A Study on Effective Elastic Modulus of Nanocomposites Li-Chun BIAN1,a,*, Jing PAN1 and Cai-Hua WANG1

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1 nd Annual Intenational Coneence on Adanced Mateial ngineeing (AM 06) A Study on ectie lastic Modulus o Nanocoposites i-chun BIAN,a,*, Jing PAN and Cai-Hua WANG Key aboatoy o Mechanical Reliability o Heay quipents and age Stuctues o Hebei Poince, Yanshan Uniesity, Qinhuangdao , PR China a e-ail addess: bianlic@6.co *Coesponding autho Keywods: lastic odulus, Cabon nanotube, Nanocoposite, Fibe, Wainess. Abstact. Nanocoposites einoced by cabon nanotubes exhibit geat adantages in engineeing applications. The wainess o nanotubes is one o the ipotant actos which aect the eectie elastic odulus o nanocposites. This pape ais to deelop a new odel to eplace the way cabon nanotubes with eectie ibe. The eectie odulus o nanocoposites with andoly oiented ibes can be pedicted based on the icoechanics. The eects o wainess and aggloeation on the eectie odulus ae also analyzed. It is shown that they can educe the stiness o cabon nanotubes, signiicantly. Moeoe, the eectie elastic odulus o nanocoposites is ey sensitie to the wainess and the aggloeation. Intoduction To-date, any eseaches hae studied on the eectie elastic odulus o nanocoposites. Fo the discoey o cabon nanotube (CNT), its eakable popeties attact uch attention o scholas [-3]. Many eects, e.g., CNT geoety and dispesion, atio o length and diaete, and inteacial bonding on echanical popeties o CNT-einoced coposites hae been inestigated in soe ways [4-6]. Shady and Gowayed [7] inestigated the eect o wainess on eectie odulus o coposite based on the act assuing the CNT as a special helical shape. It has been ound that the eect o cuatue sees oe citical with inceaseing weight action. Moeoe, the stain enegy o nanotube and the eectie ibe ae pedicted, and the constaints iposed by the atix on the deoation ae consideed in this study. As copaed with an expeiental analysis, nueical ethod is oten eployed as an eectie way o the echanical behaio o nanocoposites, since it cost less and doesn t need to establish coplex ealuation. Giannopoulos et al. [8] gies a icoechanical inite eleent appoach o the estiation o eectie Young s odulus. In the wok o Giannopoulos et al. [8], a cylindical epesentatie, which consists o einoceent, inteace and atix ateial is eployed, and a discete sping-based inite eleent ethod is used to siulate the nanotubes. Based on the inite eleent ethod, it has been ound that an incease in the aount o andoness esults in deceased Young s odulus alues and stength o coposite [9]. The ai o pesent pape is to inestigate the eectie elastic odulus o CNT-einoced coposites. We will pay oe attention to the inluence o CNT wainess and aggloeation on the coposite. A spatial way ibe with a solid cicula coss-section is used to siulate the actual CNT state in the atix. Then, we eplace the spatial way ibe with a staight ibe along the axis and a coin shape wae. Both ibe and coin shape ae assued to be isotopic, and then the eectie odulus can be deteined by the Moi-Tanaka eectie-ield ethod popose. Micoechanics Model Many expeiental esults hae shown that CNTs geneally exist in a way shape in the nanocoposites. In this study, paates a and λ ae the aplitude and waelength o spatial way ibe. The diaete and olue action o CNTs ae denoted by d and. Then, we eplace the 06. The authos - Published by Atlantis Pess 90

2 spatial way ibe with a staight ibe along the axis and a coin shape wae. The length and the olue action o staight ibe ae denoted by λ and. The adius, the olue action, and the thickness o waes ae denoted by a, and h, espectiely. Consideing a epesentatie olue V o a CNT-einoced coposite, we hae the elations: πd λ N () 4V πa h N () 4V d h (3) a whee N is the nube o a cetain CNTs with sae wainess which is identiied by the adio a/λ denoted by δ. Fo qs.,, and 3, we can obtain: λ (4) a In ode to eain the olue action, it is assued that the su o staight ibe olue and wae olue equals to the cabon nanotube olues. Then, we can expess this elation as, + (5) Suaizing equations aboe, we can obtain: (6) +δ δ (7) + δ It is assued that the spatial CNT hoizontally eectie odulus is poided by the staight ibe. Though the ules-o -ixtues, we get + x (8) + y (9) whee is the Young's odulus o atix, and ae the olue actions o atix and staight ibes, espectiely. Hee, we adopt the esult o eectie longitudinal odulus [0], 3/ (+ c) c 3c 3/ c (0) x (+ ) S -[+ - (+ c) ] S + (S + S66) 9

3 3/ (+ c) y () c 3/ 3c c (+ ) S + [(+ c) -- ] S + (S + S66) whee πa λ c ( ) 4π δ () S S - ; S T ; S 66 T G T (3) In the aboe oulas,, T, and G T ae the longitudinal, tansese and shea oduli o coposite with way ibes. Te T is the longitudinal-tansese Poisson's atio. The ules-o -the ixtues can be used to obtain these paaetes, T T ( ) (4) Accoding the intinsic elationship, we can get: G, G (5) (+ ν ) (+ν ) Fo the aboe analysis we can get the eectie elastic odulus o staight and wae, x y - - (6) Results and Discussion The espectie eectie oduli o coposites containing cued ibes and coesponding staight ibes ae pedicted. Note that these two ibes hae the sae length. To estiate the eectie elastic odulus o coposite einoced by both o way CNTs and aggloeated CNTs, the eectie elastic stiness o inclusions is calculated and then igue up the oeall popety o whole coposite syste. In this study, the paaetes used ae pesented in Table. Table. Paaetes and atios used o nueical calculations. (GPa) NT (GPa) a/λ λ/d ν ν Nt

4 The CNTs ae diided into concentated and eely scatteed ones in the atix. The egion aggloeated CNTs neaby atix is assued to be spheical in shape and the eely scatteed ones ae thought in a spatial way shape as shown in Fig.. Then, we eplace the spatial way shape with a staight ibe and a wae. So thee ae thee kinds o inclusions in the atix. Fig.. Inclusion odel Poposed o the aggloeation o CNTs. Fig. gies the aiation o eectie elastic odulus with the incease o wainess o dieent olue actions. It can be ound o Fig. that thee is alost no obious inluence o wainess on the alue o eectie elastic odulus when 0<δ<0.. Howee, the eectie elastic oduli ae ey sensitie to the wainess in the egion o 0.< δ<0.3. Moeoe, it s noted that the alue o eectie elastic odulus seas no longe change uch when δ is oe 0.3. Fig.. ectie elastic odulus esus wainess o dieent olue actions. Conclusions In this inestigation, based on the icoechanics, a new odel to eplace the way cabon nanotubes with eectie ibes has been deeloped. It can be used to pedict the eectie elastic odulus o nanocoposites with andoly oiented eectie ibes. The eects o both wainess 93

5 and aggloeation on the eectie elastic odulus ae also discussed. As a esult, it is ound that the stiness o cabon nanotubes can be educed, signiicantly due to the inluences o wainess and aggloeation o nanotubes. Moeoe, thee is a signiicant inluence o wainess and aggloeation on the eectie elastic odulus o nanocoposites. Acknowledgent This eseach was inancially suppoted by the Hebei Natual Science Foundation o China (A040339). Reeences [] H. Hedayati, Aagh B. Sobhani, Inluence o gaded aggloeated CNTs on ibation o CNT-einoced annula sectoial plates esting on Pastenak oundation, Applied Matheatics and Coputation. 8 (0) [] Ingo Alig, Peta Pötschke, Dik ellinge, Tetyana Skipa, Sen Pegel, Gaua R. Kasaliwal, Tobias Villow, stablishent, ophology and popeties o cabon nanotube netwoks in polye elts, Polye. 53 (0) 4-8. [3]. Bian, Q. Wang, Inluence o the paticle size and olue action on ico-daage o the coposites, Achie o Applied Mechanics. 83(3) (03) [4] F. Oteo, X. Matínez, S. Olle, O. Saloón, Study and pediction o the echanical peoance o a nanotube-einoced coposite, Coposite Stuctues. 94(9) (0) [5] Duc Chinh Pha, Stong-contast expansion coelation appoxiations o the eectie elastic oduli o ultiphase coposites, Achie o Applied Mechanics. 8(3) (0) [6]. Bian, H. Zhao, lastic popeties o a single-walled cabon nanotube unde a theal enionent, Coposite Stuctues. (05) [7]. Shady, Y. Gowayed, ect o nanotube geoety on the elastic popeties o nanocoposites, Coposite Science and Technology. 70 (00) [8] G.I. Giannopoulos, S.K. Geogantzinos, N.K. Aniantis, A sei-continuu inite eleent appoach to ealuate the Young s odulus o single-walled cabon nanotube einoced coposites, Coposites Pat B [9] Ian slai Aooz, Andeas chsne, Moones Rahandoust, ects o the cabon nanotube distibution on the acoscopic stiness o coposite ateials, Coputational Mateials Science. 5 (0) 4-9. [0] C.M. Kuo, K. Takahashi, T.W. Chou, ect o ibe wainess on the nonlinea elastic behaio o lexible coposites, J Copos Mate. (988)