Bending of functionally graded nanobeams incorporating surface effects based on Timoshenko beam model

Accepted Manuscript Bending of functionally graded nanobeams incorporating surface effects based on Timoshenko beam model Lihong Yang, Tao Fan, Liping Yang, Xiao Han, Zongbing Chen PII: DOI: Reference:

S2095-0349(17)30026-0 http://dx.doi.org/10.1016/j.taml.2017.03.001 TAML 133

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Theoretical & Applied Mechanics Letters

Received date: 5 December 2016 Revised date: 12 January 2017 Accepted date: 20 January 2017 Please cite this article as: L. Yang, T. Fan, L. Yang, X. Han, Z. Chen, Bending of functionally graded nanobeams incorporating surface effects based on Timoshenko beam model, Theoretical & Applied Mechanics Letters (2017), http://dx.doi.org/10.1016/j.taml.2017.03.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Bending of functionally graded nanobeams incorporating surface effects based on Timoshenko beam model Lihong Yang1,, Tao Fan1, Liping Yang2, Xiao Han1, Zongbing Chen1 1. College of Aerospace and Civil Engineering, Harbin Engineering University, Harbin 150001, China 2. Engineering Training Centre, Harbin Engineering University, Harbin 150001, China

HIGHLIGHTS: 

Analyzing the bending of functionally graded nanobeams based on Timoshenko beam theory.



Considering surface stress effects of nanobeams by adopting the Gurtin–Murdoch theories.



Deriving the governing equations by using the principle of minimum total potential energy.



Investigating the influences of gradient index and surface stress on the bending responses.

Abstract: The bending responses of functionally graded (FG) nanobeams with simply supported edges are investigated based on Timoshenko beam theory in this article. The Gurtin–Murdoch surface elasticity theory is adopted to analyze the influences of surface stress on bending response of FG nanobeam. The material properties are assumed to vary along the thickness of FG nanobeam in power law. The bending governing equations are derived by using the minimum total potential energy principle and explicit formulas are derived for rotation angle and deflection of nanobeams with surface effects. Illustrative examples are implemented to give the bending deformation of FG nanobeam. The influences of the aspect ratio, gradient index and surface stress on dimensionless deflection are discussed in detail. Keywords: Nanobeam; Functionally graded materials; Bending; Surface effect; Timoshenko beam theory

1. Introduction Nanoscale structures are widely used in various engineering field since the specific physical and mechanical properties. Surface effects have an important influence on mechanical properties of nanoscale structures owing to large surface-to-bulk ratio. Poncharal et al. [1] studied the bending modulus of carton nanotubes by experiment methods and found it increased dramatically with decreasing diameters. Some researchers focused on the size-dependence of nanostructures by experiment or theoretical methods [2-6]. The approaches of studying mechanical behavior of nanoscale structures include experiments method, atomistic simulation method and continuum mechanics method, etc. The application of continuum mechanics method becomes more and more extensive because performing the controlled nanoscale experiments is very difficult and computational capacity of computers limits the atomistic simulation methods [7]. Gurtin and Murdoch [8, 9] developed a linear elastic surface effects theory based on continuum mechanics in which the surface of nanostructure is regarded as a membrane of zero thickness, and this membrane is assumed to be fully adhered to the bulk material. Gurtin–Murdoch surface elasticity theory has an important effect on the development of continuum mechanics method. Recently, researchers in various countries implemented a large number of investigations on the influence of surface effects on mechanical

                                                             

 Corresponding author. E-mail address: [email protected]

properties of nanostructures by using Gurtin–Murdoch surface elasticity theory [10-13]. To give a more accurate analysis of nanostructure, a continuum model of surface elasticity was formulated by extending Laplace–Young equation which was established to address the surface/interface tension of fluids to solid materials by Gurtin et al. [14]. Then, the generalized Laplace-Young equation of curved surface in nanostructures was derived by Chen et al. [15]. As a new kind of composite materials, functionally graded materials (FGMs) which have the continuous variation of material properties have been applied in different field of science and technology (for instance, optoelectronics, nanotechnology, tribology and high temperature technology, etc.). FGMs are generally composed of two constituents and volume fraction of each constituent varies continuously across the functionally graded (FG) body. Many researchers implemented bending, buckling, and vibration analysis of FG beams [16-18]. In recent years, FG structures had been applied in micro/nanoelectro mechanical systems [19-26], and the mechanical analysis of FG nanostructures has become one of the attractive research hotspots. Differential quadrature method was used to investigate free vibration in axially functionally graded carbon nanotubes based on the Timoshenko beam theory [27]. FG nanobeam [28-33] and FG nanoplate [34-36] were analyzed extensively on bending, buckling, and vibration. Euler–Bernoulli nanobeams made of bi-directional functionally graded materials were investigated by Mohammad on buckling [37], bending [38], and free vibration [39] based on Eringen’s non-local elasticity theory. According to the above discussion, understanding the influence of the surface effects on mechanical behavior of FG nanobeams has an important role in the design of nanodevices. Timoshenko beam theory with considering the influence of transverse shear strain is more appropriate to analyze mechanical behavior of the moderately deep beam than the conventional Euler–Bernoulli beam theory. In the present research, we focus on the bending of FG nanobeams on the basis of Gurtin–Murdoch surface elasticity theory and Timoshenko beam theory. The material properties of FG nanobeam are assumed to vary in power law along the thickness of beam. The principal of the minimum total potential energy is adopted to determine the governing equations and the corresponding boundary conditions. The exact solution for the bending deflections is proposed under simply-supported boundary conditions. The effects of aspect ratio, gradient index and surface elastic parameters on the deflection of the FG nanobeam are discussed. The obtained solutions are also verified by conducting some illustrative examples. 2. Functionally graded materials Figure 1 shows an FG nanobeam of length L, thickness h, and width b. Both distributed transverse

presure q(x) and an axial compressive force P act on this nanobeam. FG nanobeam is assumed to be made of two different constituents and the effective material properties are assumed to vary continuously along the thickness of beam. According to the rule of mixture, the effective material properties P (i.e., bulk elastic modulus E , surface elastic modulus E s , and residual surface stress  0 ) can be expressed as

  Fig. 1. Schematic of FG nanobeam P = PV 1 1  P2V2 ,

(1)

where the subscripts 1 and 2 represent the first constituents and the second constituents, respectively. Pi  i  1, 2  is the effective material properties of constituent materials, and Vi  i  1, 2  is the corresponding volume fractions which are assumed to change in power law in z direction k

 z 1 V1  z  =1     , h 2

(2)

k

 z 1 V2  z  =    , h 2

(3)

where k is the gradient index (or the volume fraction index). Then bulk elastic modulus E  z  , surface elastic modulus E s  z  , and residual surface stress  0  z  of the FG nanobeam can be derived respectively in the following form k

 z 1 E  z  =  E2  E1      E1 , h 2 E

s

 z=

E2s



E1s



(4)

k

 z 1 s     E1 , h 2 z 

1

(5)

k

 0  z  =  02   01       01 , h 2 

(6)

where  01 and  02 are residual surface stresses of two constituents. Poisson’s ratio  of FG nanobeam is generally taken as constant. 3. The governing equations of FG nanobeam with surface effects Due to no slipping between upper (below) surface and the bulk material of FG nanobeam, there exist the continuous displacements in the whole nanobeam. Based on Timoshenko beam theory, the axial displacement u1 and the transverse displacement u3 at arbitrary point (x, z) of the nanobeam can be given as

u1  u  z ,

(7)

u3  w ,

(8)

Where u and w are axial displacements and transverse displacements for arbitrary point (x, 0) on the neutral axis, respectively, and  is the rotation angle of cross-section with respect to the neutral axis. The strain-displacement relationship of Timoshenko beam theory can be expressed as

 xx 

du d z , dx dx

(9)

dw  . dx

(10)

 xz 

Assuming residual stress in the bulk is negligible due to surface energy, the bulk constitutive equations of FG nanobeam can be given by d   du z , dx   dx

(11)

 dw    ,  dx 

(12)

 xx  E  z  

 xz   G  z  

in which  is the shear correction factor and equals 5 6 for a rectangular cross section. Shear elastic modulus  G  z  = E  z   2  2  . To analyze the surface stress effects of nanostructure, a theoretical model is developed by Gurtin and Murdoch [8, 9] based on the elasticity continuum mechanics including surface stress effects. And the following surface constitutive equations were proposed [8, 9]









s    0   0   s    2  s   0    0 us ,  ,

 s z   0 u zs, ,

(13) (14)

where  ,   x, y ，the superscript s is applied to denote the quantities corresponding to the surface layer, and  s ,  s are the Lame constants of the surface. The surface constitutive equations for FG nanobeam can be obtained from Eqs. (13) and (14) as d   du z  0  z , dx   dx

s  xx  E s  z   xx   0  z   E s  z  

 xzs   0  z 

dw , dx

(15)

(16)

where the surface elastic modulus E s  2 s   s . The generalized Laplace–Young equation can be applied to determine the influence of the residual surface stress on a deformed nanobeam [14]. The stress jump  ij   ij across a surface layer depends on the curvature k of the surface layer by







 ij



s   ij ni n j    k ,

(17)

where  ij and  ij are the stresses above and below the surface, respectively, and ni represents the 2 2 unit vector normal to the surface. The second derivative of the transverse displacement d w dx approximates the curvature of the bending nanobeam. For an unformed beam ( d 2 w dx 2 = 0), no effect of the residual surface stress on the bulk of nanobeam occurs. However, for a deformed nanobeam ( d 2 w dx 2  0 ), the residual surface stress will result in the distributed transverse loading qs(x) on the upper surface of beam along the longitudinal direction, equivalently, as [14] qs  x   H

d2 w , dx 2

(18)

where H is a parameter depending on cross-section and H =  01   02  b for a rectangular crosssection. Using the continuum elasticity theory, the strain energy of an FG nanobeam with surface effects can be given by U=

1 1 1  1 L  ij  ij dV      ijs  ij dS      ijs  ij dS     qs  x  wdx  2 V 2 S 2 S  2 0

(19) =

1 L du d  Mb  M s   Qb Nb  N s     2 0  dx dx

dw  1 L  dw      Qs dx   qs  x  wdx,   dx  2 0  dx 

where Nb =   xx dA， Qb =   xz dA， M b =   xx zdA ,

(20)

s s N s =   xx dL， Qs =   xzs dL， M s =   xx zd L .

(21)

A

A

C

A

C

C

The work done by the external forces can be expressed as 2

W=

1 L  dw  1 L P  dx  0 q  x  wdx ,  0 2 2  dx 

(22)

where P is the axial compressive force and q(x) is the distributed transverse load. Energy method is applied in present research and the minimum total potential energy principle is expressed as  U  W   0 .

(23)

Substituting Eqs. (19) and (22) into Eq. (23), the governing equations for a nanobeam are obtained as d  Nb  N s  dx

0,

(24)

d  Qb  Qs  dx

P

d2 w d2 w  H 2  q  x  0 , 2 dx dx

d Mb  M s  dx

 Qb  0 .

(25)

(26)

The boundary conditions for a nanobeam can be given by  u  0 or   N b  N s   0 , 

(27)



dw

 w  0 or   N b  N s   Qb  Qs   0 , dx     0 or   M b  M s   0 .

(28) (29)

Substituting Eqs. (11), (12), (15), and (16) into Eqs. (20) and (21), the force-displacement and the moment- displacement relationships of the FG nanobeam can be determined in the following form du d  B11 , dx dx

(30)

 dw    , Qb =   xz dA   A13  A  dx 

(31)

Nb =   xx dA  A11 A

M b =   xx zdA  B11 A

s N s =   xx dL  T0  As1 C

du d  D11 , dx dx

du d du bh d  Bs1   01   02  b   Es1  Es 2  b   Es 2  Es1  , dx dx dx 2 dx Qs =   xzs dL  T0   01   02  b  C

s M s =   xx zdL  Bs1 C

dw , dx

(32)

(33)

(34)

du d bh du bh 2 d bh  Ds1   Es 2  Es1    Es1  Es 2   S0   02   01  , dx dx 2 dx 4 dx 2 (35)

in which

 A11，B11，D11  = A E  z  1, z, z 2  dA ,

(36)

 As1，Bs1，Ds1  =2h 2 E s  z  1, z, z 2  dz ,

(37)

h2

A13   G  z dA ,

(38)

h2  h 2 0

(39)

A

T0  2

 z dz ,

h2  h 2 0

S0  2 

 z   zdz .

(40)

Substitution of Eqs. (30)- (35) into Eqs. (24)- (26) leads to the governing differential equations of the FG nanobeam bending. Then the coupled governing equations for w and φ are derived by eliminating displacement u in governing differential equations as

1

d2 w d  2  q  x  0 , 2 dx dx

(41)

d 2  dw   2     0 , 2 dx  dx 

(42)

3

in which 1 =T0  2  01   02  b   A13  P ,

(43)

 2 = A13 ,

(44)

 B11  Bs1   Es 2  Es1  bh 2   bh 2  3 =    D11  Ds1   Es1  Es 2  . A11  As1   Es1  Es 2  b 4   2

(45)

4. Analytical solution for a simply supported FG nanobeam Using Eqs. (41) and (42), an uncoupled differential governing equation can be obtained as

1 3 d 3 d   2  1   q  x  =0 . 3 dx  2 dx

(46)

An FG nanobeam subjected to the uniform transverse load q is considered in present work. The general solution of Eq. (46) can be obtained as

  C1e x  C2 e  x  C3 

q x 1   2

.

(47)

Substituting Eq. (47) into Eq. (42), the deflection of the FG nanobeam can be obtained in the following form w  C1

3q  x  q  x 2  x   e  C2 2 e x  C3 x  C4  x2 . 1 1  2 1   2  2 1   2 

(48)

In Eqs. (47) and (48),  =  2  2  1  1 3  , and the unknown constants Ci (i=1,2,3,4) can be determined by the boundary conditions of FG nanobeam. According to Eqs. (32) and (35), the bending moment of the FG nanobeam is given by bh  du  d d bh 2  d   Ds1   Es1  Es 2   S0 . M  M b +M s   B11  Bs1   Es1  Es 2     D11  2  dx  dx dx 4  dx 

By using Eqs. (24), (30), and (33), the bending moment can be rewritten as

(49)

M   3

d  4 , dx

(50)

where

 4  S0   01   02 

bh . 2

(51)

For the bending problem of beam, the axial force P is set to zero. The boundary conditions for a simply-supported nanobeams acted by uniform transverse load q are expressed as w  0   0, M  0   0 ,

(52)

w  L   0, M  L   0 .

(53)

Substituting Eqs. (48) and (50) into Eqs. (52) and (53), the unknown constants Ci can be determined as C1 =

e L  1 e L  e L

   q  4 ,   1   2   3 

(54)

C2 =

4  e L  1  q   , e L  e L  1   2   3 

(55)

C3 =

C4 = 

qL , 2 1   2 

3 2    q  4  q.  1  1   2   3   2 1   2 

(56)

(57)

5. Results and discussion The accuracy and efficiency of the analytical solution obtained above is verified by some illustrative examples. A squared section FG nanobeam (b = h = 50 nm) is selected and dimensionless deflection w  100wEI (qL4 ) is obtained for action of uniformly distributed load q. Firstly, to compare the present results with the existing data [33] obtained by Navier method, the material properties are selected to be the same as those used in the work of Simsek [33]: E1  1000 GPa , E2  250 GPa ,   0.3 . Dimensionless deflection w of FG nanobeam is obtained by the present solution and effects of aspect ratio L/h on w of FG nanobeam without surface effects are shown in Fig. 2 based on Euler– Bernoulli beam theory (EBT) and Timoshenko beam theory (TBT), respectively. In Ref. [33], the given dimensionless deflections are 2.3674 based on EBT and 2.4194 (L/h = 10) and 2.3732 (L/h = 30) based on TBT when k = 1. It is seen that there are reasonable agreement between the present results and those data given in Ref. [33].

Fig. 2. Dimensionless deflection w of FG Timoshenko nanobeam and FG Euler–Bernoulli nanobeam without surface effect

To analyze the influences of surface effects on mechanics behavior of FG nanobeams, a squared section nanobeam (b = h = 50 nm) composed of AL2O3 and Al is selected as an example. The elastic properties of Al2O3 and Al are as follows [30] E1  380 GPa, E1s  6.09 N m ,  01 =0.91 N m

(Al 2 O3 )

E2  70 GPa, E2s  5.1882 N m ,  02 =0.9108 N m (Al)

The Poisson’s ratios of these two materials are  1 = 2 =0.3 . The upper surface (z = h/2) and the bottom surface (z =  h/2) of this nanobeam is Al2O3-rich and Al-rich, respectively. The next analyses are all based on Timoshenko beam theory. The variations of dimensionless deflection w along nanobeam longitudinal direction are shown in Fig. 3. It can be seen that the maximum deflection occurs at mid-span of nanobeam and the predicted deflections for an FG nanobeam made of Al2O3 and Al in the case of considering surface elastic effects are smaller than those without considering surface elastic effects based on the non-classical TBT, that is, surface stress stiffens the FG nanobeam. And it is also observed the the influence of surface effect decreases gradually with the increase of k.

Fig. 3. Dimensionless deflection w along longitudinal direction for L/h=10

The effects of aspect ratio L/h and gradient index k on w of FG nanobeam without surface effects are shown in Fig. 4. It can be observed from Fig. 4(a) that w is decreasing with the increase of aspect ratio L/h, but the decreasing trends is not significant for any magnitude of k. For gradient index k, we can see from Fig. 4(b) that w is decreasing with an increase of k and decreases sharply when k < 2. It is also evidently seen from Fig. 4(b) that the lager the magnitude of k, the slower the decrease of w and the decreasing trends becomes approximately constant when k > 6.

(a)

(b)

Fig. 4. Variation of dimensionless deflection w of an FG nanobeam without surface effects with (a) aspect ratio L/h and (b) gradient index k for L/h =10

Figure 5 demonstrates the variations of dimensionless deflection w of an FG nanobeam with surface effect with aspect ratio L/h and the gradient index k, respectively. From Fig. 5(a), we can easily see that w gradually decreases with the increase of L/h and the decreasing trend for homogeneous materials (k = 0) is more significant. Fig. 5(b) highlights the effect of k on w when L/h is set to 10, 20, 30, 50, respectively. w is decreasing with the increase of k and the influence of k on w is smaller for shallow beam (L/h is relatively larger) than for deep beam (L/h is relatively smaller). On the other hand, it is easily observed from Fig. 5 that the smaller gradient index k, the more significant the influence of k on w . From the comparison of Fig. 4(a) with Fig. 5(a), we can also conclude that the influences of L/h on w of FG nanobeam with surface effect are larger than those of FG nanobeam without surface effect.

(a)

(b)

Fig. 5. Variation of dimensionless deflection w of an FG nanobeam with surface effects with (a) aspect ratio L/h and (b) gradient index k

(a)

(b)

Fig. 6. Dimensionless deflection w corresponding to (a) different value of residual surface stress  01 for

E1s = 6.09 N m

s

and (b) different value of surface elastic modulus E1 for

 = 0.91 N m 01

Figure 6(a) implies the influence of value of the residual surface stress  01 on bending response of an FG nanobeam. Surface elastic modulus E1s is set to 6.09 N/m, Fig. 6(a) plots the dimensionless deflection w of FG nanobeam as functions of aspect ratio L/h. It is seen that the overall stiffness of FG nanobeam is increasing with the increaseing  01 and the influence of  01 on shallow beam is larger than for deep beam. Figure 6(b) represents the variation of w with aspect ratio L/h corresponding to various value of E1s by setting residual surface stress  01 to 0.91 N/m. It can be observed that the surface elasticity modulus E s leads to the decrease of bending stiffness of the FG nanobeam for E s  0 and the increase of bending stiffness for E s  0 . So, we can conclude that the surface elasticity properties have the noticeable effects on bending behaviors of nanobeams and the structural hardening or softening introduced by surface stress effect depends on the magnitude of the surface elastic constants.

6. Conclusions In present work, bending performances of FG nanobeams with surface effects were analyzed based on Timoshenko beam theory. Non-classical Timoshenko beam models including the effect of surface stress on the bending behaviors were developed by adopting Gurtin–Murdoch surface elasticity theory. Material properties of nanobeam were assumed to vary in the thickness direction in power law. The governing equation was derived by using the principal of the minimum total potential energy and explicit formulas were derived for bending deflection and rotation angle.

Results obtained in this paper showed that aspect ratio, surface stress and gradient index k affected significantly the deflection of FG nanobeam. Deflections predicted based on Timoshenko beam model incorporating surface effects were smaller than those based on the classical Euler–Bernoulli theory. The dimensionless deflection was decreasing with the increase of aspect ratio. The effect of gradient index k on dimensionless deflection is smaller for shallow beam than that for deep beam. Dimensionless deflection was decreasing with the increase of k, and decreased sharply when k < 2. It was also concluded that the smaller gradient index k, the more significant the influence of k. In the end, one can concluded from the obtained results that the overall bending stiffness of FG nanobeam tended to increase as the value of residual surface stress  01 increase and the effect of  01 on shallow beam was larger than deep beam. The surface elasticity modulus E s leads to the decrease of bending stiffness of the FG nanobeam for E s  0 and the increase of bending stiffness for E s  0 . As the authors have known, the research for the bending of FG nanobeams with surface effects based on Timoshenko beam theory was not reported so far, so the present results will be a reference to other researchers. Acknowledge This work is supported by the National Natural Science Foundation of China (11302055) and Heilongjiang Post-doctoral Scientific Research Start-up Funding (LBH – Q14046). References [1] Poncharal P, Wang ZL, Ugarte D, et al., Electrostatic deflections and electromechanical resonances of carbon nanotubes. Science 1999; 283: 1513–1516 [2] Wong E W, Sheehan P E, Lieber C M. Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes. Science, 1997, 277: 1971-1975 [3] Kazan M, Guisbiers G, Pereira S, et al., Thermal conductivity of silicon bulk and nanowires: Effects of isotopiccomposition, phonon confinement, and surface roughness. J Appl Phys 2010; 107: 08350301– 08350314 [4] Sun CT, Zhang H. Size-dependent elastic moduli of platelike nanomaterials. J Appl Phys 2003; 93: 1212–1218. [5] Chen CQ, Shi Y, Zhang TS, et al., Size dependence of Young’s modulus in ZnO nanowires. Phys Rev Lett 2006; 96: 0755051–0755054. [6] Lim CW, He LH. Size-dependent nonlinear response of thin elastic films with nano-scale thickness. Int J Mech Sci 2004; 46:1715–26. [7] Lee HL, Chang WJ. Surface and small-scale effects on vibration analysis of a non-uniform nanocantilever beam. Physica E 2010; 43: 466–469. [8] Gurtin ME, Murdoch AI. A continuum theory of elastic material surfaces. Arch Rat Mech Anal 1975; 57: 291–323. [9] Gurtin ME, Murdoch AI. Surface stress in solids. Int. J. Solids Struct 1977; 14: 431–440. [10] Ansari R, Mohammadi V, Shojaei MF, et al., Postbuckling analysis of Timoshenko nanobeams including surface stress effect. Int J Eng Sci 2014; 75: 1–10. [11] Shaat M, Mahmoud FF, Gao XL, et al., Size-dependent bending analysis of Kirchhoff nanoplates based on a modified couple-stress theory including surface effects. Int J Mech Sci 2014; 79: 31–37.

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