RELATIONSHIP BETWEEN PROBABILISTIC AND CONTINUUM DESCRIPTIONS OF THE MECHANICAL BEHAVIOR OF BRITTLE-MATRIX COMPOSITES

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1 EATIONSHIP BETWEEN POBABIISTIC AND CONTINUUM DESCIPTIONS OF THE MECHANICA BEHAVIO OF BITTE-MATIX COMPOSITES Fanços Hld MT-Cachan, ENS d Cachan / UM CNS / Unvsty Pas 6 6 avnu du Pésdnt Wlson, F-9435 Cachan Cdx, Fanc SUMMAY : Intnal vaabls odlng th dgadaton chanss occung n bttlatx coposts a latd to coscopc quantts by usng a pobablstc appoach to alu. Two xapls a psntd. Th st on dals wth atx-cackng and th scond on s concnd wth dbondng and slp. In both cass, t s shown that n gnal th onts o od hgh than on a ndd to dscb th bhavo n a Contnuu Mchancs awok. KEYWODS: Contnuu Thodynacs, stat potntal, ntnal vaabls, atxcackng, dbondng and slp, b bakag, alu pobablty. INTODUCTION Consttutv laws o (copost) atals can b wttn wthn th awok o Contnuu Thodynacs []. Th a s to dscb th dgadaton chanss occung n bttlatx coposts by ntnal vaabls ntoducd n a stat potntal. Th stat potntal ψ s ad up o two ts, vz. a covabl pat ψ and non-covabl pat ψ. Th covabl pat s usually d to as lastc ngy dnsty and th non-covabl pat cosponds to lastc ngy dnsts assocatd wth sdual stsss nducd by th dgadaton chanss. Th stat potntal s postulatd vy otn. Th scop o th psnt study s to show that th stat potntal can b valuatd and that th ntnal vaabls dnd at a contnuu lvl can b latd to th dstbuton o coscopc quantts chaactzng th dgadaton chanss. Th psnt study dals wth atx-cackng, ntac dbondng and slp as wll as b bakag occung n bttl atcs nocd by contnuous bs. Fo th sak o splcty, undctonal achtctus a dscussd. At a coscopc lvl, th cack locatons a andoly dstbutd wthn th atal. At a soscopc lvl (.., Contnuu Mchancs lvl) th bhavo s dscbd by ntnal vaabls. Th latonshp btwn th two dscptons s dvd and th lvant vaabls a ntoducd to odl th abovntond dgadaton chanss.

2 GENEA FAMEWOK Upon loadng, a b nocd copost xpncs ultpl atx-cackng accopand by ntacal dbondng and slp. Th syst studd hn (Fg. ), contans atx cacks andoly dstbutd and chaactzd by a pobablty dnsty uncton F() whch volvs wth th appld soscopc stss. l dp l dq l d l ds () p q () a s z () pq q s Fg. : Elntay clls Th atx cacks nduc an ncnt o th Gbbs nthalpy ϕ ϕ + + ϕ F d F d ( ) ( ) wth ( ) () 0 0 wh ϕ ( ) s th Gbbs ngy dnsty ncnt latd to a cack n a cll o lngth. Ths sult assus that th only consdd ntactons a thos du to th closst nghbong cacks. Th long dstanc ntactons a tho nglctd. Dbondng and sldng nduc nlastc stans ε and hystss loops []. Applcaton o th pncpl o vtual wok ylds ε 0 0 (, z) dz F( ) d () E wh ( z, ) s th sdual stss ld nducd by sldng n th b, z th cunt poston n a cll o lngth. Th cospondng stss ld s sl-balancd and nabls to sto ngy ψ EE (, z) ψ dz F d E ( ) (3) ( ) E 0 0 wh s th b volu acton, E, E and E a th spctv Young's odul o th b, th atx and th undaagd copost ( E E + ( ) E ). Th dgadaton chanss n th copost a dscbd by ntnal vaabls. Matxcackng s odld by a daag vaabl D dnd by [3] ϕ ϕ ( D) ϕ ( D 0) E( D) (4)

3 Equatons () and (4) show that th daag vaabl D s a uncton o all th onts assocatd wth th pobablty dnsty uncton F. Th nlastc stan ε s on o th vaabls odlng ntacal dgadaton (.., dbondng and sldng). Th scond vaabl, d, s obtand o th xpsson o th non-covabl pat o th ngy dnsty ψ ψ ( ε ) E d Ths xpsson was also usd n th study o cackng n conct and ocks [4]. Th Hlholtz ngy dnsty ψ s gvn by ( ε ) E( D) E ψ ( ε ε ) + d and th assocatd ocs to th stat vaabls a xpssd as ψ ψ ψ ψ, Y, y, X (7) ε D d ε Equaton (7.) gvs th dnton o th soscopc stss, Eqn (7.) ntoducs th ngy las at dnsty, Eqn (7.3) spcs th ngy las o sdual stsss du to sldng and Eqn (7.4) chaactzs th back-stss assocatd wth cton. Th voluton laws o th ntnal vaabls can b obtand th by usng cochancal analyss (s blow) o by acoscopc xpnts (Fg. ). In patcula, th calculaton o th daag vaabl d nds th valuaton o th non-covabl ngy dnsty ψ : s o nstanc Chysochoos t al. [5] o Cho t al. [6]. (5) (6) Stss, E( D) ε Stan, ε Fg. : Msoscopc stss/stan cuv By way o an xapl, two patcula cass a now nvstgatd. APPICATIONS In ths st xapl w only consd atx-cackng. By usng th Cox odl [7], th covabl pat o th ngy dnsty ϕ ( ) can b calculatd ϕ ( ) ( ) E tanh( β) E β E (8)

4 wh β s a constant dpndnt upon th lastc and gotc popts o th atx and th b (β s nvsly popotonal to th b adus ). A chaactstc lngth assocatd to atx-cackng s dnd by ϕ ( ) ϕ (9) To calculat ϕ, th pobablty dnsty uncton F() s dvd by assung that th laws a dstbutd accodng to a Posson pont pocss. Und ths assupton, th dstbuton o agnts F() can b wttn as (s o xapl. 8) [ ] F( ) λt( )xp λt( ) (0) wh λt ( ) dnots th dnsty o bokn laws. Whn th dnsty λt ( ) s assud to b odld by a pow-law uncton o th local stss, a Wbull odl s tvd. Th avag agnt lngth s latd to λt ( ) by λt ( ) Th avag vaaton o th nthalpy dnsty bcos ( ) E ϕ E β β G () β E wh G s th Batan uncton, aplac tanso o th hypbolc tangnt [9]. Equaton () shows that th daag vaabl D (s Eqn (4)) and th noalzd psntatv lngth β (s Eqn (9)) a non-tval unctons o β tanh( β ) β β β G (3) β In patcula, ths uncton s dnt o that obtand by assung that ϕ ( ) ϕ o whch (4) Fgu 3 shows th voluton o th o dnd by () (5) Fo lag valus o β, th two lngths a vtually dntcal. On th oth hand whn β dcass, th two lngths spaat o ach oth. Consquntly, th lngth s not ncssaly th chaactstc quantty to dscb th daag vaabl D. In th ollowng, w consd that th cackng and dbondng/sldng chanss a uncoupld. Tho th pvous sults a stll vald vn though dbondng/sldng wll b addd. W also assu that th ntac toughnss s vanshngly sall and that th ntacal bhavo s appoachd by a constant ntacal sha sstanc τ [0, ]. Th quvalnt dbond lngth l d s dnd though th xpsson o th nlastc stan τld ε (6) E Th nlastc stan dpnds upon th dstbuton o cack dstancs and dbond lngths. Th valuaton o th daag vaabl d s obtand o th non-covabl pat o th

5 ngy dnsty (s Eqn (3)). Whn th nu dstanc btwn two adjacnt cacks s gat than th dbond lngth, th latt taks a unqu valu and th daag vaabl d s wttn as 3 ( ) Eld d (7) 4 E Ths sult s only vald whn th ntacal bhavo vs th hypothss o ths scton. Whn th ntacal bhavo s o coplx, th vaabl d s popotonal to th ato l / povdd th stss ld n ont o th cack tp s that o th undaagd atal. d latv o, / Fg. 3: latv o Noalzd avag lngth, β / vsus noalzd avag lngth β To llustat th pvous sults, w wll study a sngl lant n whch th atx s o sstant than th b. Ths s a classcal tst conguaton to chaactz th ntacal popts o th copost. Th atus o th acoscopc stss/stan cuv a donatd by th dbondng and sldng chanss snc th avag dstanc btwn atx cacks (at satuaton) s two o th ods o agntud gat than th b adus 3 ( β 0, 0 ). Tho th valu o th daag vaabl D s nglgbl and th psntatv lngth s dntcal to th avag cack dstanc. To dtn th quvalnt lngth l d, on nds th valuaton o th nlastc stan ε as a uncton o th appld stss. In th cas o a constant ntacal sha sstanc τ, two altnat outs can b ollowd. Th st on s analytcal. Fo an solatd bak, an xcluson zon n whch no uth bak can occu can b xhbtd. Whn a bak s locatd at z 0, no nw bak can b catd n th ntval ] ] l /, + l / [[ snc th stss lvl ans lss than th alu stss at z 0 (Fg. 4). Th lngth l s xpssd as [0] wh T dnots th stss lvl n an unbokn b. T l (8) τ Stss n b, (T,z) Tl (T) z 0 Coodnat, z Fg. 4: Dpcton o an xcluson zon

6 Gulno and Phonx [] popos an appoxaton o th agntaton pocss by assung that t ans a Posson pont pocss thoughout th whol hstoy. Th dawback o ths odl s that t s unabl to captu th satuaton o cackng. To pov th sult, on ay us a Boolan ando slands odl [, 3] whch consttuts a low bound to th nub o bokn laws. Ths odl dos not account o xcluson ovlappngs and ov-stats th zon ov whch dcts cannot bak. Th law dnsty λ t ( T) s dscbd by a powlaw uncton o th local stss T λ t ( T) T S 0 0 wh and S 0 0 a th Wbull paats. Two chaactstc paats S c and δ c can b xhbtd [4, 5] by usng th latonshp λ t ( S c ) l ( S c ) S c (9) S + 0 0τ and δ c l( Sc) (0) Tho, th dnsty o bokn dcts λ b p unt chaactstc lngth δ c vs λb ( T) γ, T T T wth () S wh γ s th ncoplt gaa uncton. Ths sult posssss th ntstng popty o asonably dscbng th satuaton pocss (s Tabl ). Bsds, th xpsson o th pobablty dnsty uncton F() dvd n th pvous xapl wll b usd as a st appoxaton. To coput th avag nlastc stan ε, Eqn () can b wttn as ε ε ( ) F ( ) d () 0 wh ε ( ) dnots th nlastc stan o a cll o lngth. Fo an ntac wth a constant sha sstanc τ, th nlastc stan ε ( ) s xpssd as τ l whn < l ( ) E ε ( ) l τ whn l ( ) E so that th nlastc stan ε bcos l τ τ l ε E E xp ( ) ( ) (4) By usng Eqn (6) and placng l d by l, E by ( ) E (th ol playd by th atx and th b a nvtd n a onolant tst), on obtans l l l xp (5) A st od appoxaton about l / 0 ylds l (6) l c (3)

7 Ths sult agan shows that th psntatv lngth l can b appoxatd by l as a st od valuaton but dvgs o hgh valus o l /. Th non-covabl pat o th ngy dnsty ψ gvn by Eqn (3) can b wttn as ψ ψ ( ) F ( ) d (7) 0 wh ψ ( ) dnots th non-covabl ngy dnsty o a cll o lngth. Whn th ntacal sha stngth s assud to b constant, ψ ( ) s xpssd as Eτ l l + whn < l ( ) E E 6 ψ ( ) 3 Eτ l whn l ( ) EE 6 so that th avag ngy dnsty ψ bcos Eτ l l ψ l + xp EE (9) ( ) By usng Eqn (3), th ollowng xpsson o th daag paat d s ound E d ( ) E l l + xp l l l + xp A st od appoxaton o d about l / 0 ylds 3 E l d (3) 4 ( ) E Ths sult s dntcal to that o Eqn (7) by nvtng E and ( ) E (th ol o th atx and th b s nvtd n th agntaton tst o a onolant). To account o xcluson, a ky ssu clos to satuaton, on ay us oth appoachs. Cutn [5] bass hs appoach on a had co sph odl [6]. Ths odl has bn shown to b an appoxaton o th so-calld xact soluton [7]. All ths appoachs a analytcal. On can also us nucal solutons basd upon a Mont-Calo sulaton. Whn th ntacal sha stngth τ s assud to b constant, a nucal soluton s not ncssaly ndd, vn though t has bn usd [4]. On ay not that whn th xcluson phnona a not as spl as thos alludd to, a nucal thod s th only vabl altnatv [8-0]. To study th voluton o th psntatv lngth l, w now us a nucal analyss o th sak o splcty. To chck th sults, a copason (Tabl ) o th bokn law dnsty λ b p unt lngth δ c s pod wth data obtand by Hu t al. [7], Cutn [5] and Hnstnbug and Phonx [4]. (8) (30)

8 Tabl : Bokn law dnsty at satuaton λ b ( ) vsus Wbull odulus by usng dnt thods Hnstnbug Cutn Hu 3 Eqn () Mth. Mth *.49 * *.49 * [4], [5], 3 [7], * : 000. Two outs can b ollowd. Th st on (d to as thod No. ), s vy clos to a nt lnt concpt. It conssts n dsctzng a lngth tot lnts o dntcal lngth dz and xpncng on alu at ost. A Posson pont pocss s usd such that th pobablty o ndng a bokn law wthn dz << tot s dzλ t. A ando gnato o good qualty [] can cat a squnc o ando alu stsss accodng to th pvous pobablty. Th agntaton hstoy s dnd by ankng th pvous squnc n ascndng od and by chckng that a dct abl to bak s not locatd n an xcluson zon o a pvously bokn law. Ths pocdu s cad out untl all th alu stsss hav bn analyzd. It s vy asy to poga but s oy-consung snc all th alu stsss nd to b stod. Ths odl appoachs th analytcal soluton obtand o an nnt lngth (s o nstanc. 8). Th ltaton o th thod s th nub o lnts (o th od o 0 5 n ou cas). Th sults o 0 alzatons p Wbull odul a shown n Tabl. Th analyzd lngth s tot 0 3 δ c and th lnt lngth s dz 0 δ c. Th scond out (d to as thod No. ) s lss dandng on oy but ay tun out to b o t-consung than th pvous on whn th nub o bokn laws ncass. Th only stsss to b stod a thos o th bokn laws. Th hypothss a dntcal to thod No. ( tot Ntot dz). Th dnc cos o th coputaton o th stss stp whch s chosn n od to bak on law, t s not locatd n an xcluson zon o an alady bokn law. Th stss stp T s such that: dz λt / t( T) T ; th dct poston s andoly slctd accodng to a uno dstbuton ov a lngth tot. Th sach o all th xcluson zons o th bokn laws can pnalz ths thod snc t has to b pod o ach nw dct. Ths odl appoachs th analytcal soluton obtand o an nnt lngth (s o nstanc. 8). Th sults o 0 alzatons p Wbull odul a also shown n Tabl o a lngth tot 0 3 δ c and an lnt lngth dz 0 3 δ c. Th sults o Tabl show that th two nucal thods lad to valus vy clos to thos ound by Hu t al. [7]. To avod nd cts, podcty condtons a nocd to choos xcluson zons. Fgu 5 shows th noalzd quvalnt lngth l / as a uncton o th noalzd covy lngth l / o a Wbull odulus 3. Th th appoachs lad to th sa sult whn l / << : l l. Howv, th a dncs whn l / ncass. Th sults o a alzaton ( λ b ( ) 05). wth thod No. (dnotd by Nu.), thos o an appoxaton (dnotd by Appox.) by a Posson pont pocss (Eqn (5)) and thos o whch th ovlappng o xcluson zons s nglctd ( l l ). Th appoxaton by a Posson pont pocss ov-stats th nub o alus and tho und-stats th nlastc stan ε and th quvalnt lngth l. On th contay, th qualty l l

9 consttuts an upp bound snc t dos not account o xcluson ovlappngs: t ovstats th quvalnt lngth l. Noalzd quvalnt lngth, l / Nu. Appox. l / Noalzd covy lngth, l / Fg. 5: Noalzd quvalnt lngth l / vsus noalzd covy lngth l /. Th dnt appoachs a usd ( 3) CONCUSIONS All ths sults show that th choc o quvalnt lngths to dscb atx-cackng, dbondng/sldng qual to th avag valus s only vald as a st od appoxaton. Th dscpton o psntatv lngths n an lntay cll s tho an potant task n tsl. I on wants to chaactz dgadaton chanss by coscopc obsvatons, avag lngths ctanly psnt lvant quantts to asu. On th oth hand, th a s to dscb th nlunc on th chancal bhavo, oth lnts o th dstbuton lngth can b o potanc (.., not only avags nd to b consdd). A ull cochancal analyss s ndd to know th xact anng o ach lngth. EFEENCES. Gan, P., Nguyn, Q.S. and Suqut, P., Contnuu Thodynacs, ASME J. Appl. Mch., Vol. 50, 983, pp Avston, J., Coop, G.A. and Klly, A., Sngl and Multpl Factu, Natonal Physcal aboatoy: Popts o Fb Coposts (97), pp at, J. and Dually, J., Modélsaton t dntcaton d l'ndoagnt plastqu ds étaux, 3 congès anças d écanqu (977). 4. Andux, S., Babg, Y. and Mago, J.-J., Un odèl d atéau cossué pou ls bétons t ls ochs, J. Méc. Th. Appl., Vol. 5, No. 3, 986, pp Chysochoos, A., Masonnuv, O., Matn, G. and Cauon, H., Plastc and Dsspatd Wok and Stod Engy, Nucl. Eng. Ds., Vol. 4, 987, pp Cho, C. Hols, J.W. and Bab, J.., Estaton o Intacal Sha n Cac Coposts o Fctonal Hatng Masunts, J. A. Ca. Soc., Vol. 74, No., 99, pp Cox, H.., Th Elastcty and th Stngth o Pap and oth Fbous Matals, B. J. Appl. Phys., Vol. 3, 95, pp Gady, D.E., Patcl Sz Statstcs n Dynac Fagntaton, J. Appl. Phys., Vol. 68, No., 990, pp Span, J. and Oldha, K.B., An Atlas o Functon, Spng, Nw Yok, NY (USA), 987.

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