Characterizing and Modeling Mechanical Properties of Nanocomposites- Review and Evaluation

Transcription

1 Journal o Minerals & Materials Characterization & Engineering, Vol. 9, No.4, pp.75-9, 00 jce.org Printed in the USA. All rights reserved Characterizing and Modeling Mechanical Properties o Nanocoposites- eview and Evaluation Hurang Hu a *, Landon Onyebueke a, Ayo Abatan b a Departent o Mechanical and Manuacturing Engineering, Tennessee State University, Nashville, TN 709, USA b Departent o Engineering Technology, Miai University, Hailton, OH 450, USA *Corresponding Author: hhu@tnstate.edu ABSTACT This paper presents a critical review o the current work o experient, theory o icro-nanoechanics, and nuerical analysis on characterizing echanical properties o nanocoposites. irst, the classiications o nanoaterials are presented. Then nanoindentation testing and the corresponding inite eleent odeling are discussed, ollowed by analytical odeling stiness o nanocoposites. The analytical odels discussed include Voigt and euss bounds, Hashin and Shtrikan bounds, Halpin Tsai odel, Cox odel, and various Mori and Tanaka odels. These icroechanics odels predict stiness o nanocoposites with both aligned and randoly oriented ibers. The ephasis is on nuerical odeling includes olecular dynaics odeling and inite eleent odeling. Three dierent approaches are discussed in inite eleent odeling, i.e. ultiscale representative volue eleent (VE) odeling, unit cell odeling, and object-oriented odeling. inally, the echanis o nanocoposite echanical property enhanceent and the ways to iprove stiness and racture toughness or nanocoposites are discussed. Key words: Nanocoposites; Mechanical properties; Multiscale odeling; inite eleent analysis (EA); Object-oriented odeling.. INTODUCTION Nanoscience and nanotechnology reer to the understanding and control o atter at the atoic, olecular or acroolecular levels, at the length scale o approxiately to 00 75

2 76 Hurang Hu, Landon Onyebueke, Ayo Abatan Vol.9, No.4 nanoeters, where unique phenoena enable novel applications. Nanotechnologies are the design, characterization, production and application o structures, devices and systes by controlling shape and size at nanoeter scale. According to Braun et al. [], ro 980s, the growth o research papers dealing with the preix called nano is exponential. Aong all the work, characterizing and odeling echanical properties o nanocoposites is one o the ost iportant subjects. Nanocoposites are coposite aterials in which the atrix aterial is reinorced by one or ore separate nanoaterials in order to iprove perorance properties. The ost coon aterials used as atrix in nanocoposites are polyers (e.g. epoxy, nylon, polyepoxide, polyetheriide), ceraics (e.g. aluina, glass, porcelain), and etals (e.g. iron, titaniu, agnesiu). Nanoaterials are generally considered as the aterials that have a characteristic diension (e.g. grain size, diaeter o cylindrical cross-section, layer thickness) saller than 00 n. Nanoaterials can be etallic, polyeric, ceraic, electronic, or coposite. Nanoaterials are classiied into three categories depending on their geoetry, as shown in ig. [,]:. Nanoparticles: When the three diensions o particulates are in the order o nanoeters, they are reerred as equi-axed (isodiensional) nanoparticles or nanogranules or nanocrystals.. Nanotubes: When two diensions are in the nanoeter scale and the third is larger, oring an elongated structure, they are generally reerred as nanotubes or nanoibers/whiskers/nanorods.. Nanolayers: The particulates which are characterized by only one diension in nanoeter scale are nanolayers/nanoclays/nanosheets/nanoplatelets. These particulate is present in the or o sheets o one to a ew nanoeter thick to hundreds to thousands nanoeters long. The nanoaterials can also be distinguished in three types as natural, incidental, and engineered nanoaterials depending on their pathway [4]. Natural nanoaterials, which are ored through natural processes, occur in the environent (e.g. volcanic dust, lunar dust, agneto-tactic bacteria, inerals, etc.). Incidental nanoaterials occur as the result o an ade industrial processes (e.g. coal cobustion, welding ues, etc.). Engineered nanoaterials are produced either by lithographically etching o a large saple to obtained nanoparticles, or by assebling saller subunits through crystal growth or cheical synthesis to grow nanoaterials o the desired size and coniguration. Engineered nanoaterials ost oten have regular shapes, such as tubes, spheres, rings, etc. U.S. Environental Protection Agency divides engineered nanoaterials into our types. They are carbon-based aterials (nanotubes, ullerenes), etalbased aterials (including both etal oxides and quantu dots), dendriers (nanosized

3 Vol.9, No.4 Characterizing and Modeling Mechanical Properties o Nanocoposites 77 polyers built ro branched units o unspeciied cheistry), and coposites (including nanoclays). igure. Various types o nanoscale aterials [4]. Coparing to the conventional icro-coposites, nanocoposites greatly iprove the physical and echanical properties. The nanoscale reinorceents over traditional illers have the ollowing advantages [5]:. Low-percolation threshold (~0. vol.%).. Large nuber density o particles per particle volue ( particles/µ ).. Extensive interacial area per volue o particles (0 0 4 /l). 4. Short distances between particles (0 50n at ~ 8 vol.%). Although any kind o aterial can be produced to appear in a nanoscaled shape and size, carbon nanotubes and nanoplatelets as shown in ig. are the two kinds o nanoparticles that gained the ost attention [6]. igure. Scheatic o (a) nanotube and (b) nanoplatelet [6].

4 78 Hurang Hu, Landon Onyebueke, Ayo Abatan Vol.9, No.4 This paper presents a thorough review o characterizing and odeling echanical properties o nanocoposites. The critical review covers the current work on the experient, theory, and nuerical analysis in this area. Nanoindentation testing and the inite eleent odeling are discussed, ollowed by analytical odeling stiness o nanocoposites. The nuerical odeling includes olecular dynaics odeling and inite eleent odeling. Three dierent approaches are discussed in inite eleent odeling, i.e. ultiscale representative volue eleent (VE) odeling, unit cell odeling, and object-oriented odeling. inally, the echanis o nanocoposites echanical property enhanceent is explored, and the ways to iprove their stiness and racture toughness are discussed.. CHAACTEIZING AND MODELING O NANOCOMPOSITES. Nanoindentation Tests and Coputing Siulations There are dierent ways to experientally characterize nanocoposites. or exaple, tensile and lexural tests (ostly conducted on Instron achines), ipact tests (conducted on pendulu ipact testing achine) [7-], and icro-copression tests [,]. Nanoindentation test is one o the ost eective and widely used ethods to easure the echanical properties o aterials. This technique uses the sae principle as icroindentation, but with uch saller probe and loads, so as to produce indentations ro less than a hundred nanoeters to a ew icroeters in size. During the past dozen years or so, it has been widely used in easuring the echanical properties o various nanocoposites [4-5] and huan enael and dentin [6-8]. Hardness (H) and elastic odulus (E) are calculated ro the load-displaceent curve obtained ro a nanoindentation test. A typical load-displaceent curve is shown in ig.. As the indenter penetrates into the specien, the loading curve clibs up. At soe point, the axiu load P ax is reached, and then ollowed by the unloading. I the aterial is perectly elastic and has no hysteresis, the loading curve and the unloading curve will be identical. h ax gives a easure o the total axiu deoration, while h represents the axiu peranent (plastic) deoration (inal penetration depth).

5 Vol.9, No.4 Characterizing and Modeling Mechanical Properties o Nanocoposites 79 igure. Typical load-displaceent curve o the nanoindentation test. The ost coonly used ethod to obtain the hardness and the elastic odulus o a aterial by nanoindentation is the Oliver-Pharr ethod [5]. According to this ethod, the nanoindentation hardness as a unction o the inal penetration depth o indent can be deterined by: Pax H = (.) A where P ax is the axiu applied load easured at the axiu depth o penetration (h ax ), A is the projected contact area between the indenter and the specien. or a spherical indenter, A = πh (where is the radius o the indenter), whereas or a pyraidal (Berkovich or Vickers) indenter, A can be expressed as a unction o h as A / 4 /8 = 4.504h / Ch Ch Ch L C8h (.) where C to C 8 are constants and can be deterined by standard calibration procedure. The inal penetration depth, h, can be deterined ro the ollowing expression: Pax h = hax ε (.) S where ε is a geoetric constant, ε=0.75 or a pyraidal indenter, and ε=0.7 or a conical indenter. S* is the contact stiness which can be deterined as the slope o the unloading curve at the axiu loading point, i.e. dp S = (.4) dh h= h ax The reduced elastic odulus E r is given by

6 80 Hurang Hu, Landon Onyebueke, Ayo Abatan Vol.9, No.4 S π E r = (.5) β A where β is a constant that depends on the geoetry o the indenter. or both a Berkovich and a Vicker s indenter, β =.04, whereas or both a conical and a spherical indenter, β =. The specien elastic odulus (E s ) can then be calculated as: υ s υi = (.6) Er Es Ei Where E i, s and υ i, s are the elastic odulus and Poisson s ratio, respectively, or the indenter and the specien. or a diaond indenter, E i is 40 GPa and υ i is The contact stiness, S*, can be derived ro the unloading curve which siply obeys the ollowing power law P ) n = B( h h (.7) where B and n are epirical constants that can be deterined by itting the experientally easured pairs o data (P, h) during unloading. Thus the contact stiness can be expressed as dp n S = = Bn( hax h ) (.8) dh h= hax Thereore, the specien s hardness H and elastic odulus equations. E s will be obtained ro this set o Indentation is a highly nonlinear proble. It involves large plastic deoration, aterial nonlinearity, and contact. In order to better understand and characterize the echanical properties and to provide guidelines or proper design o experients, inite eleent ethod is oten used to siulate the nanoindentation tests [4, 5, 8, 8-5]. It is also noted that the priary echanical properties extracted ro a nanoindentation test are the hardness and the elastic odulus. inite eleent siulation could be eployed to get other properties, such as yield stress and hardening [8, 5-58]. ig. 4(a) shows the geoetry o indentation o a cylindrical specien with a conical indenter, and 4(b) shows the Mises stress contour ro the inite eleent analysis [5]. Note that the inite eleent eshes are the two-diensional (axisyetric) eleents. ig. 5 shows a three- diensional nanoindentation inite eleent esh syste [8]. Note that because o syetry, only hal o the specien volue was odeled.

7 Vol.9, No.4 Characterizing and Modeling Mechanical Properties o Nanocoposites 8 igure 4. (a) Geoetry o indentation o a cylindrical specien with a conical indenter. (b) The Mises equivalent stress ield in the specien during indentation at h ax = 600 n. (The stress values ust be ultiplied by 0 7 to respect the scale o the proble) [5].. Analytical Modeling Stiness o Nanocoposites It is well known that coposite aterials have advantages over traditional aterials. Nanocoposites, where nano-sized reinorceents (illers) are dispersed in the base aterial (atrix), oer a novel class o coposites with superior properties and added unctionalities [59-6]. Although the applicability o continuu echanics (including icro echanics) to nanocoposites has been subjected to debate [59,6], any recent works directly applying continuu echanics to nanostructures and nanoaterials have reported eaningul results and elucidated any issues [64-7]. Thus, echanics-based orulas or predicting the echanical properties will be reviewed.

8 8 Hurang Hu, Landon Onyebueke, Ayo Abatan Vol.9, No.4 igure 5. Illustration o a three diensional nanoindentation inite eleent odel [8]. In nanocoposites, there are typically three kinds o illers. They are cylinder-like nanoibers (nanotubes), lake-like (disk-like) platelets (nanolayers, nanoclays), and spheroid-like particulates, reer to igs. and. or the iber-reinorced nanocoposites, there are two cases depending on the orientation o the ibers, i.e. aligned ibers and randoly oriented ibers, see ig. 6 below. The popular icroechanical odels or prediction o odulus o elasticity are suarized and discussed in the ollowing:.. Voigt upper bound and euss lower bound (V- odel) Assued aligned ibers, and ibers and atrix are subjected to the sae unior strain in the iber direction, Voigt [74] got the eective odulus in the iber direction as: E L = φe ( φ) E (.9) euss [75] applied the sae unior stress on the iber and atrix in the transverse direction (noral to the iber direction), and got the eective odulus in the transverse direction as: φ φ = (.0) ET E E where φ is the volue raction o iber in the two-phase coposite syste, and subscripts and respectively reer to the iber and atrix, whereas the subscripts L and T reer to

9 Vol.9, No.4 Characterizing and Modeling Mechanical Properties o Nanocoposites 8 the longitudinal and transverse directions, respectively. Equation (.9) is the parallel coupling orula, and it is also called the rule o ixtures, whereas (.0) is the series coupling orula, and it is also called the inverse rule o ixtures. a. Aligned ibers b. andoly oriented ibers c. Aligned platelets d. particulates igure 6. Scheatics o nanocoposites: (a) with aligned ibers; (b) with randoly oriented ibers; (c) with aligned platelets; and (d) with randoly oriented particulates [6]. Equations (.9) and (.0) can be extended to any two-phase coposites regardless the shape o the iller, and E L and E T represent the upper and lower bounds o the odulus o the coposite, respectively. Note that in these orulas, only three paraeters are involved, i.e. odulus o the iber and the atrix, and the iber volue raction... Hashin and Shtrikan upper and lower bounds (H-S odel) Hashin and Shtrikan [76,77] assued acroscopical isotropy and quasi-hoogeneity o the coposite where the shape o the iller is not a liiting actor, and estiated the upper and lower bounds o the coposite based on variational principles o elasticity. Depending on whether the stiness o the atrix is ore or less than that o the iller, the upper and lower bounds o the

10 84 Hurang Hu, Landon Onyebueke, Ayo Abatan Vol.9, No.4 bulk oduli, as: K K G upper lower upper K upper and K lower, and shear oduli, G upper and G lower, o the coposite are given = K = K = G φ ( φ ) (.) K K K 4G ( φ) φ (.) K K K 4G 6φ ( K G ) ( φ ) (.) G G 5G (K 4G ) 6( φ)( K ) G Glower = G φ (.4) G G 5G (K 4G ) where the subscripts and reer to the iller (iber) and atrix, respectively. The upper and lower bounds o the elastic odulus can then be calculated using the ollowing relation: 9K E = (.5) K / G Siilar to Voigt and euss odels, H-S odel only involves three paraeters... Halpin-Tsai odel (H-T odel) or aligned iber-reinorced coposite aterials, Halpin and Tsai [78-8] developed the equations or prediction o elastic constants based on the work o Herans [8] and Hill [8]. The H-T odel is a sei-epirical odel, and the longitudinal and transverse oduli are given by: ( l / d) φηl EL = E (.6) φη φη = T E T E φηt L (.7) where l and d are the length and diaeter o the iber, and η L and η T take the ollowing expressions: E E η L = (.8) E ( l / d) E E E η T = (.9) E E or aligned nanoplatelets as shown in ig. 6 (c), equations (.6) to (.9) ay still be used by replacing (l/d) with (D/t), where D and t are respectively the diaeter and thickness o the platelet (reer to ig. ).

11 Vol.9, No.4 Characterizing and Modeling Mechanical Properties o Nanocoposites 85 H-T odel takes the consideration o the iber geoetry, and has ive independent paraeters...4 Hui-Shia odel (H-S odel) Mori and Tanaka [84] developed analytical expressions or elastic constants based on the equivalent inclusion odel o Eshelby [85]. Taya and Mura [86] and Taya and Chou [87] used Mori-Tanaka approach to predict the longitudinal odulus o iber-reinorced coposites, Weng [88] and Tandon and Weng [89] urther developed equations or the coplete set o elastic constants o coposite aterials with aligned spheroidal isotropic inclusions. Based upon the results o Tandon and Weng [89], Hui and Shia [90] and Shia et al. [9] derived sipliied orulas or predicting the overall oduli o coposites with aligned reinorceents with ephases on iber-like and lake-like reinorceents, and ound that their theoretical predictions agree well with experiental results. The H-S odel presents the Young s odulus as ollows: where φ E L = E (.0) ξ φ E T = E ( 4 (.) ξ ξ Λ E ξ = φ E E ( g) α g / ( φ) α ( α 0.5) g α Λ = ( φ) α (.) (.) α [ α α cosh α] α / ( α ) g = (.4) α [ α α cos α] α / ( α ) and α is the aspect ratio o the iller, deined as the ratio o the iller s longitudinal (with Young s odulus E L ) length to its transverse (with Young s odulus E T ) length. or exaple, reer to ig., α = l / d or nanotube, α = t / D or nanoplatelet, and E L will be along axis, and E T will be along axis (or )...5 Wang-Pyrz odel (W-P odel) or a coposite aterial coposed o an isotropic atrix and randoly oriented transversely isotropic spheroids, Qiu and Weng [9] and Chen et al. [9] gave the orulas or the overall bulk and shear oduli using the Mori-Tanaka ethod. These orulas are expressed in ters o

12 86 Hurang Hu, Landon Onyebueke, Ayo Abatan Vol.9, No.4 the Eshelby tensor [85], thus are not inal. Wang and Pyrz [94] urther gave the closed and concise orulas or the overall bulk odulus and shear odulus as ollows: φϕ K = K K (.5) φ( α) φψ μ = μ μ (.6) φ( β ) The expressions or ϕ, ψ, α and β are given in the Appendix. Note that W-P odel is based on the Mori-Tanaka approach, and deals with the coposite aterials reinorced with randoly oriented and transversely isotropic spheroids. By varying the aspect ratio, the oblate spheroids can be approxiate to platelets, and the prolate spheroids can be approxiate to ibers...6 Cox odel (Shear lag odel) Shear lag odel was the irst icro-echanics odel or iber-reinorced coposites. Cox [95] analyzed a single iber o length l and radius r, which is encased in a concentric cylindrical shell o atrix having radius. He derived the longitudinal odulus as E L = η LφE ( φ) E (.7) where η L is a length-dependent eiciency actor, tanh( βl / ) η L = (.8) βl / with 4μ β = (.9) r E ln( K / φ) K is a constant that depends on the iber packing arrangeents. or soe typical iber packing arrangeents, the values o K are given in Table [96]. Table. Values or IBE PACKING K in Eq. (.9) K Cox π / =.68 Coposite cylinders.000 Hexagonal π / =0.907 Square π / 4 =0.785 It is well known that the orientation o the dispersed phase has a draatic eect on the coposite odulus. It is apparent ro their geoetry that lake-like platelets can provide equal

13 Vol.9, No.4 Characterizing and Modeling Mechanical Properties o Nanocoposites 87 reinorceent in two directions, i appropriately oriented, while ibers provide priary reinorceent in one direction. I the longitudinal odulus E L and the transverse odulus E T are known, then the eective odulus o the coposite with randoly oriented ibers and platelets in all three orthogonal directions are given by [97]: iber E = 0.84E 0. 86E (.0) D L platelet D 0.49EL T T E = 0. 5E (.). Molecular Dynaics Siulation In odeling echanical properties o nanocoposites, there are two ain approaches: one is olecular dynaics siulation using direct ethods, and the other is inite eleent siulation using continuu ethods. Molecular dynaics siulation is a technique that allows one to deterining the physical and echanical properties o aterials in nanoscale through solving Newton s equations o otion with the atos interacting through assued interatoistic potentials [98, 99]. It generates inoration such as atoic positions, velocities and orces ro which soe acroscopic properties can be derived by eans o statistical echanics. Molecular dynaics siulation usually consists o three constituents: () a set o initial conditions (e.g., initial positions and velocities o all particles in the syste); () the interaction potentials to represent the orces aong all the particles; () the evolution o the syste in tie by nuerically solving a set o classical Newtonian equations o otion or all particles in the syste [00]. In 997, Cornwell et al. used olecular dynaics to predict the elastic properties o single-walled carbon nanotubes [0]. In recent years, olecular dynaics siulation has been extensively used in predicting echanical properties o carbon nanotubes and nanotubes reinorced coposites [0-09], graphite/epoxy nanocoposites [0-], and other nanocoposites [-9]. Molecular dynaics siulation involves the proper selection o interaction potentials, nuerical integration, periodic boundary conditions, and the controls o pressure and teperature to iic physically eaningul therodynaic ensebles. The interaction potentials together with their paraeters or a orce ield which describes in detail how the particles in a syste interact with each other. Such a orce ield ay be obtained by quantu ethod, epirical ethod or quantu-epirical ethod. The criteria or selecting a orce ield include the accuracy, transerability and coputational speed. The total potential energy U ay consist o a nuber o bonded and non-bonded interaction ters: U U U U U U (.) bond angle torsion inversion non = bonded The irst our ters represent bonded interactions, i.e., bond-stretching between two bonded atos, angle-bending by three neighboring atos, angle variation between two planes ored by our neighboring atos, and angle variation o two planes ored by our atos where one ato

14 88 Hurang Hu, Landon Onyebueke, Ayo Abatan Vol.9, No.4 is bonded to other three, as shown in ig. 7 [0]. The last ter represents non-bonded interactions between two atos. It usually includes van der Waals and electrostatic interactions. igure 7. Bond structures and corresponding energy ters o a graphene cell [0]. Molecular dynaics siulations can be perored in dierent ensebles, such as grand canonical ( μ VT), icrocanonical (NVE), canonical (NVT) and isotheral isobaric (NPT). The constant teperature and pressure can be controlled by adding an appropriate therostat (e.g., Berendsen, Nose, Nose Hoover and Nose Poincare) and barostat (e.g., Andersen, Hoover and Berendsen), respectively. The sotware packages available or olecular dynaics siulations include DL-POLY developed by Daresbury Laboratory [, ], LAMMPS developed by Sandia National Laboratories [], and TINKE developed by University o Washington [4]. To deonstrate how to use olecular dynaics siulation to evaluate the echanical properties o nanocoposites, the work by Adnan et al. [5] using olecular dynaics siulation to investigate the eect o iller size on elastic properties o polyer nanocoposites will be presented below. Adnan et al. constructed the nanocoposite by reinorcing aorphous polyethylene (PE) atrix with nano sized buckinister ullerene bucky-ball. Three types o bucky-balls, C 60, C80, and C0 (subscripts denote nuber o carbon atos) with three dierent diaeters (0.7,. and.7 n, respectively) were utilized to incorporate size eect in the nanocoposites. The PE atrix was represented by united ato (UA)-CH - units. All buckyballs were inused in atrix by approxiately 4.5 vol%. Once the olecular structures were developed, the corresponding olecular echanics orce ields were deined. The PE chains were described by appropriate bond stretching, angle bending and dihedral potentials between - CH - units. The non-bonded van der Waals interactions within or between PE chains were odeled using lennard-jones (LJ) potential [6, 7]. The unctional or and paraeters o the orce ield are shown in Table.

15 Vol.9, No.4 Characterizing and Modeling Mechanical Properties o Nanocoposites 89 Table. unctional or and paraeters or the orce ield [5] Interactio n Potential unctional or Paraeters Bond Haroni k r = 700 kcal/ol U ( r) = k r ( r r0 ) c o r 0 =.5A Angle Haroni U ( ) = k [cos( ) cos( 0 )] c cosine k =.5 kcal/ol 0 Dihedral Cosine 0 = U ( φ) = kφ [ cos( φ)] k φ =.00 kcal/ol, Nonbonded Lennard- Jones σ 4ε = ( ) U ( r) r 0 ( ) r σ 6 r < r r r cut cut = PE PE: ε = 0.66kcal/ol o σ = 4.8 A PE Bucky: ε = kcal/ol o σ =.85 A r cut o = 0.7 A igure 8. Cells o dierent neat and nanocoposites odel used or siulation [5].

16 90 Hurang Hu, Landon Onyebueke, Ayo Abatan Vol.9, No.4 ig. 8 shows the cells o dierent neat and nanocoposites odel used or siulation. Periodic boundary conditions were eployed to replicate the unit cells in three diensions. Sotware package DL_PLOY (version.4) was used in the siulation. All the calculations were carried out at a teperature o 00 0 K with 0.5 s tie steps. Two ajor steps o siulation or both neat polyer and nanocoposites were perored. In the irst step, the equilibriu state o the olecular odel was obtained, and then the odel was subjected to dierent strain ields and reequilibrated. Adnan et al. applied a unior strain ield (0.5%) to the periodic cells o both neat polyer and nanocoposites. or the cases o hydrostatic tension and hydrostatic copression, they evaluated the bulk odulus K, and their results were shown in Table. Table. Evaluation o bulk odulus K or various nanocoposites [5] Syste Type Hydrostatic Copression Hydrostatic Tension K(GPa) % Gain/loss K(GPa) % Gain/loss C 60 - PE C 80 - PE It is evident ro Table that elastic properties o nanocoposites are iproved appreciably with the inusion o bucky-balls in PE atrix, and they are also signiicantly aected by the size o reinorcing bucky-balls..4 inite Eleent Modeling As a very general and powerul nuerical analysis tool, inite eleent ethod was used to predict echanical properties o coposite aterials started in early 970s [8-9]. Since then, various inite eleent odels have been developed to characterize all kinds o coposite aterials [e.g. 0-6]. In 99, Suio Iijia, a Japanese scientist, discovered carbon nanotubes (CNTs) which possess exceptionally high stiness and strength, as well as superior electrical and theral properties [7-9]. Soon ater that CNTs were used as reinorceent in developing nanocoposite aterials. In the past decade or so, there have been explosively experiental work [e.g. 7, 8, 40-55] and analytical work [e.g ], as well as inite eleent odeling work [e.g ] on developing, analyzing and characterizing CNT reinorced nanocoposites and other nanocoposites. In the ollowing, three inite eleent odeling approaches will be discussed. They are ultiscale representative volue eleent (VE) odeling, unit cell odeling, and object-oriented odeling..4. Multiscale VE odeling Liu and Chen [80] extended the VE concept used by Hyer [99] and Neat-Nasser and Hori [00] or conventional iber-reinorced coposites at the icroscale to nanoscale, and evaluated the eective echanical properties o CNT-based coposites by using a three-diensional

17 Vol.9, No.4 Characterizing and Modeling Mechanical Properties o Nanocoposites 9 nanoscale VE based on elasticity theory and solved by the inite eleent ethod. An VE is coposed o a single (or ultiple) nanoiller(s) with surrounding atrix aterial, plus proper boundary conditions to account or the eects o the surrounding aterials. It is used as a building block to asseble the coposite. Zhang et al. [0] linked continuu analysis with atoistic siulation by incorporating interatoic potential and atoic structures o CNTs directly into the constitutive law. Shi et al. [85] presented a hybrid atoistic/continuu echanics ethod to study the deoration and racture behavior o CNTs ebedded in coposites. The ethod is based on a representative unit cell divided into three distinct regions analyzed using an atoistic potential, a continuu ethod based on the Cachy Born rule and a icroechanics ethod, respectively. Li and Chou [80] proposed a ulti-scale odeling approach to study the copressive behavior o CNT/polyer coposites. They odeled the nanotube at the atoistic scale and analyzed the atrix deoration using the continuu inite eleent ethod. The van der Waals interactions between carbon atos and the inite eleent nodes o the atrix were siulated using truss rods. The ultiscale VE integrates nanoechanics and continuu echanics, thus bridging the length scales ro the nano- through the esoscale. The procedure o ultiscale VE odeling is exhibited by the work o Tserpes et al. [7] in the ollowing. Tserpes et al. proposed a ultiscale VE to investigate the tensile behavior o CNT/polyer coposites. The VE is a rectangular solid whose entire volue is taken up by the atrix, and the nanotube is odeled as a three-diensional (D) elastic bea. The D solid eleents and bea eleents are used to odel the atrix and nanotube, respectively. The VE is synthesized in two steps. irst, the behavior o the isolated nanotube is siulated using the progressive racture odel [0]. The concept o the odel is based on the assuption that carbon nanotubes, when loaded, behave like space-rae structures. The bonds between carbon atos are considered as load-carrying ebers while carbon atos as joints o the ebers. The non-linear behavior o the C-C bonds is odeled by the odiied Morse interatoic potential [0], and the nanotube structure is odeled by inite eleent ethod. Second, the nanotube is inserted into the atrix to or the VE. The atrix is odeled by solid eleents, and the nanotube is represented by D elastic bea eleents created by binding the nodes o the atrix. The synthesis o the VE is shown in ig Unit cell odeling The conventional unit cell concept is the sae as the VE [, 04]. Here we deine a unit cell as a special VE that it has a relatively big size (usually in icroeters) and contains a signiicant nuber o illers (usually in tens to hundreds or ore). Such deined unit cell is still the building block o the coposite, but as it gets ore coplicated, analytical odels are diicult to establish or too coplicated to solve, and nuerical odeling and siulation becoe a necessity.

18 9 Hurang Hu, Landon Onyebueke, Ayo Abatan Vol.9, No.4 igure 9. Synthesis o the VE [7]. The ost coon ethod used to characterize the echanical properties o nanocoposites with unit cell is the inite eleent ethod. Hbaieb et al. [77] exained the Young s odulus o nanoclay/polyer nanocoposites with both D and D unit cells using the inite eleent ethod. our unit cells were created. They are, respectively, D and D aligned and randoly oriented nanoclay particles odels, as shown in ig. 0. Two kinds o boundary conditions are considered. They are periodic boundary conditions and syetrical boundary conditions. or the D odels (both aligned and rando cases) the periodic boundary conditions are: u(e)=u(le) δ v(e)=v(le) u(te)=u(be) v(te)=v(be) δ where E, LE, TE, BE and δ and δ are the right, let, top, botto edges and the axial and transverse displaceents, respectively. The syetrical boundary conditions or the D odels are: u(le)=0 v(be)=0 u(e)=δ

19 Vol.9, No.4 Characterizing and Modeling Mechanical Properties o Nanocoposites 9 where δ is the given noral displaceent in the x direction. In addition, all edges are ree o shear traction and the top edge is ree o noral traction as well. or the D odels (both aligned and rando cases) only syetrical boundary conditions are applied, and they are given as: u(l)=0 v(b)=0 w(bk)=0 u()=δ where L, B, BK and stand or let ace, botto ace, back ace and right ace. All other aces are ree o any displaceent or traction constraints. The nuerical results indicated that D odels do not predict the elastic odulus o clay/polyer nanocoposites accurately. The Mori- Tanaka odel [89] gives reasonably accurate predictions o the stiness o the nanocoposites whose volue raction is less than 5% or aligned particles but underestiates the stiness at higher volue ractions. or randoly oriented particles the W-P odel [94] overestiates the stiness o the nanocoposites. igure 0. Mesh details o the odel or (a) D aligned particle distribution, (b) D randoly oriented-particle distribution, (c) D aligned particle distribution, and (d) D randoly oriented-particle distribution. Particle volue raction is 5%, the particle aspect ratio is 50, Ep/E=00, ν =0.5, ν p =0.. Subscripts p and represent particle and atrix, respectively [77]. ecently, Lee et al. [70] used a D unit cell odel to analyze the deoration behavior o randoly distributed Al 8 B 4 O whisker-reinorced AS 5 agnesiu alloy atrix coposite. The Al 8 B 4 O whiskers are 0 0μ long and 0.5.0μ in diaeter. The diensions o

20 94 Hurang Hu, Landon Onyebueke, Ayo Abatan Vol.9, No.4 the unit cell are 0 0 0μ which contains (ully or partially) 60 whiskers. The volue raction o the whiskers is 5%. ig. shows a typical unit cell (with the eshes o the whiskers) and an optical icrograph o the coposite. or the Young s odulus and overall elastic-plastic response o the coposite, the inite eleent odeling results are in excellent agreeent with the experiental results. igure. (a) D rando whisker-reinorced coposite odel, and (b) an optical icrograph o squeeze-iniltrated Al 8 B 4 O /Mg rando whisker coposite [70]..4. Object-oriented odeling In both ultiscale VE odeling and unit cell odeling, two basic assuptions are ade. irst, nanoillers can be idealized to siple geoetries such as spheres, ellipsoids, cylinders, or cubes. And second, nanocoposites can be reproduced by assebling a large nuber o such VEs (or unit cells). This can be a serious liitation when dealing with coplex and highly heterogeneous nanocoposites. or exaple, or highly variable and irregular angular structure o illers, using approxiation o siple geoetrical particles could not capture the coplex orphology, size, and spatial distribution o the reinorceent. Thereore, the object-oriented odeling which is able to capture the actual icrostructure orphology o the nanocoposites becoes necessary in order to accurately predict the overall properties. The object-oriented odeling is a relatively new approach. It incorporates the icrostructure iages such as scanning electron icroscopy (SEM) icrographs into inite eleent grids. Thus the esh reproduces exactly the original icrostructure, naely the inclusions size, orphology, spatial distribution, and the respective volue raction o the dierent constituents. A objectoriented inite eleent code, OO [05, 06], developed by National Institute o Standards and Technology (NIST), has been extensively used in analyzing racture echaniss and aterial properties o heterogeneous aterials [07-6] and echanical properties o nanocoposites

21 Vol.9, No.4 Characterizing and Modeling Mechanical Properties o Nanocoposites 95 [8, 78, 79, 7]. In the ollowing, a D object-oriented inite eleent odeling will be discussed, ollowed by a D odeling. igure. Typical exaple o creating OO odel o PP/organoclay nanocoposites (5 wt% in clay content): (a) original SEM iage, (b) captured SEM iage portion, (c) iage segentation using pixel selection, and (d) inite eleent esh (highlighted regions contain organoclay particles and the rest are PP atrices) [8]. Dong et al. [8] studied the echanical properties o polypropylene (PP)/organoclay nanocoposites with dierent clay contents ranging ro to 0 wt%. Their work started with the specien abrication through experiental characterization to theoretical predictions and nuerical odeling using OO. SEM icrographs ro longitudinal loading direction o the specien were captured and apped onto the inite eleent odel, as shown in ig.. The actual nano/icrostructures (their size, shape, and distribution etc.) o the PP and the organoclay were used in the coputational odel, and each phase was attributed the corresponding aterial properties. The OO odeling results or the tensile odulus show a good agreeent with the experiental data and theoretical predictions. Chawala et al. [78] used D object-oriented inite eleent odeling to evaluate the echanical behavior o SiC particle-reinorced Al coposites. or a volue o μ cell, there are about 00 SiC particles which produce 0% volue raction. They copared the

22 96 Hurang Hu, Landon Onyebueke, Ayo Abatan Vol.9, No.4 results o the Young s odulus and the stress-strain relations ro the object-oriented (icrostructure-based) odel with the results o the experient and the nuerical results ro sipliied odels (which include rectangular pris, ultiparticle-ellipsoids, and ultiparticlespheres, etc.). Soe o the results were shown in ig.. Their results indicate that D icrostructure-based odel can accurately predict the properties o particle-reinorced coposites, while the siple analytical odels can not as they do not account or the icrostructural actors that inluence the echanical behavior o the aterial. igure. Coparison between D inite eleent odels incorporating actual icrostructure and approxiation to spherical particles: (a) EM odels, (b) von Mises stress distribution in particles, and (c) plastic strain in atrix [78].. MECHANICAL POPETY ENHANCEMENT illers added to atrix can change the echanical properties o the atrix aterial. Coparing to traditional coposite aterials, nanocoposites have the ollowing characterizations:. Nanoparticles can substantially iprove the echanical properties o the host atrix aterials [40,4,8-0]. Even at very low iller volue content such as -5%, a considerable iproveent o the echanical properties can be achieved [4, -].. It is observed that or soe nanocoposites, with the sae iller volue raction, the stiness and strength increases as the particle size decreases [5,8, 4-7].

23 Vol.9, No.4 Characterizing and Modeling Mechanical Properties o Nanocoposites 97. In general, the stiness o nanocoposites tends to increase as the iller volue raction increases. This unction ay be nonlinear. There ay exist a critical volue raction beyond which the stiness starts decrease [8]. or conventional coposite aterials, icroechanics theories consider that the overall echanical properties o coposites are unctions o constituent properties, constituent volue raction, inclusion shapes and orientations, and state o dispersion. It does not consider the interactions between iller and atrix at their interace. or nanocoposites, the echanical property enhanceent not only depends on the above actors, but also depends on the interaction between the iller and the atrix.. Mechaniss o stiness and strength enhanceent It is widely accepted that there is an interphase exist between the nanoillers and the atrix aterial in nanocoposites. This interphase is a transition region, which extends nanoeters to icroeters over which the echanical and physical properties change ro the properties o iller to the properties o the atrix. Aong any researchers who studied the nanocoposites interphase behavior, Boutaleb et al. [56] investigated the inluence o interphase on the overall behavior o silica spherical nanoparticle/polyer coposites by eans o analytical and inite eleent ethods. ig. 4 shows a scheatic o a coposite aterial containing randoly located spherical nanoparticles (let) and a spherical nanoparticle coated with a graded interphase (right). The interphase is represented as a third phase around the nanoparticles. A odel o axisyetric VE with periodical boundary conditions was exained. The analysis results show that the interphase is a doinant paraeter controlling the overall nanocoposite behavior. To estiate the elastic odulus o the interphase in polyer nanocoposites, Saber-Saandari and Khatibi [9] developed a D unit cell odel to represent the three constituent phases including particle, interphase and atrix. The elastic odulus o the interphase at any point, r, is described by a power law as: ri r E ( r) E r / r ( E E r / r ) = (.) i i i ri r where E and E are atrix and nanoparticle elastic oduli, respectively, r and r i are the iller and interphase radii, and n is the intragallery enhanceent actor which depends on the cheistry and surace treatent o the particles considered. n /

24 98 Hurang Hu, Landon Onyebueke, Ayo Abatan Vol.9, No.4 igure 4. Scheatic o a coposite aterial containing randoly located spherical nanoparticles (let) and a spherical nanoparticle coated with an interphase (right) [56]. How exactly the interphase aects the nanocoposites properties is still a research topic. Soe intend to think the interphase reined the grain size o atrix leads to saller critical law size and higher strength. Soe researchers believe that nanoparticles yield dislocations around the, and these dislocations release residual stresses in the atrix. Thus the deect size along the grain boundaries is reduced. There are also soe researchers who think nanoillers ipart additional strength o their own to the atrix through the interphase. Nevertheless to say, the strengthening echanis o nanocoposites is not ully understood. Several echanical properties o nanocoposites are also iproved or the sae reason, such as hardness, wear resistance, and theral shock resistance. The interaction between nanoillers and atrix is the key to the nanocoposites properties enhanceent. There are any actors aecting that interaction, such as the iller volue (weight) raction, degree o dispersion, the iller geoetry and orientation, etc. We assue the sae volue raction and identical degree o dispersion, only the iller geoetry (aspect ratio) and orientation will be considered. We deine a reactive surace area per unit volue o iller,γ, as A γ = (.) V where A and V are surace area and volue o the iller, respectively. Table 4 shows the ajor axis and the γ value or soe typical geoetry o the nanoillers. Consider three ost coon geoetries, i.e., sphere (nanoparticles), disk (nanoplatelets, nanolayers), and cylinder (nanotubes, nanoibers). or the cuboid, i a=b=c, it becoes a cube, close to sphere; i a=b>>c, it becoes a platelet; i a>>b c, it becoes a rod, close to cylinder. Assue that the diaeter o the sphere, the diaeter o the cylinder cross-section, and the

25 Vol.9, No.4 Characterizing and Modeling Mechanical Properties o Nanocoposites 99 thickness o the disk are the sae. According to the values in Table 4, the reinorceent eiciency o the three geoetries in the ajor axis direction, ro good to poor, is spherecylinder-disk. But nanoplatelets are thought to possess better reinorceent eects than those o spherical and iber-like particles [0]. Table 4. γ value and the ajor axis or typical iller geoetries Nae Shape γ Paraeters 4 Cylinder ( ) as t << a t a t t-diaeter o cross section area, a- Disk-like length platelet ( ) as t << a t a t ectangula t-thickness, a-radius r platelet ( ) as t a b t t << a, b t-thickness, a and b- length and width 6 Sphere t t-diaeter Cuboid ( ) a b c a, b and c-lengths 4 o the three sides Cone ( ) h h t t-base diaeter h-height As the iller orientation is very iportant in reinorceent, equation (.) has to be odiied to account or the eect o orientation o the iller suraces. Now we deine an eective surace area per unit volue o iller,γ, as A γ = (.) V where A is the eective iller surace area, and it represents the portion o suraces which is noral to the direction o ajor axis (see Table 4). The value o γ or sphere, disk, and cylinder in the ajor axis is /t, /t, and 4/t, respectively. Thereore, in the ajor direction shown, the order o reinorceent eiciency, ro good to poor, is cylinder-disk-sphere. I the nanoillers are randoly oriented, the reinorceent eiciency o nanospheres is probably better than that o nanolayers, and the reinorceent eiciency o nanolayers is probably better

26 00 Hurang Hu, Landon Onyebueke, Ayo Abatan Vol.9, No.4 than that o nanocylinders. This is because sphere is isotropic, and disk is transversely isotropic, and cylinder is anisotropic. or all the geoetries o the iller, as the characteristic diension (the sallest diension) decreases, the value o γ will increase. That is, the saller the iller, the better enhanceent it will provide. This is siilar to the Hall-Petch eect on the strength o etals. Hall-Petch relates the yield stress o a etal to its average grain diaeter d as / σ = σ kd (.4) y 0 where σ and k are the constants related to the aterial o interest. The yield stress increases as 0 the grain size decreases. It is also interesting to note that just as Hall-Petch equation does not apply to extreely ine grain sizes, ine size iller enhanceent on nanocoposites ay also have a liit. Schiotz and Jacobsen [] investigated nanocrystalline copper, and pointed out that there ay be a axiu in the strengthening that can be obtained by decreasing the grain size, so that below a certain critical grain size the strength begins to decrease again as the grain size decreases.. racture Toughness Nanocoposites can not only iprove stiness and strength, but also racture toughness [- 4]. In general, the racture toughness o nanocoposites increases as the volue raction increases, and increases as the nanoiller size decreases. or silica/epoxy nanocoposites, agosta et al. [5] ound the racture toughness iproved as the volue raction o 5-n silica particles increases. Siilar results were obtained by Zhang et al. [4] with 5-n silica particles, and by Chen et al. [] with -n silica particles. Through experients and an analytical odel, Adachi et al. [] studied the ode I racture toughness o silica/epoxy nanocoposites, and ound that the toughness increased drastically as the silica volue raction increased and the particle diaeters decreased. In nanocoposites with a low volue raction o particles, the volue raction aected the racture toughness ore; and with high volue ractions, the particle size aected the racture toughness ore. Just as or stiness and strength, the toughening echanis o nanocoposites is also ainly ro the interaction between the illers and the atrix. Awaji et al. [4] observed silicon carbide/aluina nanocoposites by transission electron icroscopy (TEM), and ound that silicon carbide nanoparticles were dispersed both inside the aluina grains and on the grain boundaries. The racture toughness is iproved by the change o racture ode ro intergranular racture o onolithic aluina to transgranular racture o nanocoposites. ig. 5 shows a scheatic illustration o the toughening echanis [44]. Nanoparticles are dispersed within the atrix grains. Then sub-grain boundaries or dislocation networks are generated around the nanoparticles (ig. 5A). When the tip o a propagating large crack reaches this area, these dislocations in the atrix will operate as nano-crack nuclei in the vicinity o the

27 Vol.9, No.4 Characterizing and Modeling Mechanical Properties o Nanocoposites 0 propagating crack tip (ig. 5B). The highly stressed rontal process zone (PZ) ahead o the crack tip is then released by nano-crack nucleation, and the nano-cracks expand the PZ size, enhancing the racture toughness o the aterials [6]. igure 5. Scheatic description o the toughening echanis in nanocoposites. (A) Intra-type nano-structure, (B) PZ creation [44]. 4. CONCLUDING EMAKS Characterizing and odeling echanical properties o nanocoposites is reviewed and evaluated. Nanocoposites are ade by dispersing nanoillers (e.g., silicate and ceraic nanoparticles, CNTs, etc.) into atrix (e.g., soe polyers, ceraics, etals, etc.). Coparing with conventional coposite aterials, nanocoposites have nuerous advantages such as high echanical and physical properties, and high reinorceent eiciency. The high enhanceent o echanical properties o nanocoposites is ainly attributed to the interaction between the nanoillers and the atrix aterial through the interphase which is a transition region ro the nanoillers to the atrix, and the high value o the reactive surace area per unit volue o nanoillers. Coprehensive understand o the echaniss o echanical property enhanceent is crucial in order to achieve the longstanding goal o predicting nanoparticles nanocoposites property relationships in aterial design and optiization. Experiental characterizing and nanoechanics-based coputer odeling and siulation o echanical properties o nanocoposites are the two wings in understanding the echaniss. Many traditional siulation techniques have been eployed, and soe novel siulation techniques have been

28 0 Hurang Hu, Landon Onyebueke, Ayo Abatan Vol.9, No.4 developed to study nanocoposites. These techniques represent approaches at various tie and length scales ro olecular scale to icroscale, and then to acroscale, and have shown success to various degrees in addressing any aspects o nanocoposites. The siulation techniques developed thus ar have dierent strengths and weaknesses, depending on the need o research. Despite substantial progress ade in the past decade, there are a nuber o challenges in coputer odeling and siulation. New concepts, theories and coputational tools should be developed. In general, there are two ronts that should be pointed out. irst, there is a need to develop new and iproved siulation techniques at individual tie and length scales. Secondly, it is iportant to integrate the developed ethods at wider range o tie and length scales, spanning ro quantu doain to olecular doain, to esoscopic doain, and inally to acroscopic doain, to or a useul tool or exploring the structural and echanical properties, as well as optiizing design o nanocoposites [00]. Speciic challenges and the solution strategies are discussed in the ollowing:. In either developing new or characterizing the current exist nanocoposites, a coprehensive approach should be adopted that integrates the experiental techniques with nanoechanics-based analytical explorations and coputer odeling and siulation.. New coputational tools are specially needed in the area o ultiscale VE odeling. The ultiscale VE odeling is in nature a local-global approach. In order to catch the local nano/icro characteristics, quantu echanics or olecular dynaics needs to be explored. But the prediction o global acro-echanical properties requires the continuu echanics-based inite eleent ethod. How to transit ro local to global becoes a research issue. Ogata et al. [98] proposed a way o cobing quantu echanics, olecular dynaics, and inite eleents. In regions where the atos obey the laws o continuu echanics, the inite eleent ethod is used. However, in critical areas such as the extreity o a racture, olecular dynaics and even quantu echanics are required to obtain a ore detailed study o the racture process. The transition ro the global to local levels involves a change o scale. Xiao and Belytschko [45] proposed a way o iproving the nuerical copatibility between regions odeled by olecular dynaics and those odeled using the inite eleent ethod. The suggested ethod is introducing a broad transition region by superposing the inite eleent esh o the continuu region on the atoistic structure o the olecular dynaics region. Clearly, there is still a lot o work needs to be done in connecting the local paraeters to the global paraeters.. In object-oriented inite eleent odeling, D odeling has been extensively used in nanocoposites [e.g. 8, 79, 7], and there are also soe works on D odeling [e.g. 78]. There are still issues to be resolved in D odeling, especially advanced objectoriented D inite eleent codes.