Duality in constitutive formulation of nite-strain elastoplasticity based on F ˆ F e F p and F ˆ F p F e decompositions

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1 Intrntionl Journl of Plticity 15 (1999) 1277±1290 Dulity in contitutiv formultion of nit-trin ltoplticity bd on F ˆ F F p nd F ˆ F p F dcompoition V.A. Lubrd Dprtmnt of Mchnicl nd Aropc Enginring, Univrity of Cliforni, Sn Digo, L Joll, CA , USA Rcivd in nl rvid form 19 July 1999 Abtrct A contitutiv thory for lrg ltic±pltic dformtion i prntd by mploying F=F p F dcompoition of th totl dformtion grdint. A dulity in contitutiv formultion bd on thi nd th wll-known L' dcompoition F=F F p i tblihd for iotropic polycrytllin nd ingl crytl plticity. # 1999 Publihd by Elvir Scinc Ltd. All right rrvd. Kyword: Elto-plticity; Contitutiv qution; Finit trin; Dformtion grdint; Multiplictiv dcompoition; Rvrd dcompoition 1. Introduction Th multiplictiv dcompoition of th dformtion grdint into it ltic nd pltic prt F=F F p, originlly introducd by L (1969), h bn frquntly mployd during th pt thr dcd to tudy th contitutiv bhvior of ltopltic polycrytllin mtril nd ingl crytl. Th dcompoition i introducd by d ning t ch tg of dformtion proc tr-fr intrmdit con gurtion B p, obtind from ltoplticlly dformd con gurtion B by concptul ltic unloding to zro tr. Sinc ltic dformtion i rvrd during thi unloding proc, th intrmdit con gurtion i dformd only plticlly, i.. it di r from th originl (undformd) con gurtion B 0 by pltic prt of th dformtion grdint F p. Thu, th multiplictiv dcompoition F=F F p, whr F rprnt ltic prt of th dformtion grdint from B p to B (Fig. 1 ). Th ltopltic dformtion proc i, thrfor, imgind to tk plc in two tg. Firt, thr i pltic ow of mtril, t zro tr, from th initil con gurtion B 0 to intrmdit con gurtion B p, followd by ltic dformtion from B p to B /99/$ - front mttr # 1999 Publihd by Elvir Scinc Ltd. All right rrvd. PII: S (99)00039-X

2 1278 V.A. Lubrd / Intrntionl Journl of Plticity 15 (1999) 1277±1290 Fig. 1. () Intrmdit con gurtion B p i obtind from ltoplticlly dformd con gurtion by ltic unloding to zro tr. L' dcompoition F=F F p rult. (b) Intrmdit con gurtion B i obtind from initil undformd con gurtion by ltic tring, kping frozn mchnim of pltic dformtion. Rvrd dcompoition F=F p F rult. du to totl pplid tr. Th rulting dcompoition of dformtion grdint, F=F F p, providd ound kinmtic nd kintic bi for ltopltic contitutiv nlyi of polycrytllin mtril (.g., Lubrd nd L, 1981, Lubrd nd Shih, 1994), nd ingl crytl (.g., Aro, 1983, Hvnr, 1985,1992). In th wk of F=F F p dcompoition, thr hv bn vrl uggtion for n ltrntiv, rvrd dcompoition F=F p F (.g. Clifton, 1972, Nmt-Nr, 1979), but thi dcompoition rmind virtully unmployd in ubqunt contitutiv nlyi of ltopltic bhvior. Th prnt ppr i rult of my rcnt tudy of F=F p F dcompoition, which I nvr ttmptd to invtigt bfor, bing intrtd in F=F F p dcompoition only. Intrtingly, but prhp not urpriingly, it turnd out tht th contitutiv nlyi of ltopltic bhvior cn b dvlopd by uing th rvrd dcompoition, quit nlogouly to uing L' dcompoition. Th two formultion cn, thu, b viwd dul to ch othr, both lding to th m nl tructur of contitutiv qution, lthough om of th drivtion nd intrprttion r implr in th c of L' dcompoition. Th dcompoition F=F p F i introducd follow. An rbitrry tt of ltopltic dformtion, corrponding to dformtion grdint F, i gin imgind to b rchd in two tg. Firt, it i umd tht ll intrnl mchnim rponibl for pltic dformtion r frozn, o tht, for xmpl, th criticl forc ndd to driv diloction, or criticl rolvd hr tr on crytllin lip ytm, r ignd in nitly lrg vlu. Th ppliction of totl tr to uch mtril, incpbl of pltic dformtion, rult in pur ltic dformtion F tht crri th mtril from initil con gurtion B 0 to intrmdit con gurtion B. Subquntly, th mtril i plticlly unlockd, by unfrzing th mchnim of pltic dformtion, which nbl mtril to ow t contnt tr. Th corrponding prt of th dformtion grdint, from intrmdit con gurtion B to nl con gurtion B, i th pltic prt of th dformtion grdint F p. Thu, th rvrd multiplictiv dcompoition F=F p F (Fig. 1b).

3 V.A. Lubrd / Intrntionl Journl of Plticity 15 (1999) 1277± Uniqun nd objctivity iu Th untrd intrmdit con gurtion B p of L' dcompoition i not uniqu, bcu uprimpod rottion Q lv thi con gurtion untrd, i.. F ˆ F F p ˆ F^ F^ p; F^ ˆ F Q T ; F^ p ˆ QF p : 1 Suprimpod T dnot th trnpo. If mtril i lticlly iotropic nd rmin uch in th cour of pltic dformtion, rottion of th intrmdit con- gurtion B p h no ct on th tr rpon. Th intrmdit con gurtion cn thn b pci d uniquly by chooing convnintly F =V (th ltic lft trtch tnor). Thu, F=V F p, which w bi of L' (1969) nd Lubrd nd L' (1981) contitutiv nlyi. In th c of rvrd dcompoition F=F p F, th intrmdit lticlly dformd con gurtion B i ncrily uniqu, bcu rottion uprimpod to it would rott th tr tt, wll, nd th pltic ow from B to B would not b t contnt tr ny mor. Furthrmor, if mtril i lticlly iotropic, n initil rottion t B 0 do not ct th tr rpon, nd th rlvnt prt of th totl dformtion grdint i F=F p V. Thrfor, for contitutiv nlyi w cn writ F ˆ V F p ˆ F p V : 2 If mtril i lticlly niotropic, thn, rltiv to givn orinttion of principl dirction of ltic niotropy, thr i uniqu F tht giv ri to totl tr in B or B, which ˆ F F T Hr, i th pci c trin nrgy, E =(1=2)(F T F I) i th Lgrngn ltic trin rltiv to it ground tt (B p in th c of L' dcompoition nd B 0 in th c of rvrd dcompoition, both hving th m orinttion of th principl x of niotropy rltiv to xd frm of rfrnc). Th Kirchho tr i = F, whr dignt th Cuchy tr, nd th dtrminnt. It i lo umd tht pltic ow i incompribl nd tht it do not ct th ltic proprti. Thrfor, w now hv F ˆ F F p ˆ F p F : 4 Not tht th m ltic dformtion grdint mtrix F ppr in both dcompoition. Th rltionhip btwn th two pltic prt of th dformtion grdint i conquntly F p ˆ F 1 F p F ; 5 whr 1 dignt th invr.

4 1280 V.A. Lubrd / Intrntionl Journl of Plticity 15 (1999) 1277±1290 It i hlpful for th ubqunt dvlopmnt to bri y dicu objctivity rquirmnt for th contitunt of th two dcompoition. If rigid-body rottion Q i uprimpod to ltoplticlly dformd con gurtion B, th dformtion grdint F chng to FÃ =QF, whil th Kirchho tr bcom ^=QQ T. Sinc pltic ow from B to B occur t contnt tr, B h to b rottd by Q, wll. Conquntly, F Ã p =QF p Q T nd F Ã =QF. On th othr hnd, F p rmin unchngd, F^ p ˆ F p. Morovr, inc F ˆ V R, th ltic trtch nd rottion tnor chng to V Ã =QV Q T nd R Ã =QR. For brvity, th objctivity rquirmnt corrponding to n indpndnt rottion of intrmdit con gurtion B p of L' dcompoition r not dicud hr, inc thy hv bn xmind in dtil by Lubrd (1991) nd Lubrd nd Shih (1994). 3. Eltic unloding During ltic loding from B p to B, or ltic unloding from B to B p, th pltic dformtion grdint F p of L' dcompoition F=F F p rmin contnt. Thi grtly impli drivtion of th corrponding contitutiv qution. Indd, th vlocity grdint in B i L ˆ F : F 1 ˆ F : F 1 F F : pf 1 p F 1 ; 6 o tht during ltic unloding F : p=0 nd L=F : F 1. Howvr, in th frmwork of rvrd dcompoition, F=F p F, th pltic prt of th dformtion grdint F p do not rmin contnt during ltic unloding. In fct, upon complt unloding from n ltopltic tt of dformtion to zro tr, th con gurtion B p i rchd, nd F p =F p t tht intnt (Fig. 2). Thrfor, F : p 6ˆ 0 during ltic unloding. Thi cn lo b rcognizd from th gnrl rltionhip btwn F p nd F p. By di rntiting Eq. (5), w obtin F : p ˆ F 1 F p F ; 7 Fig. 2. Pltic prt of dformtion grdint F p do not rmin contnt during ltic unloding. Upon complt unloding to zro tr, th con gurtion B p i rchd, nd F p =F p t tht intnt.

5 V.A. Lubrd / Intrntionl Journl of Plticity 15 (1999) 1277± whr F p ˆ F : p F : F 1 F p F p F : F 1 8 i convctd-typ drivtiv (Hill, 1978) of F p rltiv to ltic dformtion. Clrly, from Eq. (7), F p ˆ 0ifḞ p =0, o tht in tht c F : p ˆ F : F 1 F p F p F : F 1 : 9 Th lt xprion d n th chng of F p during ltic unloding. From Eq. (7), furthrmor, F : pf 1 p ˆ F 1 F p F p 1 F ; 10 nd ubtitution into Eq. (6) giv L ˆ F : F 1 F p F p 1 11 Thi i dul qution to Eq. (6) of L' thory, which will b frquntly ud in th ubqunt nlyi. 4. Polycrytllin plticity W now lbort on th contitutiv formultion of nit-trin ltoplticity within th tructur of rvrd dcompoition F=F p F. Phnomnologicl polycrytllin plticity i rt conidrd. For implicity, ltic iotropy i umd throughout th cour of dformtion. Th ltic tr rpon from B 0 to B i givn by Eq. (3) with F =V. Thu, upon di rntition ˆ K : V : V 1 ; ijkl ˆ V im V V pk V ql : pq Hr, tnd for th convctd-typ drivtiv of th Kirchho tr with rpct to ltic dformtion ˆ : V : V 1 V : T: V Th trc product in Eq. (12) i dnotd by : nd th ubcript tnd for th ymmtric prt.

6 1282 V.A. Lubrd / Intrntionl Journl of Plticity 15 (1999) 1277±1290 In trm of th Jumnn drivtiv ˆ V : V 1 V: V 1 ; 14 Eq. (12) cn b rwrittn ˆ L : V : V 1 ; L ijkl ˆ ijkl 1 2 ik jl il jk ik jl il jk : 15 Th componnt of th unit tnor r th Kronckr ij. By tking ymmtric nd ntiymmtric prt of Eq. (11), w hv: V : V 1 ˆ D V : V 1 ˆ W F p F p 1 F p F p 1 ; 16 : 17 Th trin rt nd pin tnor in th con gurtion B r dnotd by D nd W. Hnc, ˆ F p F p 1 F p F p 1 ; 18 whr ˆ t : W W i th Jumnn drivtiv of th Kirchho tr with rpct to W. Conquntly, ubtitution of Eq. (18) into Eq. (15) giv ˆ L : V : V 1 M : F p F p 1 F p F p 1 : 19 Th fourth-ordr tnor M =L 1 i th intntnou ltic complinc, th invr of th ltic ti n tnor L. Th ltic trin rt D, corrponding to tr rt, i vidntly D ˆ V : V 1 M : F p F p 1 F p F p 1 : 20 In viw of Eq. (16) nd dditiv dcompoition of th trin rt into it ltic nd pltic prt, th pltic prt of th trin rt bcom D p ˆ F p F p 1 M : F pf p 1 F p F p 1 : 21

7 V.A. Lubrd / Intrntionl Journl of Plticity 15 (1999) 1277± Clrly, during ltic unloding D p =0 nd D =(V V 1 ), inc F p ˆ 0 in th unloding proc. Furthr contitutiv nlyi procd in th c of L' dcompoition. From Eq. (19), th ltic trin rt, xprd in trm of th tr rt, i D ˆ M : : 22 Thi d n rvribl prt of th trin incrmnt, rcovrd upon unloding of th tr incrmnt ocitd with th Jumnn tr rt. Th rmining, pltic prt of th trin rt giv ridul prt of th trin incrmnt. Undr uul umption of clicl plticity, thi i codirctionl with th outwrd norml to loclly mooth yild urfc f in th tr pc. In th c of iotropic hrdning D p ˆ @ : whr th clr function h ccount for th pltic dformtion hitory. Thu, D ˆ M h : ; or, by invrion, ˆ L L : : L : D; whr h 0 =h+(@f/@):l :(@f/@) Strin rt xprion in trm of V,F p nd thir rt Th ltic nd pltic trin rt wr xprd in trm of th contitunt V nd F p of L' dcompoition F=V F p by Lubrd nd L (1981), nd in mor gnrl contxt by Lubrd (1991,b) nd Lubrd nd Shih (1994). In thi ction th corrponding rltion r drivd in th frmwork of rvrd dcompoition F=F p V. Agin, for implicity, only iotropic rpon i conidrd. Firt, introduc th pin th olution of th following mtrix qution W ˆ V : V 1 V V 1 : Th corrponding Jumnn drivtiv of th Kirchho tr nd th ltic trtch V r: 26 r ˆ : ; V r ˆ V : V V : 27

8 1284 V.A. Lubrd / Intrntionl Journl of Plticity 15 (1999) 1277±1290 Applying th drivtiv to Eq. (3), with F =V, thr follow r r ˆ V V 1 V r T V 1 : V r V 1 : 28 In viw of Eq. (26) nd (27) 2, th rltionhip hold V r V 1 ˆ V r V 1 W ; 29 nd ubtitution into Eq. (28) giv ˆ L : V r V 1 : Thi idnti th ltic trin rt D ˆ V r V 1 ˆ V : V 1 V V 1 : On th othr hnd, compring Eq. (17) nd (26) yild V V 1 ˆ F p F p 1 : 32 Thrfor, from Eq. (16) nd (31) th pltic trin rt i givn by D p ˆ D D ˆ F p F p 1 V V 1 : 33 Thi i idnticlly qul to D p ˆ F p F p 1 V V 1 ; 34 inc ntiymmtric prt of th di rnc on th right-hnd id of Eq. (34) vnih by Eq. (32). If hrdning i iotropic, th principl dirction of th tr nd th ltic trtch V r prlll to tho of th pltic trin rt D p. Thu by multiplying Eq. (34) by V 1 from th lft, nd by V from th right, it follow D p ˆ V 1 F p F p 1 V : 35 Tking ymmtric nd ntiymmtric prt of Eq. (35) nlly giv:

9 D p ˆ ˆ V.A. Lubrd / Intrntionl Journl of Plticity 15 (1999) 1277± F p F p 1 ; 36 V 1 V 1 F p F p 1 V V : 37 Th xprion r in complt grmnt with th originl rult obtind by Lubrd nd L (1981) in th frmwork of L' dcompoition F=V F p. Indd, inc F p =V 1 F p V, on h V 1 F p F p 1 V ˆ F : pf 1 p ; 38 nd Eq. (36) nd (37) rduc to Lubrd nd L' rult D p ˆ F : pf 1 p ; ˆ F: pf 1 p : 39 Thi dmontrt dulity in contitutiv formultion for iotropic ltoplticity bd on th two ltrntiv dcompoition F=V F p nd F=F p V. 6. Singl crytl plticity W nxt conidr th contitutiv formultion for lrg ltopltic dformtion of ingl crytl, in which crytllogrphic lip i umd to b th only mchnim of pltic dformtion. Th rvr multiplictiv dcompoition F=F p F i mployd to compr th formultion with th wll-known formultion bd on th dcompoition F=F F p (.g. Aro, 1983, whr th nottion F=F F p i ud). Th totl tr i pplid to initil con gurtion B 0, conidring tht crytllogrphic lip i prvntd by igning in nit vlu to criticl rolvd hr tr on ch lip ytm. Th rulting dformtion F i purly ltic, nd cud by lttic trtching nd rottion tht crri B 0 into intrmdit con gurtion B. Subquntly, th criticl rolvd hr tr r rlxd to thir ctul nit vlu, which nbl mtril to ow through th crytllin lttic t contnt tr. Thi prt of dformtion grdint i th pltic prt F p, nd th dcompoition F=F p F hold. Dnot th unit lip dirction nd th norml to th corrponding lip pln in th undformd con gurtion B 0 by 0 nd m 0, whr dignt th lip ytm. Th vctor 0 i mbddd in th lttic, o tht it bcom =F. 0 in th intrmdit con gurtion B. Sinc th mtril ow through th lttic during th dformtion F p, th lip dirction rmin th m in th nl con gurtion B (Fig. 3). Th norml to th lip pln in th con gurtion B nd B i d nd by th rciprocl vctor m =m 0.F 1. In gnrl, nd m r not unit vctor, but r orthogonl to ch othr.

10 1286 V.A. Lubrd / Intrntionl Journl of Plticity 15 (1999) 1277±1290 Fig. 3. Unit lip dirction nd unit norml to lip pln in undformd con gurtion B 0 r 0 nd m 0. Th vctor 0 i mbddd in th lttic, bcoming =F. 0 in th intrmdit con gurtion B.A mtril ow through th lttic during dformtion F p, th lip dirction rmin th m. Th vlocity grdint in th dformd con gurtion B i conqunc of th lip rt : ovr n ctiv lip ytm, uch tht F p F p 1 ˆ n ˆ1 m : ; 40 whr dignt dydic product, nd F p i convctd-typ drivtiv d nd by Eq. (8). Thi convctd rt of F p ppr in Eq. (40) bcu, during ltopltic incrmnt of dformtion, both th currnt nd intrmdit con gurtion, with rpct to which pltic dformtion grdint F p i d nd, dform. Furthrmor, dicud in Sction 3, during ltic unloding from th currnt con gurtion, F p ˆ 0, o tht Eq. (40) i in ccord with th rquirmnt tht during ltic unloding th lip rt : mut lo b qul to zro. In viw of Eq. (10), Eq. (40) i quivlnt to th corrponding qution from th formultion bd on F=F F p dcompoition,whr th point of dprtur i n xprion for F : pf p 1 in trm of th um of lip contribution ( 0 m 0 ) : in th intrmdit con gurtion B p (Aro, 1983). Th ltic rpon of th lttic nd th mtril mbddd on it i dcribd by Eq. (3). It i umd tht crytllogrphic lip i n iochoric dformtion proc, nd tht ltic proprti of th crytl r un ctd by th lip. Thu,

11 V.A. Lubrd / Intrntionl Journl of Plticity 15 (1999) 1277± ˆ L : F : F 1 ; ˆ : F : F 1 F: F 1 Th componnt of th intntnou ltic ti n tnor of th crytl 2 L ijkl ˆ F im F pq : 41 F kp F lq 1 2 ik jl il jk ik jl il jk : 42 Th trin rt nd pin tnor cn b xprd from Eq. (11) nd (40) : D ˆ F : F 1 ˆ1 n P : ; 43 W ˆ F : F 1 ˆ1 n W : ; 44 whr th cond-ordr tnor r introducd: P ˆ 1 2 m m ; W ˆ 1 2 m m ; 45 commonly don in crytl plticity. Furthr nlyi procd idnticlly in th c of dcompoition F=F F p. Sinc ˆ n ˆ1 W W : ; 46 Eq. (41) bcom ˆ L : D n ˆ1 P M : W W Š : ; 47 which d n th pltic trin rt corrponding to D p ˆ n ˆ1 P M : W W Š : : 48 Phyiclly, D p yild th ridul trin incrmnt lft in th crytl upon n in nitiml loding/unloding cycl ocitd with th tr rt. Th rvribl prt i th ltic prt of th trin rt, D =M :. Th tructur of th trin rt xprion corrponding to othr choic of th tr rt h bn dicud by Lubrd (1994,1999). Eq. (47) cn b rwrittn

12 1288 V.A. Lubrd / Intrntionl Journl of Plticity 15 (1999) 1277±1290 ˆ L : D n ˆ1 K : ; K ˆ L : P W W : 49 To complt th contitutiv formultion, th lip rt : hv to b xprd in trm of th rt of dformtion or th rt of tr. Thi i don in tndrd mnnr. For rt-indpndnt mtril it i umd tht pltic ow occur on lip ytm whn th rolvd hr tr on tht ytm rch th criticl vlu ( ˆ cr ). Th gnrlizd Schmid tr cn b d nd th work conjugt to th lip rt :, uch tht (Hill nd Ric, 1972; Hill nd Hvnr, 1982; Aro, 1983) n ˆ1 : ˆ : n ˆ1 P : : 50 Thrfor, ˆ P :, it rt bing : ˆ K : F : F 1 : If th rt of chng of th criticl rolvd hr tr on givn lip ytm i d nd linr combintion of th lip rt :, th co cint bing th lippln hrdning rt h, th conitncy condition for th lip on th ytm i 51 : ˆ n h : : ˆ1 52 In viw of Eq. (51) nd (43), thi bcom K : D ˆ n g : ; g ˆ h K : P : 53 ˆ1 Thu, th lip rt cn b dtrmind from : ˆ n ˆ1 g 1 K : D; 54 providd tht th mtrix with componnt g i poitiv-d nit (Hill nd Ric, 1972). Subtitution of Eq. (54) into Eq. (49) nlly giv ˆ L n : D; 55 ˆ1ˆ1 n K g 1 K which i th m contitutiv tructur of ingl crytl plticity prntd by Aro (1983) in th frmwork of th dcompoition F=F F p. For prcribd componnt of vlocity grdint, th lip rt r dtrmind from Eq. (54), nd F by intgrtion from Eq. (11) with th incorportd Eq. (40). Th pltic dformtion grdint i thn F p =FF 1.

13 7. Concluion V.A. Lubrd / Intrntionl Journl of Plticity 15 (1999) 1277± W hv dmontrtd in thi ppr dulity in contitutiv formultion of lrgdformtion ltoplticity bd on L' dcompoition F=F F p nd rvrd dcompoition F=F p F, t lt for th contitutiv modl conidrd in thi ppr. It i hown tht for mtril tht prrv ltic proprti during pltic dformtion, th m ltic dformtion grdint F ppr in both dcompoition, whil pltic dformtion grdint F p nd F p nturlly di r. Exprion for ltic nd pltic trin rt, drivd in th frmwork of rvrd dcompoition, r givn by Eq. (20) nd (21), nd Eq. (31) nd (36). A kinmtic rprnttion of th vlocity grdint in Eq. (11) w ud in thi drivtion. Th tructur of th obtind xprion i found to b mor involvd thn of tho bd on L' dcompoition, cf. Eq. (36) nd (39). Thi i prtly bcu during ltic unloding th pltic grdint F p of L' dcompoition rmin contnt, whil F p of rvrd dcompoition chng, lbit in d nit mnnr pci d by Eq. (9). Nonthl, within th m phyicl ingrdint, both formultion ld to th m nl tructur of ltopltic contitutiv qution for iotropic polycrytllin mtril nd ingl crytl. Th r givn by Eq. (25) nd (55). In ddition to concptul importnc of dmontrting dulity, th nlyi prntd in thi ppr my b of intrt bcu it i poibl tht in om ppliction th rvrd dcompoition h crtin dvntg. For xmpl, Clifton (1972) found it to b lightly mor convnint for th nlyi of on-dimnionl wv propgtion in ltic-vicopltic olid. It hould, howvr, b pointd out tht L' dcompoition h d nit dvntg in modling plticity with volving ltic proprti. In thi c, t of tructurl tnor cn b ttchd to th intrmdit con gurtion B p to rprnt it currnt tt of ltic niotropy. Th tructurl tnor volv during pltic dformtion, dpnding on th ntur of microcopic inltic proc, thir chng bing rprntd by pproprit volution qution. Th tr rpon t ch intnt of dformtion i givn in trm of th grdint of ltic trin nrgy with rpct to ltic trin, t th currnt vlu of th tructurl tnor. Thi typ of nlyi w ud by Lubrd (1994b) nd Lubrd nd Krjcinovic (1995) in thir tudy of dmg-ltoplticity. In th c of rvrd dcompoition, howvr, th ltic rpon i d nd rltiv to th initil con gurtion B 0, which do not contin ny informtion bout volving ltic proprti or ubquntly dvlopd ltic niotropy. Additionl rmdy h to b introducd to dl with th ftur of mtril rpon, which i likly to mk th rvrd dcompoition l ttrctiv thn th originl dcompoition. Acknowldgmnt Rrch funding providd by th Alco Cntr i kindly cknowldgd. I m in prticulr indbtd to Dr. Own Richmond for dicuion nd for hi long-tnding upport of my work.

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