ;
i ¼ 1; 2; . . . ; NG;
i–k; i ¼ j:
ð26Þ
k¼1
Here NG depict the number of grid points. The higher order weighting coefficient matrices are obtained from matrix multiplication: NG X
ð2Þ
C ij ¼
ð1Þ
ð1Þ
ð3Þ
C ik C kj ; Cij ¼
k¼1
¼
NG X
ð1Þ
ð2Þ
C ik C kj ¼
k¼1
NG X
ð1Þ
ð3Þ
C ik C kj ¼
k¼1
ð20Þ
i; j ¼ 1; 2; . . . ; NG;
and
ð19Þ
2
d wðLÞ
eo a ; L
where
It should be noted that M is the nonlocal bending moment and not the classical bending moment. However it is interesting to note that for simply supported boundary condition, the boundary equations for the classical beam model and nonlocal beam models are same. This is in view of w = 0 at the boundaries. Consequently the nonlocal effects are neglected. For other boundary conditions this criteria may not apply. The derivative nonlocal boundary condition for the present study is thus reduced to
d wð0Þ
dX
4
ð17Þ
where ax is the coefficient of thermal expansion in the direction of x axis, and m is the Poisson’s ratio, respectively. The term h denotes the change in temperature. In the present study, it is assumed that axial load due to temperature change exists on the SWCNT. Here it should be noted that the Young’s modulus is assumed to independent of temperature. According Hsieh et al. [27] Young’s modulus of an SWCNT is insensitive to temperature change in the tube at temperatures of less than approximately 1100 K, but decreases at higher temperatures. Consider the SWCNT simply supported at the two ends. The CNT is considered to be of length L. The simply supported boundary conditions are specified by
wð0; LÞ ¼ 0;
w¼
ð21Þ
4
! 2 d w
Nh L2 ; EI
Nh ¼
Substituting Eq. (21) into Eq. (15), the governing differential equation of motion can be deduced into following dimensionless form
d W
2
where I = Ay dA Again substituting Eq. (14) into Eq. (12) one can obtain the final governing equation as
EI
Nm ¼
kL ; EI
K¼
! @2w @2w 2 M ¼ EI 2 þ ðe0 aÞ q kw þ N 2 @x @x
Nm L2 ; EI 4 qL Q¼ : EI
X ¼ x=L;
NG X
ð3Þ
NG X
ð2Þ
ð1Þ
ð4Þ
C ik C kj ; Cij
k¼1 ð1Þ
C ik C kj :
ð27Þ
k¼1
The grid point is chosen based on well established grid points, Chebyshev–Gauss–Lobatto points [30]:
Xi ¼
1 ði 1Þ p ; 1 cos 2 ðNG 1Þ
i ¼ 1; 2; . . . ; NG:
ð28Þ
Substitution of Eq. (23) into Eq. (22) and rearranging, one could obtain the following differential quadrature equation NG X
ð4Þ
C ij W j þ Q i þ K i W i ðNm þ Nh Þ
j¼1
NG X
ð2Þ
C ij W j
j¼1 2
w
NG X j¼1
ð2Þ C ij Q i
þ Ki
NG X j¼1
ð2Þ C ij W j
ðN m þ Nh Þ
NG X
! ð4Þ C ij W j
¼ 0:
j¼1
ð29Þ
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In the present study, we neglect the effect of externally applied transverse load Q on the SWCNT. Assuming this condition, the Eq. (29) can be easily transformed to an eigen- value problem
½KfWg ¼ Nm fWg:
1.0010 1.0000
ð30Þ 0.9990 Scale
Using Eq. (30), one can easily compute the eigen-values numerically and obtain the critical buckling load of the SWCNTs with various parameter values. The nonlocal boundary conditions are incorporated within the formulation during the determination of weighting coefficients (MWCM approach). Details of the procedure can be seen in Murmu and Pradhan [24]. The present article would thus bring out the simplicity and generality of employing DQ approach in the field of nonlocal elasticity theory.
0.9970 0.9960 0.9950 0.0
4. Results Based on the nonlocal Euler–Bernoulli beam model, the buckling properties of SWCNT in environment of temperature change are investigated and discussed here. A large aspect ratio of SWCNT is assumed which will neglect the shear deformation in the analysis. To obtain the buckling load for the single-walled carbon nanotubes embedded in an elastic medium (Fig. 1), Eq. (22) is transformed to a DQ analogous form. A computer code is developed based on the differential quadrature approach. The grid distributions were taken based on Chebyshev-Gauss–Lobatto points (Eq. (28)). Sufficient number of grid points (NG = 20) were employed to predict accurate critical load results. The effective properties of SWCNT are taken as that of Reddy and Pang [31]. The Young’s modulus E = 1000 GPa, mass density q = 2300 kg/m3, Poisson’s ratio m = 0.19 are considered in the analysis. An aspect ratio (L/2R) of 100 is taken in the analysis. The diameter is assumed as 1.0 nm. The scale coefficients are taken as e0a = 0.0 nm, 0.25 nm 0.5 nm, 0.75 nm, 0.1 nm, 1.25 nm, 1.5 nm, 1.75 nm and 2.0 nm. These values were adopted because e0a should be smaller than 2.0 nm for carbon nanotubes as described by Wang and Wang [32]. It is reported (Jiang et al. [33]) that all the coefficients of thermal expansion for SWCNT are negative at low and room temperature and are positive at high temperature. In the present study, temperature change at low or room temperatures is considered. The coefficient of thermal expansion for CNTs is thus negative and taken as – 1.6 106K1 [11]. First, the effect of surrounding elastic medium is neglected here (KW = 0). Fig. 2 illustrates the variation of scale load ratio vScale of carbon nanotubes with different scale coefficients e0a for different temperature changes, h. The scale load ratio vScale is defined as
vScale ¼
0.9980
Critical Load by Nonlocal Model Critical Load by Local Model
ð31Þ
The parameter vScale is used to give a better illustration of the nonlocal effects in thermo-mechanical buckling response of CNTs. Four different set of temperature change is considered viz, h = 0 K, 25 K, 50 K and 100 K. It is considered that the temperature change h is uniformly distributed in the SWCNT. From the figure it is found that
0.5
1.0
1.5
2.0
Scale coefficient (eoa) Fig. 2. Change of scale based critical load ratio with scale coefficients for different change in temperatures.
as the small-scale coefficient (e0a) increases the scale load ratio,
vScale decreases. This reduction in scale load ratio vScale values of CNTs with scale coefficient is attributed to the small size of SWCNT. Further the behavior of scale load ratio, vScale with (e0a) is found to be significantly effected by the temperature change h. Without any temperature change (h = 0), the scale load ratio vScale is observed to be dropping heavily with the increase of scale coefficients (e0a). This means the nonlocal effects are more prominent without thermal effect. For the case of large temperature changes (h = 50, h = 100), the difference between local buckling load and nonlocal buckling load is comparatively reduced. This implies that the large temperature change on SWCNT has a suppressing influence on the scale coefficient or nonlocality parameter. To illustrate the effect of aspect ratio on the critical buckling load of SWCNT computations have been carried out with different aspect ratios. Fig. 3 depicts the variation of scale load ratio vScale with the aspect ratio (L/d) of SWCNT for different magnitudes of temperature change. Computations have been carried out considering L/d = 40, 50, 60, 70, 80, 90. It is seen that with the increase of aspect ratio of SWCNT the scale load ratio vScale increases. Furthermore the column buckling load for the SWCNT is dependent on the temperature change h. For low aspect ratios, the differences in magnitudes of critical buckling load (vScale) for different temperature changes are larger. While for larger aspect ratios the differences in magnitudes of critical buckling load for different temperature changes is relatively smaller. In addition for larger temperature change the rate of increase of scale load ratio vScale is less compared to smaller temperature change. Fig. 4 illustrates the variation of thermal load ratio vThermal of carbon nanotubes with different temperature changes. The thermal load ratio vThermal is defined as
vThermal ¼
L
N
=0 K =25 K =50 K =100 K
Buckling Load with Thermal Effect Buckling Load without Thermal Effect
CNT
Elastic Medium
N
Fig. 1. Carbon nanotubes embedded in an elastic medium and subjected to axial load, N.
ð32Þ
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T. Murmu, S.C. Pradhan / Computational Materials Science 47 (2010) 721–726
1.000 1.027
=0 K =25 K =50 K =100 K
0.995
Thermal
1.019
Scale
0.990
=0 K =25 K =50 K =100 K
0.985
0.980
1.011
1.003
0.995 0.5
0.975 40
50
60
70
80
90
1
100
1.5
2
Winkler modulus (Gpa)
Aspect Ratio (L/d) Fig. 3. Change of scale based critical load ratio with aspect ratio for different change in temperatures.
Fig. 5. Change of thermal based critical load ratio with Winkler modulus for different change in temperatures.
6.000
6
5.000
=0 K =25 K =50 K =100 K
=0 =25 =50 =100
4
Thermal
Thermal
4.000
5
3.000 2.000
3 2
1.000
1
0.000 40
50
60
70
80
90
100
Aspect Ratio (L/d) Fig. 4. Change of thermal based critical load ratio with aspect ratio for different change in temperatures.
Here otherwise mentioned, buckling load in Eq. (32) would denote the critical buckling load (first eigen-value). From the figure it is found that the buckling load including the thermal effect is larger than that without considering the change in temperature and increases with the increase of temperature change. To illustrate the effect of elastic medium on the thermal buckling response of SWCNT, graphs have been plotted for change of thermal load ratio vThermal with stiffness of elastic medium. A Winkler-type foundation model is assumed for simulating the interaction between the SWCNT and the elastic medium. The stiffness of elastic medium is described by the Winkler modulus. The computation has been carried out using nonlocal theory. For this study we consider a value of scale coefficient e0a = 2 nm. Fig. 5 shows the variation of vThermal with Winkler modulus k. From the figure it is observed that as the stiffness of elastic medium increases (Winkler constant), the vThermal of carbon nanotubes decreases. It means that the influence of temperature change on buckling load gradually reduces in stiffer elastic medium. This reducing effect is found to be larger for the case of large change in temperature. The variations of thermal load ratio vThermal with number of modes for different temperature change are shown in Fig. 6. An aspect ratio of 100 is taken in the computation. It is clearly observed from the figure that critical buckling load with thermal effects is larger than those ignoring the influence of temperature change.
0 1
2
3
4
5
6
mode numbers Fig. 6. Change of thermal based critical load ratio with mode numbers for different change in temperatures.
The thermal effect on the buckling load becomes more significant with the increase of the temperature change h. Moreover the thermal effect on the critical buckling load decreases with the increase in the vibrational mode number n. This is due to small wave length effect for higher modes. At smaller wave lengths the interaction of wave lengths increase leading to increase in nonlocal effects at higher buckle modes. Fig. 7 illustrates the variation of critical buckling load with temperature change for critical buckling load computed from local model and nonlocal model. For the nonlocal model scale coefficient is taken as (e0a = 2 nm). Computations have been carried out for both low and high temperatures environment. For high temperature environment, the coefficient of thermal expansion ax is taken as 1.1 106 K1 [7]. Fig. 7a and b plots the variation of critical buckling load with temperature change for low and high temperature environments, respectively. For the case of low or room temperature, the critical buckling load increases with the increase of temperature change. While, for the case of high temperature the critical buckling load decreases with the increase of temperature change. Fig. 7 shows that the increase or decrease of load ratio follows a straight line path. It may be explained as follows. The reported results computed here all correspond to a uniform temperature change which creates a uniform thermal axial load. Thus, the total axial load is constant and could be treated as the
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T. Murmu, S.C. Pradhan / Computational Materials Science 47 (2010) 721–726
(a)
Critical Load (Nn)
Critical Load (Nn)
(b) 0.31
Low Temperature Environment
0.48
0.43
0.38
0.33
Local Model Nonlocal Model
High Temperature Environment
0.27
0.23
0.19
Local Model Nonlocal Model
0.15
0.28 0
20
40
60
80
0
100
20
40
60
80
100
Temperature Change (K)
Temperature Change (K)
Fig. 7. Change of critical buckling load with temperature change for local and nonlocal model.
eigen-value. Further the thermal axial load could be subtracted from the total axial load to determine the mechanical axial load (buckling load). This explains the straight lines depicted in Fig. 7. Further from the figure it is clear that small scale plays a prominent role in the buckling analysis of SWCNT in thermal environment. It is observed that the critical buckling loads considering nonlocal theory are always smaller than the critical buckling loads considering local theory. This observation is found for both the low and high temperature conditions. 5. Conclusions A nonlocal elastic beam model has been applied to analyze the buckling analysis of single-walled carbon nanotubes (SWCNT) with temperature change. Considering nonlocal effects, effect of aspect ratios, axial compression, and temperature change and mode numbers on the buckling load of SWCNT is investigated. Further the SWCNT is also assumed to be embedded in a Winkler-type elastic medium. Differential quadrature method is utilized to obtain the critical load of the SWCNT. The influence of small scale and temperature change on the buckling load of the SWCNT is discussed. It is found that the thermal effect on the buckling load is strongly dependent on the small-scale coefficient, temperature changes, the aspect ratios of SWCNT, and the buckling modes of the SWCNT. For smaller temperature change h the scale load ratio, vScale is observed to be reducing rapidly with increase in the scale coefficients (e0a). However for large temperature change h, scale load ratio, vScale reduces slowly with increase in the scale coefficients (e0a). Thus the thermal effect (change in temperature) dominates over the scale effects (nonlocal effects). With embedded SWCNT, the normalized buckling load vThermal gradually reduces with increase in stiffness of elastic medium. This reducing effect can be seen more with the case of large temperature change h. For low aspect ratios, the differences in magnitudes of critical buckling load (vScale) for different temperature change are larger. While for larger aspect ratios the differences in magnitudes of critical buckling load for different temperature change is relatively smaller. For higher buckling mode numbers, temperature change h has insignificant effect on the buckling loads. Finally, the
nonlocal model always predicts smaller critical buckling loads than the local model at all temperature considered and in both low and high temperature environment. The present study clearly shows the importance of using nonlocal elasticity in thermo-mechanical analysis of the SWCNT.
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Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Thermal effects on the stability of embedded carbon nanotubes T. Murmu, S.C. Pradhan * Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur, West Bengal 721 302, India
a r t i c l e
i n f o
Article history: Received 16 May 2009 Received in revised form 12 September 2009 Accepted 20 October 2009 Available online 20 November 2009 PACS: 61.46.Fg 46.70.p 46.15.x
a b s t r a c t In this paper, nonlocal beam model is applied to the buckling analysis of single-walled carbon nanotubes (SWCNT) with effect of temperature change and surrounding elastic medium. The SWCNT is considered to be embedded in a Winkler-type elastic medium. The small scale and the thermal effects in SWCNT are incorporated through the nonlocal and thermal elasticity mechanics, respectively. Small-scale effects on buckling load are examined considering various parameters. These parameters include temperature change, aspect ratios, stiffness of Winkler-type elastic medium and mode numbers. The present study shows that at low temperature changes and large scale coefficient, the difference between local buckling load and nonlocal buckling load is comparatively large. Further it is found that the influence of temperature change on buckling load decreases in case of stiffer elastic medium. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Nonlocal elasticity Thermal elasticity Single-walled carbon nanotubes
1. Introduction For studying the mechanical behavior of CNTs such as bending, vibration and buckling, recently elastic continuum models are being widely used. This is because, though not impossible, at current state there are difficulties in experimental characterization of nanotubes. Moreover studying the behavior of CNTs via molecular dynamic simulation consumes much time and is computationally expensive. Thus because of the simplicity and accuracy, continuum mechanics are often applied for study of mechanical response of CNTs [1–3]. The study of vibration and buckling of carbon nanotubes is of practical interest for better understanding of mechanical responses of CNTs [4–6]. Similarly the investigation of thermal effect on the mechanical properties of CNTs is of great importance and necessity. Thermal effect can induce an axial force within CNTs and may lead to bending and buckling. However, only a limited portion of the literature is concerned with the vibration and buckling analysis of carbon nanotubes considering the thermal effects. Zhang et al. [7] studied the thermal effect on the vibration of doublewalled carbon nanotubes using thermal elasticity. Wang et al. [8] studied the thermal effects on the vibration and instability of conveying fluid CNTs based on thermal elasticity mechanics. Hsu et al. [9] analyzed the frequency of chiral single-walled CNT subjected to thermal vibration and using Timoshenko beam model. Ni et al. [10] * Corresponding author. Tel.: +91 3222 283008; fax: +91 3222 282242. E-mail address: [email protected] (S.C. Pradhan). 0927-0256/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2009.10.015
conducted an analysis of buckling behavior of single-walled CNTs subjected to axial compression under a thermal environment. Yao and Han [11] performed a buckling analysis of multi-walled CNTs subjected to torsional load under temperature field. They concluded that at room or lower temperature the critical load for infinitesimal buckling of a multi-walled carbon nanotubes (MWCNT) increases as the value of the temperature change increases, while at a temperature higher than room temperature theoretical load for infinitesimal buckling of a MWCNT decrease as the value of the temperature change increases. Tylikowski [12] studied the instability of thermally induced vibrations of carbon nanotubes by double-elastic shell model. Based on thermal elasticity mechanics, Zhang et al. [13] developed elastic multiple column model for column buckling of MWCNTs with large aspect ratios under axial compression coupling with temperature change. They concluded that at low or room temperature the buckling strain including thermal effect is larger than that excluding the thermal effect and increases with the increase of temperature change. In all the references discussed above, the size effects of the CNTs were not taken into consideration. While dealing with nano-scale structures such as CNTs, the effect of small size becomes prominent and cannot be neglected. Classical continuum mechanics however fails to capture the small-scale effects. Though some thermal based works with molecular dynamics (MD) solution have been reported [14–16]. One widely used theory which accounts for the effect of small scale is the nonlocal elasticity theory initiated by Eringen [17]. Unlike the local elasticity theory, in nonlocal elasticity theory the
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small-scale effects are captured by assuming that the stress at a point as a function not only of the strain at that point but also a function of the strains at all other points of the domain. Both experimental and atomistic simulation results have shown a significant ‘size-effect’ in mechanical properties when the dimensions of these structures become small. As the length scales are reduced, the influences of long-range inter-atomic and intermolecular cohesive forces on the static and dynamic properties tend to be significant and cannot be neglected. The classical theory of elasticity being the long wave limit of the atomic theory excludes these effects. Thus the traditional classical continuum mechanics would fail to capture the small-scale effects when dealing in nano structures. The small size analysis using local theory over predicts the results. Thus the consideration of small-scale effects is necessary for correct prediction of micro/nano structures. The importance of nonlocal elasticity theory motivated the scientific community to explore the behavior of the micro/nano structures [18–25] much accurately and easily. Prompted by the lack of study on the buckling of embedded SWCNT with allowance for small scale and thermal effects, this article presents a buckling analysis of SWCNT under axial compression coupling with temperature change. In this model, the small scale and thermal effects of SWCNT are incorporated through the nonlocal and thermal elasticity mechanics, respectively. A Winkler-type foundation model is assumed for simulating the interaction of the CNT and the elastic medium (polymer matrix). A differential quadrature (DQ) approach is being utilized and numerical solutions for the critical buckling load are obtained. Influence of nonlocal small-scale effects, Winkler modulus parameter, temperature change and buckling modes of the SWCNT on the buckling load are investigated and discussed. 2. Thermal nonlocal beam model for embedded SWCNT The most general form of the constitutive relation for nonlocal elasticity involves an integral over the whole body. The basic equations for a linear homogenous nonlocal elastic body neglecting the body force are given as
rij;j ¼ 0; rij ðxÞ ¼
Z
By combinations of Eqs. (1) and (2) one can obtain
ð1 s2 ‘2 r2 Þr ¼ t;
0
0
kðjx x j; sÞC ijkl ekl ðx ÞdVðx Þ; 8x 2 V;
ð1Þ
1 2
kðjxj; sÞ ¼ ð2p‘2 s2 Þ1 K 0
where K0 is the modified Bessel function.
ð2Þ
ð4Þ
It is noted that the parameter e0 is estimated such that the relations (3) of the model could provide satisfied approximation of atomic dispersion curves of plane waves with those of atomic lattice dynamics. The above nonlocal constitutive Eq. (3) has been recently employed widely for the study of micro and nanostructure elements due to its simplicity. The nonlocal constitutive relation (Eq. (3)) can be approximated to a one-dimensional form as
rðxÞ ðe0 aÞ2
@ 2 rðxÞ ¼ EeðxÞ @x2
ð5Þ
where E is the Young’s modulus. Thus, the scale coefficient (e0a) in the modeling will lead to small-scale effect on the response of structures in nano-size. Many studies showed that the classic Euler–Bernoulli beam theory offers a simple and reliable model for an overall mechanical deformation of CNTs, provided the characteristic wavelength is much larger than the diameter of CNTs. Therefore, the present work studies the thermal effect on the buckling of SWCNTs using Eringen’s nonlocal elasticity theory [17] and classic Euler–Bernoulli beam model. In the present theory the plane cross sections of the SWCNT remain plane during flexure and that the radius of curvature of a bent SWCNT is large compared to the beam’s depth. In addition, CNT is assumed to be simply supported at both ends, with or without being embedded into an elastic medium such as polymer. The displacement function for Euler–Bernoulli beam theory is given as
u1 ¼ uðx; tÞ z
@w ; @x
u2 ¼ 0;
u3 ¼ wðx; tÞ
ð6Þ
where (u, w) are the axial and transverse displacements of the point (x, 0) on the mid-plane (i.e., z = 0) of the beam. The nonzero strains of Euler–Bernoulli theory can be written as
@u @2w z 2 : @x @x
ð7Þ
The following stress resultants are introduced for the present analysis
N¼
pffiffiffiffiffiffiffiffiffi x x=‘s
ð3Þ
r2 ¼ ð@ 2 =@x2 þ @ 2 =@y2 Þ:
eij ¼ ðui;j þ uj;i Þ: The terms rij, ekl Cijkl are the stress, strain and fourth order elasticity tensors, respectively. kðjx x0 j; sÞ is the nonlocal modulus or attenuation function incorporating into constitutive equations the nonlocal effects at the reference point x produced by local strain at 0 the source x0 |x x | represents the distance in Euclidean form and s is a material constant that depends on the internal (e.g. lattice parameter, granular size, distance between C–C bonds) and external characteristics lengths (e.g. crack length, wave length). Material constant s is defined as e0a/‘. e0 is a constant for adjusting the model in matching with experimental results and by other models. The parameter e0 is estimated such that the relations of the nonlocal elasticity model could provide satisfied approximation of atomic dispersion curves of plane waves with those of atomic lattice dynamics. The terms a and l denotes the internal and external lengths, respectively. The kernel function kðjx x0 j; sÞ is given by Eringen [17] as
e0 a ‘
where t = C:e, and ‘:’ represents the double dot product. r2 is the Laplacian operator and is expressed as
exx ¼ 0
s¼
M¼
Z
ZA
rxx dA;
ð8aÞ
zrxx dA:
ð8bÞ
A
Using the nonlocal constitutive relations Eqs. (5) and (8a) we can obtain
N ðe0 aÞ2
@2N @u ¼ EA : @x2 @x
ð9Þ
Similarly using the nonlocal constitutive relations Eqs. (5) and (8b) we can obtain
M ðe0 aÞ2
@2M @2w ¼ EI 2 : 2 @x @x
ð10Þ
The Euler Lagrange equation are expressed as
@N þf ¼0 @x @2M @ @w ¼0 þ q þ kw N @x2 @x @x
ð11Þ ð12Þ
where f and q are the axial and transverse distributed force measured per unit length and N is the applied axial force.
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Substituting first order derivative from Eq. (11) to Eq. (9) we get
N ¼ EA
@u @f ðe0 aÞ2 @x @x
ð13Þ
4
Substituting second order derivative from Eq. (12) to Eq. (10) we get
R
ð14Þ
4
d w 4
dx
2
þ qðxÞ þ kw N
d w 2
dx
2
ðe0 aÞ2
d
2
dx
qðxÞ þ kw N
2
dx
¼0 ð15Þ
where x is an axial coordinate, w(x) is deflection of the CNT, I is the moment of inertia of the cross section of the CNT, A is the area of the innermost cross section of the CNT. The term k denotes the Winkler constant of the surrounding elastic medium described as a Winklertype elastic foundation [26]. The term (e0a) denotes the small-scale coefficient accounting the small size effects. It should be noted that the Eq. (15) reduces to the conventional Euler–Bernoulli equation when scale coefficient (e0a) is reduced to zero. Here N represents the axial force on the CNT and is expressed as
N ¼ Nm þ Nh
ð16Þ
where Nm the axial force due to the mechanical loading prior to buckling. While Nh is the axial force due to the influence of temperature change. Based on the theory of thermal elasticity mechanics, the thermal axial force Nh can be written as [13]
Nh ¼
EA ax h 1 2m
! @2w @2w Mð0; LÞ ¼ EI 2 þ ðe0 aÞ2 q kw þ N 2 ¼ 0: @x @x
2
2
dx
¼
2
dx
¼ 0:
2
þ Q þ KW ðNm þ Nh Þ
d W dX
2
d
dX
Q þ KW ðNm þ Nh Þ
2
w2 ! 2 d W 2
dX
2
¼ 0:
ð22Þ
If the influence of thermal effects and nonlocal parameter are not taken into account (i.e., Nh = 0, w = 0), the equation reduces to that of classical Euler–Bernoulli beam model. Though for the solution of Eq. (22), analytical solutions can be obtained, however we adopt a differential quadrature (DQ) method for the solution purpose. Various complex advanced nonlocal problems can be handled with DQ method much easily. 3. Solution approach by DQM The differential quadrature method is proved to be a good computational method for various engineering problems [28–30]. The essence of DQ method is that a derivative of a function F is approximated as a weighted linear sum of all functional values within the computational domain. The derivative of a function F is assumed as n NG X d F ðnÞ ¼ C ij Fðxj Þ n dx x¼xi j¼1
Rðxi Þ ; ðxi xj ÞRðxj Þ
ð1Þ
ð18Þ
C ij ¼
ð23Þ
Rðxi Þ ¼
NG Y
For the sake of simplicity the following nondimensional variables are introduced:
i–j;
ð24Þ
ðxi xj Þ;
i–j
ð25Þ
j¼1 ð1Þ
ð1Þ
C ij ¼ C ii ¼
NG X
ð1Þ
C ik ;
i ¼ 1; 2; . . . ; NG;
i–k; i ¼ j:
ð26Þ
k¼1
Here NG depict the number of grid points. The higher order weighting coefficient matrices are obtained from matrix multiplication: NG X
ð2Þ
C ij ¼
ð1Þ
ð1Þ
ð3Þ
C ik C kj ; Cij ¼
k¼1
¼
NG X
ð1Þ
ð2Þ
C ik C kj ¼
k¼1
NG X
ð1Þ
ð3Þ
C ik C kj ¼
k¼1
ð20Þ
i; j ¼ 1; 2; . . . ; NG;
and
ð19Þ
2
d wðLÞ
eo a ; L
where
It should be noted that M is the nonlocal bending moment and not the classical bending moment. However it is interesting to note that for simply supported boundary condition, the boundary equations for the classical beam model and nonlocal beam models are same. This is in view of w = 0 at the boundaries. Consequently the nonlocal effects are neglected. For other boundary conditions this criteria may not apply. The derivative nonlocal boundary condition for the present study is thus reduced to
d wð0Þ
dX
4
ð17Þ
where ax is the coefficient of thermal expansion in the direction of x axis, and m is the Poisson’s ratio, respectively. The term h denotes the change in temperature. In the present study, it is assumed that axial load due to temperature change exists on the SWCNT. Here it should be noted that the Young’s modulus is assumed to independent of temperature. According Hsieh et al. [27] Young’s modulus of an SWCNT is insensitive to temperature change in the tube at temperatures of less than approximately 1100 K, but decreases at higher temperatures. Consider the SWCNT simply supported at the two ends. The CNT is considered to be of length L. The simply supported boundary conditions are specified by
wð0; LÞ ¼ 0;
w¼
ð21Þ
4
! 2 d w
Nh L2 ; EI
Nh ¼
Substituting Eq. (21) into Eq. (15), the governing differential equation of motion can be deduced into following dimensionless form
d W
2
where I = Ay dA Again substituting Eq. (14) into Eq. (12) one can obtain the final governing equation as
EI
Nm ¼
kL ; EI
K¼
! @2w @2w 2 M ¼ EI 2 þ ðe0 aÞ q kw þ N 2 @x @x
Nm L2 ; EI 4 qL Q¼ : EI
X ¼ x=L;
NG X
ð3Þ
NG X
ð2Þ
ð1Þ
ð4Þ
C ik C kj ; Cij
k¼1 ð1Þ
C ik C kj :
ð27Þ
k¼1
The grid point is chosen based on well established grid points, Chebyshev–Gauss–Lobatto points [30]:
Xi ¼
1 ði 1Þ p ; 1 cos 2 ðNG 1Þ
i ¼ 1; 2; . . . ; NG:
ð28Þ
Substitution of Eq. (23) into Eq. (22) and rearranging, one could obtain the following differential quadrature equation NG X
ð4Þ
C ij W j þ Q i þ K i W i ðNm þ Nh Þ
j¼1
NG X
ð2Þ
C ij W j
j¼1 2
w
NG X j¼1
ð2Þ C ij Q i
þ Ki
NG X j¼1
ð2Þ C ij W j
ðN m þ Nh Þ
NG X
! ð4Þ C ij W j
¼ 0:
j¼1
ð29Þ
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In the present study, we neglect the effect of externally applied transverse load Q on the SWCNT. Assuming this condition, the Eq. (29) can be easily transformed to an eigen- value problem
½KfWg ¼ Nm fWg:
1.0010 1.0000
ð30Þ 0.9990 Scale
Using Eq. (30), one can easily compute the eigen-values numerically and obtain the critical buckling load of the SWCNTs with various parameter values. The nonlocal boundary conditions are incorporated within the formulation during the determination of weighting coefficients (MWCM approach). Details of the procedure can be seen in Murmu and Pradhan [24]. The present article would thus bring out the simplicity and generality of employing DQ approach in the field of nonlocal elasticity theory.
0.9970 0.9960 0.9950 0.0
4. Results Based on the nonlocal Euler–Bernoulli beam model, the buckling properties of SWCNT in environment of temperature change are investigated and discussed here. A large aspect ratio of SWCNT is assumed which will neglect the shear deformation in the analysis. To obtain the buckling load for the single-walled carbon nanotubes embedded in an elastic medium (Fig. 1), Eq. (22) is transformed to a DQ analogous form. A computer code is developed based on the differential quadrature approach. The grid distributions were taken based on Chebyshev-Gauss–Lobatto points (Eq. (28)). Sufficient number of grid points (NG = 20) were employed to predict accurate critical load results. The effective properties of SWCNT are taken as that of Reddy and Pang [31]. The Young’s modulus E = 1000 GPa, mass density q = 2300 kg/m3, Poisson’s ratio m = 0.19 are considered in the analysis. An aspect ratio (L/2R) of 100 is taken in the analysis. The diameter is assumed as 1.0 nm. The scale coefficients are taken as e0a = 0.0 nm, 0.25 nm 0.5 nm, 0.75 nm, 0.1 nm, 1.25 nm, 1.5 nm, 1.75 nm and 2.0 nm. These values were adopted because e0a should be smaller than 2.0 nm for carbon nanotubes as described by Wang and Wang [32]. It is reported (Jiang et al. [33]) that all the coefficients of thermal expansion for SWCNT are negative at low and room temperature and are positive at high temperature. In the present study, temperature change at low or room temperatures is considered. The coefficient of thermal expansion for CNTs is thus negative and taken as – 1.6 106K1 [11]. First, the effect of surrounding elastic medium is neglected here (KW = 0). Fig. 2 illustrates the variation of scale load ratio vScale of carbon nanotubes with different scale coefficients e0a for different temperature changes, h. The scale load ratio vScale is defined as
vScale ¼
0.9980
Critical Load by Nonlocal Model Critical Load by Local Model
ð31Þ
The parameter vScale is used to give a better illustration of the nonlocal effects in thermo-mechanical buckling response of CNTs. Four different set of temperature change is considered viz, h = 0 K, 25 K, 50 K and 100 K. It is considered that the temperature change h is uniformly distributed in the SWCNT. From the figure it is found that
0.5
1.0
1.5
2.0
Scale coefficient (eoa) Fig. 2. Change of scale based critical load ratio with scale coefficients for different change in temperatures.
as the small-scale coefficient (e0a) increases the scale load ratio,
vScale decreases. This reduction in scale load ratio vScale values of CNTs with scale coefficient is attributed to the small size of SWCNT. Further the behavior of scale load ratio, vScale with (e0a) is found to be significantly effected by the temperature change h. Without any temperature change (h = 0), the scale load ratio vScale is observed to be dropping heavily with the increase of scale coefficients (e0a). This means the nonlocal effects are more prominent without thermal effect. For the case of large temperature changes (h = 50, h = 100), the difference between local buckling load and nonlocal buckling load is comparatively reduced. This implies that the large temperature change on SWCNT has a suppressing influence on the scale coefficient or nonlocality parameter. To illustrate the effect of aspect ratio on the critical buckling load of SWCNT computations have been carried out with different aspect ratios. Fig. 3 depicts the variation of scale load ratio vScale with the aspect ratio (L/d) of SWCNT for different magnitudes of temperature change. Computations have been carried out considering L/d = 40, 50, 60, 70, 80, 90. It is seen that with the increase of aspect ratio of SWCNT the scale load ratio vScale increases. Furthermore the column buckling load for the SWCNT is dependent on the temperature change h. For low aspect ratios, the differences in magnitudes of critical buckling load (vScale) for different temperature changes are larger. While for larger aspect ratios the differences in magnitudes of critical buckling load for different temperature changes is relatively smaller. In addition for larger temperature change the rate of increase of scale load ratio vScale is less compared to smaller temperature change. Fig. 4 illustrates the variation of thermal load ratio vThermal of carbon nanotubes with different temperature changes. The thermal load ratio vThermal is defined as
vThermal ¼
L
N
=0 K =25 K =50 K =100 K
Buckling Load with Thermal Effect Buckling Load without Thermal Effect
CNT
Elastic Medium
N
Fig. 1. Carbon nanotubes embedded in an elastic medium and subjected to axial load, N.
ð32Þ
725
T. Murmu, S.C. Pradhan / Computational Materials Science 47 (2010) 721–726
1.000 1.027
=0 K =25 K =50 K =100 K
0.995
Thermal
1.019
Scale
0.990
=0 K =25 K =50 K =100 K
0.985
0.980
1.011
1.003
0.995 0.5
0.975 40
50
60
70
80
90
1
100
1.5
2
Winkler modulus (Gpa)
Aspect Ratio (L/d) Fig. 3. Change of scale based critical load ratio with aspect ratio for different change in temperatures.
Fig. 5. Change of thermal based critical load ratio with Winkler modulus for different change in temperatures.
6.000
6
5.000
=0 K =25 K =50 K =100 K
=0 =25 =50 =100
4
Thermal
Thermal
4.000
5
3.000 2.000
3 2
1.000
1
0.000 40
50
60
70
80
90
100
Aspect Ratio (L/d) Fig. 4. Change of thermal based critical load ratio with aspect ratio for different change in temperatures.
Here otherwise mentioned, buckling load in Eq. (32) would denote the critical buckling load (first eigen-value). From the figure it is found that the buckling load including the thermal effect is larger than that without considering the change in temperature and increases with the increase of temperature change. To illustrate the effect of elastic medium on the thermal buckling response of SWCNT, graphs have been plotted for change of thermal load ratio vThermal with stiffness of elastic medium. A Winkler-type foundation model is assumed for simulating the interaction between the SWCNT and the elastic medium. The stiffness of elastic medium is described by the Winkler modulus. The computation has been carried out using nonlocal theory. For this study we consider a value of scale coefficient e0a = 2 nm. Fig. 5 shows the variation of vThermal with Winkler modulus k. From the figure it is observed that as the stiffness of elastic medium increases (Winkler constant), the vThermal of carbon nanotubes decreases. It means that the influence of temperature change on buckling load gradually reduces in stiffer elastic medium. This reducing effect is found to be larger for the case of large change in temperature. The variations of thermal load ratio vThermal with number of modes for different temperature change are shown in Fig. 6. An aspect ratio of 100 is taken in the computation. It is clearly observed from the figure that critical buckling load with thermal effects is larger than those ignoring the influence of temperature change.
0 1
2
3
4
5
6
mode numbers Fig. 6. Change of thermal based critical load ratio with mode numbers for different change in temperatures.
The thermal effect on the buckling load becomes more significant with the increase of the temperature change h. Moreover the thermal effect on the critical buckling load decreases with the increase in the vibrational mode number n. This is due to small wave length effect for higher modes. At smaller wave lengths the interaction of wave lengths increase leading to increase in nonlocal effects at higher buckle modes. Fig. 7 illustrates the variation of critical buckling load with temperature change for critical buckling load computed from local model and nonlocal model. For the nonlocal model scale coefficient is taken as (e0a = 2 nm). Computations have been carried out for both low and high temperatures environment. For high temperature environment, the coefficient of thermal expansion ax is taken as 1.1 106 K1 [7]. Fig. 7a and b plots the variation of critical buckling load with temperature change for low and high temperature environments, respectively. For the case of low or room temperature, the critical buckling load increases with the increase of temperature change. While, for the case of high temperature the critical buckling load decreases with the increase of temperature change. Fig. 7 shows that the increase or decrease of load ratio follows a straight line path. It may be explained as follows. The reported results computed here all correspond to a uniform temperature change which creates a uniform thermal axial load. Thus, the total axial load is constant and could be treated as the
726
T. Murmu, S.C. Pradhan / Computational Materials Science 47 (2010) 721–726
(a)
Critical Load (Nn)
Critical Load (Nn)
(b) 0.31
Low Temperature Environment
0.48
0.43
0.38
0.33
Local Model Nonlocal Model
High Temperature Environment
0.27
0.23
0.19
Local Model Nonlocal Model
0.15
0.28 0
20
40
60
80
0
100
20
40
60
80
100
Temperature Change (K)
Temperature Change (K)
Fig. 7. Change of critical buckling load with temperature change for local and nonlocal model.
eigen-value. Further the thermal axial load could be subtracted from the total axial load to determine the mechanical axial load (buckling load). This explains the straight lines depicted in Fig. 7. Further from the figure it is clear that small scale plays a prominent role in the buckling analysis of SWCNT in thermal environment. It is observed that the critical buckling loads considering nonlocal theory are always smaller than the critical buckling loads considering local theory. This observation is found for both the low and high temperature conditions. 5. Conclusions A nonlocal elastic beam model has been applied to analyze the buckling analysis of single-walled carbon nanotubes (SWCNT) with temperature change. Considering nonlocal effects, effect of aspect ratios, axial compression, and temperature change and mode numbers on the buckling load of SWCNT is investigated. Further the SWCNT is also assumed to be embedded in a Winkler-type elastic medium. Differential quadrature method is utilized to obtain the critical load of the SWCNT. The influence of small scale and temperature change on the buckling load of the SWCNT is discussed. It is found that the thermal effect on the buckling load is strongly dependent on the small-scale coefficient, temperature changes, the aspect ratios of SWCNT, and the buckling modes of the SWCNT. For smaller temperature change h the scale load ratio, vScale is observed to be reducing rapidly with increase in the scale coefficients (e0a). However for large temperature change h, scale load ratio, vScale reduces slowly with increase in the scale coefficients (e0a). Thus the thermal effect (change in temperature) dominates over the scale effects (nonlocal effects). With embedded SWCNT, the normalized buckling load vThermal gradually reduces with increase in stiffness of elastic medium. This reducing effect can be seen more with the case of large temperature change h. For low aspect ratios, the differences in magnitudes of critical buckling load (vScale) for different temperature change are larger. While for larger aspect ratios the differences in magnitudes of critical buckling load for different temperature change is relatively smaller. For higher buckling mode numbers, temperature change h has insignificant effect on the buckling loads. Finally, the
nonlocal model always predicts smaller critical buckling loads than the local model at all temperature considered and in both low and high temperature environment. The present study clearly shows the importance of using nonlocal elasticity in thermo-mechanical analysis of the SWCNT.
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