A model study of the effective Young s modulus for randomly distributed short-fiber composites

Transcription

1 FFI-rapport 0/00 A odel study o the eective Young s odulus or randoly distriuted short-ier coposites To Thorvaldsen Norwegian Deence Research stalishent (FFI 5 January 0

2 FFI-rapport 0/ P: ISBN : ISBN Keywords Youngs odul Kortierkopositt Mateatisk odellering Fierorientering Fierlengde Approved y inar Willassen Jan Ivar Botnan Project Manager Director FFI-rapport 0/00

3 nglish suary This report provides an overview o a set o acro-echanical odels or the eective Young s odulus o short ier coposites. Various odels are ound in the literature with varying coplexity, ranging ro siple rule o ixtures odels to ore sophisticated odels explicitly taking into account the ier length and the ier orientation distriution. The odels can e used as a tool or a etter understanding o the aterial properties and aterial ehavior, and are valuale additions to the knowledge otained ro experiental tests. In addition, representative and accurate aterial odels are essential in inite eleent ethod (FM analyses. FM odeling is nowadays applied in the design and developent o ost coplex structures. FFI-rapport 0/00 3

4 Saendrag Denne rapporten gir en oversikt over et sett ed akroekaniske odeller or den eektive verdien av Youngs odul or kortierkopositter. Forskjellige odeller ed varierende kopleksitet innes i litteraturen, ra enkle ikseregelodeller til er soistikerte odeller so har eksplisitte uttrykk or ierlengde og ierorientering. Modellene kan rukes so et viktig verktøy or edre å orstå aterialenes egenskaper og oppørsel, i tillegg til det so lir oservert i eksperientelle orsøk. I tillegg er representative og nøyaktige aterialodeller essensielt i eleentetodeanalyser. leentetodeodellering er per i dag anvendt i design og utvikling av de leste kopliserte konstruksjoner. 4 FFI-rapport 0/00

5 Contents Introduction Matheatical odels. Rule o ixtures odels.. D odels.. 3D odels. Fier length and ier orientation distriution unctions.3 The lainate analogy approach (AA.4 A lainate approxiation odel with an explicit FOD unction.5 The lainate-plate ethod.5. General expressions or the eective properties o the coposite.5. The longitudinal Young s odulus.5.3 Three odel cases.6 The paper physics approach (PPA 3 Suary and uture work Acknowledgeents Reerences FFI-rapport 0/00 5

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7 Introduction In this report, a set o atheatical odels or descriing the eective properties o short-ier coposite aterials has een investigated. Two categories o atheatical odels ay e deined or this purpose. The irst type o odels is typically dealing with one ier eedded in a atrix aterial, using this unit to descrie the overall aterial properties. The ai is then to estalish expressions or the interace etween the ier and the atrix, and to odel the ier stress and the interace shear stress distriutions o the ier. Both analytical expressions and nuerical siulations are eployed. The second type o odels are ore concerned with inding average values or the aterial, so that the coposite can e descried acroscopically y an isotropic or, ore generally, an anisotropic aterial odel. This category typically includes rule o ixtures odels, as well as ore sophisticated variants. In addition, this latter category o odels distinguishes etween aligned and randoly distriuted short-ier coposites. A rando distriution ay in any situations e ore adequate. This report is restricted to only considering acro-echanical odels or randoly oriented short-ier coposites. Dierent approaches or descriing the eective properties have een suggested and presented in the literature. The ain interest here is to estalish the paraeter values or descriing the elastic properties o the coposite aterial. These values are required in, or exaple, inite eleent ethod (FM analyses. One ain paraeter is the odulus o elasticity, i.e. Young s odulus, in the direction o the applied load. In this report, the odeling approaches where algeraic expressions are availale will e covered. Soe o the odels in this survey are ipleented in Matla, showing the elastic properties as a unction o the ier volue raction. For the siplest odels, the inluence o ier length and ier orientation is included iplicitly. For the ore sophisticated odels, explicit expressions or length variation and ier orientation are included. Hence, the nuer o possile paraeters that can e tuned depends on the coplexity o each odel. With the odels at hand, we are ale to run a lot ore test cases than would e easile experientally. Matheatical odeling hence ecoes an additional tool in the analysis. It should, however, e entioned that no experiental tests are perored to veriy or support any o the atheatical odels. Future studies on discontinuous ier coposites will, however, naturally include an experiental part. Parts o the contents o this report has already een presented at the 4 th uropean Conerence on Coposite Materials []. Future work, which is also expressed in the conerence proceeding paper, will investigate whether the short-ier odels are applicale to odel nanocoposites. For that reason, soe reerences to papers where short-ier odels are applied or nanocoposites are included here. FFI-rapport 0/00 7

8 Matheatical odels. Rule o ixtures odels The irst type o odels or descriing the acro-echanical properties o short-ier coposites is ased on the assuption that the iers are randoly oriented, and that the overall aterial properties can e expressed y soe weighted su o the aterial properties o the constituents o the coposite aterial, that is, the iers and the atrix. I at all included in the odel, the ier length variation is assued to e expressed y soe average, or ean, value. This type is oten reerred to as rule o ixtures odels. The inal expressions or the coposite aterial are algeraic relations, estalished ro experiental tests. The Young s odulus, and in soe cases also the Poisson s ratio, is given. Macroscopically the aterial is now treated as isotropic. Also, assuing linear elastic properties, a linear stress-strain relation can e deined or the coposite. Most expressions are presented or two-diensional (D cases. Due to the icroscopic anisotropy, it is oten ore coplicated to peror tests or 3D cases. Soe o the works ound in the literature or this odel type are, however, also descried or 3D cases. Models descriing the properties and assuptions in ore detail will e shown next... D odels One o the earliest studies on randoly oriented short-ier coposites is ound in the paper y Cox [], which considered odeling o cellulose ier aterials, i.e. paper. His theory is oten reerred to as the shear-lag analysis or the paper physics approach, see e.g. [3]. The paper physics approach is ased on the undaental point o calculating the orce ro iers crossing an aritrary line in the test specien. The Young s odulus is then ound ro the inal expression o the orce. The paper physics approach is ore thoroughly descried in Section.6. Based on a discussion o the orientation o the iers, and y assuing that the aterial at hand acroscopically can e descried as isotropic, Cox suggested a very siple odel or the eective elasticity odulus, V C ( V. (. 3 In the aove expression, V is the ier volue raction, is the Young s odulus o the iers, and is the Young s odulus o the atrix. Fro assuing a linear elastic aterial, and knowing the eective shear odulus, G C, o the coposite, the Poisson s ratio ay e ound ro 8 FFI-rapport 0/00

9 C C, (. G C Many later odels are ased on the pioneer work y Cox. A siilar odel as the Cox odel or descriing the elastic properties o paper has een presented y Horio and Onogi [4]. Instead o including the elastic odulus o the iers and the atrix, they applied the elastic odulus in the direction o the paper production achine and in the cross direction, that is, and, respectively. The elastic ean value through the angular distriution is given as, C / (. (.3 A slightly ore sophisticated odel or randoly oriented reinorceents in thin resin ils, i.e. a D orientation distriution, ay e estalished. The elasticity odulus and shear odulus o the coposite can e expressed y eploying a relation containing the aterial properties o the two ain constituents o the coposite, see e.g. [5-8], 3 5 C T, ( GC T, ( where and T are the longitudinal and transverse odulus o an aligned short-ier coposite. The oduli can e expressed atheatically, ( l/ d ηv η V, (.6 T η TV, (.7 η V T where ( / η, ( / ( l/ d (.8 ( / ηt ( /. (.9 FFI-rapport 0/00 9

10 The relations in (.6 and (.7 are reerred to as the Halpin-Tsai equations or aligned short-ier coposites. In the aove expressions, V is the ier volue raction, is the elastic odulus o the iers, is the elastic odulus o the atrix, l is the ier length, and d is the ier diaeter. The aspect ratio is deined as the ier length divided y the ier diaeter ( l/ d. For the randoly oriented short-ier coposite under consideration, the ier aspect ratio and ier volue raction is the sae as in the corresponding aligned short-ier coposite. Furtherore, ro assuing an isotropic, linear elastic aterial, the Poisson ratio is given in (.. Also, note that, in this latter rule o ixtures odel, the ier length and the ier diaeter are oth explicitly included in the odel, and hence the geoetric properties o all iers are assued equal. Typically soe kind o average, or ean, value is applied or the ier length. This ay not e the case or real-lie coposites. Note that the oduli and can alternatively e deterined experientally, and put directly into (.4 and (.5. T The Halpin-Tsai equations have ecoe one o the coonly used odels or descriing the eective properties o randoly oriented short-ier coposites. For exaple, Fu and auke [3] eployed this odel or coparison with their extended lainated analogy approach (AA odel, urther descried in Section Moreover, these equations have also een used or other aterials, or instance in the work y Qian et al. [6] or estiating the elasticity odulus o a coposite consisting o ulti-walled caron nanotues (MWCNTs dispersed in a polystyrene (PS atrix. In this case the nanotues play the role o the iers, where d is the outer tue diaeter. As reported in the review paper y Colean and co-workers [9], the Halpin-Tsai equations are known to it experiental data or low ier volue raction coposites. For high volue ractions the stiness is, however, underestiated. Colean et al. [9] also showed a rule o ixtures odel, which is ased on the pioneer work y Cox []. One general expression is estalished, which include discontinuous ier coposites with either aligned iers or randoly oriented iers. This rule o ixtures expression is written as, C ηη 0 V ( V, (.0 where η 0 or aligned iers, η 0 3/8 or iers uniorly distriuted in a plane, and η 0 /5 or 3D uniorly distriution [0]. Furtherore [], al tanh d η, (. al d 0 FFI-rapport 0/00

11 With 3 a. (. lnv In the aove expressions, and is the Young s odulus o the ier and the atrix, respectively. Moreover, l is the ier length, d is the diaeter, and raction. V is the ier volue A siilar odel as the previous one, is also shown y Fu and auke [], C χχ V ( V, (.3 where the χ and χ generally are unction expressions or the ier orientation and the ier length, respectively, ut ay also e set to constant values. Such unctions will e descried in ore detail in later sections. They are oten applied in coination with the lainate analogy approach, see Section.3, where an in-plane ier orientation is presued. Moreover, the product o the actors, that is, χχ, is denoted the ier eiciency actor. Christensen and Waals [3] and Christensen [4] presented expressions or the eective stiness or randoly oriented ier coposites ased on a geoetric average process. In their odel, the iers are assued to have any orientation in a given plane. The short-iers are assued to e suiciently long, so that they can e treated as continuous iers, and such that the end eects ay e neglected. For the D case, the eective properties can e written as C ( µ µ µ µ C µ (.4 where µ µ 3 G (3 3 G K 8 ( G K G ( 6 G K 8 ( G K (.5 The quantities in the aove expressions are ased on and taken ro the work y Hashin and Rosen [5], Hashin [6;7], and Hill [8;9], FFI-rapport 0/00

12 V ( V 4 V( V G ( V V K 3 G G G V G V ( ( V G VG k G /3 k G /3 G G V ( V ( k G /3 k G /3 ( V G VG k G /3 k G /3 G V k 3 V k k ( / 3( G G k (4 / 3 G G ( V G ( V ( ( 3( V G3 G G [ k (7 / 3 G]( V G G ( k (4 / 3 G k 3( k (.6 Note that the expression or the transverse shear odulus, G 3, is a lower ound. An exact value is not availale. Furtherore, in case the iers are very sti copared to the atrix aterial, the expressions in (.6 ay e sipliied. Under certain restrictions, given in the reerred papers, it ay e shown that, FFI-rapport 0/00

13 4 V ( V G( V ( V VG k G /3 ( V V K G G 3 3 G V ( k (4 / 3 G k 3 V ( V G ( V V ( k (4 / 3 G G [ k (7 / 3 G ]( V G V ( V( k G /3 VG k G /3 (.7 Please note that the expressions in (.6 and (.7 also can e applied or 3D cases, see Section.. or ore details. Another odel or D cases is presented y Weng and Sun [0]. This odel is also reerred to and applied in the paper y Chon and Sun []. In this case, the eective Young s odulus or the coposite is coputed ro C ( µ µ µ µ C µ (.8 where 3 G (3 3 G3K µ 8 ( G K G ( 6 G3K µ 8 ( G K (.9 FFI-rapport 0/00 3

14 And V ( R V ( R ( R/ R V( R V( R R K 3 G V k 3 V k k ( G G /3 k 4 G /3 G ( V G ( V ( ( G G G V G V G3 G G G G k 3( k 3( V [ k (7 / 3 G]( V ( k (4 / 3 G (.0 In the aove expressions, the odulus and the Poisson nuer are presented in the paper, whereas the other expressions are taken ro the paper y Christensen and Waals [3], which again is ased on the work y Hashin and Rosen [5], Hashin [6;7], and Hill [8;9]. Hence, this odel is very siilar to the Christensen-Waals odel. In (.0, the paraeter l R, l (. where l l lis the su o the length o the atrix, l, and the length o the ier, l. In their odel a unit is deined, consisting o a cylindrical short-ier eedded in and surrounded y a cylindrically shaped atrix tue. Hence, this theory also takes into account the length o the iers. The rest o the paraeters are given earlier. The Weng-Sun odel is in their paper [0] copared to other odels and also experiental results, and is said to e the ost suitale odel or the given experiental data. For coparison, the rule o ixtures odel shown in the paper y Christensen and Waals [3] is included, V ( V. C (. 4 FFI-rapport 0/00

15 This odel ay at irst e a natural choice. It is, however, stated in their paper that this odel produces asurd results. It does neither take into account the rando orientation nor the ieratrix interaction eects, which is the case or the Cox odel and the ore sophisticated odels. A siilar expression is also oten applied or the Poisson s ratio o the coposite. Most o the aove odels or the eective Young s odulus o randoly distriuted short-ier coposites have een ipleented in Matla or a varying ier volue raction, V. However, only those odels where the elastic odulus is a unction o the elastic properties o the constituent aterials, are displayed in the plot, as shown in Figure.. In all cases, and 6.5. Moreover, or the Halpin-Tsai odel the aspect ratio is set to 0. For the (advanced Christensen-Waals odel and the Weng-Sun odel, 0.35 and 0.0. Finally, or the Weng-Sun odel, l l. Please note that ecause the dierent odels require dierent aterial paraeters, the odel results ay not e directly coparale. For exaple, y choosing a dierent value or l in the Weng-Sun odel, the curve will change, while the reaining odels, which do not explicitly depend on this quantity (or have a siilar input paraeter, will not e altered. Nevertheless, ro the curves we oserve that the Young s odulus when applying the Christensen-Waals rule o ixtures odel (in green gives a uch steeper increase and higher stiness copared to the other odels. The ore sophisticated Christensen-Waals odel (in red is very close to the odel where the Halpin-Tsai expressions (in agenta are applied. The other odels overall predict a lower increase in the Young s odulus as a unction o the ier volue raction, and lower stiness values or large ier volue ractions. FFI-rapport 0/00 5

16 Figure. ective Young s odulus or the coposite aterial versus ier volue raction, D case. All values are noralized with the Young s odulus o the atrix aterial... 3D odels As descried in the previous section, the paper y Cox [] presents one o the earliest odels or randoly oriented short-ier coposites. For 3D orientation o the iers, ased on the description o the D case presented aove, he suggested that V C ( V, (.3 6 where V, as eore, is the ier volue raction, is the Young s odulus o the iers, and the Young s odulus o the atrix. In the sae way as or the D case, the Poisson ratio is given in (.. Breton et al. [] used this odel also or estiating the elasticity odulus o ulti-walled caron nanotues (MWCNT eedded in an epoxy resin. In this case, the MWCNTs take the role o the short-iers. Another odel or the eective elasticity odulus o short-ier coposites, with a rando orientation in 3D, have een presented y avengood and Goettler [7], 6 FFI-rapport 0/00

17 4 C T. ( The longitudinal and transverse elasticity oduli in the latter expression can e expressed y the Halpin-Tsai expressions in (.6 and (.7. In [7], the corresponding shear odulus or 3D cases is not explicitly given, neither is the Poisson s ratio. Hence, the paper does not provide enough inoration or descriing all the aterial properties, especially needed in atheatical and nuerical odeling. Moreover, Kardos [8] reerred to this odel in his paper, stating that ore work need to e done or 3D rando orientation to otain a general orat, as is the case or D cases. Fidelus et al. [3] have presented a rule o ixtures odel or prediction o Young s odulus or 3D randoly dispersed caron nanotue coposites, λ V ( V. (.5 C NT NT NT As Fidelus and co-workers used this odel or caron nanotues, V NT in their odel reer to the volue raction o the tues, and NT and denote the elastic odulus or the nanotues and the atrix, respectively. Moreover, λ is here the so-called Krenchel s coeicient, which or 3D randoly oriented rods with high aspect ratio, ay e set to a constant value. However, a siilar expression should also e applicale to short-ier coposites. As entioned in the previous section, Christensen and Waals [3] and Christensen [4] presented expressions or the eective stiness o randoly oriented ier coposites, oth or D and 3D cases. For the 3D case, the eective properties o the coposite can e written as, C [ (4 8 4 K ][ (4 4 K 6( G G ] [ (8 7 K3 ( G G3] (4 6 6 K 4( G G C 4 (6 4 4 K 4( G G (.6 It can also e shown, see [4], that in this case GC [ ( K3 6( G G3 ] 5 [ kc 4( K3 ] 9 (.7 The paraeters in (.6 and (.7 are given in (.6. In a siilar way as in the D case, or sti iers (copared to the atrix, the paraeters are given in (.7. FFI-rapport 0/00 7

18 Weng and Sun [0] also presented a 3D variant or the eective properties o 3D randoly oriented short-ier coposites. The expressions or the eective Young s odulus and Poisson s ratio are taken ro the Christensen-Waals odel in (.6. As or the D case, descried in the previous section, the paraeters involved are given in (.0. Finally, the Colean odel [9] presented in (.0 ay also e applied or 3D uniorly distriution. As already entioned in Section.., the only change is that η 0 (now equals /5. Soe o the 3D odels or the eective Young s odulus o randoly distriuted short-ier coposites have een ipleented in Matla, and plotted as shown in Figure.. In the sae way as or the D case in the previous section, the dierent odels require dierent aterial paraeters, and hence the odel results ay not e directly coparale. Figure. ective Young s odulus or the coposite aterial versus ier volue raction, 3D case. All values are noralized with the Young s odulus o the atrix aterial.. Fier length and ier orientation distriution unctions In the previous section, rule o ixtures odels or the eective properties o randoly distriuted short-ier coposites have een presented. In these odels, the coposite properties are estalished ro assuing that all orientations are equally proale. Moreover, the ier length and diaeter, deining the aspect ratio o the iers, are assued to e constant. To extend 8 FFI-rapport 0/00

19 this a it urther, unctions have een presented in the literature to also include variations in ier length and ier orientation. The variation in ier length is denoted the ier length distriution (FD unction, whereas the variation in ier orientation is called the ier orientation distriution (FOD unction. Soe coonly applied FD and FOD unctions will e presented next. Fu and auke [3;4;5] put up a two-paraeter Weiull distriution or odeling the FD, ( l ( / n( l/ n ex p l[ n ( / ], (.8 or l > 0, where n and are shape paraeters. Another or o the Weiull distriution unction is the so-called Tung distriution, ( l al exp( al, (.9 also valid or l > 0. Setting and a n into the Tung distriution, gives the Weiull distriution unction. The cuulative distriution unction, can then e given as l F( l ( l dl exp( al, (.30 0 and the ean ier length as ean ( / ( /. (.3 0 l l l dl a Γ The Tung distriution has recently een applied y Fu et al. [6] or analyses o caron nanotues polyer coposites. For the FOD, Fu and auke [3;4;5] used a two-paraeter unction proposed y Xia et al. [7], g( θ in {sin( θ} {cos( θ} p q θax p q {sin( } {cos( } d θ θ θ θ. (.3 The angle θ is deined as the angle etween the ier direction and the direction o the applied load. During production, the iers are typically distorted, oved, and roken, or exaple in an injection olding process, and hence the ier length and the orientation o the iers vary in a short-ier coposite. Although considering a 3D distriution, the ier orientation can, as an approxiation, still e odeled y the single angle θ descriing the direction o the ier within the coposite plane. An algeraic expression or the integral o the aove FOD unction is, however, only availale or certain ier orientation ranges. FFI-rapport 0/00 9

20 Fu and auke [3;8] presented the sae proaility unctions as descried in [3;4;5] or characterization o the anisotropy o the elastic odulus o isaligned short-ier reinorced polyers (SFRP. In addition to the aove two unctions, they included another ier orientation distriution unction g( φ, ased on the work y Xia et al. [7], or the variation in another direction than considered y the g( θ unction in (.3, g( φ in {sin( φ/ } {cos( φ/ } s t φax s t {sin( / } {cos( / } d φ φ φ θ. (.33 While the angle θ is deined as the angle etween the -axis (the axis in the direction o the externally applied load and the local ier axis, the angle φ is deined as the angle etween the -axis and the projection o the ier on to the -3-axes plane. A sketch o the coordinate directions is ound in [8]. Note that in the latter expression the angle varies ro 0 to. Moreover, a siilar expression or this angle range can e expressed also or the g( θ unction in (.3. In that case, to satisy the periodic conditions, p ust equal q, and s ust e equal to t. Chin et al. [9] presented two possile approaches or odeling the FD. The irst approach was the two-paraeter Weiull distriution, which is expressed as c c l c ( l exp[ ( l/ ]. (.34 Here, and c are shape paraeters. Fro this, one ay ind an expression or the ean ier length, which y deinition is equal to the expected ier length, lean lexp Γ c, (.35 where Γ is the gaa unction. The unction in (.34 is, in act, identical to the unction presented y Fu and auke [3;4;5], given in (.8, with c and n. The second unction presented y Chin et al. or the FD is a log-noral distriution unction, which ay e expressed as, ( l exp[ (ln l µ / s ], (.36 sl where s and µ are odel paraeters, and l is, as eore, the ier length. Note that the ier length, l, in the denoinator o the aove expression is issing in quation (7 in [9]; this is proaly a isprint. 0 FFI-rapport 0/00

21 In this case, the ean ier length (and the expected ier length ay e expressed as, s lean l exp exp[ µ ]. (.37 Moreover, the ost proale length (odal length ay e expressed as l od exp[ µ s ]. (.38 Fro rearranging the expressions in (.37 and (.38, the odel paraeters ay e expressed as, s (ln lean ln lod 3 ln lean ln lod µ 3 (.39 For the FOD, Chin et al. [9] proposed a odiied version o the odel y Kacir et al. [30]. To e ore speciic, the proaility density unction and the cuulative distriution unction o Chin et al. read λe g( θ e e F( θ e λθ λθ (.40 respectively. It is seen that F( θ, which is the accuulated percent o iers oriented etween zero and ± θ, satisies the requireent F(, which eans that the relative proaility o the ier angle to e etween 0 and / equals unity. That is, since all iers are oriented in the range ro 0 to /, the cuulative distriution unction ust equal 00 percent or all values o λ.this is, however, not the case or the FOD unction proposed y Kacir et al. [30], expressed as, ρθ ( λe λθ. (.4 That is, the orientation distriution unction is not noralized, and thereore it is only suitale or cases where λ is large. FFI-rapport 0/00

22 The integral o the Chin et al. FOD unction in (.40 is θax θax λθ λe λθin λθax g( θ dθ dθ e e. θin θin e e (.4 For an angle variation o θ ro zero to /, the latter integral ecoes unity. To get a etter understanding o how the ier orientation is aected y the values o λ, the proaility density unction and the cuulative distriution unction in (.40 are plotted in Figure.3 and Figure.4, respectively, or dierent values o λ. As can e seen, a large value o λ indicates that the iers are aligned, whereas a sall value o λ indicates a rando distriution. Siilar plots are shown in [9]. Figure.3 Orientation density curves or various values o λ. FFI-rapport 0/00

23 Figure.4 Cuulative distriution o iers or various values o λ. Chen and Cheng [3] used a slightly ore coplex theory or estiating the elasticity properties o a plane and transversely isotropic aterial case. They adopted the single exponential ier orientation unction proposed y Kacir et al. [30], deined y (.4. In addition to the ier distriution, the elastic oduli o the short-ier coposite were assued to e a unction o the elastic oduli o the constituent aterials, the ier volue raction, and the aspect ratio. Dierent expressions or the coponents o the stiness atrix were applied in cases o planar orientation o the iers and transversely isotropic situations. In the latter case, the distriution unction ρθ ( was assued axisyetric with respect to one ajor axis o the test specien considered, and the aterial then ecae transversely isotropic in planes noral to this axis. Unortunately, all paraeters are not deined in the paper, and hence it is diicult to ipleent this odel. In later sections, soe odels, where the aove distriution unctions are applied in estiating the eective Young s odulus or the coposite, will e descried. FFI-rapport 0/00 3

24 .3 The lainate analogy approach (AA A lainate analogy approach has een eployed in any papers or estiating the eective elasticity odulus o short-ier coposites. One o the earliest approaches, and proaly the irst, was proposed y Halpin and co-workers [3-34]. In this approach, the short-ier coposite is odeled atheatically as a lainated coposite. General lainate theory is applied, see e.g. [5]. The eective properties o the quasi-isotropic short-ier coposite can in this odel e expressed as, C 4 U ( U U U 5 5, (.43 C U U U 5, (.44 GC U5. (.45 Here, U U 5 3Q 3Q Q 4Q 8 Q Q Q 4Q (.46 with Q Q Q Q 66 T T G T T T (.47 In the aove expressions, is the longitudinal elasticity odulus and T is the transverse elasticity odulus, is the ajor Poisson s ratio and T is the inor Poisson s ratio, and inally, is the in-plane shear odulus. G 4 FFI-rapport 0/00

25 The elasticity oduli and the shear odulus are in this odel calculated ro the Halpin-Tsai expressions, which are expressed as ( l/ d ηv η V, (.48 T η TV, (.49 η V T and G ηsv G, (.50 η V S where ( / η, (.5 ( / ( l/ d ( / ηt ( /, (.5 and ( G / G ηs ( G / G. (.53 As can e seen, the shear odulus ay e expressed in a siilar way as the elasticity oduli; or convenience, we have repeated the expressions or the elasticity oduli here (already given in (.6 - (.9. Moreover, or the longitudinal Poisson s ratio a rule o ixture expression is used, resulting in V ( V. (.54 The inor Poisson s ratio ay e expressed as, T T. (.55 FFI-rapport 0/00 5

26 The Halpin et al. odel presented in this section and the 3D rule o ixtures odels introduced in Section.. are shown in Figure.5. The sae set o aterial paraeter values are used or all odels in the plot. As also entioned in the discussion o Figure., the length o the (short iers are not explicitly included in all the odels, and thereore the results are not directly coparale. However, as can e seen in the plot, the Halpin et al. odel (lue, dashed line is very close to the Halpin-Tsai rule o ixtures odel (in agenta. Figure.5 ective Young s odulus or the coposite aterial versus ier volue raction, 3D case. All values are noralized with the Young s odulus o the atrix aterial..4 A lainate approxiation odel with an explicit FOD unction The AA descried in Section.3, which here will e reerred to as the original AA, is ased on the Halpin-Tsai expressions developed and estalished ro epirical test results. In this odel, the ier aspect ratio, i.e. the ier length divided y the ier diaeter, is set to soe constant value, and the variation in ier orientation is iplicitly contained in the odel. As already entioned, odels that explicitly include distriution unctions or ier length and ier orientation have een presented, and a odel descried y Choy et al. [35], where a FOD unction is explicitly included, will now e descried. In the Choy et al. odel, the aterial properties are given reerred to the local ier syste or each ply in a layered coposite. These axes generally ake angles with respect to the (ixed 6 FFI-rapport 0/00

27 gloal coordinate syste or the coposite. The relationship etween the coponents o the stiness atrix in the gloal, or o-axis, syste and the local ier syste, can e expressed as Q U U cos θ U cos 4θ ' 3 Q U U cos θ U cos 4θ ' 3 Q U U ' 4 3 Q U U ' cos 4θ cos 4θ ' Q6 U sin θ U3 sin 4θ ' Q6 U sin θ U3 sin 4θ (.56 where U U U U U Q 3Q Q 4Q 8 Q Q Q Q Q 4Q66 8 Q Q 6Q 4Q66 8 U U4 66 (.57 Please note that the expressions or the coponents reerred to the local ier syste, Q ij, are not explicitly given in [35]. However, it ay e assued that the standard lainate theory, as descried in e.g. [5], and with the coponents given in (.47, is applied. Now, using the syntax in [35], reerring to [36;37], a set o odiied rule o ixtures expressions or unidirectional laina can e written. The Young s odulus along the ier direction,, and the longitudinal Poisson s ratio,, o the unidirectional coposite ay e expressed as η V η V V V, (.58 ηv V. (.59 η V V FFI-rapport 0/00 7

28 The longitudinal shear odulus, G 6, the transverse shear odulus, G 4, and the plain strain ulk odulus, k, ay e written G 6 V V G 6 η V 6 η6v G, (.60 G 4 V V G 4 η V 4 η4v G, (.6 V k V k k k η V η V k. (.6 Finally, the transverse Young s oduli are related to the aove paraeters, 4kG 4 3 k G, (.63 4 where 4k. (.64 Moreover, η V βα V βα tanh βα, (.65 / with β [ G V / ( ( ] and α l/ d, and G 3 4 G η4 4( G / G η6 G k ηk ( 6 4 (.66 8 FFI-rapport 0/00

29 In all the aove expressions or this odel, reers to the iers, whereas reers to the atrix. Fro a stress-strain relation or the lainated coposite, the stiness atrix ay e expressed as ij h/ ' ' ij,,,, 6 h, (.67 h/ A Q dh i j where h is the thickness o the lainate and explicitly included. ' dh λθ hλe dθ, with the FOD unction λ / e Sustituting (.56 into (.67, we get A U UV UV 3 A U UV UV 3 A U UV 4 3 A U UV A6 0 A (.68 Here, V h/ λ / ' λ ( e cos θ dh λ / h λ e h/, (.69 ( 4( and V h/ ' λ cos 4θ dh h λ h/. (.70 6 The in-plane elastic oduli can then e calculated ro the in-plane stiness. Moreover, ro assuing unior out-o-plane stresses, also the out-o-plane elastic oduli can e calculated ro the aove expressions. More details can e ound in the paper..5 The lainate-plate ethod As an even urther developent o the lainate approxiation approach descried in the previous sections, the odeling can e extended y also explicitly including a ier length distriution (FD unction. One such odel, which is ased on the original lainate analogy approach (AA, is reerred to as the lainate-plate ethod [3;4;5;9]. The real coposite is in this odel, as in the original AA, replaced y a layered short-ier odel coposite. The ters set y the authors or the real coposite and the odel coposite will e adopted in this report. FFI-rapport 0/00 9

30 In the lainate-plate ethod, it is assued that the overall ier orientation can e descried y a proaility unction or the in-plane ier orientation. Hence, in the siulated coposite it is assued that no iers are pointing out o the plane. The loading direction is in the in-plane direction. Thus, the out-o-plane direction is perpendicular to the loading plane. The siulated, or odel, coposite is irst deined y a set o lainae, each with a given ier length. Hence, the ier length is constant or each lainate. Second, each lainate with a given ier length is again split up in to a set o lainae, where each lainate has a given ier orientation. In this way, all iers in each lainate have the sae length and orientation. Fro this, odels or aligned shortier coposites with a constant ier length, such as the Halpin-Tsai odel, ay e applied in the odeling, see Section... A sketch o the siulated coposite is shown in Figure in [3]. A lainate-plate odel, very siilar to the one just descried, has also een presented y Xia et al. [7]. In their odel, the lainate is uild up o a core layer and two (outer skin layers. The skin layers are split up into layers with iers o equal length and with in-plane ier orientation. ach layer is then again split up into a set o lainae with in-plane unidirectional ier orientation. The core layer, on the other hand, is irst split up into a set o lainae, with a given ier length or each lainate, ut with out-o-plane ier orientation. Next, each lainate is again split up into lainae with unidirectional ier orientation projected onto the in-plane plane. A sketch o the odel coposite in this case is shown in Figure in [7]. This particular odel will, however, not e urther discussed in this report. In the next section, general expressions are given or estiating the eective Young s odulus using the lainated-plate ethod in [3;4;5;9]. These expressions contain the longitudinal elasticity odulus, as well as a ier length distriution unction (FD and a ier orientation distriution unction (FOD. The expressions applied or the longitudinal elasticity odulus will e given in Section.5.. The expressions or the distriution unctions are already given in Section.. To recapitulate, the FD and FOD unctions given in (.9 and (.3, respectively, are applied y Fu and auke [3;4;5] in estiating the Young s odulus o a isaligned shortier coposite. I including these proaility unctions, the ier length and the ier orientation can explicitly e taken into account. Siilarly, the FD and FOD unctions ro (.36 and (.40, respectively, are eployed y Chin et al. [9] or the sae purpose; Kacir et al. [30] used the sae FD, ut the FOD was replaced y the unction in (.4. The lainate-plate ethod ay hence e seen as an extension o the odels or the eective properties o a randoly oriented short-ier coposite descried in sections.,.3, and General expressions or the eective properties o the coposite In the sae way as descried in Section.4, aterial properties are given reerred to the local ier syste or each ply in a layered coposite. These axes generally ake angles with respect to the gloal coordinate syste or the coplete coposite. 30 FFI-rapport 0/00

31 The relationship etween the coponents o the stiness atrix in the gloal, or o-axis, syste and the local ier syste, can e expressed as [5], ' 4 4 Q n n 4n ' 4 4 Q n n 4n Q ' 4 4 Q n n n 4n Q ' Q66 n n n ( n ' Q 6 n n n n ( n n Q ' Q 6 n n n n ( n n. (.7 Q 66 Here, cosθ and n sinθ, and the coponents o the right hand side vector Qij are given in (.47. Fro the siulated coposite, the overall stiness properties are ound y suing the contriution ro each layer, where the layer aterial properties are ultiplied y the layer thickness. However, when introducing continuous distriution unctions or the variation in ier length and ier orientation, the overall stiness atrix is expressed y an integral, that is [3], lax θax ' ij ij ( ( θ θ, (.7 lin θin A Q l g dld where 0 lin l lax <, and 0 θin θ θax /. Inserting the expressions in (.7 into (.7, the integrand o each coponent o the stiness atrix, A, can e written as a product o a ier length dependent unction and a ier orientation ij dependent unction. This yields, lax lin T lax lin lax lin T T θax 4 ( cos ( A l dl θg θ dθ in θax 4 ( sin ( l dl θg θ dθ in ax T ( cos sin ( l dl g d T θin lax θax 4 ( cos sin ( lin θin G l dl θ θg θ dθ θ θ θ θ θ θ θ (.73 FFI-rapport 0/00 3

32 lax lin T lax lin lax lin T T θax 4 ( sin ( A l dl θg θ dθ in θax 4 ( cos ( l dl θg θ dθ in ax T ( cos sin ( l dl g d T θin lax θax 4 ( cos sin ( lin θin G l dl θ θg θ dθ θ θ θ θ θ θ θ (.74 lax lin T lax lin lax lin T T T T T T θax ( cos sin ( A l dl θ θg θ dθ lax lin lax lin in θax ( cos sin ( l dl θ θg θ dθ in θax 4 ( cos ( l dl θg θ dθ ( l dl θ θ θ in θax 4 sin g( in θ θ dθ 4 θax ( cos sin ( θin G l dl θ θg θ dθ θ (.75 lax 66 lin T lax lin lax lin T T θax ( cos sin ( A l dl θ θg θ dθ lax lin lax lin lax lin in θax ( cos sin ( in ax T ( cos sin ( l dl g d T θin l dl θ θg θ dθ θax 4 ( cos ( G l dl θ g θ dθ in θax ( cos sin ( θin θ in θ θ θ θax 4 ( sin ( θ θ θ θ G l dl θ θg θ dθ G l dl θg θ dθ θ (.76 3 FFI-rapport 0/00

33 The stiness coponents aove are applicale to dierent longitudinal elasticity odulus unctions, as well as various ier length and ier orientation distriution unctions. Furtherore, the eective coposite engineering tensile properties are given y [3] AA A, (.77 A AA A, (.78 A G A, ( A A A A A A. (.80 As an extension o the integral expression in (.7, Fu and auke [8] included an additional unction or the ier orientation distriution, see Section.. In this case, the stiness atrix or the coposite is expressed as lax θax φax ' ij ij ( ( θ ( φ θ φ, (.8 lin θin φin A Q l g g dld d with ( ΘΦ, eing the loading direction, or the direction o the easureent. This latter odel is discussed no urther in this report, ut ore details can e ound in the reerred paper [3]..5. The longitudinal Young s odulus As indicated in the previous section, the two irst coponents o the right hand side vector,q, in (.7 contain the longitudinal elasticity odulus,. Furtherore, the is generally a unction o the ier length, and ust thereore e kept within the integral expression or the coposite stiness atrix. The other aterial paraeters or a unidirectional layer do not depend on the ier length. Dierent expressions or the longitudinal Young s odulus are ound in the literature, and two o the will e descried in the ollowing. An oten eployed approach or odeling the longitudinal elasticity odulus is using the Halpin- Tsai expression, given in (.6. A second approach is applying the shear-lag odel y Cox [], reerred to in [3], tanh( βl / V ( V βl /, (.8 FFI-rapport 0/00 33

34 where V is the ier volue raction, odulus o the atrix, as eore, and is the elasticity odulus o the iers, is the G β ( r ln( R/ r /, (.83 where G is the shear odulus o the atrix, and R is a paraeter or the ean separation o the iers to their length. For a hexagonal packing o the iers, the logarithic actor in the denoinator can e expressed y R ln ln, r 3V (.84 while a square packing o the iers leads to R ln ln. r V (.85 These two odel approaches or the longitudinal elasticity odulus will e applied in the ollowing..5.3 Three odel cases With the aove general expressions and odels or calculating the eective Young s odulus, we now descrie three odels, naely the Fu-auke odel [3;4;5], the Chin et al. odel [9], and the Kacir et al. [30] odel. In each case, the distriution unctions or the ier length and the ier orientation deine the odel. Furtherore, or each odel the expression or the longitudinal Young s odulus applied ay e varied. Two variants have een descried in the previous section, that is, the Halpin-Tsai expression and the Cox shear-lag expression. Also, taking into account that the shear-lag odel includes two dierent ways o packing the iers, we then end up with three dierent expressions or each odel, resulting in three dierent integral expressions or the stiness atrix. In addition to this, diverse assuptions are ade or the solution o the integral The Fu-auke odel The FD and FOD unctions applied in the Fu-auke odel [3;4;5] are given in (.9 and (.3, respectively. These unctions are sustituted into the expressions or the coponents o the stiness atrix in (.73 to (.76. Due to the cutting process (ro a ier tow and the production process o the short-ier coposite, the ier length is assued to e in the range ro l in to the cut length o the iers, l ax. For the ier length dependent integral, an explicit expression or the solution is, however, not availale. This is at least the case or Matla, which is 34 FFI-rapport 0/00

35 applied or the calculations; oth the FD unction and the longitudinal Young s odulus are generally depending on the ier length. Moreover, or the ier orientation dependent integral, a solution is not availale or all ranges o ier orientations. Fro assuing that the orientation o the iers are in the range ro zero to /, i.e. 0 θin θ θax /, the solution o this integral can e explicitly expressed y the Γ unction. Due to the challenges o solving the ier length dependent integral, is, as a irst approxiation, assued to e constant. In this way, we can ove the ter containing the aterial paraeters outside the integral, and liit the integration to the FD unction. An analytical expression or the integral o the FD unction alone is availale, and given in Section.. The ier length in the expression or the constant longitudinal elasticity odulus is set to either the ean ier length or the odal ier length. In the Fu-auke odel, the ean ier length is calculated ro ean ( / ( /. (.86 0 l l l dl a Γ d Moreover, the ost proale ier length, i.e. odal length, can e ound ro setting 0 dl, giving l od a a /. (.87 In the aove expressions the shape paraeters a and are tuned such that the odal and ean values are within the ier length interval, i.e. larger than or equal tol in, and saller than, or equal to, l ax. Fro these assuptions, the coponents o the lainate stiness atrix can e written as Γ ( q Γ ( q A exp( al exp( al Γ( q Γ ( q in ax T Γ ( Γ ( q exp( alin exp( alax Γ( Γ ( q T T Γ ( q Γ( Γ ( q exp( alin exp( alax Γ( q Γ( Γ ( q T T Γ ( q Γ ( Γ ( q 4G exp( alin exp( alax Γ ( q Γ ( Γ ( q (.88 FFI-rapport 0/00 35

36 Γ ( Γ ( q A exp( al exp( al Γ( Γ ( q in ax T Γ ( q Γ ( q exp( alin exp( alax Γ( q Γ ( q T T Γ ( q Γ( Γ ( q exp( alin exp( alax Γ( q Γ( Γ ( q T T Γ ( q Γ ( Γ ( q 4G exp( alin exp( alax Γ ( q Γ ( Γ ( q (.89 Γ ( q Γ ( Γ ( q A exp( al exp( al Γ( q Γ( Γ ( q in ax T Γ ( q Γ ( Γ ( q exp( alin exp( alax Γ( q Γ( Γ ( q T T Γ ( q Γ ( q exp( alin exp( alax Γ ( q Γ ( q T T Γ ( Γ ( q exp( alin exp( alax Γ( Γ ( q T T Γ ( q Γ ( Γ ( q 4G exp( alin exp( alax Γ ( q Γ ( Γ ( q (.90 Γ ( q Γ ( Γ ( q A exp( al exp( al Γ( q Γ( Γ ( q 66 in ax T Γ ( q Γ ( Γ ( q exp( alin exp( alax Γ( q Γ( Γ ( q T T Γ ( q Γ ( Γ ( q exp( alin exp( alax Γ ( q Γ ( Γ ( q T T Γ ( Γ ( q exp( alin exp( alax Γ( Γ ( q T T Γ ( q Γ ( q G exp( alin exp( alax Γ ( q Γ ( q Γ ( q Γ ( Γ ( q G exp( al in exp( al ax Γ ( q Γ ( Γ ( q Γ ( Γ ( q G exp( alin exp( alax Γ ( Γ ( q (.9 In the second case, the elasticity odulus is assued to e a unction o the ier length, and can hence not e oved outside the integral. Because no analytical solution o the ier length 36 FFI-rapport 0/00

37 dependent integral is availale, nuerical integration is used. One solution is eploying the Sipson s rule, which reads, a a ( x dx ( a 4 ( 6. (.9 a Please note that or the Fu-auke odel, only the eective Young s odulus in (.77 is valid or the real coposite, whereas the rest o the expressions are valid or the odel coposite The Chin et al odel In the Chin et al. [9] odel case, the FD and FOD unctions eployed are given in (.36 and (.40, respectively. For the sae reasons as descried or the Fu-auke odel, see Section.5.3., we assue that the orientation o the iers are in the range ro zero to /, i.e. 0 θ θ θ /. In this way, an explicit solution o the integral is availale. Also, no in ax explicit solution o the ier length dependent integral is availale. As or the Fu-auke odel in the previous section, we irst apply a constant value or, such that the aterial paraeters can e put outside the integral expression. An analytical solution o the integral o the FD unction is given in Section.. For the calculations, the odal and ean ier length is taken ro the Fu-auke odel. These values are applied in the expression or the longitudinal elasticity odulus, as well as or the paraeters or the log-noral distriution unction. Hence, the values o a and are equal to the values in the Fu-auke odeling. Fro these assuptions, the coponents o the stiness atrix in this case ecoe, 4 λ 6λ 4e 4 ( ln µ in ( µ ln ax T 4 A er l er l ( λ 0λ 64( e T ( ln er µ lin er ( µ ln lax T 4 4 e λ 6e λ 4e 4 ( λ 0λ 64( e T e ( ( ln er µ lin er ( µ ln lax T ( λ 6( e ( e 4 G er ( µ ln lin er ( µ ln lax λ ( λ 6( e (.93 FFI-rapport 0/00 37

38 A ( ln er µ lin er ( µ ln lax T 4 4 e λ 6e λ 4e 4 ( 0 64( λ λ e 4 T λ 6λ 4e 4 er ( µ ln l in er ( µ ln lax T λ 4 ( λ 0λ 64( e (.94 T e ( ( ln er µ lin er ( µ ln lax T ( λ 6( e ( e 4 G er ( µ ln lin er ( µ ln lax λ ( λ 6( e e ( ( ln µ in ( µ ln ax T A er l er l ( λ 6( e e ( er ( µ ln lin er ( µ ln lax T T ( λ 6( e T ( ln er µ lin er ( µ ln lax T 4 λ 6λ 4e 4 4 ( 0 64( λ λ e T er ( ln l µ in er ( µ ln lax T 4 4 e λ 6e λ 4e 4 ( λ 0λ 64( e (.95 e ( µ in µ ax 4 G er ( ln l er ( ln l ( λ 6( e 38 FFI-rapport 0/00

39 e ( 66 ( ln µ in ( µ ln ax T A er l er l ( λ 6( e e ( er ( µ ln lin er ( µ ln lax T T ( λ 6( e T e ( ( ln er µ lin er ( µ ln lax T ( λ 6( e 4 λ 6λ 4e 4 ( µ ln in ( µ ln ax 4 G er l er l ( λ 0λ 64( e (.96 e ( µ in µ ax G er ( ln l er ( ln l ( λ 6( e 4 4 e λ 6e λ 4e ( µ ln in ( µ ln ax λ 4 G er l er l ( λ 0λ 64( e Instead o applying a constant value or the longitudinal elasticity odulus, which is calculated y inserting the odal or the ean ier length, one ay ind an averaged odulus where the FD unction is included. Chin et al. [9] suggest that the ean longitudinal elasticity odulus could e calculated ro ( 0 ( l ( l dl 0 ( l dl. (.97 This approach has, however, not een studied urther in this report. A third approach is applying nuerical integration or the ier length dependent part, in the sae way as descried or the Fu-auke odel The Kacir et al odel The odel y Kacir et al. [30] is ipleented in alost the sae way as the Chin et al. odel. e The only dierence is the lack o the constant actor ( in the denoinator o the FOD unction. Because the exponential ter is approaching zero or λ 0, which is the value typically applied or getting the ain ier orientation in the direction o the applied load, the two odels produce the sae result or the eective Young s odulus or the coposite. FFI-rapport 0/00 39

40 .6 The paper physics approach (PPA Cox conducted a pioneer work on what is coonly reerred to as the paper physics approach (PPA []. Several others have later applied and reerred to this approach, see e.g. [3;38] and the reerences therein. In the PPA, the key point is calculating the orce across a scan line in the test specien. This involves irst calculating the nuer o iers with length l and orientation θ that crosses this (iaginary scan line. Second, the axial orce in a ier o length l and orientationθ is calculated, and the load-direction coponent o this orce is ound. Third, the calculated loaddirection axial orce is ultiplied y the nuer o iers crossing the scan line. Finally, this quantity is integrated over the ier length and ier orientation distriution, to calculate the total orce sustained y the iers crossing the scan line. According to Fu and auke [3], which reer to the paper y Jayaraan and Kortschot [38], the elastic odulus o the short-ier coposite in the load direction can, in the context o the PPA, e expressed as, _ χχ V ( V. (.98 In this expression, χ and χ are ier length and ier orientation actors, respectively, and given as, lax tanh( βl / χ l ( ldl l βl/ (.99 ean lin and θax (cos (sin (cos g( d, (.00 θin χ θ θ θ θ θ where 0 lin l lax and 0 θin θ θax /. For illustration and or relating to the lainate-plate ethod in Section.5, we apply the sae FD and FOD unctions or the PPA. In the irst case, inserting the distriutions unctions o the Fu-auke odel in (.9 and (.3, the integral expressions in (.99 and (.00 ecoe lax tanh( βl / χ al exp( al dl l βl/ (.0 ean lin 40 FFI-rapport 0/00

41 and χ θ θ ax in θ θ θ θ θ θ 4 p q cos cos sin {sin( } {cos( } d θax p q {sin( } {cos( } d θ θ θ θ (.0 in Γ ( q Γ ( q p Γ ( q Γ ( p Γ ( q p Γ ( q Γ( q Γ ( q Γ( q Γ( where the latter expression is ased on the act that 0 θin θ θax /, as also assued or the Fu-auke odel, see Section In the second case, applying the Chin et al. odel in Section.5.3., the distriution unctions ecoes, lax tanh( βl / exp[ (ln / ]d χ l s l l µ βl/ s (.03 ean lin and χ θ θ ax in 4 λθ cos θ cos θsin θ λe dθ e 4 λ 6λ 4e 4 ( e e e 4 λ ( 0 64( ( 6( λ λ λ λ (.04 where the latter expression again is ased on the act that 0 θin θ θax /. Both odels include the Cox shear-lag unction, where the value o β depends on the packing o the iers. FFI-rapport 0/00 4

42 3 Suary and uture work In this report, dierent odels are descried or calculating the eective odulus o randoly oriented short-ier reinorced coposite aterials. The survey is not at all coplete, ut gives soe exaples on possile odeling approaches. The overall purpose o this study is to estalish a undaent or urther research within the ield o discontinuous ier coposites. The next step will e to apply the short-ier odel expressions to nanocoposites. I required, adjusted variants, or new odels, will e estalished. At this point in the study o discontinuous ier coposites, suicient experiental tests have not een perored. This will e included in later studies. Acknowledgeents The author would like to thank Harald Osnes (University o Oslo and FFI and Bernt B. Johnsen (FFI or useul discussions, reading the docuent, and or suggesting iproveents. 4 FFI-rapport 0/00