Prediction of the stiffness and stresses for carbon nano-tube composites based on homogenization analysis

Faculty WbMT

Dept. of Marine Technology

Mekelweg 2,2628 CD Delft

T h e Netherlands

Reports of Research Institute for Applied Mechanics, Kyushu University No. 129 (37-45) September 2005 37

Prediction ofthe Stiffness and Stresses for Carbon Nano-Tube

Composites Based on Homogenization Analysis

Dong-Mei. LUO*', Wen-Xue WANG*', Yoshihiro TAKAO*' and Koichi KAKIMOTO*'

(Received July 29, 2005)

Abstract

Nuinerical prediction of the macroscopic stiffiiess and microscopic stresses for carbon nanotube polymer composites is performed based on the homogenization theory. A new solution method is proposed for the homogenization analysis. The conventional inhomogeneous integral equation related to the microscopic mechanical behavior in the basic unit cell is replaced by a homogeneous integral equation based on a new characteristic function. According to the new solution method, the computational problem o f the characteristic function subject to initial strains and periodic boundary conditions is reduced to a.simple displacement boundary value problem without initial strains, which simplifies the computational process. The effects of various geometry parameters including straight and wavy nanotubes on the macroscopic stif&iess and microscopic stresses are presented. Numerical results are compared with previous results obtaihed from the Halpin-Tsai equations, Mori-Tanaka method, which proves that the present method is valid and efficient.

Key words : Carbon nanotube composite, Homogenization theory. Solution method, Macroscopic Stiffoess, Microscopic stresses

1. Introdiiction

The extremely high strength and stiffness combining with high aspect ratio tnake carbon nanotube (CNT) become attractive as reinforcement of polymer matrix composites. Many researches o f ^experimental and analysis have been carried to develop CNT polymer composites, e.g. as reviewed by Andrews and Weisenberger [ I ] . In order to promote the development o f excellent CNT ploymer composites, the prediction of the macroscopic stiffness and microscopic stresses plays an important role in the design and the application of CNT ploymer composites in practice. On the other hand, it is extremely difficult to analyze a CNT polymer composite with complex material heterogeneity involviiig with individual nanotubes precisely due to the huge computational time and cost. Hence, many researches have been focused on exploring an approximate but simple and efficient analysis method.

Halpin-Tsai equation [2] is a popular method for predicting the macroscopic properties o f traditional fiber-reinforced composites. The modified Halpin-Tsai equation is utilized to predict the elastic properties of CNT polymer composites in [ 3 ] . Mori-Tanaka method •1 Research Institute for Applied Mechanics, Kyushu

Uni-versity

developed in (e.g. [4,5]) is also a well-known method to predict the macroscopic material properties of various composites and is applied to the prediction o f macroscopic stiffness of nano-composites with layered silicate [6] and long wavy CNT polymer composites[7]. However, both of these two methods have a common shortcoming that they cannot accurately reflect the interactions between neighboring fibers or nanotubes because of the limitation o f their analytical models. Recently, a shear lag model is developed to study the macroscopic stiffness [8]. It is a very simple analysis, but it also cannot reflect .the interacfions between neighboring nanotubes accurately.

In parallel to the above analytical methods, different approaches to composites analysis have been also developed based on the analysis of a basic cell by finite element method (FEM) in the past years. In general, these approaches can be roughly classified into the averagerfield method and the homogenization method. The averagerfield method (e.g. [9]) has been developed based on the: physical viewpoint that the macroscopic material properties obtained from experiments represent the properties of volume average. In contrast, the homogenization method (e.g. [10]) has been developed based on the mathematically tnuiti-scale perturbation theory. In [7,11], the average-field method is used to

predict the macroscopic properties o f straight CNT and long wavy CNT polymer cotiiposites. However, it is realized that it is difficult to impose exact periodic condition along a basic unit cell with asymmetric and complicated ralcrostructures in the average-field method [12]. I n the: case of homogenization method, a characteristic function of the third order tensor is introduced to relate the microscopic displacements to the macroscopic displacements, which make it possible to express the exact periodic conditions formally along the boundary o f a basic cell. However, since the integral equation related to the characteristic function is inhomogeneous, two computational processes of imposing initial strains and periodic displacement conditions are needed to obtain the characteristic function in the conventional solution method. It is obviously inefficient because there are six independent sets o f components o f t h e characteristic function need to be solved for a general three-dimensional unit cell.

In this paper, numerical predictions o f the macroscopic stiffness and microscopic stresses for CNT polymer composites are performed based on the homogenization theory. A new solution method is proposed for the homogenization analysis. According to the new solution method, the computational problem of the characteristic function subject to initial strains and periodic boundary conditions is reduced to a simple displacement boundary value problem without initial strains, which simplifies the computational process. The effects o f various geometry parameters including straight and wavy nanotubes on the macroscopic stiffness and microscopic stresses are presented. Numerical results are compared with previous results obtained from the Halpin-Tsai equations, Mori-Tanaka method.

2.

Formulation

Consider a linearly elastic body with a periodic microstructure, as shown in Fig. I . Q denotes the open subset of three-dimensional space occupied by the body, r the boundary o f Y the open subset o f the space occupied by the basic unit cell, Sy the boundary o f Y. The sub-domain 7, may represent an inclusion in the unit cell to describe a composite, or a void to describe a porous material. Define S,,,, as the interface when yj and Y^ are different materials, or as the intemal boundary o f 7, when Y^ represents a void> For the sake of simplicity, it is assume that Sy,^ is a traction-fi-ee surface i f Y^ represents a void. A brief review o f the basic equations o f homogenization theory is given in the following paragraphs.

Matrix Y,

(fl) aobe\ connguralion (b) Unit cell Fig. I A material with periodic microstructures.

In the construction o f the homogenization theory, the displacements M , ( X ) are assumed as an asymptotic expansion with respect to a parameter 7 that is a scaling factor o f the microscopic/macroscopic dimension, i.e.

M , ( x ) = w ° ( x , y ) + 77M;(x,y) + 7 ' t / ; ( x , y ) + - ( I ) Where x = (x^,Xj,x^) md y = (.y,,yi,y,) represent the macroscopic and the microscopic coordinate systems, respectively, which are related to each other by

y, = — (2)

Then, based on. the elasticity, the strain-displacement and stress-strain relations can be expressed as

2 aXj etc, (3)

Where denotes the elastic constants tensor. Applying the chain rule o f differentiation o f a fiinction with implicit variables to the: partial differentials o f (3) leads to

1 .du", du] 1 rdu", du] du] du) .

"hi^dy^ dy,' l^-^dx^ dx

l^^dx, dx/ 'dy^ dy,'^ du\. .dul

dy^ dy,

(4)

\ ^ dul ,dul 5«' „ ,du\ du'

(5) According to the elasticity, the virtual displacement equation can be expressed as:

8u^ dv,

(6)

Where Ty denotes the region of the boundary F with specified tractions 7 j , v, is the virtual displacement and V, = 0 on the where u, =U, , f , is the

Reports of Research Institute for Applied Mechanics, Kyushu University No. 129 September 2005 39

body force. By inserting (1) into the above virtual displacement equation, applying the chain rule o f differentiation to the partial differentials o f Wjandv,, and equating the terms with the same power o f 7 , we can derive a series o f equations related to the displacements u, , , and so on as follows.

« ; ( x , y ) = < ( x ) (7) dulix) , dul(x,y)']dv,(y) dx, —dY = 0 f r 1 r . fdulix) du\{x.

v , e r (8)

Theoretically, solving all the above equations together with specified boundary conditions w i l l yields the f u l l solution for u", , u], uf,--. However, the first order approximation o f « , ( x ) is usually o f interest for many practical applications. Then only two equations of (8) and (9) related to u, and need to be solved.

In the conventional solution method for the homogenization analysis, it is assumed that

« ; ( x , y ) = - ; } r r ( x . y ) ^ ^ + « ; ( x ) (11)

Where , called as characteristic function, is an unknown Y - periodic tensor o f the third order. It is noted that x^" may also be considered as a symmetric tensor o f the second order for each k (A=l,2,3). Inserting (11) into (8) leads to

r < ( x . y ) a v , ( y ) r

Then the periodic boundary conditions o f

M j ( x , y ) require

w I ( x . y ) | / . „ , = " » ( x , y )

« t ( x , y ) | = « * ( x , y )

" * ( x , y ) | / ^ 5 = " l ( x , y ) | / „ . , (14) Inserting ( I I ) into the above equations leads to

- ; i r r ( x , y ) | , . „ , = - z r ( x , y ) ,

- ; i r r ( X , y ) | = - ; } r r ( x . y )

- ; i r r ( x , y ) | , . „ , = - ^ r ( x , y ) | / « , . ( i s ) Hence, the characteristic function can be completely determined from (12) and (15).

On the other hand, inserting ( I I ) into (9) leads to I i i > ^ ( x ) ^ ^ r f O = £ i , ( x ) v , ( x ) r f O + (x)v,(x)dr, x e O

(x) = ^ { ( £ , „ £ , „ i j ^ ) ^ !

Equation (16) describes the macroscopic equilibrium. Where denotes the homogenized stiffness and is usually called as the macroscopic stiffness. Therefore the basic equations o f a homogenization problem in the sense o f first order approximation are reduced to the integral equation (12) subject to periodic conditions o f (15) and the integral equation (16) subject to specified boundary conditions. Both o f the integral equations can be solved separately by the use o f FEM. We can firstly obtain ( x , y ) by solving (12) and then solve (16) to obtain macroscopic M°.(x). I f only the homogenized elastic constants D , ^ is o f interest, we can solve (12) and calculate (17) to obtain ;}r,*'(x,y) and D"„ . Hence, solving (12) is an important step in the homogenization analysis.

Equation (16) describes the macroscopic equilibrium. Where D^^J denotes the homogenized stiffness and is usually called as the macroscopic stiffness. Therefore the basic equations o f a homogenization problem in the sense of first order approximation are reduced to the integral equation (12) subject to periodic conditions o f (15) and the integral equation (16) subject to specified bound.ary conditions. Both o f the integral equations can be solved separately by the use o f FEM. We can firstly

obtain ;ir,*'(x,y) by solving (12) and then solve (16) to obtain macroscopic ul(x). I f only the homogenized elastic constants D,", is of interest, we can solve (12) and calculate (17) to obtain ; i r , " ( x , y ) and D"y . Hence, solving (12) is an important step in the homogenization analysis.

Where x" is also a symmetric tensor o f the second order for each k (A= 1,2,3) and is expressed by

(20)

The symbol S^j is the Kronecker delta. Then the derivation o f Xot *^2n be expressed by

dXol 1

(21)

Inserting (21) into (12) and using E^y = Eij^. lead to

-dY = 0 (22)

Similarly, by the use o f (21) the homogenized elastic constants can be rewritten as

3y, )dY

dx" ₍₂₃₎

Consequendy, it is seen that (12) has been transformed into a homogeneous integral equation (22) in terms o f the new characteristic function %"(.x,y). That is, the original problem with initial strains and periodic conditions is reduced to a simple displacement boundary value problem. Hence, the calculation process o f imposing the initial stresses is reduced during the solution o f every set (z">Xi>Xi)- Furthermore, the periodic conditions for a rectangular parallelepiped unit cell in terms o f can be easily expressed as follows by the substitution o f (19) into (15).

lxr(^.y)-Xo".(y)]\,.,> = [ j r ( x , y ) - ; i r o 7 ( y ) ] | Ixr ( X , y ) - Xol ( y ) ] | = [xr ( X . y ) - Xol ( y ) ] |

lxr(^>y)-x.1.(y)]\ = [xr(^<y)-x.1(y)]\

(24)

3. Calculation Models

In the present numerical analysis, two kinds o f regular and staggered CNT arrays are calculated, as shown in Fig, 2. The unit cell with one CNT is used for the regular array

and the unit cell with a complete CNT and four quart CNT is used for the staggered array. Furthermore, the CNT in the unit cell may be straight or wavy i n order to investigate the effect o f the waviness o f CNT on the macroscopic stiffness, as. listed in Table 1. That is, five models with straight CNT, wavy CNT and mixed CNTs are calculated. The details o f geometrical parameters related to neighboring CNT are depicted in Fig. 3. The diameter D o f the CNT is taken as an unit, Hf and Tf describe the half o f the distance between neighboring CNTs. The waviness o f a wavy CNT is expressed by a sinusoidal function

Unit ceil Unit cell Fig. 2 Two nanotube arrays.

Table 1 Calculation models

Models (Array iUpperCNT 'MiddleCNT , RS iregular jstrai^t i straight

SS ;staggered i straight • straight

RW .regular iwavy j wavy 1

SWl ^staggered 1 straight [wavy I

2 H f / h ,-.--^-J^

(a) x-y plane of regular array

X

Reports of Research Institute for Applied Mechanics, Kyushu University No. 129 September 2005 41

(d) x-z middle plane o f staggered array for wavy CNTs

Fig. 3 Geometrical parameters

z = AsiniZTVC / L) (25)

and the wavy plane o f the wavy CNT is assumed to coincides with the x-z plane.

4. Numerical Results

In this section, numerical results are presented to demonstrate the validity and efficiency o f the new solution method for the prediction o f the macroscopic stiffness and microscopic stresses o f CNT polymer composites. Finite element analysis >is perfonned by the use o f a commercial finite element code ABAQUS. In the calculation, the CNT are considered as a transversely isotropic fiber [13] and the effective stiffiiess constants are Cii=457.6GPa, Ci2=Ci3=8.4GPa, C22=C33=14.3GPa, C23=5.5GPa, C44=C35=27.0GPa, and C « = 4 . 4 G p a . The Young's rriodulus and Poisson's ratio o f t h e matrix are 3.8 GPa and 0.4, respectively.

The variation o f macroscopic stiffness with the aspect ratio is shown in Fig. 4 for the case o f straight CNTs. Where, SS and RS denote the results o f staggered array CNT and regular array CNT, respectively. The results obtained from Mori-Tanaka method and Halpin-Tsai equation are also depicted for a comparison. The ratio o f Tf to Hf is taken as a parameter. From these results o f four stiffness constants, it is seen that E l l is sensitive to the aspect ratio, and that the others are slightly influenced by the aspect ratio, except for small aspect ratio. The staggered array gives high E,, than the regular array, especially for relatively large Tf/Hf, but the differences between the two arrays for the other

constants are small. The values o f E n obtained from Mori-Tanaka method and Halpin-Tsai equation are close to the present ones with T[=Hf. It is interesting that a small Tf gives high E n , which is useful for the design and fabrication o f CNT composites. The present results except for E n predict lower stiffness than Mori-Tanaka method and Halpin-Tsai equation. Figure 5 shows the variation o f macroscopic stiffness with the fiber volume fraction. A l l constants increase with increasing fiber volume fraction. Also only the results for the models with straight CNT are presented. Similarly, a small.Tf/Hf

ives high En and the staggered array gives high E n lan the regular anay, especially for large T/Hf.

The effects o f waviness of the CNT on the macroscopic stiffness are presented i h Fig. 6 and Fig. 7. Figure 6 shows the variations o f the sfif&iess with fiber volume fraction and waviness A / D . It is seen that large waviness reduces E i , but improves G ^ due to the wavy plane coinciding with the x-z plane. The other stiffness constants, that are not presented here, have little variation with thewaviness.

The microscopic stresses at the surface o f the effective fiber and along the fiber axial direction o f the CNT are presented in Fig. 8 when the composite is subjected to a uniform tension. Only two stress components are depicted due to the limitation o f pages. The stresses are normalized by the average tensile stress. The upper two figures show the distributions of the fiber axial stress and the shear sfress in x-y plane in the case o f straight fibers with regular array. The axial stress is almost uniform except for the region near to the two ends, while high shear stress appears in the end regions. These results are similar to the results in many previous papers. The second two figures describe the stress distributions in the case o f straight fibers with staggered array. It is seen that high axial stress appears in the middle region diie to the influence o f neighboring fibers (referring to Fig.3). Also high shear stress occurs in the regions near to the two ends and it is a few higher than that in the case o f regular array. The third two figures present the stress distributions in the case o f wavy fibers with regular array. The effect o f the waviness on the axial stress is apparent and the maximum value occurs at the center region o f the fiber, although the variation o f the shear stress in x-y plane is not clear. Finally, the lower two figures give the stress distributions in the case of wavy fibers with staggered anay. The distribution o f the axial stress is similar to that in the case of wavy fibers with regular array, but the variation o f t h e shear stress with the waviness is more obvious. These stress results are Useful for the understanding to the microscopie damage and the macroscopic strength o f CNT polymer composites.

-m—SS,Tf=0.1Hf RS,"n=0.1Hf -k— SS,TT=Hf . . . ^. - - RS.Tf=Hf —SS,TT=10Hf ---o---RSJ=10Hf H Mo-Ta — ^ — H p - T s 10 100 1000 A s p e c t ratio ( a ) E n / E „ . •w—SS,Tf=0.1Hf RS,Tf=0.1Hf -k— SS.Tf=Hf . . . A- - - [RS,Tf=lHf •è—SS,Tf=10Hf . . . o . . . RS,T=10Hf - 1 - ^ IVIo-Ta — K — Hp-Ts ratio (b) E j j / E „ «—SS,Tf=0.1Hf *r^SS,Tf=Hf •—SS.Tr=10Hf — iVb-Ta — ^ E O 1 1 a..: RS,Tf=0.1Hf A- - - RS.Tf=Hf « — R S T = 1 0 H f h^-Ts 4 3 2 1 r 1 V ^ 2 0 % 10 100 A s p e c t ratio 1000 Hi—SS,Tf=0.1Hf -jt^SS,TF=Hf -•—SS,TT=10Hf H IVIo-ta — ^ RS,TT=0.1Hf A- - • RS,TT=Hf e RS.T=10Hf Hp-Ts 1.6 1.5 E l - 4 § 1 . 3

S l. 2

Fig. 4 Variations o f macroscopic stiffiiess with the fiber aspect ratio.

-3i«-5k 1000 -m—SS,Tr:<l.1Hf RS.Tf^.lHf -k—SS,Tr=Hf ..^A--RS,TT=Hf —SS,TT=10Hf - . - o - . - I ^ T = 1 0 H f H IVb-Ta Hp-Ts 0.05 0.1 0.15 Fiber w l u m e fraction ( a ) E | , / E „ 0.2 - • SS,'IT=0.1Hf . - - o - - - RS,TT=0.1Hf -k— SS.-IT=Hf - - . A- • - RS,Tf=Hf -•—SS.Tr=10Hf e RS,T=10Hf -4 M6..Ta — ^ — l - i p - T s 0.05 0.1 0.15 0.2 Fiber w l u m e fraction (b) E^j/E^

Reports of Research Institute for Applied Mechanics, Kyushu University No. 129 September 2005 4,3 — s s, - n . = o. i H f . . - o - . . RS,n=o,iHf - * SS,Tf=Hf „ . . - A- • - RS,TT=Hf - • — SS,TT=1 OHf - - - c ... RS;T=1 OHf - I Mo-Ta — - H— H p- T s - • SS,Tf =0.1 Hf RS,"n=0.1Hfl •-.A-• • SS,TT=Hf ..-A---RS,Tf=Hf - • - ^ SS,TT=1 Olrtf •' - ^ e-- ^ RS,T=1 OHf

I - .Mo-Ta ' ' )(• . Hp-Ts

0 0.05 0.1 0.15 0.2 Fiber volume fraction

0.05 0.1 0.15 Fiber vblCime fraction

0.2

. • ( c ) G , j / G „ <d) G i 3 / G „

Fig. 5 Variations of macroscopic stifhess with the fTber volume fraction.

Aspect ratio=20

0.05 •. 0.1 0.15 Fiber \olume fraction

••„ ., 0.1 Fiber volume fraction

0.2

(a)-Eu/En, . (b) G,3/G„. \

Fig. 6 Variation of macroscopic stifSiess with the fiber volume .fraction in the case of SW2 model.

0.4 0.8 : A / D

( a ) E n / E „ (b) É33/E„

Q) O) 0.2 0.1 Q> a? Si a> 0

I

Reports of Research Institute for Applied Mechanics, Kyushu University No. 129 September 2005 45

5.

Summary

A new solution method is proposed for the homogenization analysis. Numerical prediction o f the macroscopic stiffness and microscopic stresses for CNT polymer composites is performed based on the new solution method. The effects o f various geometry parameters including straight and wavy CNT on the macroscopic stiffness and microscopic stresses o f the cotnposites are presented. Numerical results of macroscopic stiffness are compared with previous results obtained from the Halpin-Tsai equations, Mori-Tanaka method, which proves that the present method is valid and efficient.

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