Modelling of Electronic Textiles

Transcription

1 Moding of Ectronic Txtis M.T.J. Fontyn MT0.07 uprvisors: Dr. P.C.P. Boutn Dr. Ir. R.H.J. Prings Prof. Dr. Ir. M.G.D. Grs Phiips Rsarch, Eindhovn Photonic Matrias and Dvics Univrsity of Tchnoogy, Eindhovn Dpartmnt of Mchanica Enginring Mchanics of Matrias March, 00

2 Abstract A mod is prsntd for prdiction of strsss and strains in ctronic txti. Charactristic structura bhaviour of txti is impmntd in th mod by using a combination of truss mnts and a continuum matria. Th rotation of th truss mnts rsuts in a non-inarity and voving anisotropy in th rspons, which is cosy ratd to that of th tru matria. Tnsi tsts ar prformd in thr orintations to dtrmin matria paramtrs usd in th mod. Ths tsts ar prformd in combination with imag procssing to acquir goba and oca dformation information of th txti. Tsts ar prformd in ths and othr orintations and compard and with numrica tnsi tsts as vaidation. A study on th ffct of intgration of stiff two-dimnsiona componnts in fabric is prformd. Th infunc of componnt stiffnss, siz, shap and th distanc btwn th componnts on issus as strss concntrations and substrat rspons ar studid making us of anaytica and numrica cacuations. Infunc of componnt siz with rspct to ths issus is drivd anayticay. i

3 Contnts Introduction. Probm dscription. Rsarch Goa..3 Currnt stat of art...4 tratgy foowd 3.5 Outin 4 Charactristics of th usd txti. Matria.. 5. Mchanica rspons. 6 3 Moding of txti 3. tructur of th mod Rspons in diagona dirction Finit mnt moding Vaidation 5 4 Paramtr idntification 4. Tst stup and quipmnt 7 4. Optica mthod Fitting xprimnta rsuts. 0 5 Numrica tnsi tsts 5. Marc-Mntat mod and 90 dgrs (Warp and wft) Dgrs (Diagona) 5.3. Ovra comparison Contraction of th samp and orintation of th yarns Out of pan dformation Prdicting strain of yarns Infunc of truss stiffnss and 75 dgrs and 60 dgrs ii

4 6 Moding of txti with Componnts 6. Unit c mod Rsuts rfrnc mod Rsuts of paramtr variations 6.3. Variation of componnt stiffnss Variation of componnt siz Anaytica mod, dformation in diagona dirction Gnra Infunc truss proprtis Infunc truss spacing Comparison of anaytica and numrica mod Infunc of componnt pattrn Discussion Concusion 50 Rfrncs 53 Appndics A.. 55 B.. 57 C. 6 D. 64 E. 65 E. 73 iii

5 trsss and strains σ Γ Enginring strain Tru strain Enginring strss Tru strss har strain Matria paramtrs E v G Young s moduus Poisson s ratio har moduus Dimnsions h A Lngth in unit c/mnt Thicknss of samp and mod Cross sction ara L iz of unit c with componnt D Componnt siz d pacing btwn trusss = / Forc F Forc in truss Vctors v i Tnsors E α Warp Wft Diagoba train tnsor trss tnsor Haf of th ang btwn warp and wft yarns. 0 Dgrs 90 Dgrs 45 dgrs iv

6 Introduction inc th invntion of incandscnt ight bub ight sourcs and othr ctronics hav undrgon trmndous dvopmnts which hav mad ctronics appicab at xtrmy sma scas and for many purposs. Ovr th ast dcad thr is a growing trnd towards fxib ctronics and ctronic componnts attachd on or intgratd in fxib substrats ik txtis. Many appications ar imaginab with fxib ctronics, as thy can b intgratd in for xamp coths or furnitur. Howvr, intgration of ctronics in fxib substrats inducs nw changs concrning riabiity and mchanica faiur which hav to b ovrcom. As a first stp towards addrssing ths changs th mchanica bhaviour of a txti substrat with and without componnts is invstigatd.. Probm dscription Txtis ar compx matrias which consist of on or mor ayrs of yarns that ar wovn in a mor or ss priodic structur. Txtis on which ctronic componnts ik LEDs ar attachd and in which th conductiv wirs ar an intgra part of th (wovn) substrat ar so cad ctronic txtis. Txti is usd as a compiant substrat on which rativy stiff componnts such as.g. LEDs and ICs ar fixatd. An xamp of ctronic txti with LEDs is shown in figur.. Th conductiv wirs usuay hav matria proprtis that ar diffrnt from thos of th non-conductiv yarns in th txti. Particuary th stiffnss mismatch btwn componnts and substrat rsuts in strss concntrations and may compromis th intgrity of th substrat and th connctions of th componnts. This may caus probms concrning riabiity and manufacturing. Th matria usd in this study is an orthogonay wovn mutiayrd txti with a matrix of conductiv wirs intrwovn with th warp and wft yarns. Th conductiv Eitx wirs ar poystr wirs with a mta coating. A matrix structur is cratd in which th conductiv warp yarns ar wovn in th ayr at th opposit sid of th ayr with th conductiv wft yarns and ths ayrs ar sparatd by a ayr of non-conductiv yarns. Via connctions ar mad by intrwaving conductiv warp and wft yarns at spcific points within th matrix. Fig.. LEDs attachd to txti with conductiv yarns. ourc: Phiips Ectronics.

7 In ordr to b ab to prdict dformations and strss concntrations nar componnts quaitativy as w as quantitativy, and to prdict th infunc of matria and gomtric proprtis of th componnts, it is ncssary to hav a mod of th txti substrat. This mod woud aow on to optimiz componnt proprtis such as gomtry th usd matria, and th componnt ocation with rspct to othr componnts and conductiv wirs, without producing and tsting a possib variations.. Rsarch goa Th ong trm rsarch goa towards which th prsnt study contributs is th dvopmnt and optimization of txti substrats and th fixation of componnts on thm in ordr to dvop robust and durab appications with ctronic txtis. Th goa in this study is th dvopmnt of a numrica matria mod which dscribs th matria bhaviour of th txti propry in a quaitativ and quantitativ way. Th mod is to b usd to prdict strsss and strains around componnts attachd to a txti. Essntia paramtrs in th dscription of th txti s bhaviour wi b obtaind from xprimnts..3 Currnt stat of th art Txtis ar quit an od fid of rsarch. Howvr, during th ast dcad numrica moding of txti has bcom a possibiity. Initiay ths matrias wr modd as a continuum, but dvopmnts in computr scinc mad muti sca moding and moding of gomtrica dtais possib. Th smar th sca, th mor dtaid th mod can b. Moding at th micro v concrns th intrna structur of th yarns in trms of fibrs and th intraction btwn th fibrs and prhaps a fw yarns in a rativy sma part of th txti. An xamp of micro sca moding is th muti chain mnt moding as don by Zhou (004). This mod dscribs th dformation of yarns by moding th mutua contact btwn fibrs in a yarn and th contact btwn ths fibrs and thos in th orthogona undrying yarns. Msoscopic mods ar mosty FEM mods in which th yarns ar modd without gomtrica dtais such as fibrs. An xamp of a msoscopic program is Wistx, which mods a 3- dimnsiona unit c of th txti and computs th shap using th princip of minimum bnding nrgy (Vrpost & Lomov, 005), or FEM mods, which mod th comprssion of soid yarns undr shar oading or frictiona contact mods (Lin t a., 008, Chrouat t a., 008). For spcia appications such as impacts,.g. for armours, aso visco-astic mods ar avaiab. (Liu t a., 006) At th macroscopic v txti is modd as a continuum in which no yarns and fibrs ar distinguishd, but th txti is modd as a uniform matria. Non-orthogona continuum mods with a covariant coordinat systm hav bn dvopd by Xu t a. (003) and Png t a. (004). Ths orthotropic matria mods ar mainy inar astic. Aso isotropic hyprastic mods ar avaiab (Ruíz & Gonzáz, 006). A macroscopic mod has bn dvopd arir at Phiips Rsarch (Fron, 008b). This mod is an orthotropic astopastic mod. It uss a Hi48 yid critrion combind with a Nadai- Ludwik hardning aw. Unoading bhaviour is inar astic as no pastic dformation occurs during unoading. Th constitutiv mod has its origin in sht mta forming. An advantag of th mod is that it is a continuum mod, thus computationay fficint. A disadvantag of th mod is th arg numbr of paramtrs, nin in tota, which hav to b dtrmind

8 xprimntay and do not a hav a car physica maning for txti. A major imitation of th mod is that th non-inarity of th oading curvs can not b fittd dirction indpndnt as th powr of th hardning aw dtrmins th (isotropic) hardning bhaviour in a dirctions. Onc th powr of th hardning aw is fittd th ovra stiffnss can b fittd by changing th yid strss but th inarity of th curv cannot b tund. As rsut of this th oading bhaviour cannot b dscribd propry in a dirctions, which is iustratd in figurs. and.3. In txti th non-inarity is strongy dpndnt on th orintation. Fig.. Fit with Nadai-Ludwik hardning in warp dirction. Fig..3 Insufficint dscription of non-inar oading bhaviour..4 tratgy foowd For a bttr dscription of th dirction dpndncy of th non-inarity a nw matria mod is dvopd in which th structura bhaviour of th txti is incorporatd. As a mod at macroor mso-v is th most appropriat for this study th mod must not contain much gomtrica dtai. In contrary to th continuum mod dvopd by Fron th nw mod consists of a combination of truss mnts which mods th intraction of yarns, and a compiant continuum matria rsuting in a mod which is a mix of a msoscopic and a macroscopic dscription. Combinations of truss and continuum mnts ar usd bfor by Chrouat (008), but without th ink to physica or masurab proprtis of txti. For th nw mod no mor than 4 paramtrs ar ncssary as th anisotropy and non-inarity paramtrs ar capturd naturay by th truss mnts. Masurmnts on txti ar prformd in thr dirctions, i.. 0, 90 and 45 dgrs with rspct to th warp yarns in ordr to acquir input paramtrs for th matria mod. Loca strain data from ths xprimnts is obtaind with optica anaysis and input paramtrs for th mod ar idntifid making us of ast squar fits and an anaytica drivation of th mod rspons in 45 dgrs dirction. Tnsi tsts at othr orintations ar prformd and aso ths tsts ar compard with numrica tnsi tsts to vaidat th mod. Finay numrica cacuations ar prformd to mod txti with stiff componnts attachd to it and study th ffct of componnt proprtis and componnt pattrns on th rsuting strsss and strains in th substrat. 3

9 .5 Outin In chaptr th proprtis of th txti usd in this study and its mchanica rspons ar dscribd. Th mod is prsntd in chaptr 3 and th rspons of it is drivd anayticay. Th xprimnta mthods and paramtr fitting mthod ar dscribd in chaptr 4 and th mod is vaidatd on xprimnts in chaptr 5. In chaptr 6 a numrica study to th rspons of txti with componnts is prsntd. Th ast chaptr contains concusions, discussion and rcommndations. 4

10 Charactristics of th usd txti. Matria Thr ar many typs of txti, among thm knittd, wovn and non-wovn txtis can b distinguishd. Ony wovn txtis ar within th scop of this study, so knittd and non-wovn txtis wi not b discussd. Within th catgory of wovn txtis orthogona and nonorthogona wovn txtis can b distinguishd. Both ar quit common, but th orthogona wovn txtis ar mor natura choic for th conductiv matrix structurs dsird in ctronic txtis. Wovn txtis can ithr b mono-ayrd or muti-ayrd, but hr too muti-ayrd structurs ar ndd for ctronic txtis. During production of th txti, th warp yarns ar strtchd in a oom and rpatdy iftd in a crtain combination and ordr whi th wft yarns pass through. Th ordr of passing and ifting dtrmins th structur and numbr of ayrs of th txti. Procssing conditions ik yarn tnsion and yarn tmpratur hav a major infunc on th proprtis of th txti. Howvr, th production of txti is not within th scop of this study. Fig.. A top viw of a backit mutiayrd orthogona wav. Fig.. Cross sctions of imprgnatd txti. Th txti usd in this study is an orthogonay wovn thr ayrd txti. This txti is shown in figur.. In this imag th orthogona wft and warp dirctions ar cary visib. Figur. shows cross sctions of th txti at two orintations. To mak th cross sction imags th matria was first imprgnatd with a siicon rubbr to fix th yarn and fibr positions during cutting. In ths imags th thr diffrnt ayrs can b cary distinguishd and it is cary visib that th warp yarns ar dns and straight bunds of fibrs and th wft yarns ar oos bunds curvd around th warp yarns. Fig..3 Connction btwn th diffrnt ayrs. Fig..4 Cross sction of on conductiv Eitx wir. 5

11 Th connction btwn th diffrnt ayrs is mad by wft yarns wrappd around th warp yarns of th opposing ayr at svra pacs. In figur.3 such connction has bn visuaizd by indicating th warp yarns with circs and a wft yarn with a dashd curv. Ths connction points ar aso visib as th ight spots in figur.. In th usd txtis aso conductiv yarns ar intgratd. Figur.4 shows a cross sction of an imprgnatd txti with such a conductiv Eitx wir. Th conductiv yarns ar cary thickr than th othr yarns. om proprtis of th txti ar istd in tab.. Th numbr of warp yarns is.9 pr miimtr and th numbr of wft yarns is 3. pr miimtr in ach ayr, which quas 5.7 warp yarns and 9.6 wft yarns pr miimtr in th txti. Th masurd thicknss is about 0.6 mm, howvr, it cannot b masurd accuraty as th matria consists of oos bunds of fibrs and yarns as visuaizd in figur.. Th thicknss of th samps has bn masurd with a Hidnhain sty with a fat tip at svra pacs and th found vaus ar avragd. Proptris txti Orthogona pain wav, Matria Thr ayrs Poystr Ara Dnsity g/m Thicknss 0.6 mm Warp Yarns /mm /ayr.9 Wft Yarns /mm /ayr 3. Tab. Proprtis of th txti usd in this study.. Mchanica rspons To iustrat th mchanica bhaviour of th txti tnsi tsts hav bn prformd in warp, wft and diagona dirction. For a fair comparison it is prfrab to tst samps with a ngth which is much argr than th width so that th ffct of inhomognous dformations nar th camps on th tota masurd dformation is ngigib. At th sam tim th samp width must b arg nough so that boundary ffcts at th cut dgs hav imitd infunc. amps usd in this study ar imitd to a ngth of 00 mm du to th dimnsions of th avaiab matria. This is smar than dsird, howvr it is not unrasonaby sma as th txti has a dnsity of 5.7 and 9.7 yarns pr miimtr in rspctivy warp and wft dirction. Th width of samps is 6 mm and th tstd ngth is 60 mm. Th samps wr oadd in 5 cycs with maximum forcs of 5N, 0N, and 5N (3x). Th unoading rspons givs an imprssion of th inastic bhaviour of th matria. 6

12 Fig..5 trss-strain rspons in warp and wft dirction. Fig..6 trss-strain rspons in diagona dirction. To rduc viscoastic ffcts th strain rats wr kpt ow by using xtnsion rats of 0.0 mm/s in warp dirction, 0.0 mm/s in wft dirction and 0.0 mm/s in diagona dirction. Th duration of th tsts was approximaty 400 to 600 sconds in a dirctions. Th rsponss obtaind in th tnsi tsts, xprssd in nginring strsss and strains, ar shown in figurs.5 and.6. For th cacuation of th nginring strss in th txti th initia cross sction ara is usd. Th initia width tims th thicknss is usd as rfrnc ara for th nginring strss. It has to b mntiond that th nginring strss is not a masur for th ra strss in th fibrs and yarns. Th curvs shown in figurs.5 and.6 cary dmonstrat th anisotropy of th matria. Th ovra strain in wft dirction is about thr tims as high as in warp dirction at an qua nginring strss of nary MPa. Th stiffr rspons in warp dirction is du to th fact that th warp yarns ar initiay virtuay straight and hav a highr fibr dnsity compard to th wft yarns. Th ovra strain at maximum oad in diagona dirction is about thirty tims highr than in warp dirction. Aso notab ar th high non-inarity of th oading bhaviour in diagona dirction and th argr rsidua dformation compard to th warp and wft dirctions. Fig..7 Thr diffrnt dformation mods can b distinguishd in a tnsi tst in diagona dirction: tnsion in rgion A, shar in both warp and wft dirction in ara C, and shar in ithr warp or wft dirction in aras B and B. Dformations as rsut of tnsion in rgions A ar rativy sma. 7

13 Th strss-strain curvs of th tnsi tsts ar basd on th masurd camp dispacmnt and th initia samp dimnsions. Howvr, th nginring strain is an avrag vau as th dformation btwn th camps is non-uniform, particuary in diagona dirction. In figur.8 an imag of a samp tstd in diagona dirction is shown. tructura ffcts can b obsrvd in this imag. Th trianguar aras markd A bhav as rativy stiff aras du to th constraind introducd by th camps. Th oading in ths aras is mainy tnsion. In th midd of th samp, which is markd C, th dformations ar arg. In figur.8 a cos-up of ara C is shown in th undformd stat. Th dformd stat of this ara is shown in figur.9. As rsut of th prscribd ongation, th yarns in this ara rotat towards th tnsi dirction. Th dominating dformation mod is thrfor shar. Th strongy non-inar oading bhaviour obsrvd in figur.6 is causd by this dformation mod. Th transitiona aras B and B ar oadd in both tnsion and shar. Fig..8 Undformd txti. Yarns ar orthogonay orintd. Fig..9 Dformd samp, yarns rotatd towards oading dirction Th diffrnc btwn th rsponss in warp, wft and diagona dirctions is arg. This is not surprising as txti is in fact a structur. Th arg dformations and th non-inar rspons in diagona dirction ar causd by th arg rotation of th yarns. Incorporating this structura bhaviour in a matria mod coud rsut in a bttr dscription of th mchanica bhaviour without using many fitting paramtrs. 8

14 3 Moding of th txti In this chaptr th matria mod usd in our numrica study is dscribd in dtai. Anaytica drivations of th strss-strain bhaviour ar prsntd, which ar usfu for th paramtr idntification and to gt an imprssion of th rspons of th mod. This anayticay drivd rspons is aso compard with numrica cacuations to vaidat som assumptions mad in th drivation. 3. tructur of th mod Th dvopd mod is a combination of astic truss mnts, which rprsnt th tnsi rspons of th yarns in th txti, and astic continuum mnts to mod othr ffcts such as friction btwn th yarns and comprssion of th yarns. Figur 3. iustrats how th mod is constructd. Th truss mnts ar connctd at th cornrs of th continuum mnts and ar aignd with th dgs of th continuum mnts. Th connctions at th cornrs, which can b compard with pin joints, ink th dispacmnts of th continuum mnts and th truss mnts without constraining th rotations. Th combination of a continuum matria and truss mnts impis that th mod is party a constitutiv mod and party a structur. Mry introduc th truss mnts incrasd stiffnss in warp and wft dirction. Thy do not ncssariy rprsnt individua yarns so th mod is an intrmdiat soution btwn a msoscopic and a continuum mod. Fig. 3. Th matria mod is a combination of continuum mnts and truss mnts which ar connctd at th cornr nods. trtching in diagona dirction rsuts in rotation of th trusss with rspct to ach othr, mimicking th rativ rotation of yarns. 9

15 Th mod is kpt rativy simp by choosing inar astic matria proprtis for th trusss as w as for th continuum, but if it is dsird, non-inar (visco-) astic or pastic proprtis can b impmntd. Th stiffnss of th trusss is significanty highr than that of th continuum. As a rsut, th trusss act as a mchanism which rstricts th dformation of th continuum. Dformation of th continuum bcoms arg compard to that in th trusss in oading dirctions diffrnt from th warp and wft dirction. Non-inar rspons is obtaind by using arg dformation thory with ogarithmic strsss and strains in th continuum matria. Contrary to yarns in txti th trusss can withstand comprssion oading if no additiona critrion is impmntd. Ony numrica cacuations of txti with componnts wi b prformd with additiona critrion. Figur 3. shows th mod s undformd and dformd stat in diagona dirction, and shoud b compard with th undformd and dformd matria as shown in figurs.8 and.9. In this diagona dformation mod, th strss-strain rspons is govrnd by th compiant continuum matria as th rativy stiff trusss hardy dform. Not that in this mod th (atra) strains in th continuum ar dtrmind by th orintation of th truss mnts. In othr dformation dirctions comprssion or ongation of th stiffr truss mnts wi b ncssary and thus a rativy stiff rspons is obsrvd. Th rotation of th truss mnts rsuts in a noninarity and voving anisotropy in th rspons, which is cosy ratd to that of th tru matria. Thr spcific dformation mods can b distinguishd, in ach of which th rspons is dominatd by on of th thr sts of mnts, so thir proprtis can b fittd indpndnty. In figur 3. ths thr mods ar iustratd, with th rd coor indicating th mnts which dominat th strss-strain rspons. Warp Wft Diagona Fig 3. Diffrnt mnts dtrmin th strss-strain rspons in ths thr spcific dformation mods. In warp and wft dirction th trusss in rspctivy warp and wft dirction govrn th strssstrain rspons, sinc th continuum mnts ar compiant compard to thm and th atra trusss ar oadd ony indircty via contraction of th continuum. On th othr hand, in th diagona dirctions th truss proprtis hav itt infunc on th rspons as thy can b sn as rigid bars in this mod. As a rsut th strss rspons is govrnd by th continuum mnts rspons to th dformation imposd on thm by th truss ntwork. This xpctation wi b vrifid by anaytica drivations and numrica cacuations in th nxt sctions. To dtrmin th matria proprtis of th truss mnts and th continuum matria, th cross sction ara of th truss mnts and th thicknss and moduus of th continuum hav to b dfind. Th stiffnss of th truss mnts, which is a product of th Young s-moduus and th cross sction ara, dtrmins th rspons. This mans in princip that th ara or moduus can b chosn arbitrariy. This is aso th cas for th thicknss of th continuum matria. As ong as 0

16 ony th in pan stiffnss is important and out of pan dformations (bnding) ar not considrd. In figur 3.3 it is iustratd how th cross sction aras ar dfind. Th dimnsions of th continuum matria ar chosn qua to th dimnsions of th samp, which mans that th thicknss of th continuum is qua to th masurd samp thicknss h. Th sum of th cross sction aras of th individua truss mnts is st qua to th cross sction ara of th samp, and thus qua to th cross sction ara of th continuum matria. Th cross sction ara of an individua truss is thrfor dfind to b qua to th spacing btwn th trusss, d, mutipid by th samp thicknss h. Not that this mans that th Young s moduus of th trusss is diffrnt from th Young s moduus of th yarns, vn though th stiffnss of th yarns is accountd for corrcty. Fig 3.3 Th sum of th aras of th mnts T i quas th ara of th samp A. A T i n = dh i= A T i = 3. Rspons in diagona dirction In this sction th rspons of th mod is drivd anayticay in diagona dirction. Th drivation of th rspons in diagona dirction is ncssary to obtain th proprtis of th continuum matria in th mod. Th drivation is basd on th assumption that th truss mnts ar stiff compard to th continuum matria in ordr that th trusss can b assumd to b rigid. This assumption is numricay vrifid in paragraph 3.4. Fig 3.4 Th unit c with truss mnts can b simpifid as it is symmtric in dirctions. In figur 3.4 a quartr unit c, is shown in undformd and dformd stat. As rsut of symmtry ony a quartr of this mnt has to b dscribd. This quartr c consists of a continuum matria with a Young s moduus E c and Poisson s ratio ν and on truss. Th truss is rigid, thus th ngth of th truss rmains constant. This nabs cacuations of strsss and strains dircty as function of th strain in tnsi dirction. and ar th initia dimnsions of th quartr c. Ths dimnsions ar dircty ratd to th truss orintation. In undformd stat, as warp and wft yarns ar orthogonay orintd, ar and qua. Initia dformation can b dscribd with vaus for and which ar not qua.

17 Th ngth of th truss is assumd to b constant as th truss is assumd to b rigid: [ ( )] + [ ( + )] = + + [3.] Dfinition ogarithmic strains in th continuum: = n = n ( + ) ( + ) [3.] [3.3] In initia undformd situation hods: = = [3.4] (Not that this assumption dtrmins a truss spacing and cross sction of ) Making us of rations [4], [5] and [6] quation [3] can b writtn as function of ogarithmic strains: xp( ) + xp( ) = [3.5] Th strain in atra dirction can b xprssd as function of. [3.6] = n( xp( )) [3.7] Th in-pan strains ar known, so th strsss σ andσ and th strain 3 in th continuum matria can b cacuatd with th pan strss quations for arg strains. As rsut of symmtry th and orintations or th continuum do not rotat. E σ = c ( v ) + [3.8] v E c σ = ( v + v v = ( + v 3 ) ) [3.9] [3.0] Th strsss and strains in th continuum ar xprssd with th formuas abov. Howvr, dsird is a strss-strain ration in which th contribution of th truss mnts on th rsuting strss is incudd. Th nginring strsss of th combination of th continuum matria and th truss can b xprssd as function of th tru strss in th continuum and th sti unknown forcs in th truss. F xp( ) = σ xp( + 3 ) + [3.] h

18 F xp( ) = σ xp( + 3 ) + [3.] h Th first trms in ths quations ar th tru strsss in th continuum mutipid by th actua strains, to obtain nginring strss. Th scond part of ths xprssions ar th contributions of th sti unknown forc F in th truss acting and dirction, dvidd by th cross sction ara of th continuum x h. Th ratio xp( ) / is th ratio btwn th forc in th truss and its componnt in or dirction. As in uniaxia tnsion no xtrna forcs act on th mod in atra dirction has to b zro. Making us of this quiibrium th forc F in th truss in atra dirction can b xprssd as function of th known strsss and strains in th continuum: F = σ h xp( + ) [3.3] 3 ubstitution of this forc in quation [] rsuts in th foowing xprssion for th nginring strss in tnsi dirction: [ σ xp( σ ] [3.4] = xp( + 3) ) Th first trm btwn th squar brackts is th dirct contribution of th continuum to th ovra strss in tnsi dirction, th scond part is th contribution of th trusss and is strongy non-inar. By using quations [3.], [3.7], [3.8], [3.9], [3.0] and [3.4], indicatd by bu numbrs th ovra nginring strss strain rspons of th mod in diagona dirction can b cacuatd. Th rspons is inary dpndnt on th Young s moduus of th continuum matria and non inary dpndnt on Poisson s ratio. Not aso that th strss approachs infinity whn approachs, i.., whn th truss mchanism is fuy strtchd. Th atra strain rspons, dscribd by quation [3.7] and is shown in figur 3.5. Th rsuting strss in tnsi dirction is shown in figur 3.6 for a Poisson s ratio of 0.5, normaisd by th Young s moduus E c of th continuum. Th charactristic bhaviour whn approaching th strain imit of 0. 4 is cary visib in both figurs. It is aso instructiv to driv th inarizd strss strain rspons, i.. that for sma strains for which and th rotation of th trusss can b ngctd. This drivation is givn in appndix C and rsuts in: in E c = = + v 4G [3.5] whr G is th shar moduus of th continuum. This xprssion shows that for sma strains th diagona rspons of th mod is govrnd by th shar dformation of th continuum. Th inar drivation of this formua is writtn in appndix C. This inar drivation has aso bn pottd in figur 3.6. It cary capturs th ary rspons of th non-inar mod, but fais to captur th ocking ffct du to rotation of th trusss. 3

19 Fig. 3.5 Latra strain as function of tnsi strain. Fig trss-strain rspons in diagona dirction Th infunc of Poisson s ratio on th rspons, at a constant shar moduus of on is visuaizd in figur 3.7. Th incrading non-inarity with incrasing Poisson s ratio is ratd to th strain in out of pan dirction of th continuum matria, in which th matria can xpand without any rstriction. Fig. 3.7 Rspons in diagona dirction for diffrnt vaus of ν. Fig. 3.8 Unit c as impmntd in Marc-Mntat. 3.3 Finit mnt moding A unit c of th fu mod is constructd in Marc Mntat 008r by using on four-nod quadriatra sh mnt of typ 39 and two truss mnts typ 9 as iustratd in figur 3.8. Dispacmnts of th truss mnts ar coupd to th dispacmnts of th continuum matria as thy shar th sam nods. Th bottom ft nod is fixd in a dirctions. Dispacmnt of th top ft nod is fixd in horizonta dirction and th dispacmnt of th bottom right nod is fixd in vrtica dirction. Horizonta dispacmnt of th bottom right and top right nod and vrtica dispacmnt of th top ft and top right nod ar coupd. Th shap of th unit c rmains rctanguar and can ongat and contract without rstriction, thus a uniaxia strss stat is cratd. Both th truss mnts and th continuum matria hav inar astic matria proprtis so th mod is computationay fficint (Chrouat 008). Not that th bhaviour is non-inar as arg dformation thory is usd. 4

20 Th cross sction aras of th trusss ar qua to cross sction ara of th continuum mnt, as xpaind in 3.. Numrica Moding in Marc Mntat imits th Poisson s ratio btwn 0 and nary 0.5, howvr this imitation dos not xist in th anaytica drivations prsntd in th prvious sction. oution contro is st to non-positiv dfinit as th combination of truss and continuum mnts coud rsut in non-positiv dfinit matrics. Larg strain anaysis and updatd Lagrang mthod ar usd. Th dfaut stting for th convrgnc toranc of 0. is changd to 0.0. Th oad incrmnts stp siz is st to automatic, with a maximum rativ stp siz of Vaidation First th anaytica drivd rspons in diagona dirction is compard with th rspons of th numrica unit c impmntd in Marc-Mntat. To chck if th Marc Mntat cacuation matchs with th anaytica cacuation th trusss in th marc Mntat mod ar first mad xtrmy stiff compard to th continuum. For th proprtis of th continuum a Young s moduus of MPa and a Poisson s ratio of 0.4 ar chosn. Th Young s moduus of th trusss is st to x 0 6 MPa and toranc of th soution contro in Marc-Mntat is st to %. In figur 3.9 th strss strain rspons of th Marc-Mntat mod is compard with th anaytica cacuation for rigid trusss. Th rsuts of th Marc Mntat cacuation and th anaytica drivation ar nary idntica, so th Marc-Mntat mod has bn impmntd in Marc Mntat corrcty. Fig. 3.9 Marc-Mntat cacuation with stiff trusss compard with anaytica cacuation. To chck for which stiffnss vaus th assumptions of rigid trusss in diagona oading, and th assumption of ngigib infunc of th continuum in warp and wft dirction ar vaid, two sris of cacuations with diffrnt vaus for th Young s moduus ar prformd in Marc Mntat. An nginring strain of % is prscribd in warp/wft dirction and an nginring strain of 0% in diagona dirction. In warp/wft dirction th stiffnss of th trusss is varid from 0.0 to 40 tims th Young s moduus of th continuum matria. Th nginring strss for th prscribd strain is cacuatd with Marc-Mntat. In figur 3.0 th rativ contribution of th continuum as function of th rativ truss stiffnss is pottd, normaizd on a truss stiffnss of zro. At a truss stiffnss of 30 tims th stiffnss of th continuum or mor, th infunc of th continuum on th tota rspons is owr than 5%. 5

21 Fig. 3.0 Marc-Mntat cacuations warp/wft dirction. Fig. 3. Marc-Mntat cacuations diagona dirction. Rativ contribution of th continuum. At nginring stain of 0%, Poisson s ratio of Th rsuts of h cacuations in diagona dirction ar pottd in figur 3.. Hr th strss is normaizd by dividing th vaus by th rsut of th cacuation with vry stiff trusss. Th curv rachs a vau just abov 0.5 for compiant trusss. In cas of sma strains and a truss stiffnss of zro this vau woud b xacty 0.5 as th (shar) strain in th continuum is doubd as rsut of th trusss. Th diffrnc is ony 5% btwn th rspons with rigid trusss and trusss with a moduus of truss mnts of 30 tims of that th continuum. In chaptr 4 it wi turn out that th diffrnc in stiffnss is significanty highr than a factor 30, so th assumptions in warp and wft and diagona dirction ar corrct and no additiona corrctions ar ncssary with fitting th matria paramtrs. 6

22 4 Paramtr idntification Th txti is tstd at orintations of 0, 45 and 90 dgrs with rspct to th warp yarns to obtain th input paramtrs for th matria mod. Locay masurd strain data is rquird to dtrmin th dsird paramtrs. Thrfor a camra stup and imag anaysis softwar is usd to masur strains in uniformy dforming parts of th samps. Th advantag of optica mthods is that thr is no physica contact with th samp. From th oading data of th tnsi tstr in combination with strain data, obtaind by imag anaysis, th mod paramtrs ar drivd basd on tsts at 0, 45 and 90 dgrs. 4. Tst stup and quipmnt Figur 4. shows th tst stup. Th usd tsting quipmnt is an Instron 5566 tnsi tstr and a pc with Instron softwar. Loca strains ar obtaind with an AVT stingray back and whit camra in combination with Labviw imag anaysis softwar. During tsting imags ar stord on a scond PC togthr with th actua tim, camp position and th oad masurd by th tnsi tstr. Th samp is it from bhind through a narrow window to rduc gar and disturbing rfctions. To kp th camra focussd on th cntr of th samp th camra is attachd to a inar guid which movs with haf of th dispacmnt of th moving camp. Th advantag of th moving focus point is that a argr part of th rcordd ara is usfu for optica anaysis. Fig. 4. Tst stup, showing tnsi tstr, samp and th camra mountd on a inar guid. 7

23 amps ar cut from th txti at diffrnt orintations. Th tstd siz is qua for a samps, i.. 60 x 6 x 0.60 mm. Th tstd samp orintations ar 0, 45, 90 5, 30, 60 and 75 dgrs. Tsts at th first thr orintations ar usd for paramtr idntification. Ony strains masurd at th cntr of th samp ar important for acquiring th paramtrs of th matria mod. Thrfor th camra capturs ony th ara of intrst on th samp to obtain imags with maximum rsoution of this ara. As th structura bhaviour of th samps is intrsting as w (athough not for paramtr idntification) tsts ar aso prformd whr th ntir samp is rcordd, so that ffcts such as sip at th camps and faiur of yarns ar capturd. Proprtis of th tst quipmnt and samps ar istd in th tab 4.. Tst quipmnt Instron 5566 tnsi tstr 00 N oad c; Compianc 5 N/mm; acc. 0^-5 Camps with st jaws, srratd facs Loading cycs 5,0,5,5,5 N Camra: tingray AVT F0B. 64 x 34 pixs Puy for camra fixation, ½ x camp dispacmnt Tab. 4. Tst quipmnt. A samps ar oadd by fiv oading-unoading cycs with maximum oads of rspctivy 5, 0, 5, 5 and 5 N. Th unoading bhaviour givs insight in th dgr to which inastic bhaviour occurs, but is not usd for paramtr idntification. At ach orintation at ast thr samps ar tstd. Th first samp provids information usd to stimat an appropriat strain rat, th intrva of th capturd imags and th data output sttings on th tnsi tstr. Th othr samps provid th data usd for th paramtr idntification. Th xtnsion rat of tnsi tstr is st to a vau so that th tota tst duration is about 500 ± 00 sconds, as shown in tab 4. Tst conditions, dispacmnt controd Extnsion rat Avrag tst duration Warp 0.0 mm/s 400 s Wft 0.0 mm/s 550 s Diagona 0.0 mm/s 570 s Tab. 4. Avrag xtnsion rats and tst durations for warp, wft and diagona orintation. 8

24 4. Optica mthod Th ara of th samp which is rcordd is indicatd by a rd rctang in figur 4.. Th mthod which is usd to obtain dformation information of this ara is a pattrn rcognition mthod dvopd at Phiips Rsarch and programmd in Labviw. This mthod changs th rcordd imags into binary imags and tracks th cntrs of gravity of manuay sctd bright spots in th txti. Mor about this mthod can b found th rport of Fron (008b). In figur 4.3 it is iustratd how th spots ar sctd for strain cacuation. Ths spots ar sctd in th first imags of th sris and ar trackd by th softwar in th foowing imags. Thir positions ar writtn to a data fi. For a propr tracking of ths spots it is important that th contrast btwn th dots and th surrounding txti is as high as possib and th tim intrva btwn subsqunt imags is not too arg. Avrag ight intnsity has to b as constant as possib for th individua imags, but aso for a imags in th sris. Fig. 4. Rd box is indicating th ara from which optica data is rcordd during th tnsi tsts. Fig Tracking bright spots with pattrn rcognition mthod. 9

25 For a high contrast th samp is it from th back by a bright ight shining trough a narrow window in ordr that no ight can ntr th camra objctiv dircty from th ight sourc and to rduc gar and disturbing rfctions. Th mor spots ar trackd th bttr inaccuracis and inhomognitis ar avragd out, but nin sports sms to b sufficint. Th data fis with th dot positions ar importd in Matab togthr with th fis containing corrsponding oad and camp positions of th tnsi tstr. trains ar cacuatd by cacuating th avrag rativ chang in distanc btwn th sctd dots. It has to b mntiond that th rsuts can b infuncd by out of pan dformation of th samps. Thrfor som car has to b takn in th intrprtation of th rsuts. 4.3 Fitting xprimnta rsuts Aftr anayzing th data acquird with th optica mthod, oca strain data from th uniformy dformd parts of th samps is avaiab in tnsi dirction and in atra dirction. This oca strain data is usd to obtain th paramtrs for th mod and is compard with th goba strains to dtct ffcts such as faiur or sip of yarns from th camps. In figur 4.4 nginring strss is pottd as function of th goba nginring strain drivd from th camp dispacmnt of th tnsi tstr for a samp orintation of 45 dgrs. Figur 4.5 shows th nginring strss as function of th ocay masurd nginring strain. Th shap of th curvs is simiar but th ocay masurd strain in th cntr of th samp is argr than th nginring strain masurd on th camps. This is du to that fact that dformation of th samp is non-uniform and rativy high at th cntr of th samp as th rotation of th yarns, thus ongation of th txti, is constraind nar th camps. This is visib in figur 4.. Fig. 4.4 Diagona orintation, goba nginring strain. Fig. 4.5 Diagona orintation, oca nginring strain. In th figurs 4.6 and 4.7 th oca strains and th goba strains in wft and diagona dirction ar pottd as function of tim. Th goba strains masurd by th tnsi tstr ar corrctd for th compianc of th oad c. In wft dirction th curvs coincid prfcty, which mans that th oca and goba strains in tnsi dirction ar idntica at this orintation. Th atra contraction is ss than 8% of th strain in tnsi dirction, so th dformation of th samp is quit uniform. Th match of th curvs aso mans that th optica mthod is accurat, athough som nois is visib in atra dirction as this dformation is sma compard to th spatia rsoution 0

26 of th camra imags. Figur 4.7 visuaizs th diffrnc btwn oca strain and goba strain in diagona dirction. Fig. 4.6 Wft orintation, oca strain vrsus goba strain. Fig. 4.7 Diagona orintation, oca strain vrsus goba strain. Figur 4.8 shows oca oading-unoading data in wft dirction. In th fitting procdurs ony th oading data is usd so th othr data is rmovd by a simp script which rmovs data from a oad which is owr than th highst prvious oad. What rmains is shown in figur 4.9. Fig. 4.8 qunc of fiv oading-unoading cycs. Fig Loading part of th cycs is usd for data fitting. For th paramtr idntification in warp and wft dirction ast squar fits of a inar curv ar appid on th ocay masurd data. Th curvs ar fittd on two tsts for both th warp and wft dirction. Th fit dos not hav to pass th origin as ony th sop of th curv is a masur for th moduus. Th found vaus ar shown in tab 4.3. Matria paramtrs mod Warp mnts E = 88.7 MPa Wft mnts E = 30.3 MPa Continuum matria E c = 0.0 MPa, ν = 0.49 Tab 4.3 Proprtis for warp, wft and continuum matria in mod.

27 Fig. 4.0 Last squar fit in wft dirction. Fig 4.. Last squar fit in warp dirction. Contrary to th warp and wft dirctions th anaytica mod is usd for fitting th xprimnts in diagona dirction. In figur 4. oca strss strains curvs of two tsts ar shown. Th curvs do not coincid as a rsut of diffrnt initia stats of th txti. Anaysis on th imags showd that th ang btwn th warp and wft yarns in tst was initiay 90.5 dgrs. In tst th initia ang was 89.5 dgrs, rsuting in a stiffr bhaviour. Ths diffrncs in initia ang corrspond to initia stains of and rspctivy for tst and. Whn a corrction for ths initia strains is mad by shifting th curvs horizontay thy coincid prfcty as is visib in figur 4.3. A fitting script has bn writtn which convrgs to th Young s moduus for which th ast squar sum of th oca fitting rrors is minimizd. Th Poisson s ratio dtrmins th inarity of th curv and is st manuay. Th bst fit for this matria is obtaind with a Poisson s ratio of approximaty 0.5, but it is fittd with a ratio of 0.49 as this simpifis th impmntation in Marc-Mntat. Th vau found for th Young s moduus is 0.0 MPa. Fig. 4. Tnsi tsts without corrction. Fig. 4.3 Last squar fit on both tsts aftr corrction.

28 5 Numrica tnsi tsts Th input paramtrs for th matria mod hav bn obtaind from ocay masurd strains in th prvious chaptr and hav bn impmntd in a numrica Marc Mntat mod. Th Marc- Mntat mod is dscribd in dtai bow. Th purpos of th mod is not ony moding of uniform strss conditions. Aso th non-uniform structura bhaviour of th matria must b capturd propry as th mod is to b usd to prdict strsss around componnts. Proprtis of th mod such as th pin joints btwn th wft- and warp trusss, which fix a transations btwn intrscting trusss, may rsut in a rspons of th mod which dviats from th rspons of th txti. To vaidat th rspons of th mod, th tnsi tsts in warp, wft and diagona dirction, as w as tnsi tsts at orintations of 5, 30, 60 and 75 dgrs, ar compard in this chaptr with numrica tnsi xprimnts in th corrsponding dirctions. Th comparison is mad by confronting both th oad-dispacmnt bhaviour and th shaps of th dformd samps with thos of th numrica mod. ubsqunty, whr diffrncs ar obsrvd, th rspons of th mod and tstd samps is studid in mor dtai. 5. Marc Mntat mod To buid a Marc-Mntat mod th mnt siz must b dfind. Whr dtaid moding is rquird moding accuracy and cacuation ffort ar oftn conficting. For moding oca ffcts th msh must b rativy dns, but as ony th goba rspons is important for th numrica tnsi tsts prsntd in this chaptr, a rativy coars msh suffics. In appndix D it is shown that th ffct of mnt siz on th goba bhaviour is sma. For th tnsi tst mods an mnt siz of 0.5 mm x 0.5 mm and a thicknss of 0.6 mm is chosn for th sh mnts of typ 39. Th truss mnts, of typ 9, hav a ngth of 0.5 mm and a cross sction ara of 0.3 mm. This mnt siz impis trusss pr miimtr, which is ss than th numbr of yarns in th txti, but accurat nough to mod th 6 mm wid samp. Th continuum mnts ar squar, which mans that th dnsity of trusss is qua in warp and wft dirction. Not that ths vaus in princip can b diffrnt in th mod. Th thicknss of th continuum matria quas th samp thicknss of 0.6 mm so th corrsponding truss cross sction ara is 0.5 x 0.6 = 0.3 mm. For th orintations of 0, 5, 30, 45, 60, 75 and 90 dgrs, diffrnt Marc-Mntat mods ar mad. Th mod dimnsions ar 60 x 6 mm for a orintations and ar th sam as th tstd dimnsions. A convnint way to crat th mods with diffrnt orintations in Marc-Mntat is starting with on arg sht of.g. 00 mm x 00 mnts, rotating it to th right orintation and subsqunty rmoving a mnts outsid th 60 x 6 mm rctang. Th boundary conditions ar appid as foows. Th dispacmnts of th nods on th top and bottom dg ar prscribd in y-dirction and fixd in x- and z-dirction, rprsnting th infunc of th dgs of th camps. Out of pan dformations in th mod ar supprssd as ths dformations mak th simuations unncssariy compicatd. Ony in 45 dgrs a cacuation with aowd out of pan dformation ar prformd to study th infunc of this ffct. An xamp of a mod, in 45 dgrs is shown in figur 5. As dformations ar arg an updatd Lagrang procdur is usd in Marc Mntat. Th soution contro aows non-positiv dfinit matrics as thy may indd bcom non-positiv dfinit du to arg dformations. Furthrmor an automatic tim stp procdur is sctd with a maximum rativ incrmnt siz of 0.0, and a rativ forc toranc of 0.5%. This toranc is 3

29 sma but provd to b ncssary whn th oad instad of dispacmnt is prscribd in combination with a vry non inar rspons. Proprtis of th mods ar shown in tabs 5. and 5.. Dimnsions 60 x 6 mm Numbr of mnts ± 9000 Numbr of incrmnts ± 40 Toranc soution contro 0.5 % Cacuation tim ± 70 s Tab 5. Proprtis of th Marc-Mntat mod Fig. 5. Marc-Mntat tnsi tst mod Matria proprtis Emnt ngth Cross sction/ thicknss Warp mnts Wft mnts 88.7 MPa 30.3 MPa 0.5 mm 0.5 mm 0.3 mm^ 0.3 mm^ Continuum mnts 0.0 MPa nu = x 0.5 mm 0.6 mm Tab 5. Emnt proprtis of th Marc-Mntat mod 4

30 Th rsuts of th Marc Mntat cacuations ar compard bow with th xprimnts in corrsponding pairs of orintations of 0 and 90, 5 and 75, and 30 and 60 dgrs. Th 45 dgr orintation is discussd sparaty. Imags of th dformd samp and th mod ar shown ony for th dirctions wr thy provid usfu information. Th rsuts in 0, 90 and 45 dgrs ar particuary intrsting as ths dirctions hav bn usd to fit th mod paramtrs and 90 dgrs (Warp and wft) In figurs 5. and 5.3 th numricay and xprimntay dtrmind oading curvs in warp and wft dirction (0 and 90 dgrs) ar pottd. Th oad-dispacmnt data obtaind from th tnsi tstr is pottd in bu and th cacuatd oad-dispacmnt bhaviour is pottd in rd. Th rspons of th mod is virtuay inar sinc th truss mnts which rprsnt th yarns ar inar astic and th strains ar sma. Th xprimnta rspons of th samps is non-inar but th match is rasonab. Dformd samps and mods ar not shown as th dformations in ths orintations ar so sma that th dformd samp and mod cannot b distinguishd from th undformd stat. Fig. 5. Tnsi tst in warp vrsus numrica tnsi tst. Fig. 5.3 Tnsi tst in wft vrsus numrica tnsi tst Dgrs (Diagona) 5.3. Ovra comparison Th oading curvs in th 45 dgrs dirction ar shown in figur 5.4. Th Marc-Mntat cacuation prdicts a 5 % smar utimat dispacmnt than masurd in th xprimnt but th ovra match of th oading curvs is good. Whn th dformations of th samps ar studid in dtai som intrsting ffcts can b obsrvd. In figur 5.5 th goba strain masurd at th camps is pottd togthr with th oca strain masurd in th cntr of th samp. Th oca strain in tnsi dirction is pottd in bu and th oca strain in atra dirction in grn. Figur 5.6 shows an imag of th dformd samp, in which th ara in which oca strains ar masurd is indicatd by th ttr C. trains in this ara ar highr than th ovra strain in th samp as th aras indicatd by A bhav rativy stiff. Th goba strain is dircty coupd to th prscribd camp dispacmnts, thus th goba strain rspons, pottd with th rd curv in figur 5.5, is prfcty (picwis) inar in tim. Howvr, th oca strain in tnsi dirction incrass in tim in a non-inar fashion. Th diffrnc btwn oca and goba strain dcrass 5

31 in th ast thr oading cycs, which hav th sam maximum oad. This is th rsut of sip of th samp in th camps of th tnsi tstr. Fig. 5.4 Tnsi tst compard with numrica tnsi tst. Fig. 5.5 Loca strain vrsus goba strain. To visuaiz this ffct othr samps ar tstd with a camra covrag of th ntir samp to rcord th dformation of th samp nar th camps. A carfu study of th sris of imags showd that th yarns sip in th camps of th tnsi tstr during oading. Th sip occurs at th cornrs of th samp. Figur 5.6 shows a dformd samp at a maximum oad of 5N in th third oading cyc, i.. th first oading cyc at 5N. Th ocations whr sip of th yarns occurrd ar indicatd by grn circs. Th aras A, B and C, with diffrnt dformation mods, can b cary distinguishd at highr strains by th diffrnc in intnsity of ight transmittd through th samp. Figur 5.7 shows th dformd Marc Mnta mod at th sam ongation. Comparing th mod with th tstd samp it can b immdiaty sn that th siz of th compiant ara is argr in th tstd samp than in th Marc Mntat mod. As rsut of sip th siz of aras A dcrass and consqunty th siz compiant ara C incrass. To indicat th ffct of sip th actua siz of ths stiff aras is indicatd by rd triangs and th initia siz by bu triangs. Th ratio btwn th initia siz and actua siz of ths aras wi b usd to stimat th ffct of sip on th tota ongation of th samp. Fig. 5.6 Zons in which yarns sip out of th yaws ar indicatd by grn circs. This sip rsuts in a dcras of th siz of th stiff aras A and an incras of th compiant ara C. Nar th boundaris of th samp, particuary in ara C, th atra strain is owr than avrag as a rsut of sip of th yarns. 6

32 Fig. 5.7 Loca strains in th mod ar argr in ara C in th numrica mod than in th tstd samp at th sam goba strain. It is assumd that th strain in aras A can b ngctd, so that th additiona ongation of th samp as a rsut of sip can b stimatd by cacuating th chang in ratio of th ngths of aras A and C. This is an uppr bound stimation as yarns which sip may sti carry oad. Th initia ngth of ara C is 60-6 = 34 mm. As rsut of sip 6-.3 = 4.7 mm (undformd) ngth is addd to th compiant ara. Without sip of yarns th dispacmnt is xpctd to b - (60-6)/(60-.3) = % smar than th masurd ongation. Possib ffcts of th chang of th with and ngth of aras A on th dformation of ara C ar unknown and ar not takn into account. Th Marc-Mntat cacuation prdicts a bhaviour that is ony 5% stiffr and thus w bow th uppr bound of %. It has to b mntiond that som car has to b takn with intrprting th magnitud of this ffct, as th magnitud of sip in this samp might diffr from th in th samps from which oca strains or mod paramtrs ar obtaind Contraction of th samp and orintation of th yarns Enginring strss is usd for mod fitting, so th initia cross sction ara is usd for strss cacuation in both th mod and th samps. Th advantag is that th atra contraction of th samp dos not hav to b masurd. This dos not infunc th quaity of fit whn th mod and th samp hav th sam rspons in contraction. To chck if this is tru th ocay masurd atra contraction and th ang btwn th yarns of tstd samps ar compard with anaytica cacuations and th Marc-Mntat mod. In figur 5.8 th atra strain as a function of th strain in tnsi dirction is pottd in rd for th tstd samp and in bu an anaytica cacuation for rigid trusss is shown. A comparison btwn th masurd and cacuatd ang rsuts in an idntica pot at a diffrnt sca so this pot is not shown. At highr strain vs som dviation btwn th mod and samp was xpctd as rsut of hindrd rotation and ongation of th yarns (Hamia t a. 009). Howvr th curvs match prfcty, which indicats that yarns can sti rotat fry. Th atra contraction of th Marc-Mntat mod shown in figur 5.7 sms 5 á 0 % highr than th contraction of th xprimnta samp in figur 5.6. This is th rsut of boundary ffcts in th tstd samp. Th atra strain masurd in th cntr of ara C (in th samp usd for th strain masurmnts) is qua to th atra strain in th Marc-Mntat mod. Figur 5.9 shows cos ups of th boundaris of th mod and th tstd samps. Th ang of th yarns and th trusss is th sam at th intrior of th samp but at th fr boundaris of th samp it dviats. Warp and wft yarns can dtach and sip at th boundaris in th xprimnt, but not in th mod. 7

33 Fig. 5.8 Latra contraction as function of ongation. Fig. 5.9 Cos up of mod and boundary ffcts in samp Out of pan dformation During tsting a sma amount of out of pan dformation is obsrvd, but not studid in dtai. Imags from tsts and of th mod with out of pan dformation ar shown in figurs 5.0, 5. and 5.. Th maximum obsrvd out of pan dformation is about 0.5 mm for this samp gomtry. Th strain at which out of pan dformation starts is not xacty th sam in ach tst and is. ip of yarns rducs th out of pan dformation as th stiff aras nar th camps bcom smar. Th ffct of th out of pan dformation on atra contraction is visib in figur 5.3. This dformation introducs som nois on th masurmnts but its ffct on th masurd atra strain is sma as th magnitud of dformation in out of pan dirction is aso sma. Fig. 5.0 OOP dformation, sidit. Fig. 5. OOP dformation, backit. Fig. 5. OOP df. mod at argr strain. 8

34 In Marc-Mntat out of pan dformation can b initiatd by positioning randomy chosn nods sighty,.g mm, out of th mid-pan of th samp. This somtims rsuts in out of pan dformation with two fods, as shown in figur 5.. This ooks simiar to th out of pan dformation which occurrd in som tnsi tsts as shown in figur 5.0. Th out of pan dformation pattrn is dpndnt on th initia position of th nods in th Marc-Mntat mod. Dformations with on fod and thr fods in out of pan dirction ar aso found, which is shown in figur 5.4. No significant infunc of out of pan dformation on th oaddispacmnt bhaviour was found with Marc-Mntat. Th tota ongation of th mod with out of pan dformation is 6.59 mm and 6.3 mm with supprssd out of pan dformation. Th diffrnc in atra strain btwn simuations with and without out of pan dformation is about 3%. Th infunc of out of pan dformation on th goba proprtis is random and sma in both th mod and th tstd samps. Th dformation in th mod might b tunab by changing th thicknss and th moduus of th continuum matria as th bnding stiffnss has infunc on wrinking bhaviour (Hamia t a., 009). Fig 5.3 Latra strain is infuncd by OOP dformation. Fig 5.4 OOP dformation dpnds on random initia dformation Prdicting strain of yarns Th rsuts shown in figurs 5.8 and 5.3, whr it is shown that th atra contraction of th samp and th ang btwn th yarns match th cacuatd vaus, impy that th infunc of strains in th yarns can b ngctd. It is chckd if th assumption of rigid yarn bhaviour is in this compiant ara right and th infunc of strains in th yarns is chckd and an attmpt is mad to masur th stains in th yarns Th anaytica drivation prsntd in chaptr 3 can b usd for an stimation of th maximum strain in th trusss with th paramtrs obtaind from th xprimnts. Th forc transmittd by th rigid truss can b cacuatd by quation [3.3]. Whn th truss is dformab but th ongation of th truss is sma compard to th dformation of th continuum, th strain in th truss can b approximatd with inar astic thory. Th truss cross sction ara is dfind as: A truss = h [5.] 9

35 Using sma dformation thory w hav for th truss strain: truss = σ xp( + 3) [5.] E truss This strain can b writtn as function of th goba strain by substituting formuas [3.], [3.7], [3.9] and [3.0]. Th prdictd strain in th warp and wft yarns is shown in figur 5.5 and th opticay masurd strains ar shown in figur 5.6. This strain is masurd by anaysing th chang in distanc btwn th bright spots on ins orintd in dirction of th warp and wft yarns. Fig. 5.5 Estimation of avrag strain in truss mnts. Fig. 5.6 Masurd strain in warp and wft yarns. am ordr of magnitud as prdiction. Masurmnt is inaccurat. Th stimatd strain in th trusss is 0.5% in warp dirction and about % in wft dirction at a maximum ovra strain of 3%, which is in th sam ordr of magnitud as th masurd strain shown in figur 5.6. Th strain in th truss mnts is aso sma at arg dformations. Howvr, du to arg dformations of th txti in diagona dirction, it is difficut to dtrmin th rativy sma yarn strains accuraty. Thrfor th ffct of dformation of th trusss on th tota strain is aso stimatd with Marc Mntat. Th uniform dformation of a mod with rigid trusss is compard with a dformd mod with trusss with proprtis as drivd from th xprimnts. Th comparison is mad at a strss qua to th maximum strss in th xprimnts and is shown in tab 5.3. Th rativ diffrnc in tota strain is sma, about 3 %, so th assumption that th dformation of th yarns can b ngctd sms to b justifid. trss Continuum tiffnss truss Tota strain 0.96 MPa 0. MPa, ν= MPa [-] 0.96 MPa 0. MPa, ν= MPa [-] Tab 5.3 Comparison in tota strain btwn mod with xtrmy stiff trusss and dformab trusss. 30

36 5.3.5 Infunc of truss stiffnss Evn if th yarns can b rgardd as rigid in th uniformy dforming ara this dos not man that th dformation of yarns can b compty ngctd in th stiff aras nar th camps of th tnsi tstr. Th Marc Mntat mod aows to chck th infunc of truss stiffnss on th dformation of ths stiff aras. Two Marc-Mntat mods at an nginring strss of approximaty MPa or a forc of 5 N ar shown in figur 5.7. Th mod on th ft hand sid has truss proprtis as drivd from th xprimnts. Truss stiffnss in th mod at th right hand sid is xtrmy high with as stiffnss of 0 6 MPa. Th continuum proprtis ar th sam in both mods. Th ongations of th mods ar 6.36 mm and.4 mm, which impis a diffrnc of 4. mm. Th back dots in th mod on th ft indicat th intrsction of th trusss rprsnting th yarns passing through th cornrs of th mod. Th distanc btwn ths dots, and thus th ngth of th compiant ara C, is ony 0.8 mm argr than th ngth of th compiant ara in th mod with th stiff trusss. Fig. 5.7 Two Marc Mntat mods at a oad of 5N with ongations of 6.36 mm and.4 mm. Th mod at th ft has truss proprtis as drivd from th xprimnts. Th truss stiffnss in th mod at th right is vry high ( x 0 6 MPa). Continuum proprtis ar th sam in both mods. Th back dots in th mod at th ft indicat th intrsction of th trusss, rprsnting th yarns passing th cornrs of th mod. Th distanc btwn ths dots is nary qua to ngth of th compiant ara in th mod on th right hand sid. 3

37 Th infunc of truss proprtis on dformation of this compiant ara is rativy sma. Howvr, th dispacmnt of th back dots rativ to th dgs of th mod is.7 mm. Thus th strain of th aras A at th cntr of th mod is 3% in tnsi dirction and contributs to th tota ongation of th mod by 3.4 mm. This is 0% of th tota ongation of th mod. This shows that th strains in trusss do hav a significant infunc on th tota ongation of th mod and this is mainy causd by dformation in th stiff aras A. Dformations nar th boundaris of aras A ar strongy infuncd by th truss proprtis, rsuting in significant ffct on th tota ongation of th mod. Th trusss in th mod strtch up to 9% in ara A nar th boundaris of aras B. trains of 5% ar obsrvd in th fw mnts at th cornrs of th mod. In th txti ths yarns woud sip, or brak. It can b concudd that in compiant ara C th yarns bhav vry stiff with strains of maximum %. Th assumption that trusss can b assumd rigid for th anaytica drivation in chaptr 3 is corrct as th infunc of ths strains on th bhaviour of th mod is sma. Howvr, ongation of th trusss has a significant infunc on th goba dformation whn trusss ar ocay oadd as rsut of non uniform dformation. Propr fitting in of th proprtis wft and warp dirction is thrfor important and 75 dgrs In figur 5.8 and 5.9 th rsuts in th 5 and 75 dgr dirctions ar shown. Th numrica oading curvs ar vry inar, but in th xprimnts th non-inarity bcoms significant. Th strains ar or 3 tims highr than th strains in warp and wft dirction. This is th rsut of th fact that ony a fw yarns transmit th major part of th forc from camp to camp, so th dformation in ths yarns is significanty argr than in warp and wft dirction. Fig dgrs orintation, numrica vrsus xprimnta. Fig dgrs orintation, numrica vrsus xprimnta and 60 dgrs Figur 5.0 dpicts th rsuts for 30 dgrs. Th Marc Mntat cacuation prdicts a maximum dispacmnt which is about 9% owr than in th xprimnt. Th xprimnta curv is not vry smooth and sms to show a kink at a oad of 7.5 N. In 60 dgrs dirction, shown in figur 5., th match is bttr. 3

38 Fig dgrs, numrica vrsus xprimnta. Fig dgrs, numrica vrsus xprimnta. imiar to th tst in 45 dgrs dirction, yarn sip has an infunc on th tnsi tsts in 30 and 60 dgrs, but th ffct on th goba dispacmnt is vn argr in ths dirctions. ip in 30 dgrs dirction is argr than in 60 dgrs dirction and this rsuts in a rativy arg incras of th siz of th compiant shar band, which is iustratd in figur 5.. ip is dominant at th cornrs which ar opposit in th stiff warp dirction. Th wft yarns ar much mor compiant and sm to sip ss in th camps. As rsut sip affcts th tst in 30 dgrs dirction mor than that in 60 dgrs dirction. Th shar band visib in th mod in figur 5.3 has an qua siz as th bu band shown in figur 5. and is 40% smar than th shar band in th xprimnta samp at maximum oad, indicatd by rd ins. A 40% stiffr bhaviour is xpctd, but th diffrnc btwn th tst and cacuation is smar than that. Thus may b du to th fact that th dgs of th argr shar band ar not vry w dfind and not vry car and th sippd yarns sti carry oad. Fig dgrs. ip of yarns rsuts in incras of th siz of th shar band. 33

39 Fig dgrs, Marc-Mntat mod at a oad of 5N. 34

40 6 Moding of txti with componnts Our mod is usd to invstigat th ffct of componnts on th txti. Issus such as componnt shap, siz and thir distribution on th txti substrat hav infunc on th ovra stiffnss of th ctronic txti and th maximum strsss in th yarns and componnts. Ths proprtis can b ratd to comfort and riabiity of th ctronic txti. Howvr, riabiity is a compx proprty which is not straightforward dfind and not asy to masur and to mod. As masur for riabiity th maximum strsss and strains in th trusss, th quivant strss in th componnt ar compard. Th infunc of th componnt stiffnss, componnt siz and componnt distanc on ths vaus is studid making us of Marc-Mntat and th trnds obsrvd in ths simuations ar xpaind using a simpifid anaytica mod. Aso th ovra stiffnss of th ctronic txti is of our intrst as this important for th comfort of th product. 6. Unit c Mod Th considrd ctronic txti can b sn as a txti substrat with a rguar matrix of ctronic componnts as shown in figur.. uch rguar structurs can b subdividd into sma idntica unit cs from which th who structur can b buit up. for xamp th structur shown in figur 6.3, which consists of cs as iustratd in figur 6.. This c is a unit c and is an xamp of a c usd in this chaptr. This c contains thr ngth scas. Th first is th siz of th c, L, th scond th componnt diamtr, D, and th third is th truss spacing d. As bfor, th cross sction ara of th individua trusss is st qua to th truss spacing d tims th thicknss of th mod h. Th c siz L is qua to th cntr to cntr componnt distanc. Two of ths ngths, d and D ar varid with rspct to L. imiar hods for th orintation of th componnt, trusss and unit c. Th two indpndnt variabs at a givn truss orintation ar th orintation of th componnt and of th unit c with rspct to th yarns. Th first variab, th orintation of th componnt is not of intrst whn round componnts ar usd. Th scond variab, th orintation of th unit c dtrmins th orintation of th componnt pattrn on th txti and is indicatd by ψ in figur 6. and 6.3. Fig. 6. Thr diffrnt ngth scas. Fig. 6. Diagonay orintd pattrn. Fig. 6.3 Componnt pattrn aignd with yarns. Th stiffnss of th trusss in warp and wft dirction is qua to 80 MPa to achiv symmtric dformation of th unit c with rspct to th diagona dirction. This stiffnss is approximaty th stiffnss of th txti in warp dirction as obtaind in chaptr 4. Aso th proprtis of th 35

41 continuum matria ar qua to thos drivd in chaptr 4. Componnts ar impmntd in th mod, at th cntr of th unit c, by ocay incrasing th stiffnss of th continuum. As rsut th componnt thicknss is qua to th mod thicknss of 0.6 mm. Yarns in txti hav virtuay no stiffnss in comprssion as rsut of bucking. In th mod this bhaviour is mimickd by using th inar Mohr-Couomb yid option in Marc-Mntat, with a vau of α = / 3 and a yid strss of x 0-9 MPa. (Marc-Mntat manua, 008 r). On mod is dfind as rfrnc mod and ony on paramtr is varid at onc with rspct to this rfrnc mod. Th proprtis of this rfrnc mod ar istd in tab 6.. Rfrnc mod Matria proprtis/ vau Dimnsions Trusss wft 80 Mpa in tnsion 0 Mpa in comprssion d/l = 0.0 Trusss warp 80 Mpa in tnsion 0 Mpa in comprssion d/l = 0.0 Continuum matria 0.0 Mpa, ν =0.49 Componnt 800 Mpa D/L = 0. Componnt pattrn orintation - ψ = 0 dgrs Mod thicknss Tab 6. Proprtis of th rfrnc mod. 0.6 mm With priodic boundary conditions th mchanica bhaviour of an infinity arg rptitiv structur can b modd with ony on c. This priodicity is achivd by couping th dispacmnt of th opposit boundaris as shown bow th unit c in figur 6.4. Ths conditions nforc that th distanc btwn opposit points on th boundary is qua to th distanc of th cornr nods in both horizonta and vrtica dirction. As rsut, th opposit boundaris aways hav th sam shap and ngth, so th c rmains rptitiv. Iustrations of an undformd and a dformd structur of nin unit cs ar shown in figurs 6.5 and 6.6. Fig. 6.4 Unit c with coupd Fig. 6.5 Undformd structur of unit cs. Fig. 6.6 Dformd structur of unit cs. boundary dispacmnts. Th avrag strain or strss ar prscribd on th c with priodic boundary conditions via th dispacmnts and forcs of th cornrs C, C and C4. Hr w considr ony uniaxia strss stats in 0, 45 and 90 dgrs with rspct to th c. For th 0 dgr cas th strss is appid via a horizonta forc on nod C. Rigid body motion is supprssd by imposing a zro dispacmnt on nod C and a zro horizonta dispacmnt on C4. Th vrtica dispacmnt of C and C4 is fr. imiar conditions ar usd for th 90º oading. At 45 dgrs a forc at 45 º is appid to C and C4 and rigid rotation is supprssd by couping th horizonta dispacmnt of 36

42 cornr C4 to th vrtica dispacmnt of cornr C. Not that this is ony possib with symmtric unit cs, with th sam stiffnss in warp and wft dirction. In othr orintations than 0, 90 or 45 dgrs a uniaxia strss stat cannot b achivd in a straightforward fashion. 6. Rsuts rfrnc mod For a comparison of th numrica cacuations thr diffrnt vaus ar studid as a function of th varid proprtis. Th fist is th ovra strss strain bhaviour of th unit c, i.. th ovra stiffnss of th c. Th scond is th maximum strss in th truss mnts, which can b ratd quantitativy to maximum strss in th yarns. Th third obsrvd vau is th quivant strss at th cntr of th componnt. Th rfrnc mod oadd in 0 or 90 dgrs is shown in figur 6.8. In ths orintations th mod is oadd in truss dirction and th contribution of th continuum is sma, rsuting in th inar strss strain rspons as shown in figur 6.7. Th strsss in th yarns and componnt ar ow, up to a maximum of approximaty two tims th appid strss in cas of round componnts with siz D/L =0.5. Thrfor this oading orintation is not studid furthr. Fig. 6.7 Linar strss strain rspons in 0/90 dgrs. Fig. 6.8 trss in trusss in 0 dgrs oading. Th strss-strain rspons of th rfrnc mod oadd at 45º is shown in figur 6.9. Th maximum strain is approximaty 8.37 % at a strss of 0. MPa. This is sighty ss than th strain of 9.57%, in txti without componnt at an qua strss of 0. MPa. Th quivant strss distribution in th continuum matria of th mod is shown in figur 6.8. trsss in th compiant continuum around th componnt ar ow. Th strsss in th continuum in th componnt ar significanty highr than th appid strss. In cas of squar componnts strsss ar rativy high at th cornrs on th diagona in tnsion dirction. For practica rasons th vaus at th cntr of th componnt instad of th avrag strss in th componnts ar compard in th sris of cacuations with varid paramtrs. Whn th dformd mod is obsrvd mor in dtai it can b sn that th unit c contains four quadrangs, btwn th cornrs of th componnt and th cornrs of th c, that ar dformd in diagona dirction as uniform txti. Ths four quadrangs ar sparatd by bands btwn th sids of th componnt and th cntrs of th unit c boundaris. This dformation pattrn is discussd mor in dtai in paragraph

43 Fig. 6.9 Ovra strss-strain rspons rfrnc mod. Fig. 6.0 Equivant strss in continuum, squar componnt. Dformd unit cs of th rfrnc mod ar shown in figur 6. and 6., indicating th strsss in th trusss. Figur 6. shows th mod with a round componnt and figur 6. for a squar componnt. Th maximum strsss in th trusss occur in th trusss which ar dircty connctd to th componnts and ar coourd yow in th figurs. Both imags us th sam coour sca. Th maximum strss in th trusss is about 30% highr in th mod with round componnt and is in th sam ordr of magnitud as th strsss at th cornr of th componnts with vaus of 0 to 0 tims th appid strss. Fig. 6. trss in yarns, round componnt of D/L = 0.. Fig. 6. trss in yarns, squar componnt of D/L = Rsuts of paramtr variations vra paramtrs ar varid with rspct to th rfrnc mod. First th componnt stiffnss is varid. ubsqunty th componnt distanc or rativ componnt siz D/L is varid. Th variations ar istd in tab 6. 38

44 Variations Componnt stiffnss MPa Componnt siz D/L = Load Tab 6. Variations on rfrnc mod Variation of componnt stiffnss 0. MPa; φ = 45 dgrs Th componnt stiffnss is varid from 0. MPa, i.. th stiffnss of th continuum in th uniform txti, to 300 MPa to study th ffct on th goba rspons of th c, th maximum strss in th trusss and th strain in th componnt. In figur 6.3 th ovra strss strain curvs for two diffrnt vaus of th componnt stiffnss ar shown, togthr with th strss-strain curv of th txti without componnt. This pot shows that a componnt of siz D/L = 0. has ony itt infunc on th goba rspons, vn for a rativy stiff componnt. Figur 6.4 shows th ration btwn th ovra c dformation and componnt stiffnss for a round and a squar componnt at a strss of 0. MPa. It bcoms car that a round componnt rsuts in a sighty mor compiant rspons than a squar componnt. Goba strain rachs th asymptotic vau at a componnt stiffnss of about x 0 4 tims th stiffnss of th continuum matria. Th rfrnc mod is cos to this asymptotic vau with a stiffnss ratio of 4000, indicatd by Rfrnc in this figur. Fig. 6.3 trss-strains rspons for diffrnt vaus of componnt stiffnss and for th mod without componnt. Fig 6.4 Ovra strain dividd by strain uniform txti. Th maximum strss in th trusss as function of th componnt stiffnss is pottd in figur 6.5. It is rmarkab that th curvs for round and squar componnts intrsct at a componnt stiffnss which is cos to th truss stiffnss. Figur 6.6 shows th quivant strss at th cntr of th squar componnt as function of th componnt stiffnss. Equivant strss in th componnt incrass with componnt stiffnss but sms to convrg to a maximum of approximaty 5 tims th appid strss. 39

45 Fig. 6.5 Maximum strss in truss mnts. Fig. 6.6 Equivant strain at cntr componnt Variation of componnt siz Th sam vaus ar compard as function of th rativ componnt siz D/L. Figur 6.7 shows th goba strss strain rsponss of mods with diffrnt componnt siz. Th strsss and strains ar normaizd by th maximum strain of th uniform mod. Ovra stiffnss incrass cary with incrasing componnt siz. In figur 6.8 th maximum ovra strains for diffrnt componnt ar pottd as function of th rativ componnt siz, aso normaizd on strain of th mod without componnt. A nary inar dpndnc btwn componnt siz and maximum strain is shown for both round and squar componnts. Th goba rspons is nary th sam for squar and round componnts. Fig. 6.7 Ovra rspons with round componnt, normaizd on uniform mod without componnt (componnt siz 0). Fig. 6.8 Compianc as function of rativ componnt siz for both round and squar componnts. Figur 6.9 shows th maximum strss in truss as function of th rativ componnt siz for squar and round componnts. It is car that th strain in th trusss dcrass with incrasing componnt siz for both round and squar componnts, but th strsss ar highr for th mod with round componnt. Th quivant strss in th cntr of th squar componnts is shown in figur 6.9. This strss dcrass non-inar with incrasing componnts siz, and is rativy high for componnts smar than D/L <

46 Fig. 6.9 Maximum strss in truss mnts. Fig. 6.0 Equivant strain in cntr componnt. 6.4 Anaytica mod, dformation in diagona dirction For both round and squar componnts th sam typica dformation of th diagonay oadd unit c is obsrvd as iustratd in figur 6.. In this figur a dformd mod with a round componnt of siz D/L = 0. is shown. Cary four diffrnt aras with sharp boundaris can b distinguishd. In ara 4 th dformation is rstrictd by th rativy stiff componnt and this rgion is assumd to b rigid. Not that in ara 4 th trusss in th top ft and bottom right cornrs ar sighty comprssd. In ara th shar dformation occurs in vrtica dirction and in ara 3 in horizonta dirction. Ara dforms idntica to th uniform mod in diagona dirction. Th shar strains in ara and 3 hav haf th magnitud of th strains in ara, so a (shar) strains can b xprssd as a function of th avrag or ovra strain of th c. Not that ths aras can b compard to th aras obsrvd in th tnsi tsts in diagona dirction in chaptr 4 and 5. Ara 4 can b ratd to ara A, ara to ara C a and aras and 3 to aras B and B as indicatd in figur 5.6 Fig. 6. C wit round componnt. This typica dformation pattrn is th sam for round and squar componnts. 4

47 For a simpifid mod a try is givn to driv th ovra strss strain rspons of th unit c anayticay for sma dformations. A diffrnt unit c is considrd to iustrat th strsss and strains that occur in th c. This c is indicatd by a back rctang in figur 6. and shown in dtai in figur 6.3. Th trusss at th intrfacs ar iustratd by th back ins in figur 6.3 To justify th drivations som assumptions hav to b mad. It is assumd that strsss and strains within th diffrnt aras ar uniform, and that th shar strains in ara and 3 hav xacty haf th magnitud of th shar strain in ara and can b xprssd as function of th ovra strain of th unit c On of th assumptions of th simpifid mod is that th trusss ar xtrmy stiff. tiff trusss rsut in strss concntrations in ths trusss at th boundaris btwn th diffrnt aras, as obsrvd in chaptr 5. Ths strss concntrations ar visib as rd and bu coourd trusss in figur 6.. Contrary to th cacuations with varid proprtis, for this simpifid mod it is assumd that trusss hav stiffnss in comprssion. It is assumd that th diffrncs in shar strsss btwn aras, and 3 ar compnsatd by forcs in ths trusss, as iustratd in figur 6.3. Thus th rsuting (maximum) forcs or strsss in th trusss can b drivd as function of th ovra dformation of th unit c. Th shar strss in th componnt can b ratd to th shar strss in aras and 3 and th forcs in th trusss, thus aso to th ovra dformation of th c. For th strsss and strains in horizonta and vrtica dirction it is assumd that th contribution of th continuum can b ngctd and that strsss in th trusss and componnt ar continuous. Fig. 6. Considrd ara. Numrica cacuation with stiff trusss. Fig. 6.3 Dfinition of aras with th trusss at intrfacs. Th dtaid drivation can b found in appndix E. Th avrag strss and strain tnsors of th unit c can b xprssd in th oca contributions of th 4 aras as function of th c siz L and componnt siz D: () () (3) (4) [( L D) E + D( L D) E + D( L D E + D E ] E = ) [6.] L () () (3) (4) [( L D) + D( L D) + D( L D + D ] = ) [6.] L 4

48 As th dformations of th individua aras can b xprssd as function of th ovra dformation of th c and thir individua stiffnsss ar known th ovra stiffnss of th c can b drivd. Th ovra compianc of th c is xprssd as: L L D L D = + + E L D DL G L D truss [6.3] trains in trusss ar rativy sma compard to th shar strain in th compiant continuum so th ovra strss strain ration can b simpifid to quation: Compianc L D 4G L 0. 5D [6.4] Maximum strss in trusss xprssd as function of th appid strss: L L L D = + L( L D) + D 4d L 0. 5D σ max [6.5] Equivant strss in componnt can b cacuatd with quations dividd by th appid strss: (4) q L 4 3 = + [6.6] 4 [ L( L D) + D ] D ( L 0.5D) 6.5 Gnra 6.5. Infunc truss proprtis Bfor th anaytica drivations ar compard with th sris of cacuations as discussd in sction 6.3 th drivations ar compard with som Marc-Mntat cacuations. First th drivations ar compard for th numrica mod with th assumptions as dscribd in paragraph 6.4. ubsqunty th infunc of comprssion of trusss and truss stiffnss ar chckd. Th Marc-Mntat cacuation with th assumptions mad in paragraph 6.4, i.. is shown in figur 6.7. Th arg strain option is usd in Marc-Mntat and th appid strss is.0 x 0-3 MPa, % of th strss appid in th arg strain cacuations. Th ovra strain of th marc mnta mod is 3.89 x 0-3 and is nary qua to th anayticay cacuatd strain of 3.88 x 0-3, cacuatd with quation [6.4]. Th strsss in th trusss ar rativy high compard to th appid strss in both in comprssion and tnsion. Th maximum strss is th strusss is.356 x 0 - MPa and th minimum strss is -.35 x 0 - MPa. Aso th anayticay cacuatd maximum strss, using quation [6.5], is nary th sam as th strss in th numrica mod with a vau of.363 x 0 - MPa. 43

49 In th sris of arg strain cacuations with componnts comprssion strsss in th trusss ar not aowd. To chck th infunc of this critrion a cacuation is prformd in which th trusss hav no stiffnss in comprssion and is iustratd in figur 6.8 Th ffct of this critrion on ovra rspons is ngigib. Th minimum strss is th trusss is nary zro and th maximum strss is raisd so that th diffrnc btwn maximum and minimum strss at th intrfac rmains th sam. A bttr approximation for th maximum strss is: σ max L L D = d L 0. 5D [6.7] Th maximum strss of.580 x 0 - MPa matchs with th anayticay drivd vau.6053 x 0 - MPa, using quation abov. As th ovra rspons of th c in which no comprssion forcs in th trusss ar aowd is th sam as for th simpifid mod, and aso th maximum strss in th trusss can b cacuatd, th anaytica prdictions can b usd for th sris of cacuations. Fig. 6.7 Vry stiff trusss. Comprssion forc in trusss aowd. Fig. 6.8 Vry stiff trusss. Comprssion forc in trusss not aowd. In th simpifid mod th trusss ar xtrmy stiff so th strsss ar concntratd in sing chains of trusss. Howvr in th sris of cacuations with componnts th truss stiffnss is much owr, so dformation in trusss bcoms important. As rsut of ongation of th trusss th strsss ar not concntratd in a sing chain of trusss but th strsss ar smard out ovr nighbouring trusss, which is visib in figurs 6.9 and As rsut th maximum strss is significanty owr. Maximum strss is ss concntratd at th cornrs of th componnts. As rsut owr shar strsss ar xpctd in th componnt. 44

50 Fig. 6.9 Rativ truss spacing of d/l =0.0. Fig Rativ truss spacing of d/l = Infunc of truss spacing Th gt mor insight in th mod, and to stimat th vau of th Marc-Mntat prdiction of maximum strsss in th trusss, th truss spacing is varid with rspct to th rfrnc mod. Th truss dnsity of th mod is varid from L/d =0 to L/d = 40. Th sum of cross sction aras of a trusss rmains constant. Figur 6.9 shows th rfrnc mod with a truss spacing of d/l = 0.0. Figur 6.30 shows th mod with a truss spacing of In this situation th maximum strss in th trusss nar th componnt is highr as th cross sction of th yarns is smar, but incrass ss than with factor as rativy mor strss is carrid by nighbouring yarns. Th strss in th trusss at a distanc from th componnt rmains th sam. Fig. 6.3 st componnt of strss in truss mnts, = 0. MPa. Fig. 6.3 Maximum strss scas inar with powr of Figur 6.3 shows a pot of th maximum strss in th yarns as function of th truss spacing L/d Whn trusss ar vry stiff and strss woud b concntratd at a sing chain of trusss and th maximum strss woud sca inar with ovr th cross sction ara of th trusss, thus aso L/d as th forc in th trusss rmains th sam. Th numricay cacuatd strss scas amost inar with ovr d to th powr of 0.43 as shown in figur 6.3. Th vau of = 0.57 coud b sn as a masur for th strss intnsity for a truss stiffnss of 80 MPa. It can b concudd that strsss prdictd with th numrica mod 45

51 cannot b transatd straightforwardy to strsss in yarns in txti whn th yarn dnsity is diffrnt from th truss dnsity. 6.6 Comparison of anaytica and numrica mod Th anaytica xprssions ar compard with numrica cacuations for round and squar componnts as a function of componnt siz. Th anaytica drivations ar basd on sma dformation thory, so dviations ar xpctd at argr strains. But quaitativ comparisons wi giv insight in th trnds obsrvd in th cacuations. Figur 6.33 shows th prdictd goba strain dividd by th strain of th uniform mod as function th componnt siz. Th back and rd marks ar th rsuts of Marc Mntat cacuations for round and squar componnts at 0. MPa. Th stiffr rspons for ths arg strain cacuations with componnts is du to th non-inar bhaviour of th mod, according to which stiffnss incrass with strain. Howvr, th trnd is prdictd vry w. Th prdiction of th maximum strss is pottd in figur Th bu in is th anaytica prdiction (Equation [6.7]) for rigid trusss that hav no stiffnss in comprssion. This prdiction dos not match with th numrica vaus quantitativy as trusss ar not rigid in th mod and strssd ar smard out ovr nighbouring trusss. Th numricay cacuatd strsss sca by th powr of ovr 0.43 with th anaytica prdiction. Fig Compianc of unit c as a function of componnt siz. Fig Maximum strss in truss as a function of componnt siz. Figur 6.35 shows th quivant strss at th cntrs of th squar componnts, dividd by th appid strss, as function of th componnts siz togthr with th anayticay cacuatd avrag quivant strss in th componnt. Th vaus match w xcpt for th argst and smast componnts. Th trnd is catchd w. 46

52 Fig Equivant strss at cntr of squar componnt. 6.7 Infunc of componnt pattrn To study th infunc of th componnt pattrn som cacuations with diffrnt pattrns ar prformd. Thr situations ar iustratd in figur 6.36, with th maximum strsss in th trusss indicatd abov th iustration. Th componnt fraction is 4 % in th first two situations, and % in th third situation. Ths thr configurations ar xpctd to giv an qua ovra strss strain rspons whn trusss ar undformab. Th ratio of aras,, 3 and 4 is th sam for th first two configurations. Th intrsctions btwn componnts in th third configuration bhav as stiff aras. Th scond configuration can in fact b sn as 6 smar unit cs, with rativy arg yarns. Maximum strsss in th trusss ar owr in this configuration, so pacmnt of mor smar componnts on txti instad of fwr arg componnts woud b bnficnt for th maximum strsss in th yarns Max strss in truss Max strss in truss Max strss in truss.3 Fig Thr configurations which woud hav an qua goba stiffnss with inarizd dformation thory and vry stiff trusss. Figur 6.37 shows th oad dispacmnt curvs of th thr configurations iustratd abov, pottd as soid ins. Th thirst situation is in fact a mod with a componnt pattrn which is rotatd 45 dgrs with rspct to th yarn, but with a componnt fraction of % instad of 4%. Th dashd in rprsnts a cacuation with a componnt fraction of % with an orintation of 0 dgrs. This givs a mor compiant rspons as configuration 3, so a 0 dgrs pattrn orintation is prfrab for th ovra rspons. 47

53 Fig Th soid ins ar th ovra strss-strain curvs of th thr cs iustratd in figur At owr strains th oad dispacmnt curv for th third cas matchs th oad dispacmnt curvs of th first two cass. At highr strains, whn strain in th yarns bcoms important, th curv tnds towards th dashd curv, which can b sn as an owr bound cacuation. 6.8 Discussion - Th cacuations of th variation of componnt stiffnss show that a componnt stiffnss of about 0.5 tims th fabric stiffnss in yarn dirction, i.. 0 MPa (with a componnt thicknss that is qua to th fabric thicknss of 0.6mm), bhav as rigid whn th txti is dformd in diagona dirction. This mans that componnts of virtuay a soid matrias bhav as rigid in this oading dirction. - tiffr componnts rsut in highr strsss in th yarns and in th componnts. - A componnt pattrn aignd with th yarns is th most advantagous configuration to achiv a compiant ovra strss-strain bhaviour. - It can b concudd strsss in yarns n componnts concntrat in a fw yarns and at th cornrs of squar componnts. For riabiity it is important that strss concntrations to not occur in th conductiv yarns or at ctronic contacts of th componnts. Pacmnt and shap of componnts with rspct to th conductiv yarns and ctronic connctions hav to b w considrd. Round componnt cornrs can rduc strss concntrations in th componnt and at th intrfac btwn txti and componnt at strsss concntrat at four instad of two points. - Equivant strss in componnts and ovra stiffnss of th ctronic txti ar conficting issus. A smar componnt fraction is bnficnt for th ovra rspons but unfavourab for th strsss in th componnts. At th sam componnt fraction mor smar componnts ar bnficia for th maximum strsss in th yarns. Th optimum dpnds on th dsird stiffnss or comfort, and th imitations of th componnt matria, yarn strngth or th connction or th intrfac btwn th txti and th componnt. 48

54 - iicon rubbr,.g. PDM, with a Young s Moduus in th rang of 0.5 to 0 MPa, is an xamp a suitab non-rigid matria and coud b usfu to rduc strsss in yarns at dgs of componnts or ctronic contacts or via connctions. For xamp by ocay imprgnating th fabric or surrounding a sma componnt with ow moduus siicon rubbr, or by crating a rubbr ik intrfac btwn th soid componnts and th txti. 49

55 7 Concusion For th dscription of th mchanica bhaviour of txti an anisotropic msoscopic matria mod is prsntd which consists of truss mnts and a continuum matria. Th truss mnts rprsnt/mod th structura rspons of th matria, but do not rprsnt individua yarns, and th continuum matria mods th rsistanc against rativ rotation of th yarns. Th matria proprtis ar inar astic and arg dformation thory is usd. As a rsut of th cos ration btwn th structura bhaviour of th mod and txti ony four paramtrs ar ndd to mod th rspons. Dspit th ow numbrs of paramtrs, th mod dscribs th matria rspons bttr than th continuum mod which was usd for this purpos prviousy (Fron, 008a). Anothr advantag is that th mod is asy to undrstand and th paramtrs hav an intuitiv maning. Conductiv or stiffr yarns can b modd asiy by incrasing th stiffnss of trusss in th mod. This mod is impmntd in Marc Mntat with th xprimntay masurd and ocay obtaind proprtis and provd to mod both th oca as goba rspons w. Aso th structura bhaviour of th txti is capturd by th mod. tructura dformations of th mod ar th sam as th structura dformation in th txti. Quaitativ prdiction of strsss in yarns is raistic and faiur of yarns can b prdictd quaitativy. In txti in tnsi tsts is faiur of yarns is obsrvd at ocations whr th mod prdicts maximum strsss. Out of pan dformations ar obsrvd in this mod but th shap and th numbr of fods ar dpndnt on th initiation. Moding of stiff componnts can b asiy impmntd by incrasing stiffnss of th continuum matria in th mod. Th rsuts in structura bhaviour, can b compard with structura ffcts obsrvd nar camps in th tnsi tsts. Trnds in ffcts of componnt siz on th goba bhaviour ar dscribd anayticay for th simpifid cas and transatd to ffcts obsrvd in th Marc Mntat cacuations. Th mod aso sti has som imitations: - Locations of maximum strss in yarns can b prdictd with th mod, quantitativ strss prdiction and strss intnsity is not studid in dtai. For quantitativ strss prdiction ach yarn has to rprsntd by trusss, or th ffct of yarn/truss spacing on th strss has to b studid mor in dtai. - Truss mnts ar fixd at th nods but in ra txti siding btwn warp and wft yarns is possib. This coud rsut in ovrstimation of th stiffnss nar boundaris. - Th truss mnts ar inar astic. Th proprtis of trusss in warp and wft dirction hav significant infunc on th diagona dirction and probaby aso othr dirctions in cass that th strss is ocaisd in a fw yarns. In fact this bhaviour is dominatd by a fw yarns that ar subjct th arg strains. Fitting th warp and wft dirction at highr strains coud rsut in bttr prdictions in orintations cos to warp and wft dirction. - Fibrs in txti can buck asiy. Truss mnts in th mod, which cannot transmit comprssion oading dform pastic in comprssion. This wi b a probm in cas of cycic oading. 50

56 In this study componnts ar modd two-dimnsionay Th mod can aso b usd for moding of 3 dimnsiona componnts, for xamp siicon rubbr componnts, as shown in figur 7.. Fig. 7. Tnsi mod with thr- dimnsiona compiant componnt. Pasticity, moding unoading Th purpos of th continuum matria is in fact moding of th rsistanc against rotations of yarns. This rsistanc is partiay astic comprssion of th yarns but an important part is friction. Howvr, in th mod as prsntd, th continuum matria is inar astic. This rsuts in fuy rvrsib oading-unoading bhaviour, whr in practic th intrna friction in th txti dtrmins th oading bhaviour. To impmnt th friction in th unoading bhaviour of th mod th continuum matria can b mad pastic with isotropic hardning bhaviour. Thrfor two nw paramtrs hav to b impmntd, th yid strss, and a Young s moduus diffrnt from that usd in th astic mod. Th hardning paramtr H rpacs th moduus as drivd for th astic mod. Drivation of th right pasticity paramtrs is not studid in dtai, but this work is just to indicat th possibiitis. Th E moduus is important whn th strsss ar bow yid strss, which is a vry sma part in th oading unoading cyc. 5

57 Figur 7. shows a cos up of th oading curv of a tnsi tst in diagona dirction. A kind of yid strss is obsrvd and this yid strss and th Young s moduus E can b dtrmind as iustratd in th figur. H is dfind as: E H = + c σ y Whr E c is th astic Young s moduus as drivd in chaptr 4. Th pastic strss strain rspons with ths paramtrs is shown in figur 7.3. Th oading bhaviour is sighty stiffr than xpctd. Fig. 7. Estimation of Young s moduus and yid strss. Fig. 7.3 Pastic oading and unoading bhaviour. 5

58 Litratur Chrouat, A., Radi, B., E Hami, A. (008). Th frictiona contact of th composit fabric s shaping, 008 Chow, C.L., Yang, X.J. (004). A gnraisd Mixd Isotropic-kinmatic Hardning Pastic Mod Coupd with Anisotropic Damag for sht mta Forming. Intrnationa Journa of damag mchanics. 3;8 Eid, B. & Gruttmann, F. (003). Eastopastic orthotropy at finit strains: mutipicativ formuation and numrica impmntation. Computationa Matrias cinc 8 (003) Institut für Wrkstoff und Mchanik im Bauwsn (IWMB), TU Darmstadt, Axandrstr. 7, 6483 Darmstadt, Grmany Fron, M.J.M. (008a). Mchanics of ctronic txtis. Exprimnta anaysis and numrica moding. Tchnica rport Phiips Rsarch Eindhovn. Fron, M.J.M. (008b). Mchanics of ctronic txtis. Dformation anaysis using imag procssing. Tchnica rport Phiips Rsarch Eindhovn. Hahm, J., Kim, K., Yin, J., (000). Hardning of st shts with orthotropy axs rotations and kinmatic hardning. Intrnationa journa of Koran socity of prcision nginring vo no,. Jun 000. Hamia, N., Boiss, P., abourin, F., Brunt, M. (009). A smi-discrt sh finit mnt for txti composit rinforcmnt forming simuation. Intrnationa Journa for numrica mthods in nginring 009, 79:443:466. Lin, H., Long, A.C., hrburn, M., Cifford, M. J. (008). Moding of mchanica bhaviour for wovn fabrics undr combind oading. choo of Mchanica and Manufacturing Enginring, Univrsity of Nottingham Liu, G.R., Ching, T.W., Tan, V.B.C. (006) Moding baistic impact on wovn fabric with L- DYNA. Computationa mthods Pag Dpartmnt of Mchanica Enginring, Nationa Univrsity of ingapor Png, X.Q. & Cao, J. (004). A continuum-basd non-orthogona constitutiv mod for wovn composit fabrics. Composits part A: Appid cinc and manufacturing, voum 36, issu 6, Jun 005, Ruíz, M.J.G. & Gonzáz, L.Y.. (006). Comparison of hyprastic matria mods in th anaysis of fabrics. CAD/CAM/CAE Laboratory, EAFIT Univrsity, Mdín, Coombia. Vrpost, I. & Lomov,.V. (005). Virtua txtti composits softwar Wistx: Intgration with mico-mchanica prmabiity and structura anaysis. Kathoik Univrsitit Luvn. Xu, P., Png, X.Q., Cao, J., (003). A Non-orthogona constitutiv mod for charactrizing wovn composit. Composits part A: Appid cinc and manufacturing, Voum 3 issu. Fbruary () 53

59 Zhou, G., un, X., Wang, Y. (004). Muti-chain digita anaysis in txti mchanics. Composits scinc and tchnoogy. 64:

60 Appndix A Non inar anaytica drivation in diagona dirction for = = Th ngth of th truss is assumd to b constant as th truss is assumd to b rigid: [ ( )] + [ ( + )] = + + [A ] Dfinition ogarithmic strains in th continuum: = n ( + ) [A ] = n ( + ) [A 3] In initia undformd situation hods: + + = [A 4] = = [A 5] (Not that this assumption dtrmins a truss spacing and cross sction of Making us of rations [4], [5] and [6] quation [3] can b writtn as function of ogarithmic strains: = + [A 6] xp( ) + xp( ) = [A 7] Th strain in atra dirction can b xprssd as function of. [A 8] = n( xp( )) [A 9] Th in-pan strains ar known, so th strsss σ andσ and th strain 3 in th continuum matria can b cacuatd with th pan strss quations for arg strains. As rsut of symmtry th and orintations or th continuum do not rotat. E σ = c ( v ) + [A 0] v E c σ = ( v + v ) [A ] 55

61 v = ( + v 3 ) [A ] Th strsss and strains in th continuum ar xprssd with th formuas abov. Howvr, dsird is a strss-strain ration in which th contribution of th truss mnts on th rsuting strss is incudd. Th nginring strsss of th combination of th continuum matria and th truss can b xprssd as function of th tru strss in th continuum and th sti unknown forcs in th truss. F xp( ) = σ xp( + 3) + [A 3] h F xp( ) = σ xp( + 3 ) + [A 4] h Th first trms in ths quations ar th tru strsss in th continuum mutipid by th actua strains, to obtain nginring strss. Th scond part of ths xprssions ar th contributions of th sti unknown forc F in th truss acting and dirction, dividd by th cross sction ara of th continuum x h. Th ratio xp( ) / is th ratio btwn th forc in th truss and its componnt in or dirction. As in uniaxia tnsion no xtrna forcs act on th mod in atra dirction has to b zro. Making us of this quiibrium th forc F in th truss in atra dirction can b xprssd as function of th known strsss and strains in th continuum: = 0 [A 5] F = σ h xp( + ) [A 6] 3 ubstitution of this forc in quation [] rsuts in th foowing xprssion for th nginring strss in tnsi dirction: [ σ xp( σ ] [A 7] = xp( + 3) ) Th nginring strss xprssd as function of th strains i is: E = xp( + 3) [ + v + xp( ) [ v + ] [A 8] v 56

62 Appndix B Non inar anaytica drivation in diagona dirction. i i σ i F i i Enginring strain Tru strain Enginr strsss in continuum + trusss Tru strss in continuum Raction forcs of truss Th ngth of th truss is assumd to b constant ( u ) + ( + u ) = + + [B ] Dfinition ogarithmic strains: ( + ) + u u = n n = n = + [B ] + u u = n n = n( + ) = + [B 3] + u = n = n( + u ) n( ) n( + u ) = n( ) + + u [B 4] = n = n( + u ) n( ) n( + u ) = n( ) + [B 5] Actua ngth and hight of th mnt: + u = + u = ( n( ) + ) = ( n( ) + ) = [B 6] [B 7] ubstitut [6] and [7] in [] 57

63 58 + = + [B 8] + = [B 9] ( ) + = + = [B 0] can b writtn as function of ( ) + = n [B ] and ar known, so th pan strsss in th continuum can b drivd as function of. ) ( σ v v E + = [B ] ) ( σ + = v v E [B 3] Th strain in out of pan dirction can b cacuatd with th foowing formua: E v 3 σ σ + = [B 4] (or as function of th strain) E v v 3 + = [B 5] Th nginring strsss of th combination of th continuum matria and th truss can b xprssd as function of th tru strss in th continuum and th sti unknown raction forcs of th truss. t F 3 + = σ [B 6] t F 3 + = σ [B 7]

64 Th ratio btwn th raction forcs in - and -dirction is dtrmind by th position of th truss, so by th ratio of th strains in - and -dirction. F = F [B 8] In uniaxia tnsi no xtrna forcs act in atra dirction, so thr is quiibrium btwn th atra strss in th continuum and th atra forc appid by th truss. o quation [7] is qua to 0. = 0 [B 9] Making us of this quiibrium th raction forc of th truss in atra dirction can b xprssd as: F = [B 0] 3 σ Th raction forc of th truss in tnsi dirction is F = [B ] 3 tσ Th rsuting nginring strss in tnsi dirction is 3 3 = σ + σ [B ] Th first part of th xprssion is th contribution of th continuum in tnsi dirction, th scond part is th contribution of th truss. Th formua can b rwrittn as: 3 = σ + σ [B 3] Th axia forc in th truss can b xprssd as a function of σ or σ and th actua ang f th truss. Th ang of th truss is tan [B 4] 59

65 60 3 tan sin tan sin σ = = t F F [B 5] Th nginring strss xprssd as function of th strain is: [ ] = ) xp( 3 v v v E [B 6] xprssd as function of, with ) xp(x ncssariy writtn as x to sav spac. ( ) ( ) ( ) ( ) ( ) = n n n n n v v v E v v [B 7]

66 Appndix C Linar drivation in diagona dirction L is assumd to b constant ( + u ) + ( ) + + [C ] = u Enginring strains ar dfind as: u = [C ] u = [C 3] + u + u = + [C 4] u + u + + u + u 0 [C 5] = u u = [C 6] u u = [C 7] train as function of = [C 8] Pan strss quations: E c = ( + v ) v [C 9] E c = ( v + ) v [C 0] trsss in th continuum matria in - and -dirction: 6

67 6 ) ( v v E = + F [C ] ) ( v v E = + F [C ] u and ar dircty a rsut of dispacmnt u transmittd by truss L Th raction forc in truss L is a rsut of Dformations ar sma, so th nxt assumptions ar mad: u + [C 3] u + [C 4] Y-raction forc transmittd by truss to x dirction. F= ) sin( ) sin( α α F = [C 5] F = ) cos(α F [C 6] Contribution of truss mnt to th strss in -dirction: ) tan( F F α = [C 7] = tan α [C 8] F F = [C 9] Th tota strss in -dirction sum of strss in continuum and truss. c c = [C 0] = ) ( ) ( v v v E [C ]

68 This drivation can b simpifid furthr whn = E E = = v + v c ( v) [C ] E v c = ( v ) = E [C 3] + v E = = 4G [C 4] + v 63

69 Appndix D Msh siz as ony itt infunc on goba rspons. With rativy cours mshs th goba rspons can b modd propry, as ong as th dimnsions ar corrct 64

70 Appndix E Fig 6.0 Th typica dformation pattrns. Abov for round componnt, th sam for squar componnts trains in trusss whn ths comprssion forcs ar aowd. Fig 6. Considrd ara. Fig 6. Dfinition of aras with th trusss at intrfacs. 65