NONLINEAR CYLINDRICAL BENDING OF NANOCOMPOSITE PLATES
Gratis
NONLINEAR CYLINDRICAL BENDING OF NANOCOMPOSITE
PLATES ADDA Bedia El Abbas(1) , ZIDI Mohamed
(1)
, KACI Abdelhakim
(1) , BOURADA Mohamed
(1) , TOUNSI Abdelouahed
(1)
Bmfgenie@yahoo.fr
(1)
The most important features of carbon nanotubes are their extremely high stiffness combined with excellent resilience. For example, it has been reported that carbon nanotubes possess very high elastic modulus and sustain large elastic strain up to 5% (Qian et al., 2000). Therefore, the introduction of carbon nanotubes into polymers may improve their applications in the fields of reinforcing composites, electronic devices and more.
Laboratoire des Matériaux et Hydrologie, Université de Sidi Bel Abbes, Algérie
Abstract — The nonlinear cylindrical bending of simply supported nanocomposite plates
reinforced by singlewalled carbon nanotubes is investigated. The plates are subjected to uniform pressure loading in thermal environments and their geometric nonlinearity is introduced in the strain–displacement equations based on VonKarman assumptions. The material properties of SWCNTs are assumed to be temperaturedependent and are obtained from molecular dynamics simulations. The governing equations are reduced to linear differential equation with nonlinear boundary conditions yielding a simple solution procedure. Numerical results are presented to show the effect of the material distribution on the deflections and stresses.Keywords—
A. Plate; A. Nanocomposites; C. Analytical modeling; functionally graded materials
1. INTRODUCTION
Most studies on carbon nanotubereinforced composites (CNTRCs) have focused on their material properties (Odegard et al., 2003; Hu et al., 2005; Fidelus et al., 2005; Bonnet et al., 2007; Han and Elliott, 2007; Zhu et al., 2007). Several investigations have shown that the addition of small amounts of carbon nanotube can considerably improve the mechanical, electrical and thermal properties of polymeric composites (Fidelus et al., 2005; Bonnet et al., 2007; Han and Elliott, 2007; Zhu et al., 2007). Even though these studies are quite useful in establishing the stress– strain behavior of the nanocomposites, their use in actual structural applications is the ultimate purpose for the development of this advanced class of materials. As a result, there is a need to observe the global response of CNTRCs in an actual structural element. Wuite and Adali (2005) presented a multiscale analysis of the deflection and stress behavior of CNT reinforced polymer composite beams. The micromechanics models used in the study include straight CNTs aligned in one direction, randomly oriented CNTs and a two parameter model of agglomeration. Vodenitcharova and Zhang (2006) studied the pure bending and bendinginduced local buckling of a nanocomposite beam reinforced by a singlewalled carbon nanotube. It is wellknown that the applications of CNTs to nanocomposites have been hindered due to the weak interfacial bonding between CNTs and matrix. Functionally graded materials (FGMs) are inhomogeneous composites
 = _{11} _{1} _{11} η , _{CN} _{m} _{m} _{CN} E= _{22} _{22} ^{2}
 = _{12} _{12} ^{3}
 = _{11} _{11}
 zk = ε ε
 = = _{,} _{11} ^{2} _{, ,} _{11}

 = _{,} _{11} ^{2} _{, ,} _{11}
 pa pa pa p C q pa pC C a p p a q p a q A B pa pC A
 − − +

 pa pa pa p C q pa pC C a p p a q p a q A B pa pC A
 side coordinate
 thickness coordinate
 deflection
 axial stress
 load parameter
 _{1,00 0,75 0,50 0,25 0,00 0,25 0,50 0,75 1,00} _{Nondimensional Length, x/a} Ltd . [15] Vodenitcharova, T., Zhang, L.C., (2006),
 _{*} walled carbon nanotube”, Int. J. Solids Struct., dimensional length for V = . _{CN} 11 , T = 300 K .
 1,00 0,75 0,50 0,25 0,00 0,25 0,50 0,75 1,00 ^{0,0} ^{0,2} ^{0,4} ^{0,6} ^{0,8} ^{1,0} ^{1,2} ^{N} ^{on} ^{di} ^{m} ^{ens} ^{iona} ^{l D} ^{ef} ^{le} ^{ct} ^{ion,} ^{w} ^{/h} _{Nondimensional Length, x/a} ^{FG: V} ^{CN} ^{*} ^{=0,11} ^{UD: V} ^{CN} ^{*} ^{=0,11} ^{FG: V} ^{CN} ^{*} ^{=0,14} ^{UD: V} ^{CN} ^{*} ^{=0,14} ^{FG: V} ^{CN} ^{*} ^{=0,17} ^{UD: V} ^{CN} ^{*} ^{=0,17}
 _{*} _{CN}
 ^{6} _{11} × ^{CN} α
η
are the Young’s modulus, shear modulus and thermal expansion coefficient, respectively, of the carbon nanotube, and ^{m}
E , ^{m} G and ^{m}
α are corresponding properties for the matrix. ^{CN} ^{12} ν
and ^{m}
ν are Poisson’s ratios, respectively, of
the carbon nanotube and matrix. To account for the scaledependent material properties Eq. (1) includes j
(
and ^{CN} ^{22}
3 , 2 , 1 = j
)
CN
V
which is called the CNT efficiency parameter, and will be determined later by matching the elastic modulus of CNTRCs observed from the MD simulation results with the numerical results obtained from the rule of mixture. and _{m}
V (1d) (1b)
(1c) (1e) (1a)
α
G _{12 ,} ^{CN} _{11} α
characterized by smooth and continuous variations in both compositional profile and material properties and have found a wide range of applications in many industries (Suresh and Mortensen, 1998). The static bending, elastic buckling, postbuckling, linear and nonlinear free vibration of FGM structures have been extensively investigated (Benatta et al. 2008; Sallai et al. 2009; Na and Kim 2009; Wu et al. 2007; Ke et al. 2009). By using the concept of FGM, Shen (2009) suggested that the interfacial bonding strength can be improved through the use of a graded distribution of CNTs in the matrix and examined the nonlinear bending behavior of simply supported, functionally graded nanocomposite plates reinforced by SWCNTs subjected to a transverse uniform or sinusoidal load in thermal environments. In this study, a nonlinear cylindrical bending behavior of functionally graded nanocomposite plates reinforced by the SWCNT is studied. The material properties of the FGCNTRC are assumed to be graded in the thickness direction and estimated though the rule of mixture in which the CNT efficiency parameter is determined by matching the elastic modulus of CNTRCs obtained from MD simulation with the numerical results calculated from the rule of mixture. The Classical Plate Theory (CPT) and the von Karmantype nonlinear strains are used to construct the problem governing equations. Numerical results are presented to show the influence of material properties, plate geometry and mechanical loading on the resulting transverse deflection and stress state.
G
Fig. 1 shows the CNTRCs of thickness h where the distribution of CNTs is uniform across the thickness direction in Fig. 1a (UD CNTRC) and is nonuniform and graded along the thickness direction in Fig. 1b (FG CNTRC), respectively. It is assumed that the CNTRC is made from a mixture of SWCNT and an isotropic matrix. We first determine the effective material properties of CNTRC. It was pointed out by Han and Elliott (2007) that the material properties of the SWCNT and CNTRC are anisotropic. According to the rule of mixture, the effective Young’s modulus, the shear modulus and the thermal expansion coefficient of CNTRC can be expressed as (Shen, 2009) _{m} _{m} _{CN} _{CN}
E
V E
V E
V E
V E
, _{m} _{m} _{CN} _{CN}
V G
E _{11 ,} ^{CN} E _{22 ,} ^{CN}
V G
, _{m} _{m} _{CN} _{CN}
V V α α α
, _{11} _{12} _{22} _{12} _{22}
) ) 1 ( α 1 ( α ν α ν α ν − + + + = ^{CN m} ^{m} _{m} _{CN} ^{CN}
V V
where ^{CN}
2. MATERIAL PROPERTIES OF FUNCTIONALLY GRADED CNTRC PLATES
Q
(2a) (2b) (2c)
ν
and ^{21}
where (
w u,
) are the middisplacements in the (
z x,
) directions, respectively. Also, _{x}
ε and _{x} k
are the midplane strains and midplane curvatures. The subscript commas represent differentiation. The constitutive relations for nonzero strains are given by
) ( _{11} _{11} T Q _{x x} ∆ − = α ε σ
where
T T T − = ∆
is temperature rise from some reference temperature T at which there are no thermal strains. Using the material properties defined in Eq. (1), the stiffness coefficients, ij
are the carbon nanotube and matrix volume fractions and are related by
1 CN CN
, can be expressed as
,
1 _{21} _{12} ^{11} ^{11} ν ν − =
E Q
have their usual meanings, in particular for an CNTRC layer they are given in detail in Eqs. (1) and (2). Using Eq. (5) in conjunction with the strain components in Eq. (3), the constitutive relation for the CNTRC plate may be written as
(2d) (3) (4)
ν
=
1 = + _{CN m}
V V We assume the volume fraction _{CN}
V follows as: _{*}
2
V h z V
− = in which _{CN m CN m CN CN} _{CN} _{CN}
W W W
V ) / ( ) / ( ^{*} ρ ρ ρ ρ − +
where _{CN}
(5) (6)
W
is the mass fraction of nanotube, and _{CN}
ρ and _{m} ρ are the densities of carbon
nanotube and matrix, respectively. In such a way, the two cases of uniformly distributed (UD), i.e. ^{*} _{CN CN}
V V =
, and functionally graded (FG) CNTRCs will have the same value of mass fraction of nanotube.
The Poisson’s ratio is assumed to be uniformly distributed, i.e. _{m} _{m} _{CN} _{CN}
V V ν ν ν + = _{12} ^{*} _{12}
where ^{11} E , ^{12}
3. GOVERNING EQUATIONS FOR NONLINEAR CYLINDRICAL BENDING OF FGCNTRC PLATES
ε
are the resultants normal forces and bending moments, respectively, defined in the usual manner, i.e.,
The fundamental equations of large deflection analysis of pressure loaded FGCNTRC plates are briefly outlined in this section. Based on Kirchhoff’s hypotheses and von Karman large deflection theory, the straindisplacement relations in the case of the cylindrical bending may be written as
(7) (8) (9)
, _{x x x}
where
, ,
) are defined by the relations
M
and ^{T} ^{x}
N
The equivalent thermal loads ( ^{T} ^{x}
, 1 ( ^{h} _{h} _{x x x} M dz z N σ
− = ^{2 /} _{2 /} ) ,
( ) ∫
M
1 _{,} _{2} _{, , xx x x x x} − w k w u = + =
and _{x}
N
= ^{2 /} _{2 /} ^{2} _{11} _{11} _{11} _{11} , , , 1 , ^{h} _{h} Q dz z z D B A _{x}
∫ −
( ) ( )
where _{11} A , _{11} B and _{11} D are given by
B A M N _{11} _{11} _{11} _{11} ε
_{T} _{x} ^{T} _{x} _{x} _{x} _{x} _{x} M N k D B
=
−
2
− =
= + +
−
A B N N A B M −
M p q C px p D
( ) ^{T} ^{x} ^{T} ^{x x x}
Using Eq.(20) in Eq.(16),
) ( ) ( , ) ( = ± = ± = ± M a u and a a w _{x}
− q w N w D
which, on applying the boundary condition
A B _{xx x xxxx}
which can be put in the form _{,} _{2} _{,}
= − q w p w _{xx xxxx}
where
) / ( _{11} ^{2} _{11} _{11} ^{2} A B D N p ^{x}
− = and
) / ( _{11} ^{2} _{11} _{11} A B D q q
where _{1} C and _{2} C are constants, which must be determined using BCs at either edges of the plate. Assume that the origin of the coordinate system is located at the plate midspan; accordingly, the simply supported BCs yield
Eq. (18) is a linear fourthorder ordinary differential equation whose solution is readily available. The general solution is as follows: _{2} _{2} _{2} _{1}
− + + = _{2} ^{2} _{1} _{11} _{11} ^{2} ^{11} _{11} ^{11} ) cosh(
(12) (11a) (11b)
C px C x w − + =
(13) (14a) (14b) (15a)
(21) (22) (23) (24)
(19) (20)
( ) ∫
− ∆ = ^{2 /} _{2 /} _{11} _{11} ) ,
, 1 ( ^{h} _{h} ^{T} _{x} ^{T} _{x} T dz z Q M N α
Use of Hamilton’s principle yields the Euler– Lagrange equations as (Reddy, 2003) _{,}
= _{x x} N _{, ,}
= + + q w N M _{xx x xx x}
where
q is the transverse loading on the plate.
Equation (11a) leads to
constant _{x x} N N = =
Therefore, Equation (11b) becomes . _{, ,} = + + q w N M _{xx x xx x} From Equation (7) _{T} _{x x x x}
N k B A N − + = _{11} _{11} ε _{T} _{x x x x}
M k D B M − + = _{11} _{11} ε
Now, substitution of _{x}
ε and _{x} k values from
Eq. (4) gives _{T} _{x xx x x x x}
2 ) cosh( ) ( x p q
1 T
, gives
1 Equations (15) yield
(15b) (16) (17) (18)
N w B w u A N N − −
A B M N A B N pa C ^{T} _{x} ^{T} _{x} _{x}
1 p q
1 ) cosh(
− = _{4} _{11} ^{11} _{11} ^{11} _{1}
2
) ( = ±a M _{x}
8
M
(25)
− + + = _{,} _{11} _{11} ^{2} ^{11} _{11} ^{11} Plugging in this value of x
B N N A B M −
B A N N ^{T} _{x x} (10)
6 ) sinh( _{3} _{1} _{2} _{1} ^{2} ^{1} ^{2} _{4} ^{3} ^{2} _{2} _{11} ^{11} _{1} _{11} ^{11} _{11} = − + −
4
[ ] ) sinh( ) cosh( ) 2 sinh(
Integrating of Eq.(15a) with respect to x, and application of the boundary condition ) ( = ±a u leads to
C − =
2 ^{1} ^{2} ^{2} ^{2} C pa p a q
) cosh(
gives
= ±a w
in Eq. (11b), gives ^{, ,} ^{11} ^{11} ^{2} ^{11}
( ) ^{T} ^{x xx} ^{T} ^{x x x} M w D A
2
x xx x x x M w D w u B M − −
The boundary condition ) (
η . For short fiber
A B M N A B N pa C ^{T} _{x} ^{T} _{x} _{x}
composites ^{1}
η
is usually taken to be 0.2 (Fukuda and Kawata, 1974). However, there are no experiments conducted to determine the value of _{1}
η for CNTRCs. Recently, Shen
Again, Eq.(29) is a transcendental equation and Eqs. (27)  (29) are to be solved numerically.
B a A N N ^{T} _{x x}
6 ) sin( _{3} _{1} _{2} _{1} ^{2} ^{1} ^{2} _{4} ^{3} ^{2} _{2} _{11} ^{11} _{1} _{11} ^{11} _{11} = + − + + − − + −
4
8
) sin( ) cos( ) 2 sin(
C − − = and [ ]
2 ^{1} ^{2} ^{2} ^{2} C pa p a q
) cos(
1 p q
1 ) cos(
− = _{4} _{11} ^{11} _{11} ^{11} _{1}
which on applying the boundary conditions of Eq.(21), gives
C px C x w + + =
2 ) cos( ) ( p x q
is negative, the solution of Eq. (18) becomes _{2} _{2} _{2} _{1}
N
and a numerical technique must be used to obtain a solution. If _{x}
N
Equations (23)(25) contain three unknowns quantities: _{1} C , _{2} C and _{x}
(2009) gives an estimation of CNT efficiency parameter _{1}
(28) (26)
3. NUMERICAL RESULTS AND DISCUSSION
( ) GPa 0047 .
η by matching the Young’s moduli _{11} E
). /( ^{4} _{22} = ^{4} E h qa q ^{CN n} (29) (27)
α σ σ ,
10 / a T E q h ^{CN CN} _{x} ^{x}
∆ + =
−( ) [ ] ^{2} ^{22} ^{22} ^{6} ^{2}
, / h w w =
, / h z z =
, / a x x =
. The results obtained from the analysis are presented in dimensionless parametric terms of deflections and stresses as follows:
m 5 . = a
and
5 = h
The plate geometry is chosen so the mm
η is properly chosen, as shown in Table 2.
of CNTRCs obtained by the rule of mixture to those from the MD simulations given by Han and Elliott (2007). Through comparison, we find that the Young’s moduli obtained from the rule of mixture and MD simulations can match very well if the CNT efficiency parameter _{1}
51 . E 3 T ^{m} − = , in which T T T ∆ + =
and
GPa 1 . 2 = ^{m}
and K 300 = T (room temperature). In such a way,
K / 10 .
45 ^{6} × = ^{m}
α
and
E
Numerical results are presented in this section for FGCNTRC plates subjected to a transverse uniform load. We first need to determine the effective material properties of CNTRCs. Poly {(mphenylenevinylene)co [(2.5dioctoxypphenylene) vinylene ]}, referred to as PmPV, is selected for the matrix, and the material properties of which are assumed to be
at
K 300 = T
. (10, 10) SWCNTs are selected as reinforcements. It has been shown (Elliott et al. 2004; Jin and Yuan 2003; Chang et al. 2005) the material properties of SWCNTs are anisotropic, chiralityand sizedependent and temperaturedependent. Therefore, all effective elastic properties of a SWCNT need to be carefully determined, otherwise the results may be incorrect. From MD simulation results the size dependent and temperaturedependent material properties for armchair (10, 10) SWCNT can be obtained numerically (Shen, 2009). Typical results are listed in Table 1. It is noted that the effective wall thickness obtained for (10, 10)  tube is h=0.067 nm, and the wide used value of 0.34 nm for tube wall thickness is thoroughly inappropriate to SWCNTs. The key issue for successful application of the rule of mixture to CNTRCs is to determine the CNT efficiency parameter _{1}
45 ^{6} × ∆ + = T ^{m} α
1
( ) K / . 10 0005
34 . = ^{m} ν ,
1 = q as an example. The non
dimensional deflection is shown in Fig. 2 under thermal environmental condition K 300 = T . A same size uniformly distributed CNTRC plate is also considered as a comparator. In this figure, UD represents uniformly distributed CNTRC plate and FG represents functionally graded CNTRC plate. It can be seen that the linear solution overestimates the deflections of both FG and UD CNTRC plates and the deflections of FG CNTRC plate are larger than that of the UD CNTRC plate.
Figure 3 presents the effect of nanotube volume fraction on the nondimensional deflection of CNTRC plates for different values of the nanotube volume fraction
) , 17 . , 14 . ( 11 . ^{*} = _{CN}
V
subjected to a uniform pressure under thermal environmental condition
K 300 = T
in nonlinear analysis. It can be seen that the plate has higher deflection when it has lower volume fraction. It can be also found that deflections of FGCNTRC plate are larger than those of the UDCNTRC plate.
The nonlinear cylindrical bending of functionally graded carbon nanotube reinforced composite plates subjected to transverse loads is studied using a simple analytical solution. The material properties of FGCNTRC are assumed to be graded in the thickness and estimated though the rule of mixture. Navier equations according to the large deflection theory can be expressed as linear equations for the deflection, leaving nonlinear boundary conditions. This linearity of the differential equations greatly simplifies the large deflection analysis. From this study it is concluded that because of extension bending coupling, the large deflection plate theory must be used even for deflection that are normally considered small. In comparison with the nonlinear analysis, the linear solution overestimates. The results show also that the plate has higher deflection when it has lower volume fraction and the deflections of FG CNTRC plate are larger than those of the UD CNTRC plate.
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4. CONCLUSION
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ion, ct Linear FG _{2,0} le
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ef _{Linear UD} l D _{1,5} _{Nonlinear UD}
“Fundamentals of functionally graded
iona ens _{1,0} m
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di on _{0,5} N
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Figure. 2: Nondimensional deflections of
“Bending and local buckling of a CNTRC plate subjected to a uniform nanocomposite beam reinforced by a single ^{2} transverse load q =
1 KN/m versus non
43, 3006–3024.
300 5.6466 7.0800 3.4584 500 5.5308 6.9348 4.5361 700 5.4744 6.8641 4.6677
0.11
Rule of mixture _{11} E (GPa) _{11} E (GPa) _{1} η
V MD (Han and Elliott, 2007)
K 300 = T .
PmPV/CNT composites reinforced by (10,10) tube under
Table 2: Comparisons of Young’s moduli for
K) / 10 (
Figure. 3: Effects of nanotube volume fraction
E _{22} (TPa)
Temperature (K) ^{CN} E _{11} (TPa) ^{CN}
= ^{CN} ν ).
properties for (10, 10) SWCNT (L= 9.26 nm; R = 0:68 nm; h= 0:067 nm; 175 . _{12}
Table 1: Temperaturedependent material
1 = q and K 300 = T .
on the nonlinear bending behavior of CNTRC plates under uniform pressure ^{2} KN/m
94.8 94.57 0.149 0.14 120.2 120.09 0.150 0.17 145.6 145.08 0.149