Multiple GS systems

Table 4.1 shows results of the convergence study with regard to the asymptotic solutions for the lowest frequency parameters (^*) of simply-supported, cross-ply laminated plates in cylindrical bending, in which the stacking sequences of the laminated macroplates considered are [0⁰/90⁰], [0⁰/90⁰/0⁰/90⁰], [0⁰/90⁰/0⁰/90⁰/0⁰], [0⁰/90⁰]6, and [0⁰/90⁰/0⁰/90⁰/0⁰/90⁰/0⁰]5, as counted from the top layer to the bottom layer. It can be seen in Table 4.1 that the convergent solutions for each laminate are obtained at the ¹⁰-order level. The results also show that the convergent solutions are in excellent agreement with the solutions obtained using the 3D elasticity theory [56], the state space differential quadrature method [57], and the mixed higher-order shear deformation theory (HSDT) [58], in comparison with which the current asymptotic solutions are also shown to be more accurate than the solutions obtained using the 2D HSDT [58].

4.6.2 Nanobeams

Reddy [59] investigated the free vibration behavior of simply-supported, nanobeams using various nonlocal beam theories, such as the nonlocal EBBT, TBT, Reddy’s beam theory (RBT), and Levinson’s beam theory (LBT). The current asymptotic formulation of the nonlocal plane strain theory can also be used for the free vibration analysis of nanobeams by letting the Poisson’s ratio () equal zero, which is suitable for narrow beams.

Table 4.2 shows results of the convergence study with regard to the asymptotic solutions for the lowest frequency parameters of simply-supported nanobeams, the material properties of which are

E=3x10⁷ Pa, 1, and 0.0 2.0 nm . ² The frequency parameter (ˆ) in this case is defined as

2

ˆ L m0/ (EI)

  , in which ₀ ^h

m hdz





 . The geometric parameters of the nanobeam are L_x=10 nm and Lx/H=100, 20, and 10. It can be seen in Table 4.2 that the asymptotic solutions of the lowest frequency parameters of the nanobeam converge rapidly and that the convergent solutions are obtained at the    ² , ⁴ , and ⁶ order levels, for the very thin (Lx/H=100), thin (Lx/H=20), and moderately thick (Lx/H=10) nanobeams, respectively. The convergent solutions of the lowest frequency parameters of the nanobeam are in excellent agreement with the solutions obtained using the nonlocal TBT, RBT, and LBT. The nonlocal EBBT cannot provide accuracte solutions either when the nanobeam becomes thicker or when the value of the nonlocal parameter becomes greater.

4.6.3 Multiple GS systems

In this section, the cylindrical bending free vibration behavior of simply supported, embedded or non-embedded double, triple, and five GS systems is investigated using the current asymptotical nonlocal elasticity theory. The material properties of the GS layer are E 1.02TPa, 0.16, and

2250 kg / m3

 . The dimensionless frequency parameter  is defined as   L /E .

Table 4.3 shows the  ⁸ ordersolutions for the frequency parameters of simply-supported, double, triple, and five GS systems in cylindrical bending at vibration modes ˆn=1-3, in which

H/L

x=0.02; H=0.34 nm; Cw=100; Kw=0, and the values of the nonlocal parameter  are taken to be 0, 1, and 2 nm². It can be seen in Table 4.3 that the frequency parameters for each case can be clearly separated into two groups according to their magnitude orders. The group having lower frequency parameters comprise the flexural modes, while the other group having higher frequency

49

parameters comprise the extension modes. In the former, the frequency parameters differ from one another, while in the latter, the frequency parameters are identical to one another. This is due to the fact that the interactions between adjacent layers are modelled as the Winkler spring connected in the thickness direction, when there is no any connection applied to the in-plane directions. The results also show that the lowest frequency parameter of each case occurs at vibration mode ˆn=1.

The frequency parameters decrease when the value of the nonlocal parameter becomes greater, which also indicates that the small length scale effect will reduce the gross stiffness of the multiple GS system.

Figures 4.3 and 4.4 show distributions of the normalized modal deflections along the

x-direction for the GSs constituting the double and triple GS systems, respectively, in which the

vibration modes are considered at ˆn= 1 and 2. The material properties of each GS are identical to those used in Table 4.3, except the value of  is fixed to be =1 nm². The geometric parameters of each GS are H/Lx=1/20, and H=0.34 nm. It can be seen in Fig. 4.3 that for the vibration modes at

ˆn= 1 and 2, the modal deflection distributions of each GS along the x direction corresponding to the lowest frequency parameters are in-phase (synchronous), while those corresponding to the second lowest frequency parameters are out-of-phase (asynchronous). These observations are also found in the triple GS systems. The results shown in Fig. 4.4 indicate that for the vibration modes at ˆn= 1 and 2, the modal deflection distributions of each GS along the x direction corresponding to the lowest frequency parameters are in-phase (synchronous), while the modal deflection distributions of the first and third GS layers along the x direction corresponding to the second lowest frequency parameters are out-of-phase (asynchronous), and the middle GS remains almost unchanged (stationary). In addition, the modal deflection distributions of the first and third GS layers along the

x direction corresponding to the third lowest frequency parameters are in-phase (synchronous),

while the modal deflection distributions of the middle GS are out-of-phase with those of the first and third GS layers.

Figure 4.5 shows variations in the lowest frequency parameters of the triple GS system with the length-to-thickness ratio for different values of the nonlocal parameter, in which Cw =50; Kw=0; 

=0, 1, and 2 nm²

. It is shown that the small length scale effects on the lowest frequency parameters

are significant when the length-to-thickness ratio becomes smaller. The small length scale effects on the frequency parameters of the triple GS system are more significant with the increases in the nonlocal parameters in the in-phase vibration corresponding to the lowest frequency parameter as compared to those in the out-of-phase vibration corresponding to the second and third lowest frequency parameters. Again, the results also show that the small length scale effects will reduce the gross stiffness of the triple GS system because the lowest frequency parameters of the triple GS system decrease when the value of the nonlocal parameter becomes greater.

Figure 4.6 shows variations in the lowest frequency parameters of the triple GS system with the Winkler stiffnesses Cw and Kw, in which =1 nm² . It can be seen in Fig. 4.6 that the lowest frequency parameters of the triple GS system increase when the values of Cw and Kw become greater.

At the first stage, when Cw<Kw, the lowest frequency parameters of the triple GS system monotonically increase with increases in the value of Cw, and then, when Cw>Kw, this increasing trend gradually slows down.

50

Table 4.1

Results of the convergence study with regard to the asymptotic solutions for the lowest frequency parameters (^*) of simply-supported, laminated plates in cylindrical bending without foundation models.

Theories [0⁰/90⁰] [0⁰/90⁰/0⁰/90⁰] [0⁰/90⁰/0⁰/90⁰/0⁰] [0⁰/90⁰]6 [0⁰/90⁰/0⁰/90⁰/0⁰/90⁰/0⁰]5

Current ⁰ 0.0868060 0.132702 0.179718 0.143562 0.150483

Current ² 0.0811959 0.100995 0.121058 0.111454 0.115400

Current ⁴ 0.0817496 0.112896 0.149865 0.122257 0.127716

Current ⁶ 0.0816891 0.108004 0.134366 0.118266 0.122972

Current ⁸ 0.0816959 0.110093 0.143012 0.119796 0.124867

Current ¹⁰ 0.0816951 0.109184 0.138116 0.119200 0.124099

HSDT [58] 0.0816460 0.110153 0.141136 0.122924 0.134013

Mixed HSDT [58] 0.0816126 0.109414 0.139862 0.119337 0.129748

SSDQ [57] 0.0816958 0.109462 0.139892 0.119369 0.127757

3D exact [56] 0.0816952 0.109461 0.139891 0.119368 0.127756

The lay-up of the laminate is counted from the top.

51

Table 4.2

Results of the convergence study with regard to the asymptotic solutions for the lowest frequency parameters (ˆ) of simply-supported, nanobeams without foundation models.

L/H

 Current

order

0



Current

order

2



Current

order

4



Current

order

6

 EBBT TBT RBT LBT

100 0.0 9.8692 9.8682 9.8682 9.8682 9.8696 9.8683 9.8683 9.8685

0.5 9.6343 9.6334 9.6334 9.6334 9.6347 9.6335 9.6335 9.6337

1.0 9.4155 9.4146 9.4146 9.4146 9.4159 9.4147 9.4147 9.4149

1.5 9.2109 9.2100 9.2100 9.2100 9.2113 9.2101 9.2101 9.2103

2.0 9.0191 9.0182 9.0182 9.0182 9.0195 9.0183 9.0183 9.0185

20 0.0 9.8595 9.8352 9.8353 9.8353 9.8696 9.8381 9.8381 9.8433

0.5 9.6248 9.6012 9.6013 9.6013 9.6347 9.6040 9.6040 9.6091

1.0 9.4062 9.3831 9.3832 9.3832 9.4159 9.3858 9.3858 9.3908

1.5 9.2018 9.1792 9.1793 9.1793 9.2113 9.1819 9.1819 9.1868

2.0 9.0102 8.9881 8.9882 8.9882 9.0195 8.9907 8.9907 8.9955

10 0.0 9.8293 9.7338 9.7352 9.7352 9.8696 9.7454 9.7454 9.7657

0.5 9.5954 9.5022 9.5035 9.5035 9.6347 9.5135 9.5135 9.5333

1.0 9.3774 9.2863 9.2877 9.2876 9.4159 9.2973 9.2974 9.3168

1.5 9.1736 9.0846 9.0859 9.0858 9.2113 9.0953 9.0954 9.1144

2.0 8.9826 8.8954 8.8967 8.8966 9.0195 8.9059 8.9060 8.9246

52

Table 4.3

The current convergent solutions for the frequency parameters of simply-supported, non-embedded double, triple, and five GS systems in cylindrical bending, at different vibration modes.

No. layers (Nl)

Natural frequencies

ˆ1

n nˆ2 nˆ3

0 nm2

 1 nm² 2 nm²  0 nm² 1 nm² 2 nm² 0 nm² 1 nm² 2 nm²

N

l=2 ^1 0.05769 0.05673 0.05582 0.23033 0.21605 0.20413 0.51666 0.45186 0.40659

2 0.11545 0.11497 0.11452 0.25142 0.23836 0.22758 0.52811 0.46440 0.32739

3 3.18258 3.12958 3.07916 6.36504 5.97030 5.64094 9.54727 8.34991 7.51331

4 3.18257 3.12958 3.07916 6.36504 5.97030 5.64094 9.54727 8.34991 7.51331

N

l=3 ^1 0.05769 0.05673 0.05582 0.23033 0.21605 0.20413 0.51666 0.45186 0.40659

2 0.09127 0.09066 0.09009 0.24129 0.22765 0.21633 0.52339 0.45903 0.41418

3 0.13537 0.13497 0.13459 0.26117 0.24862 0.23830 0.53279 0.46973 0.42602

4 3.18258 3.12958 3.07916 6.36504 5.97030 5.64094 9.54727 8.34991 7.51331

5 3.18258 3.12958 3.07916 6.36504 5.97030 5.64094 9.54727 8.34991 7.51331

6 3.18257 3.12958 3.07916 6.36504 5.97030 5.64094 9.54727 8.34991 7.51331

N

l=5 ^1 0.05769 0.05673 0.05582 0.23033 0.21605 0.20413 0.51666 0.45186 0.40659

2 0.07239 0.07162 0.07090 0.23481 0.22077 0.20908 0.51845 0.45508 0.41026

3 0.10119 0.10064 0.10013 0.24521 0.23180 0.22070 0.52520 0.46109 0.41646

4 0.12813 0.12770 0.12730 0.25749 0.24475 0.23426 0.53100 0.46770 0.42378

5 0.14634 0.14596 0.14561 0.26701 0.25475 0.24469 0.53567 0.47299 0.42962

6 3.18258 3.12958 3.07916 6.36504 5.97030 5.96306 9.54727 8.34991 7.51331

7 3.18258 3.12958 3.07916 6.36504 5.97030 5.96306 9.54727 8.34991 7.51331

8 3.18257 3.12958 3.07916 6.36504 5.97030 5.96306 9.54727 8.34991 7.51331

9 3.18257 3.12958 3.07916 6.36504 5.97030 5.96306 9.54727 8.34991 7.51331

10 3.18258 3.12958 3.07916 6.36504 5.97030 5.96306 9.54727 8.34991 7.51331

53

Figure 4.1 Configuration and coordinates of an embedded multiple GS system.

54

Figure 4.2 Schematic for the interactive forces applied to each individual GS of a multiple GS system.

(a) (b)

(c) (d)

Figure 4.3 Distributions of the modal deflections of each individual GS along the x direction for a double GS system, corresponding to (a) the lowest frequency parameter of the

vibration mode

_ˆn

=1, (b) the second lowest frequency parameter of the vibration mode

_ˆn

=1, (c) the lowest frequency parameter of the vibration mode

_ˆn

=2, and (d) the second lowest

frequency parameter of the vibration mode

_ˆn

=2.

55

(a) (b)

(c) (d)

(e) (f)

Figure 4.4 Distributions of the modal deflections of each individual GS along the x direction for a triple GS system, corresponding to (a) the lowest frequency parameter of the vibration mode

_ˆn

=1, (b) the second lowest frequency parameter of the vibration mode

_ˆn

=1, (c) the third lowest frequency parameter of the vibration mode

_ˆn

=1, (d) the lowest frequency parameter of the vibration mode

_ˆn

=2, (e) the second lowest frequency parameter of the vibration mode

_ˆn

=2, and (f) the third lowest frequency parameter of the vibration mode

_ˆn

=2.

56

(a)

(b)

(c)

Figure 4.5 Variations in the frequency parameters of a triple GS system with the

length-to-thickness ratio of each GS for different values of the nonlocal parameter, (a) the

lowest frequency parameter, (b) the second lowest frequency parameter, and (c) the third

lowest frequency parameter.

57

Figure 4.6 Variations in the lowest frequency parameters of a triple GS system with the value

of C

w

for different values of K

w

.

58

Chapter 5 Conclusions

In chapter 2, on the basis of the multiple time scale method and Eringen nonlocal elasticity theory, in this article we developed a 3D asymptotic nonlocal elasticity theory for the free vibration analysis of simply-supported, single-layered nanoplates and GSs embedded in the elastic medium.

The interaction between the nanoplates (or GSs) and their surrounding medium is modelled by using a two-parameter Pasternak foundation.

In the illustrative examples, it is shown that these asymptotic nonlocal elasticity solutions converge rapidly, and are in excellent agreement with the 3D exact local elasticity solutions available in the literature in the cases of 0 nm², and the nonlocal FSDT and HSDT solutions in the cases of 1 and 2nm². The results also show the natural frequency parameters decrease when the nonlocal parameter becomes greater, which means the small length scales will soften the nanoplate. In contrast, the Winkler stiffness (K_W ) and shear modulus (K_G) of the medium will stiffen the nanoplates and GSs. The natural frequency parameters will increase when the values of

KW and K_G become greater. Moreover, the small length scale effect on the natural frequency parameters of nanoplates and GSs with higher-frequency modes is more significant than those with lower-frequency modes.

The results also show the through-thickness distributions of modal in-plane displacement and stress components appear to be linear functions, while they are parabolic function distributions for the out-of-plane displacement and transverse shear components and higher-order polynomial function distributions for the transverse normal stress components. The above results can be used to examine the correctness of the kinetic and kinematic assumptions of various 2D classical and refined nonlocal plates and to provide the basis for their associated modifications.

In chapter 3, based on the Eringen nonlocal elasticity theory combined with the perturbation method, we developed a 3D asymptotic nonlocal elasticity theory for the buckling analysis of simply-supported, single-layered nanoplates and GS embedded in an elastic medium, and subjected to uni- and bi-axially compressive loads. In the illustrative examples, it is shown that these asymptotic nonlocal elasticity solutions converge rapidly, and are in excellent agreement with the accurate local and nonlocal elasticity solutions available in the literature. The results show the lowest critical load parameters will occur at buckling modes in the order of (mˆ, nˆ)=(1,1), (2,1), (3, 1) and then (4,1), as the length-to-width ratio is increased from L /_x L_y=0.5 to 5, while in the bi-axial load cases (_p 1), it always occurs at the buckling mode (mˆ, nˆ)=(1,1). The lowest critical loads of the nanoplates and GS will decrease when the nonlocal parameter becomes greater, which means the small length scales will soften the nanoplates and GS. In contrast, the Winkler stiffness (K_W ) and shear modulus (K_G) of the medium will stiffen the nanoplates and GS.

In chapter 4, the multiple time scale method is successfully extended to a cylindrical bending vibration analysis of a simply-supported, multiple GS system embedded in an elastic medium. The results show that the small length effects always reduce the gross stiffness of the multiple GS system, such that the frequency parameters will decrease when the value of the nonlocal parameter increases. The small length scale effects on the frequency parameters of the multiple GS system are significant when the length-to-thickness ratio of each individual GS becomes smaller. The modal deflection distributions along the x direction for each individual GS layer corresponding to the lowest and second lowest frequency parameters of the double GS system are in-phase (synchronous) and out-of-phase (asynchronous), respectively. For the triple GS system, the modal deflection distributions along the x direction for each individual GS layer corresponding to the lowest frequency parameters are in-phase (synchronous), while those corresponding to the second and third lowest frequency parameters appear as hybrid in- and out-of-phases for the GS layers. The small length scale effects on the frequency parameters of the triple GS system are more significant with increases in the nonlocal parameters in the in-phase vibration corresponding to the lowest frequency

59

parameter as compared to in the out-of-phase vibration corresponding to the second and third lowest frequency parameters.

Because benchmark solutions for cylindrical bending problems in single-layered graphene sheets (GS) and multiple-GS systems have not been found in the literature, on the basis of the nonlocal plane strain elasticity theory the authors thus first apply the multiple time scale method to the current issue. The novelty of the current approach is that there are no kinematic and kinetic assumptions assumed, a priori, while these assumptions are strongly required for various nonlocal two-dimensional (2D) classical, advanced, and refined multiple-plate theories. For the former, when the lower-order solutions have no enough accuracy the solutions can be modified order-by-order until the satisfactory accuracy is yielded. However, for the latter, there is no way to modify their results except changing the kinematic and kinetic assumptions, which results in a different formulation and that the whole derivation needs to start again. In addition, the current solutions can provide a reference for assessing the accuracy of various nonlocal 2D multiple-plate theories.

60

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[52] E. Pan, Smart Mater. Struct. 25 (2016) 095013 (17pp).

[53] E. Pan, J. Compos. Mater., 37 (2003) 1903-1920.

[54] A.S. Sayyad, Y.M. Ghugal, Appl. Comput. Mech. 6 (2012) 185-196.

[55] R. Ansari, H. Rouhi, Solid State Commun. 152 (2012) 56-59.

[56] A. Messina and K.P. Soldatos, A general vibration model of angle-ply laminated plates that accounts for the continuity of interlaminar stresses, Int. J. Solids Struct. 39 (2002) 617-635.

[57] W.Q. Chen and K.Y. Lee, On free vibration of cross-ply laminates in cylindrical bending, J.

Sound Vib. 273 (2004) 667-676.

[58] A. Messina, Two generalized higher order theories in free vibration studies of multilayered plates, J. Sound Vib. 242 (2001) 125-150.

[59] J.N. Reddy, Nonlocal theories for bending, buckling and vibration of beams, Int. J. Eng. Sci. 45 (2007) 288-307.

106年度專題研究計畫成果彙整表

計畫主持人：吳致平 計畫編號：106-2221-E-006-036-MY3 計畫名稱：嵌入式奈米石墨烯板三維非局部彈性力學漸近理論之發展與應用

成果項目 量化 單位

質化

（說明：各成果項目請附佐證資料或細 項說明，如期刊名稱、年份、卷期、起 訖頁數、證號...等） 　　　　　　　

國

內 學術性論文

期刊論文 0

研討會論文 0 篇

專書 0 本

專書論文 0 章

技術報告 0 篇

其他 0 篇

國

外 學術性論文

期刊論文 4

篇

[1] Wu, C.P. and Li, W.C., "Three-dimensional static analysis of nanoplates and graphene sheets by using Eringen's nonlocal elasticity theory and the perturbation

method," CMC-Computers, Materials,

& Continua, SCI, vol. 52, pp. 73-103, 2016.

[2] Wu, C.P. and Li, W.C., "Free vibration analysis of embedded single-layered nanoplates and graphene sheets by using the multiple time scale method,"

Computers and Mathematics with Applications, SCI, vol. 73, pp.

838-854, 2017.

[3] Wu, C.P. and Li, W.C.,

"Asymptotic nonlocal elasticity theory for the buckling analysis of embedded single-layered

nanoplates/graphene sheets under biaxial compression," Physica E, SCI, vol. 89, pp. 160-169.

nanoplates/graphene sheets under biaxial compression," Physica E, SCI, vol. 89, pp. 160-169.

在文檔中 嵌入式奈米石墨烯板三維非局部彈性力學漸近理論之發展與應用 (頁 56-72)