Boundary conditions for unit cells from periodical microstructures and their implications

Transcription

1 Bondar conditions for nit cells from periodical microstrctres their implications Shgang Li To cite this ersion: Shgang Li. Bondar conditions for nit cells from periodical microstrctres their implications. Composites Science Technolog, Elseier, 29, 68 (9), pp <1.116/j.compscitech >. <hal > HAL Id: hal Sbmitted on 27 Sep 21 HAL is a mlti-disciplinar open access archie for the deposit dissemination of scientific research docments, hether the are pblished or not. The docments ma come from teaching research instittions in France or abroad, or from pblic or priate research centers. L archie oerte plridisciplinaire HAL, est destinée a dépôt et à la diffsion de docments scientifiqes de niea recherche, pbliés o non, émanant des établissements d enseignement et de recherche français o étrangers, des laboratoires pblics o priés.

2 Accepted Manscript Bondar conditions for nit cells from periodical microstrctres their implications Shgang Li PII: S (7)153-4 DOI: 1.116/j.compscitech Reference: CSTE 3664 To appear in: Composites Science Technolog Receied Date: 4 December 26 Reised Date: 27 March 27 Accepted Date: 3 March 27 Please cite this article as: Li, S., Bondar conditions for nit cells from periodical microstrctres their implications, Composites Science Technolog (27), doi: 1.116/j.compscitech This is a PDF file of an nedited manscript that has been accepted for pblication. As a serice to or cstomers e are proiding this earl ersion of the manscript. The manscript ill ndergo copediting, tpesetting, reie of the reslting proof before it is pblished in its final form. Please note that dring the prodction process errors ma be discoered hich cold affect the content, all legal disclaimers that appl to the jornal pertain.

3 BOUNDARY CONDITIONS FOR UNIT CELLS FROM PERIODICAL MICROSTRUCTURES AND THEIR IMPLICATIONS Shgang Li School of MACE, The Uniersit of Manchester Sackille Street, Manchester M6 1QD, UK Abstract The most important aspect in formlating nit cells for micromechanical analsis of materials of patterned microstrctres is the deriation of appropriate bondar conditions for them. There is lack of a comprehensie accont on the deriation of bondar conditions in the literatre, hile the se of nit cells in micromechanical analses is on an increasing trend. This paper is deoted to generation of sch an accont, here bondar conditions are deried entirel based on considerations of smmetries hich are present in the microstrctre. The implications of the bondar conditions sed for a nit cell are not alas clear therefore hae been discssed. It has been demonstrated that nit cells of the same appearance bt nder bondar conditions deried based on different smmetr considerations ma behae rather differentl. The objectie of the paper is to inform sers of nit cells that to introdce a nit cell one need not onl mechanicall correct bondar conditions bt also a clear sense of the microstrctre nder consideration. Otherise the reslts of sch analses cold mislead. 1 Introdction Micromechanical analses hae been on an increasing trend in order to nderst the behaior of modern materials ith sophisticated microstrctres, e.g. fibre or particlate reinforced composites, tetile composites, etc.. Unit cells are often resorted to in order to facilitate sch analses. The introdction of a nit cell is sall based on certain assmptions, sch as a reglar pattern in the microstrctre, hich is sometime a reasonable approimation or an idealisation otherise. A reglar pattern offers certain smmetries hich can then be emploed to derie the bondar conditions for a nit cell introdced for micromechanical analsis. Seeral acconts on sstematic se of smmetries for the deriation of the bondar conditions for nit cells hae been presented b the athor in [1-3]. In the literatre, there are man acconts here simplistic bondar conditions hae been imposed to nit cells in an intitie manner, sometimes, rather casall ithot mch jstification. In [4], bondar conditions hae been so introdced that bondar effects hae been broght in a significant effort has been made there to inclde more cells to 1

4 form larger nit cells to dilte the bondar effects. This old be absoltel nnecessar, had the bondar conditions been deried appropriatel. In man pblications [4-9], to name bt a fe, bondaries hae been assmed to remain flat or straight after deformation to delier simple bondar conditions, hich cannot sall be flfilled hen the material is sbjected to the macroscopic shear deformation. Sch simplistic bondar conditions are correct in a fe special cases, e.g. sqare nit cells ith reflectional smmetric microstrctre hen the are sbjected to a deformation corresponding to a macroscopic direct strain. Een so, the do not come ithot implications on the patterns of the microstrctres, hich do not seem to hae been gien an attention hitherto. Another confsing isse is ho man bondar conditions need to be prescribed at an gien part of the bondar of a nit cell. Sometime, onl one displacement has been prescribed bt in some other times more are prescribed. In [1], eqilibrim or compatibilit conditions ere imposed. No clear eplanations can be fond in the literatre. This paper is deoted to the isses as raised aboe, in particlar, the confsing isses associated ith the se of reflectional smmetries. Becase of the natre of the smmetries emploed, the bondar conditions obtained for the nit cells are often in a form of eqations relating displacements on one part of the bondar to those on another. This ma impose restrictions on the applications of these bondar conditions hence the nit cells. For instance, hen finite elements are emploed for the micromechanical analsis, as is often the case, the mesh to be generated mst possess identical tessellation beteen the parts of bondar hich are related throgh those eqation bondar conditions. This cold sometimes be difficlt to achiee for 3-D problems, sch as in particle reinforced or tetile composites. The constraints in eqation form ma not be aailable in some codes. It is therefore desirable to aoid sch eqation bondar conditions hereer it is possible. 2 Sfficient necessar nmber of bondar conditions for nit cells In the literatre, it is often fond, e.g. in [11], that the nmber of bondar conditions prescribed at the same part of the bondar aries from case to case. Withot appropriate jstification, it is presented as a rather confsing matter. As a reslt, incorrect sages are often fond, e.g. in [4]. Some of the confsions reslt from the se of finite elements hich is sall based on a ariational principle of some kind in hich some bondar conditions, called natral bondar conditions, are satisfied atomaticall as a part of the ariation process shold not be imposed. Almost all commercial FE codes are displacement-based bilt on the basis of the minimm total potential energ principle or the irtal displacement principle. In sch codes, traction bondar conditions 2

5 are natral bondar conditions the sers do not need to impose an constraints in order to hae them satisfied apart from an appropriate prescription of tractions, if an, as part of the load to the strctre nder analsis. It shold be emphasised that natral bondar conditions shold not be imposed prior to the ariation process. It does not help to obtain a more accrate reslt bt, rather on the contrar, it ma preent the total potential energ to reach its minimm in the soltion space hence lead to less accrate reslt. To illstrate, a simpl spport beam as shon in Fig. 1 nder niforml distribted load is considered as a simple eample. Assming an approimate deflection pattern as a qadratic fnction 2 a + b + c, after imposing the ero deflection bondar conditions at both ends, the minimisation of the total potential energ reslts in a non-dimensional deflection ( 1 ) 2 as opposed to the eact soltion of ( 1 )( 1 ) +. The maimm deflection is 2% less than the eact soltion hile the moment- natral bondar conditions hae onl been satisfied approimatel as far as the assmed deflection fnction allos. The approimate total potential energ is -2/6 as opposed to the eact soltion of -21/6. Hoeer, if one imposes the moment bondar conditions (natral bondar conitions) at both ends to the qadratic deflection fnction prior to ariation, the beam old not deflect at all, gien 1% error in deflection the total potential energ is hich is obiosl a mch orse approimation to -21/6 than -2/6. Figre 1 A simpl spported beam nder niforml distribted load According the theor of continm mechanics for the deformation problem of materials in 3-D space, at an gien point on the bondar, there reqire three prescribed bondar conditions in an logical combination of displacements tractions. For instance, for a bondar perpendiclar to the -ais, the three bondar conditions can be a prescription of the folloing or or or here, are displacements in, directions respectiel are direct shear stress components ith sbscripts in their conentional sense. In terminolog of partial differential eqations, displacement bondar conditions are bondar condition of the first kind 3 (1)

6 traction ones are the second kind. There cold be a third kind corresponding to elastic spport phsicall, hich are hoeer irreleant to the present topic hence dropped from present considerations. When the bondar conditions are imposed in a form of eqations relating the displacements or tractions on one part of the bondar to those of another part of the bondar, the eqation bondar conditions shold be imposed to. Instead of three bondar conditions on one part of the bondar, there are si bondar conditions for to parts of the bondar. (2) Bearing in mind that traction bondar conditions are natral bondar conditions in conentional FE analsis, the ill be left ot of the prescription list. For eample, if a part of bondar is sbjected to prescribed it is sfficient necessar to prescribe onl on this part of bondar for the FE analsis. Appling the same argment to eqation bondar conditions, it is obios that eqation constraints hae to be imposed to all three displacements to be both sfficient necessar, hereas eqations for tractions can shold be left ot, as in [2,3]. Becase of the eistence of natral bondar conditions hich do not need to be imposed, the same part of the bondar nder different loading cases ma be sbjected to different nmbers of bondar conditions, especiall hen reflectional smmetries are emploed. Another sorce of confsion is the natre of smmetr. The loading deformation can be smmetric as ell as antismmetric. Distingish one from another is essential in derie appropriate bondar conditions associated ith smmetr. For eample, hen a deformable bod smmetric abot the -plane (=) is nder smmetric loading, e.g. stretching in the -direction, there is one bondar condition on the smmetr plane, i.e. = 4

7 hile the to remaining traction bondar conditions shold not be prescribed. Hoeer, hen the same bod is sbjected to antismmetric loading, e.g. shear in the - plane, there ill be to bondar conditions on the smmetr plane, i.e. = = there is onl one traction bondar condition in this case, hich shold not be imposed. Another potential sorce of confsion is associated ith bending problem, hen elements, sch as beams, plates shells, are inoled, here displacements are in a generalised sense, i.e. nodal rotations are considered as displacements. Bondar conditions described in terms of bending moments shear forces are generalised traction bondar conditions hence natral bondar conditions in the terminolog of ariational principles. To conclde this section, it is clear that at a bondar of a nit cell, the nmber of bondar conditions to be imposed before a finite element analsis can be condcted is not definite. It depends on the natre of the smmetries adopted in the definition of the nit cell. Hoeer, one thing remains definite, hich shold gide the introdction of bondar conditions, i.e. onl displacement bondar conditions shold be imposed, not the natral bondar conditions. When bending is inoled, displacements shold inclde generalised displacements, sch as nodal rotations. 3 Selection of nit cells their implications For argment s sake, a microstrctre in 2-D space of a sqare laot as shon in Fig. 2a is considered first, hich can be perceied as, bt not restricted to, a transerse cross-section of a UD composite or an in-plane pattern of a tetile composite. As the onl smmetries aailable are translations, in directions, respectiel, the nit cell as shon in Fig. 2b ill hae to be sbjected to eqation bondar conditions as gien in [2] in terms of displacements hile ignoring the traction ones. Hoeer, if the material nder consideration allos one to idealise it into a microstrctre as shon in Fig. 3a, the repetitie cell as shon in Fig. 3b old be the nit cell of smallest sie if onl translational smmetries are emploed. Obiosl, the sie of the nit cell can be redced to that as shon in Fig. 3c after the aailable reflectional smmetries abot aes hae been tilised. As a reslt, bondar conditions can be obtained ithot eqations relating the displacements on the opposite sides. Haing sed the reflectional smmetries, one needs to bear in mind to isses 5

8 associated ith a nit cell as gien in Fig. 3c, hich can be easil oerlooked. Firstl, the microstrctre of the material the nit cell as in Fig. 3c represents is gien in Fig. 3a not as in Fig. 2a althogh the appearances of the nit cells in Fig 2b Fig. 3c look identical. Secondl, some macroscopic strain states, in particlar, associated ith shear, are antismmetric nder the reflectional smmetr transformations. Appropriate considerations shold be gien to the antismmetric natre hen bondar conditions are deried from the smmetr transformations. As a reslt, the nmber of bondar conditions for some load cases old be different from that for the other cases. (a) Fig. 2 Sqare packing (b) (a) (b) (c) Fig. 3 Sqare packing ith frther reflectional smmetries Sometimes, for patterns like the one as shon in Fig. 4a, nit cells of onl a qarter of the sie as shon in Fig. 4c are seen in the literatre. It, in fact, reslts from eactl the same consideration as 6

9 in Fig. 3. A qarter is sfficient onl becase the presence of the reflectional smmetries abot the aes ithin the repetitie cell as shon in Fig. 4b. Man reglar shapes of the inclsion possess these smmetries, sch as diamond, rectangle circle, bt there are also shapes hich do not possess sch smmetries, e.g. that in Fig. 2a. The conditions implied b a qarter sie nit cell are the eistence of reflectional smmetries, as in Fig. 3. The bondar conditions for a qarter sie nit cell shold be deried in eactl the same a as that for the nit cell in Fig. 3c. (a) (b) (c) Fig. 4 An eqialent laot to that in Fig. 3 The ltimate nit cells obtained from Fig.3 Fig. 4 share the same appearance. Hoeer, the are nder different bondar conditions associated ith different microstrctres. An obios conseqence of the difference in the microstrctre is that the one in Fig.4a is macroscopicall orthotropic hile that in Fig. 3a is not necessaril the case, as ill be seen later throgh an eample. Users of nit cells shold be aare of the difference decide if the difference bears an significance for their particlar applications hile choosing the nit cell to be emploed. Another reglar pattern often encontered in the literatre is heagonal one. Argment similar to aboe for the sqare pattern applies to a large etent. The onl difference is that there are more as to epress the translational smmetries as discssed fll in [1,2]. Whether the repetitie cell sed to epress the translations smmetries can be frther redced in sie depends on the eistence of the other smmetries, inclding reflectional rotational. Withot sch additional smmetries, the smallest sie old be a complete heagon as shon in Fig. 5, if one is prepared to emplo translations in direction not perpendiclar to each other. Otherise, to trade for translations in directions onl, one ill hae to deal ith nit cell a sie bigger as shon in the rectangle in 7

10 Fig. 5, hich is obiosl not a niqe choice. When analsing these nit cells, in general, eqation bondar conditions ill hae to be emploed. Fig. 5 Heagonal packing Smallest nit cell possible bt ith eqation bondar conditions Simplest nit cell ithot eqation bondar conditions Fig. 6 Heagonal packing ith reflectional smmetries Hoeer, if the microstrctre possesses additional smmetries, tpicall, reflectional smmetries ithin the repetitie cells, the sie of the nit cells can be frther redced as shon in Fig 6. The trapeim nit cell is apparentl the smallest in sie. Hoeer, it ill inole some bondar conditions in form of eqations. The shaded rectangle nit cell can be deried from eqation bondar conditions, hich remains as the simplest nit cell as far as the bondar conditions are concerned. Unit cells of other shapes as sggested in [1,2] ill definitel bear more complicated 8

11 bondar conditions. The bondar conditions for the rectanglar nit cell can be deried in the same as as those as presented earlier for the sqare pattern. It shold noted that those nit cells of redced sies as shaded in Fig. 6 cannot be obtained ithot reflectional smmetries abot ertical horiontal aes. 4 Bondar conditions for a nit cell from 3-D microstrctre ith reflectional smmetries Pt the case as illstrated throgh Fig. 3 into a 3-D scenario. The bondar conditions can be deried in general as follos, assming the periods of translational smmetries in the, directions are 2b, 2b 2b, respectiel. This is general enogh to encapslate reglar packing laots sch as simple cbic ith b = b = b = b, bod centred cbic ith 4 face centred cbic ith b = b = b = b close packed heagonal ith 2 4 b = b = b = b, 3 b 4 = b, b = 2b 2 b = 2 3b, respectiel, here b is the characteristic radis, i.e. the radis of largest sphere hich can be accommodated, as defined in [3] for each packing. Obiosl, the sie of the nit cells for bod centred cbic, face centre cbic close packed heagonal packing are no longer as compact as in [3]. Assme that there eists an intermediate repetitie cell eqialent to that in Fig. 3b hich can represent the material fll sing translations smmetries onl this cell is defined in the domain b b b b b b (3). The materials is sbjected to a set of macroscopic strains {,,,,, } γ γ γ hich can be introdced as si etra degrees of dom (d.o.f.) in a FE analsis, e.g. as si indiidal nodes, each haing a single d.o.f., or si degrees of dom of a special node. Upon an of these etra d.o.f. s, a concentrated force can be applied in order to impose a macroscopic stress to prodce a macroscopicall niaial stress state. The translational smmetries reqire: 9

12 = 2b = b = b = = b = b = = b = b = = b = b = = b = b = = b = b (4) nder translation in -direction; = 2b γ = b = b = = b = b = 2b = b = b = = b = b = = b = b = = b = b (5) nder translation in -direction, = 2b γ = b = b = 2b γ = b = b = 2b = b = b = = b = b = = b = b = = b = b (6) nder translation in -direction. The form of man of the aboe eqations is not niqe, especiall, those associated ith shear, depending on the a rigid bod rotations are constrained. For instance, the first to eqations in (5) can be replaced b = = 2b + 2b γ = b = b = b = b = b γ = 2 b + b γ = b = b = b = b ithot affecting the reslts. The lack of niqeness contribtes to the likelihood of confsion hen introdcing bondar conditions for nit cells. The se of bondar conditions deried from translational smmetries alone as shon in (4)-(6) has been illstrated fll in [3]. It is the interest of the present paper to derie appropriate bondar conditions hen frther reflectional smmetries are present in the intermediate repetitie cell as defined aboe. To appl frther reflectional smmetries abot, planes, the problem has to be considered separatel for indiidal loading cases epressed in terms of macroscopic stresses {,,,,, } as presented in the folloing sbsections. In deriing the bondar conditions in the folloing sbsections, the principle of smmetries ill be emploed, hich states that smmetric stimli, i.e. loads, reslt in smmetric responses, 1

13 inclding displacements, strains stresses, hile antismmetric stimli prodce antismmetric responses 4.1 Under Consider first the -faces of the nit cell, i.e. those perpendiclar to the -ais. as a stimls is smmetric nder reflection abot -plane (perpendiclar to -ais). Responses, are smmetric hile, are antismmetric. On the smmetr plane (=), the smmetr conditions reqire = = = = = = = = = hich can be re-epressed as = = = = = = = = The onl bondar condition in effect on side = is = = = = = = = = = = = = = = = =. = = (9) hile traction bondar conditions are natral bondar conditions shold not be imposed. (7) (8) In the aboe, the conditions on =, = = do not ield an constraints the shold hence be left, as indicated b paper.. The same notation ill be adopted for the rest of the Considering the opposite faces at = ± b, appling the smmetr condition, one has = = b = b = = b = b = = b = b = = b = b = = b = b = = b = b In conjnction ith the translational smmetr conditions as gien in (4), one obtains. (1) 11

14 = b = b = = b = b = = b = b = b = b = b = b = = =. (11) The onl sriing bondar condition on side = b is = b b =. (12) The bondar condition aboe introdces an etra d.o.f. into the sstem. In an FE analsis, in order to impose a macroscopic stress, an appropriate concentrate force can be applied to this d.o.f.. The macroscopic effectie stress can be orked ot from the concentrated force easil as discssed in [3] hile the obtained nodal displacement at this etra d.o.f. gies the effectie macroscopic strain directl. Consider no the to opposite faces at = ± b. The stimls is also smmetric nder reflection abot -plane (perpendiclar to -ais). Responses, are smmetric hile, are antismmetric. Hence, on the smmetr plane (=) the smmetr conditions reqire = = = = = = = = = = = = = = = = = = (13) hich can be reritten as = = = = = = = = = = = = = b = b =. (14) Leaing the natral bondar conditions aside, a single bondar condition on side = is obtained = =. (15) The smmetr conditions on the to opposite faces at = ± b reqire 12

15 = = b = b = = b = b = = b = b = = b = b = = b = b = = b = b. (16) The aboe in conjnction ith (5) lead to = b = b = b = = b = = b = b = b = = b = b = b = = (17) hich reslt in a single bondar condition for side = b = b b = (18) The bondar condition aboe introdces another etra d.o.f. into the sstem. To impose a niaial macroscopic stress, d.o.f. shold be left, i.e. the etra d.o.f. gies this macroscopic strain directl. =. The nodal displacement at Appling the same argments, the bondar conditions on sides = = b can be obtained as = = = b b =, respectiel (19) here the etra d.o.f. as introdced throgh translational smmetr conditions (6). To impose a macroscopic stress alone, shold be left, i.e. etra d.o.f. gies this macroscopic strain directl. =. The nodal displacement at the To smmarise, nder a macroscopic stress, the bondar conditions on the three pairs of the sides of the nit cell are gien b (9), (12), (15), (18) (19) = = = = b b = = = = = b b = (2) = = b b here etra d.o.f. is sbjected to a concentrated force associated ith shold be left to prodce a macroscopicall niaial stress state., hile 13

16 4.2 Under With similar considerations as gien aboe, the bondar conditions for the nit cell nder are identical to those in (2). The onl difference is that it shold be the etra d.o.f. that is sbjected to a concentrated force associated ith, hile left to prodce a macroscopicall niaial stress state. The nodal displacements at those etra d.o.f. s gie the corresponding macroscopic strains directl. 4.3 Under The bondar conditions are again identical to those in (2). Hoeer, the etra d.o.f. shold be sbjected to a concentrated force associated ith, hile macroscopicall niaial stress state. left to prodce a 4.4 Under The natre of shear stresses is slightl more complicated than their direct conterparts. With respect to a reflectional smmetr, one of the three shear components is smmetric hile other to are antismmetric. Under the reflection abot the -plane, the stimls is smmetric. The responses, are smmetric hile, smmetr plane, =, the smmetr conditions reqire = = = = = = = = = hich can be reritten into = = = = = = = = are antismmetric. Hence, on the = = = = = = = = = = = = = = = =. Ignoring the traction bondar conditions, the onl bondar condition to be imposed is = =. (23) (21) (22) 14

17 On the opposite faces at = ± b, the reflectional smmetr conditions are similar to (21) bt on = ± b instead of =. In conjnction ith the translational smmetr conditions (4), the lead to the bondar condition = =. (24) b Consider no the pair of sides parallel to the -plane. The stimls is antismmetric abot - plane (=). The responses, are smmetric hile, are antismmetric. Hence, on the smmetr plane, the smmetr conditions reqire = = = = = = = = = = = = = = = = = =. (25) Rerite = = = = = = = = = = = = = = =. (26) From (26), the bondar conditions on = are obtained as = =. (27) = = Notice that there are to displacement bondar conditions in this case as opposed to the = plane on hich there is onl one bondar condition as gien in (24). The hae to be imposed in order to define the nit cell properl nder this loading condition. There is one traction bondar condition = = hich has been ignored as a natral bondar condition in an FE analsis. Appling the reflectional smmetr to the to opposite faces at = ± b, one obtains = = b = b = = b = b = = b = b = = b = b = = b = b = = b = b. (28) Combining the aboe ith (5), one obtains 15

18 = b = = b = b = b = = = b = b = b = = = = b = b (29) hich lead to the folloing bondar conditions for side = b = =. (3) = b = b Similarl, the bondar conditions on = antismmetric abot -plane = = = = = b can be obtained, bearing in mind that is = = b = b b γ = (31) here the etra d.o.f. γ is introdced throgh the translational smmetr conditions (6), hich can be associated ith = instead of b = if the rigid bod rotation of the nit cell is constrained b differentl. There ill be no difference hatsoeer as far as the deformation is concerned. The same applies to the consideration on the to sbseqent loading cases ithot frther eplanations. To impose a macroscopic stress, a concentrate force can be applied to the d.o.f. γ. The nodal displacement at γ obtained after the analsis gies the corresponding macroscopic strain directl. Since is not constrained on = = b, these faces do not hae to remain flat after deformation. As a smmar, all bondar conditions for the nit cell nder macroscopic stress are as follos = =, = = = = b = =, = = (32) = b = b = =, = = = & = b γ. = b = b Notice that there are different nmbers of conditions on different sides. In general, smmetr reslts in one condition hile antismmetr to. The same applies to the sbseqent shear loading cases here details of deriation are omitted. 4.5 Under After considering all smmetr conditions, the bondar conditions for the nit cell can be obtained 16

19 as = = = = = = = b = b = = = = (33) b = = = = = b γ & =. = b = b 4.6 Under The corresponding bondar conditions are = = = = = = = b = b = = = = = b γ & = (34) = b = b = = = =. b It has been shon in this section that ith reflectional smmetries additional to translational ones, nit cells can be formlated ith rather conentional bondar conditions (2) for loading in terms of microscopic stress, or, (32) for, (33) (34) for hich do not inole eqations associating the displacements on opposite sides of the nit cell. The price to pa is the fact that nder different loading conditions, different bondar conditions ma hae to be emploed. 5 2-D problems The 3-D presentation of bondar conditions can be easil degenerated to 2-D problems in the - plane, for argment s sake, inclding plane stress, plane strain, generalised plane strain problem anticlastic problem. The appl to the rectanglar nit cell as obtained from the heagonal laot as shon in Fig. 6 as ell as to the sqare one as from Fig. 3. The are gien as follos ithot detailed deriations. 5.1 Under When a 2-D nit cell, in the - plane, is sbjected to macroscopic stresses or, the bondar conditions are the same as belo = = b = = b 17

20 = = = b b =. (35) The difference is a concentrated force needs to be imposed to the etra d.o.f. or to achiee these to macroscopicall niaial stress states, respectiel. 5.2 Under The bondar conditions for a nit cell nder macroscopicall niaial shear stress plane are as follos = = = = b in the - = = = b b γ = (36) A concentrated force at the etra d.o.f. γ deliers the macroscopicall niaial shear stress states. As on bondar =, diplacement is not constrained in an form there is no restriction hether the side shold remain straight after deformation. The same applies to all other sides. 5.3 Generalised plane strain problem macroscopicall niaial stress state For generalise plane strain problems, an etra d.o.f., in addition to, γ, has to be introdced, hich can be dealt ith in the same manner as other etra d.o.f. s corresponding to macroscopic strains. This etra d.o.f. shold be left hen appling macroscopicall niaial stress states bt constrained for as is antismmetric nder the reflectional smmetr hile is smmetric. It shold be pointed ot that neither plane stress nor plane strain is capable of reprodcing the macroscopicall effectie niaial stress states nder hich effectie properties are measred eperimentall according to their definitions. For UD composites, the generalised plane strain problem is the onl 2-D formlation hich is capable of achieing macroscopicall effectie niaial stress state. When appling a macroscopicall niaial stress state, the bondar conditions are the same as in (35). Hoeer, the concentrated force shold be applied to the etra d.o.f. hile leaing. The nodal displacements at these etra d.o.f. s, 18, gie the macroscopic strains directl, hich can be sed to ork ot the effect Yong s modls in -direction

21 Poisson ratios associated ith direction. ACCEPTED MANUSCRIPT 5.4 Under in an anticlastic problem The anticlastic problem in the - plane inoles onl one displacement. When macroscopicall niaial shear stress can be obtained as = = is applied, from sbsection 4.5, the bondar conditions for the nit cell = b b γ = (37) hile edges = =b are left. A concentrate force can be applied to the etra d.o.f. γ to delier a macroscopicall niaial stress state the nodal displacement at γ gies this macroscopic strain directl. Similar argments appl to the macroscopicall niaial stress state the corresponding bondar conditions are obtained = = hile edges = =b are left. from sbsection 4.6, = b b γ = (38) 6 Deformation of the sides of nit cells As eamples of the applications of the bondar conditions deried aboe, seeral eamples of nit cells hae been analsed. Particlar attention in this section ill be paid to the deformation of the sides of the nit cell, hich do not alas remain flat/straight after deformation, hen bondar conditions hae been deried imposed rigorosl D nit cell for particle reinforced composites ith simple cbic particle packing A nit cell for simple cbic packing as presented in [3] a mesh as generated there ith a spherical geometr for particle appropriate constitent material properties in the eamples. The same ill be adapted the reflectional smmetries in the nit cell as presented in [3] ill be made se of frther. As a reslt, onl an octant is reqired as the nit cell for the present analsis. Withot losing generalit, onl macroscopicall niaial stress states The bondar conditions are as gien in (2) (32), respectiel. are eamined here. 19

22 Under macroscopicall niaial stress state, the reslts are identical to the reslts as shon in [3] hen the corresponding octant is taken ot of the nit cell in [3] for comparison. This shold not ndermine the nit cells formlated in [3] as the present one are onl applicable if the particle possesses reqired reflectional smmetries. It can be noted that all sides remain flat after deformation. This is imposed b the bondar conditions as in (2). The same is epected hen the nit cell is nder other macroscopic stress or or an combination of,. When the nit cell is sbjected to a macroscopicall niaial shear, the on Mises stress contor plot is shon in Fig. 7. The reslts obtained here also agree identicall ith those in [3], althogh the corresponding contor plot as not shon in [3]. According to bondar conditions as gien in (32), onl the -faces, i.e., = =b, hae to remain flat after deformation, hile the bondar conditions on the remaining faces impose no restriction in this regard. As a reslt, the remaining to pairs of faces, i.e. -faces -faces, do not hae to remain flat. Fig. 7 illstrated the cred trend for these to faces. The cratre of these faces redces as the disparit of properties beteen the particle the matri redces. In fact, flat faces are epected hen the particle matri share identical properties. The same obseration applies to the nit cell hen it is sbjected to either of the to remaining macroscopic stresses,. The onl difference is that the faces remaining flat after deformation become -faces -faces instead, respectiel, hile other faces arp after deformation, in general. Figre 7 Deformation of a nit cell for particle reinforced composite ith a simple cbic packing It shold be noted that hile (2) applies to an or an combination of macroscopic direct stress state, (32) is onl applicable to. Bondar conditions (33) (34) hae to be sed for, respectiel. One has to trn back to the nit cell as proposed in [3] if an combination of 2,

23 has to be applied D nit cell for UD fibre reinforced composites ith sqare fibre packing Appling bondar conditions as gien in Section 5 aboe, 2-D nit cells can be analsed pertinent to UD fibre reinforced composites ith circlar fibre cross-section. The eamples here correspond to the cases as pblished in [2]. Hoeer, nit cells of smaller sies hae been sed here, taking adantage of reflectional smmetries present in the problem. As in [2], generalise plane strain problem applies to the problem for macroscopic stress states,,, -ais being along the fibre hile the anticlastic problem for macroscopic stress states can be analsed sing heat transfer as an analog to aoid 3-D modelling. Once again, perfect agreement in reslts can be obtained beteen the nit cells presented here those in [1,2] for both sqare packing heagonal packing. Similar obserations to those in their 3-D conterparts in the preios sbsection can be made on the deformation of the sides of the nit cells. Under direct macroscopic stress states, all the sides of a sqare nit cell remain straight after deformation. Hoeer, nder other loading conditions or for heagonal nit cells, sides ma not remain straight after deformation as shon in Fig.8 nless the fibre bears the same elastic properties as the matri. For macroscopic stress states, the sides ma look straight from the perspectie along the -ais (fibre direction) bt the - plane itself arps into a cred srface. The sides are in fact cred in space. (a) (b) (c) Figre 8 Cred edges in deformed nit cells (a) sqare nit cell nder macroscopic transerse shear (1MPa) (b) heagonal nit cell nder macroscopic transerse shear (1MPa) (b) heagonal nit cell nder macroscopic transerse tension (1MPa) The objectie of the eamples in Figs. 7 8 is to demonstrate that the sides of the properl 21

24 established nit cells do not alas remain flat/straight after deformation. Intitiel formlated nit cells assming flat/straight sides after deformation are incorrect in general. The nderling principle for the formlation of nit cells is the principle of smmetries, hile intition is often sbject to limitations. Once the smmetries present in the microstrctre in the material hae been made proper se of, correct nit cells can be obtained. The bondar conditions deried in this a ill not reslt in an bondar effect as presented in [4]. In fact, if one analsis an assembl of cells, e.g. the one as shon in Fig. 3b, the reslts obtained ill be eactl the same as those from the analsis of that in Fig. 3c, proided that the bondar conditions hae been imposed correctl in both cases. 7 Effects of microstrctres implied b different nit cells The nit cells as sketched in Fig. 2b 3c bear the same geometr bt are sbjected to different bondar conditions. The are therefore different nit cells. The differences do not alas reslts in different reslts, especiall hen the inclsions (fibre or particle) possess sfficient reflectional smmetries. Hoeer, it cold be badl rong if one is sed blindl in place of the other, in particlar, hen the inclsions do not sho the reqired smmetries. The prpose of this section is to illstrate sch differences as nothing in the literatre seems to sggest that the implications hae been fll recognised. Assme a 2-D microstrctre inoling inclsions of an elliptical cross-section inclined at 3. The ellipse is of 2:1 aspect ratio occpies a olme fraction of 4%. The elastic properties of the inclsion the matri are assmed as list in Table 1. The same mesh as shon in Fig. 9 ill be sed for both nit cells corresponding to microstrctres as shon in Fig. 2 3, respectiel. The on Mises stress contor plots at deformed configrations nder macroscopic stress states (=1MPa) are presented compared in Fig. 1. It is obios that differentl assmed microstrctres as implied b the to different nit cells reslt in different stress distribtions microscopicall. The differences are een more prononced hen effectie properties are etracted from these nit cells compared as listed in Table 2 here properties η ij are defined as the ratio of shear strain γ j to the direct strain i hen the nit cell is sbjected to a macroscopicall niaial direct stress state i µ ij as the ratio of shear strain γ j to shear strain γ i hen the nit cell is sbjected to a macroscopicall pre shear stress state i [12]. These properties in the material s principle ais anish for orthotropic material, as is the case for the nit cell corresponding to Fig. 3c. Material represented b the nit cell corresponding to Fig. 2b is not orthotropic bt monoclinic in 22

25 general relatie to the - aes as shon in Fig. 9. The differences as illstrated here ill disappear hen the ellipse is replaced b a circle bt this is not a sfficient reason for ignoring the differences. When a nit cell is sed, the ser oght to be clear abot the implications of the nit cell adopted on the microstrctre of the materials, e.g. the one in Fig. 2 or the one in Fig. 3, hich are apparentl different enogh from each other. Fig. 9 Mesh for the nit cell ith an reinforcement of an elliptical cross-section (a) (b) (c) Figre 1 Defromation on Mises stress contor plots (a) nit cell corresponding to Fig. 2b nder (b) nit cell corresponding to Fig. 3c nder (c) nit cell corresponding to Fig. 2b nder (d) nit cell corresponding to Fig. 3c nder 23 (d)

26 Table 1 Properties of the constitents for the nit cells Properties Inclsion Matri E 1 GPa 1 GPa ν.2.3 Table 2 Effectie properties corresponding to the nit cells Effectie properties Unit cell in the sense of Fig. 2b Unit cell in the sense of Fig. 3c E GPa 4.65 GPa E GPa GPa E GPa 1.89 GPa G GPa.671 GPa G GPa.7144 GPa G GPa GPa 23 ν ν ν η η η µ Conclsions Unit cells for micromechanical analses hae to be introdced ith de consideration of the microstrctres implied b the nit cell. Bondar conditions for nit cells representing microstrctres of periodic patterns shold follo entirel from the smmetries present in the microstrctre the nit cell represents rather than from one s intition. The smmetries inclde translations, reflections rotations. Using translations alone leads to bondar conditions in form of eqations relating displacements on opposite sides of the bondar of the nit cell. Frther se of reflection smmetries, if the eist, can aoid sch eqation bondar conditions, making the application of bondar conditions easier. Hoeer, sers mst be aare of the differences in the microstrctres implied b the bondar conditions for the nit cell. Althogh nit cells ma look 24

27 identical geometricall, different bondar conditions imposed old associate the nit cell ith rather different microstrctres. It has been illstrated in this paper that sch differences in the microstrctres ma reslt in rather different effectie properties of the composites represented b the nit cells. References [1] Li S. On the nit cell for micromechanical analsis of fibre-reinforced composites. Proc Ro Soc Lond A 1999; 455: [2] Li S. General nit cells for micromechanical analses of nidirectional composites. Composites A, 21; 32: [3] Li S, Wongsto A. Unit cells for micromechanical analses of particle reinforced composites. Mech Mater 24; 36: [4] Fang Z, Yan C, Sn W. A. Shokofeh W. Regli, Homogeniation of heterogeneos tisse scaffold: A comparison of mechanics, asmptotic homogeniation, finite element approach. Applied Bionics Biomechanics 25; 2:17-29 [5] Peng X, Cao J. A dal homogenisation finite element approach for material characterisation of tetile composites. Composites, Part B 22; 33:45-56 [6] Llorca J, Needleman A, Sresh S. An analsis of the effects of matri oid groth on deformation dctilit in metal-ceramic composites. Acta Metall Mater 1991; 39: [7] Adams DF, Crane DA. Finite element micromechanical analsis of a nidirectional composite inclding longitdinal shear loading. Compters Strctres 1984; 18: [8] Nedele MR, Wisnom MR. Finite element micromechanical modelling of a nidirectional composite sbject to aial shear loading. Composites 1994; 25: [9] Zahl DB, Schmader S, McMeeking RM. Transerse strength of metal matri composites reinforced ith strongl bonded continos fibres in reglar arrangement. Acta Metall Mater 1994; 42: [1] Yeh JR. Effect of interface on the transerse properties of composites. Int J Solids Strctres 1992; 29: [11] Aitharaj VR, Aerill RC. Three-dimensional properties of oen-fabric composites. Composites Science Technolog1999; 59: [12] Lekhniskii SG. Theor of Elasticit of an Anisotropic Elastic Bod. San Francisco: Holden Da,