Micromechanical models for stiffness prediction of alumina trihydrate (ATH)

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1 Microechanical odels or stiness prediction o aluina trihydrate (ATH) reinorced poly (ethyl ethacrylate) (PMMA): eect o iller volue raction and teperature. O. A. Stapountzi, M. N. Charalabides, J. G. Willias Mechanical ngineering Departent, Iperial College London SW7 AZ, UK The elastic odulus o aluina trihydrate (ATH) reinorced poly (ethyl ethacrylate) was easured at dierent iller volue ractions and teperatures. The experiental values showed a % increase as the iller contents increased ro % vol. to 49% vol., and a % decrease with rising teperature ro to 9 C. The indings were conired by analytical odelling predictions ro the literature. Four analytical odels were in very good agreeent with the experiental data, including the Halpin-Tsai, the Generalised Sel Consistent, the Lielens and Nielsen odels. Aongst the the best odel or the volue ractions used here was that o Lielens. Keywords: A. Polyer atrix coposites, A. Particle reinorced coposites, B. Mechanical properties, C. Modelling.. Introduction Coposites ade ro aluina trihydrate (ATH) illed poly (ethyl ethacrylate) (PMMA) are being used as engineering aterials in structural applications because o their enhanced properties and low production costs []. They are usually ade with high volue ractions o iller to give a large increase in the elastic odulus which is iportant in several applications. It is useul to know i these increases can be predicted ro the large nuber o theories available [-4]. Soe o the theories are liited to low volue raction coposites. There is soe interest in predicting the eects o teperature on odulus or design purposes. However there is also an incentive to easure the properties over a range o

2 teperatures because the atrix values change and can be easured separately. The properties o the iller do not change. Thus when aterials with varying volue ractions o iller are tested over a range o teperatures, they provide data over a very wide range o atrix property values which can be copared to predictions ro the various theories.. Analytical odels The theories dealing with the prediction o the elastic perorance o two-phase coposites can be classiied into two ain groups: constitutive relationships and seiepirical equations. They all consist o analytical expressions that relate the eective oduli o coposite aterials to the volue ractions and the elastic oduli o their constituents, i.e., the atrix and particle inclusions. The atrix and inclusions are assued to be linearly elastic and hoogeneous, the atrix is isotropic, the particles isotropic or transversely isotropic and identical in shape and size. The subscripts, and, are used to denote the atrix and iller particles (inclusions). For each odel discussed below, only the relevant equations will be outlined with no details on their derivation or brevity purposes. For ore details on the speciic odels, the reader is directed to the original publications or [4].. Constitutive relationships The basis o alost all existing constitutive odels or elastic odulus prediction o particle illed coposites lies in Hill s theory [] or average stress and strain in a representative volue V o the two-phase coposite aterial. In the representative volue concept, V represents the volue o the inclusions and V the volue o the atrix, such that V = V + V. The constituent volue ractions and can be expressed by qn. : = V V, = V V and + = ()

3 The undaental expression or the eective elastic stiness o coposite aterials is []: ( ) C = C + C - C A () where C and C are the elastic stiness tensors or the atrix and inclusions respectively and A is the strain-concentration tensor which relates the average strain in the inclusions, ε, with the average strain in the representative volue, ε, through: _ e _ = A e () The theoretical prediction o the elastic coponents is dependent upon the deterination o the strain concentration tensor o the inclusions, A. The irst attept to characterise the coplicated strain ields occurring in and around an inclusion contained in an ininite atrix was provided by shelby [6]. shelby considered the case o a single inclusion ebedded in an ininite ediu which is subjected to conditions o unior deoration at large distances ro the inclusion. Hill earlier showed that the average strain within the representative volue is identical to the unior strain applied on the surace o the volue, ε, i.e., ε _ = o ε (average strain theore). shelby used this theore to show that the strain o concentration actor, _ A = ε εo = A S, o an ellipsoidal inclusion surrounded by an ininite atrix subjected to unior deoration, i.e., u()= S ε x, at long distances ro the o inclusion, is expressed by: - ( ) ù - A é S = êi + SC C - C ë ú (4) û where S stands or the shelby s tensor. xpressions or the coponents o the shelby s tensor, S, in ters o the atrix Poisson s ratio, n, and the aspect ratio, a, o spheroidal inclusions shown in Fig, can be ound in [7].

4 qn. 4 can be cobined with qn., to predict the coponents o the elastic stiness tensor or coposites illed with spherical or spheroidal inclusions at dilute concentrations. The Mori-Tanaka (M-T) theory [9] is based on the cobination o the average unior strain inside an ellipsoidal inclusion due to shelby, and the theory o the average stress in the atrix [9]. The M-T strain concentration tensor is obtained by: é( ) - A - + ù M-T = AS ê I A S ë ú () û (a) (b) (c) d / d / d / l / l w / w d / t / d / t / d t Fig. (a) Spherical inclusions (isotropic) a =, (b) ibre-like (prolate spheroid) inclusions, a = l d, and (c) disk-like (oblate spheroid) inclusions, a = t w ; (b) and (c) are transversely isotropic in the - plane. The M-T ean-ield theory has been eectively applied to study the elastic behaviour o ibre-reinorced coposites at relatively low volue tractions. To overcoe the coplications associated with the description o the internal icrostructure o coposite aterials, any authors have concentrated on inding the upper and lower bounds o the elastic oduli. The ain eature o the bounding techniques is the evaluation o the energy content o the aterials undergoing deoration, which is in turn associated with an assued stress/strain ield in the coposite systes []. Variational principles o the elasticity theory, e.g., iniu or axiu stress/strain conditions and iniu energy assuptions are eployed in such odels. The siplest bounding odels 4

5 are the Voigt and Reuss [] approxiations, known as the rule o ixtures and the inverse rule o ixtures, respectively. Closer bounds or the elastic oduli have been developed or continuous ibre coposites by Hill [, ] and Hashin and Shtrikan [8]. The Hashin-Shtrikan approach is based on the introduction o a hoogeneous and isotropic reerence aterial. The ethod is based on evaluating the dierences between the coparison aterial and the actual coposite. In order to calculate the upper bounds, the coparison aterial is chosen to have stiness values higher than or equal to the rigidity o the inclusions (hard phase), and or the lower bounds the reerence aterial has stiness saller than or equal to the stiness o the atrix (soter phase). Weng [] extended the Hashin and Shtrikan ethod and obtained detailed expressions or the elastic stiness tensor o aligned ellipsoidal inclusions. By choosing the atrix or iller as reerence aterial, and with the use o the convenient strain concentration actor M, ie: ( - ) - A = M é ù êë I + M úû (6) the lower bound o the convenient strain concentration actor, - ( ) ù M lower - M é lower = êi + S - ë C C C ú (7) û and the upper bound o the convenient strain concentration tensor M - ( ) ù - M é upper = êi + S - ë C C C ú (8) û The convenient strain concentration tensors,, is: upper, is: M lower and M upper are substituted into qn 6 to derive the lower and upper bound strain concentration tensors A lower and A upper. The latter can then be substituted in qn to derive lower and upper bound values or the stiness tensors o the coposite. The lower bounds obtained using the above considerations coincide

6 with the predictions o the M-T theory. The upper bounds predicted by qns 6, and 8 represent precise estiates o the coposite oduli or very high particle concentrations, i.e., at axiu particle packing conditions. This observation, inspired Lielens et al. [4], to use a ixture rule between the upper and lower bound to provide predictions o the elastic oduli o coposites with interediate iller volue ractions. The Lielens convenient strain concentration actor M L is given by - é - - ù ML = ê( - FL)( Mlower ) + FL( M upper ) ú (9) ë û where the ixture unction, F L, ust increase onotonously with the volue raction o the inclusions, thereore or =, F L = and or =, F L =. Lielens et al., suggested the ollowing or the ixture unction: F L + = () % vol. range. qns 9 and give iproved results at reinorceent volue ractions in the 4-6 The last class o constitutive odels that are considered in this work are the ones based on the sel-consistent ethod. In the sel-consistent ethod, proposed by Hill [], a single inclusion o ellipsoidal geoetry is supposed to be ebedded in an ininite hoogeneous ediu, the latter possessing the eective properties o the coposite. Thereore sel-consistent schee assues that the inclusion is in act ebedded in a coposite aterial with unknown properties, rather than the atrix. Fro qn. 4, the selconsistent concentration tensor is given by substituting the atrix tensor, corresponding elastic tensor o the coposite aterial, C, yielding: - - ( )( ) ù C, with the A = é S-C êi + SC C - C ë ú () û 6

7 The cobination o qns and provides the basic sel-consistent orula or the - overall stiness. The ters C and ( C ) in qn., represent properties o the ininite ediu and they are initially unknown [, ]. The sel-consistent odels have oten been ound to over-predict the elastic oduli o particle illed coposites, especially at higher volue ractions []. Thereore, another approach has been developed, called the generalized sel-consistent (GSC) theory. It was proposed by Kerner [6] or spherical particles and by Herans [7] or cylindrical ibre coposites. The GSC schee also uses an ebedding ethod, except that in this case the inclusion is ebedded in the atrix aterial; these concentric volues o inclusion and atrix are theselves ebedded in the coposite ediu [4]. For illers that are uch ore rigid than the polyer atrix ( G G, where G is shear odulus), Kerner s equation is reduced to: é ( - v ) ù G = G + or G G ê ( 8- v)( - ë ) ú û c () or up to interediate iller volue ractions []. qn. has been used in various studies to predict the elastic odulus o particle illed polyers directly by substitution o the ter with c. G c. Sei-epirical equations The results oered by Kerner and Herans have been the oundation o the Halpin- Tsai (H-T) sei-epirical equations [8]. Halpin and Tsai discovered that Heran s and Kerner s equations could be expressed in a copact or as: Pc P = + sh - h h = P P P- P + s () where, is the iller volue raction, and P represents the bulk and shear oduli while s, is a easure o the reinorceent packing geoetry and loading conditions. For s, the 7

8 H-T equations reduce to the inverse rule o ixtures, while or s, the H-T theory gives the Voigt upper bound. Assuing that the particles are aligned with the loading direction, Halpin and Kardos [9] proposed s = a in order to copute the longitudinal odulus o short-ibre systes. The odulus in the transverse direction was ound not to be sensitive to the aspect ratio with s =. The H-T equations have been ound to under-predict the stiness values o coposites with relatively high iller loadings [, ]. This has been credited to the act that Heran s solution or continuous ibres does not consider any liit in axiu packing raction in an actual coposite syste. To account or this axiu, Lewis & Nielsen [] proposed a odiication to the original or o the H-T equations, by introducing a new paraeter y( ) in qn : Pc P + s Nh P P- = h = - hy P P + s ( ) N (4) where y ( ) is a unction o the volue raction o the inclusions and the axiu packing o the inclusions ; s N = k -, with k being the instein s coeicient. The values o k and consequently s N, are related to the bonding between iller and atrix []. For a atrix with v =., illed with dispersed spheres, and i there is no slippage in the interace, the coeicient is k =., ( k =, i there is slippage). For v. reduced and the shape actor s N is calculated directly by:, the k values are s N 7- v = () 8 - v An epirical orula is proposed by Lewis and Nielsen [] or the actor y ( ) as: ( ) y é - ù ax = + êë ax úû (6) 8

9 Nielsen and Landel [] provide tabulated values or a range o particle shapes and packing ractions, e.g. ax =.74 or dispersed spheres with hexagonal close packing..4. Orientation o non-spherical inclusions In all o the above theories it is assued that the inclusions are aligned. However, realistically all coposites contain soe level o iller disorientation. Coposites with a rando orientation o particles would be expected to give lower odulus values. Van s et al. [] used the lainate theory to provide siple approxiations or the stiness o coposites with randoly oriented ibre-like or disk-like inclusions. The odulus o a short ibre reinorced coposite with D randoly oriented ibres was generated +.86 (7) D ibre where and are the longitudinal and transverse oduli respectively. A unidirectional plate reinorced with disk-like particles, is already D randoly orientated and in this case the odulus o the coposites is derived +. (8) D disk. xperiental The coposite aterials used in the experients were supplied by.i. DuPont Neours & Co. (Inc.) in the or o sheets. The atrix constituent was slightly cross-linked PMMA. The plaques were all ade with the sae ATH iller with volue ractions ranging ro % vol. to 49% vol. The degree o unctionality and degree o cross-linking o the polyeric atrix ored during the casting process at each iller level was kept constant. To iprove the interace between iller and atrix, the anuacturer used an adhesion prooting additive. Unilled PMMA sheets were tested to provide the elastic properties o the atrix. The ATH particles were studied both as powder and as contained in the coposites. Gibbsite 9

10 was supplied by Alcan Cheicals USA, Division o Alcan Aluiniu Corporation. The echanical tests were perored or the noinal teperature range o ºC - 9 ºC. Three-point bend tests were carried out or the coposite aterials and or unilled PMMA. The tests were perored according to Method I Procedure A o the ASTM D79M-9 standards or plastics [], utilising siple bea speciens with rectangular cross sections. The thickness o the speciens was taken equal to the thickness o the aterial sheets, ie, and the width and length were. and respectively. The rate o crosshead otion or each group o aterials was set at.6 /in coputed according to the standards, where the strain rate is. in -. A iniu o ive replicates were used or each test. Tensile tests were perored or the estiation o the elastic odulus and Poisson s ratio o unilled PMMA. Tensile tests were also conducted on coposites along perpendicular directions o the aterial sheets to exclude potential directional dependence o their elastic odulus. The tests were perored according to the ASTM D68M-84 standard [4] or plastics. For PMMA, the tests were carried out using Type M-I tensile bone-shaped speciens with a thickness o 7. and gauge length and width equal to and respectively. For the coposites, Type M_II tensile speciens were used with a thickness o 4 and gauge length and width equal to and 6 respectively. The elastic odulus and Poisson s ratio o PMMA were easured in tension using etal-oil strain gauges. The test speed was set at /in, resulting in strain rates o. in -. For the coposites tests, an extensoeter was attached onto the surace o the saples to record the strain within the gauge length. The test rate was set at /in, giving strain rates o.4 in -. Optical icroscopy was used to provide an indication o the dispersion o particles into the polyer atrix and to characterise their shape, as they are contained in the

11 coposites. Saples with cross-sectional areas o x were cut ro both the suraces and through the thickness planes. A succession o polishing steps was ollowed which aied to reove any daage produced by sectioning. A iniu nuber o ten iages at rando locations o each o the saples was acquired, and quantitative statistical easureents were perored using the iage processing sotware IageJ []. Scanning electron icroscopy (SM) was used to observe the orphology and shape characteristics o the ATH powder. To iniise charging o the saples, a thin layer o gold was deposited on the suraces prior to observation. Laser light scattering (LLS) was used to assess the aspect ratio o the particles and conir the indings o the optical icroscopy. A particle analyser, Malvern Mastersizer (. μ to ) was utilised. The particles were dispersed in distilled water, and the aount o water ( l) and the quantity o powder (9. g) used were easured beore and ater the experient. The ratio between the actual volue concentration presented to the easuring device and the volue concentration calculated by the Malvern Mastersizer was used to provide an estiate o the aspect ratio o the particles [6]. 4. xperiental results Fig. shows the results o the lexural odulus o the coposites plotted against iller volue raction and teperature ro to 94 C. The standard deviation o the odulus values presented in the graphs is less than % o the ean values (ive replicates, tested our ties each). As they were so sall copared to the sybol size, they are not shown. The odulus or the coposites shows a % increase as the iller content increases ro % to 49% vol. Parallel trends o dependence o the elastic odulus on the aount o the iller are seen at all teperatures. The odulus o the coposites decreases by % with rising teperature ro to 9 C. The dierence in the odulus values obtained ro the tensile tests along

12 perpendicular directions o the coposite sheets was less than %, indicating rando orientation o the iller particles in the coposites. lastic odulus C C C 8 C 6 C 4 C C 6 C 7 C 8 C 94 C ATH volue raction b. Fig. Flexural odulus vs. iller volue raction or ATH-PMMA coposites. The results or the tensile odulus o PMMA are shown in Fig., along with the values ro the lexural tests. The tensile data presented by Ferreira et al. [7] or PMMA (strain rate =.4 in - ) are given or coparison and veriication o the results. Very good agreeent is observed within the three sets o data. Modulus 6 4 Tension:. in - [this work] -point bend:. in - [this work] Tension:.4 in - [7] [6] Teperature ( C) a. Fig. Modulus o unilled PMMA as a unction o teperature.

13 The Poisson s ratio data coputed ro the transverse vs. axial strain plots are presented in Fig. 4. The results are copared to the values calculated [4] ro pressurevolue-teperature (PVT) data presented by Zoller and Walsh [8] on PMMA. Good agreeent is observed between the results, with the Poisson s ratio o increasing gradually with teperature Tension:. in - [this work] PVT data [8] [49] Poisson's ratio Teperature ( C) b. Fig. 4 Poisson s ratio o unilled PMMA as a unction o teperature Typical optical icrographs o the ATH-PMMA suraces containing iller loadings o 9% and 49% vol. are presented in Fig.. The lighter phase represents the ATH particles while the grey regions correspond to the PMMA atrix. The digital photographs were converted into 8-bit greyscale iages and transored into binary iages (see Fig 6a). The IageJ acility shape descriptors [], was then utilised to obtain the aspect ratio o the particles. This is deterined by itting an ellipse to each particle and the aspect ratio is obtained ro the ratio o the ajor to the inor axis (see Fig. 6b). The calculations were based on the constituent areal ractions assuing that it was well represented by the regions shown in the optical pictures.

14 a. b. Fig. Typical optical icrographs o the ATH-PMMA a. 9 % vol., b. 49 % vol. a. b. Fig. 6. ATH-PMMA (% vol.): a. binary iage, and b. itting ellipse procedure. Fig. 7 shows histogras o the aspect ratio o the ATH-PMMA coposites or 9% and 49% vol. ractions. Siilar histogras were obtained or the interediate ractions [4]. The aspect ratio o the particles is in the range ro to 7. A considerable portion o the particles are spherical or nearly spherical with 9 % o the values being below a =. The ean aspect ratio is ound to be in the range o.7 to.86, the edian (iddle o the distribution: % o the values above the latter and % o the values below) between. to.8, and the ode (ost requent value) in the range ro. to.9. 4

15 7 6 Histogra 9 % vol. 7 6 Histogra 49 % vol. Frequency ( x ) 4 Interval =. Count: 988 Mean:.8 Median:.6 Mode:. SD:.66 Percentage (%) Frequency ( x ) 4 Interval =. Count: 9 Mean:.79 Median:.6 Mode:.8 SD:.66 Percentage (%) Aspect ratio a Aspect ratio d. Fig. 7 Histogras o the aspect ratio or ATH-PMMA, 9 % and 49 % vol. Fig. 8 shows typical SM icrographs o the ATH powder. The ATH particles reseble aggregates o platelet and prisatic crystals, with sizes varying ro around μ up to μ or ore. Finally, the laser light scattering results conired the average aspect ratio o the ATH particles, giving an average aspect ratio o. with standard deviation o 6. Fig. 8 SM icrographs o the ATH powder at dierent agniications:. Analytical Modelling The elastic odulus and Poisson s ratio v o PMMA obtained ro the tensile tests at dierent teperatures (Figs and 4) are used as input in the analytical equations given in section. The elastic odulus and Poisson s ratio v o the ATH particles are

16 derived ro the literature [9, ] with values o 76 GPa and.4 respectively. The properties o the iller particles are assued to be constant with teperature. Valuable coparison between the theories was achieved by assuing, in the irst instance, that the iller particles are spherical, ie a =. The shelby tensor S or spherical particles is utilised or the M-T and Lielens odels. The shape actor in the H-T equations is taken as the one or spherical particles, i.e., s = a =. The Nielsen [] solutions are obtained or the axiu packing o ax =.74, suggested or dispersed spheres [], with no interacial slip. Fig. 9 shows the predictions o the elastic odulus c as a unction o iller volue raction, assuing a =, together with the experiental data or the coposites (% - 49% vol.), at roo teperature, C. xp. C Nielsen Lielens GSC H-T M-T c 4 6 φ Fig. 9 Coparison o analytical predictions to the experiental data at C Very good results are provided by the Nielsen ( y ) and H-T odels. These odels are the ost widely used odels or the prediction o the odulus o particle reinorced coposites due to their siple or. The GSC odel is also ound to be in very good agreeent with the experiental data. The sel consistent equations are not shown since they over-predict the odulus signiicantly. The M-T odel predictions are ound to be lower than 6

17 the experiental data as it was derived or relatively low particle concentrations. The Lielens odel, which is proposed as an iproveent to the M-T odel or higher iller volue ractions, provides a superior prediction to the M-T odel. The eect o altering the particle aspect ratio within the range ro - 7, on the odulus predictions, is next exained or the Lielens and H-T odels. The Lielens odel allows or two dierent particle shapes to be odelled: ibre-like and disk-like inclusions. The solutions,, and, along directions x i =,,, or both these shapes are shown in Fig, together with the solutions or randoly oriented particles ibre-like D given by Van s or D ibre (qn. 7), and disk shaped D disk (qn. 8) inclusions. Fig. shows that the aspect ratio has an iportant eect on the odulus or coposites with aligned ibrelike inclusions and on the oduli and or coposites with aligned disk-like inclusions, while the eect on the transverse direction is inial. The rando solutions D iber D disk, are interediately aected by the change in the aspect ratio values with the, and reinorcing eect being greater or disk-shaped inclusions. The Lielens odel ( or D iber D disk ) provides an excellent description o the experiental data or the range o aspect ratios ro to and good prediction or aspect ratios until about ; this is in agreeent with the values established by the optical icroscopy aspect ratio distributions. The H-T longitudinal, transverse and rando Van s predictions or s = a are shown in Fig. The predictions (parallel and transverse) or either ibres or disks or s = a (Fig.a.) are identical and in this case only the rando solutions, D iber or D disk, dier. For aspect ratios above, the H-T odel starts to deviate slightly ro the experiental data, siilarly to the Lielens pattern. 7

18 4 Lielens xp. C α= α=. α= α= α= α=7 Lielens xp. C α= α=. α= α= α= α=7 l d α = l/d t α = t/w w 4 6 φ 4 6 φ xp. C Lielens α= α=. α= α= α= α=7 4 Lielens xp. C α= α=. α= α= α= α=7, l d α = l/d, t α = t/w w 4 6 φ 4 6 φ xp. C Lielens α= α=. α= α= α= α=7 Lielens xp. C α= α=. α= α= α= α=7 l w D iber d α = l/d D disk t α = t/w 4 6 φ 4 6 φ Fig. Lielens odel: ect o aspect ratio on the predictions or ibre-like and disk-like inclusions and coparison with experiental data, at C. 8

19 lastic odulus Halpin-Tsai xp. C α = sphere and ibre-like or disk-like transverse α =. ibre-like or disk-like parallel α = ibre-like or disk-like parallel D Halpin-Tsai D disk xp. C α = D iber α =. α =. α = α = 4 6 φ a. 4 6 φ b. Fig. H-T odel: ect o α on the predictions or ibre-like and disk-like inclusions. The Lielens, GSC, H-T and Nielsen y odels are shown against the experiental data at dierent teperatures in Figs and. For siplicity reasons, the aspect ratio o a = is used. Siilar results can be obtained or the rando solutions using the Lielens and H-T odels or aspect ratios up to. The our odels are in very good agreeent with the experiental data at all teperatures, with the Lielens and Nielsen odels giving an excellent description. The H-T and GSC odels slightly deviate ro the test data or volue ractions higher than 4 % vol. As entioned in Section the odiication o the H-T odel by Lewis & Nielsen was proposed in order to accoodate higher particle volue ractions into the solutions, so the better agreeent in this case was expected. c Lielens xp.data Model C C C 8 C 6 C 4 C C 7 C 8 C 94 C c GSC xp.data Model C C C 8 C 6 C 4 C C 7 C 8 C 94 C φ a φ b. 9

20 Fig. lastic odulus against iller volue raction and teperature: a. Lielens and b. GSC analytical odelling predictions or a =. c Halpin -Tsai xp.data Model C C C 8 C 6 C 4 C C 7 C 8 C 94 C c Nielsen φ ax =.74 xp.data Model C C C 8 C 6 C 4 C C 7 C 8 C 94 C φ a. φ b. Fig. lastic odulus against iller volue raction and teperature: a. Halpin-Tsai, and b. L&N analytical odelling predictions or a =. 6. Conclusions The odulus o the ATH-PMMA coposites increases by % as the ATH iller contents increase ro % vol. to 49 % vol., and decreases by 6 % with rising teperature ro to 94 C. This elastic behaviour o the coposites is odelled by easuring only the elastic properties o the PMMA atrix and deterining the shape o the ATH illers in the coposites via icroscopy. Under the assuption o constant iller elastic properties and by altering the teperature ro to 94 C, it is as i at every teperature a dierent coposite is tested as the atrix aterial sotens. The Lielens odel is the ost representative o the elastic behaviour o the ATH- PMMA coposites. The predicted values are in excellent agreeent with the experiental data at dierent teperatures and various iller contents. It provides an exceptional description o the experiental data or the range o aspect ratios ro to and good prediction or aspect ratios until about ; values that represent the ATH particles precisely.

21 Moreover, the Lielens odel is based on constitutive odelling equations and can be used to predict all the elastic constants o the coposites at the sae tie. This akes it superior to the other odelling equations, i.e., the Halpin-Tsai, GSC and Nielsen odels which involve a high degree o epiricis. 7. Acknowledgeents The authors would like to acknowledge. I. DuPont Neours &Co. (Inc.) or unding this work and supplying the aterials. 8. Reerences.) Duggins, R. B., U.S. Patent,847,86, 6/4., 974.) Tucker III, C. L. and Liang,., Copos Sci Technol, vol. 9, 6-67, 999.) Fu, S-Y., Feng, X-Q., Lauke, B. and Mai, Y-W., Coposites Part B, 9, 9-96, 8. 4.) Stapountzi O. A., PhD thesis, Stiness and racture properties o Aluina Trihydrate illed Poly (Methyl Methacrylate) coposites, Iperial College London, April 8.) Hill, R., J Mech Phys Solids, vol., 7-7, 96 6.) shelby, J. D., Proc Roy Soc A, vol. 4, 76-96, 97 7.) Mura, T., Microechanics o Deects in Solids, The Hague: Martinus Nijho, 98 8.) Hashin, Z. and Shtrikan, S., J Mech Phys Solids, vol., 7-4, 96 9.) Mori, T. and Tanaka, K., Acta Metallurgica, vol., 7-74, 97 ) McCullough, R. L., In: Microechanical aterials odelling, Delaware Coposites Design ncyclopedia - Volue, Whitney, J. M. and McCullough, R. L., editors, Technoic Pub. Co, Lancaster, PA, 99.) Hill, R., Proc Phys Soc A, vol. 6, 49-4, 9.) Hill R., J Mech Phys Solids, vol., 99-, 964.) Weng, G. J., Int J ng Sci, vol., 8-9, 99

22 4.) Lielens, G., Pirotte, P., Couniot, A., Dupret, F., Keunings, R., Coposites A, vol. 9, 6-7, 997.) Hill, R., J Mech Phys Solids, vol., -, 96 6.) Kerner,. H., Proc Phys Soc B, vol. 69, 88-8, 96 7.) Herans, J. J., Proc Kon Ned Akad v Wetensch B, vol. 6, -9, ) Halpin, J. C., J Copos Mater, vol., 7-74, ) Halpin, J. C. and Kardos, J. L., Polyer ng Science, vol. 6, 44-, 976.) Lewis, T. B, Nielsen, L.., J Appl Poly Sci, vol. 4, , 97.) Nielsen, L.. and Landel, R. F., Mechanical properties o polyers and coposites, CIP, 994.) Van s, M., Xiqiao, F., Van Turnhout J., Van der Giessen,., Al-Malaika S, Golovoy AW, editors, Specialty polyer additives: principles and applications, CA Malden, MA Blackwell Science,.) ASTM D79M-9, Annual Book o ASTM Standard, Aerican Society or Testing and Materials, 99 4.) ASTM D68M-84 Standard test ethod or tensile properties o plastics, 99.) IageJ or Microscopy, MBF_IageJ/ v.8 -.9d, User Manual, 7 6.) Malvern Instruents ( 7.) Ferreira, J. M., Costa, J. D., Capela, C., Theor Appl Fract Mech, vol. 6, -6, ) Zoller, P. and Walsh, D., Standard pressure-volue-teperature data or polyers, Technoic Pub. Co., 99 9.) Nie, S. and Basaran, C., Int J Solid Struct, vol. 4, ,.) Liu, H., Hu, J., Xu J., Liu, Z., Shu, J., Mao, H. K. and Chen, J., Phys Che Miner, vol., 4-46, 4