Micromechanical Modeling of Concrete at Early Age

Transcription

1 Micromchanical Modling of Concr a Early Ag A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIERSITY OF MINNESOTA BY aira Tulubkov IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Advisr: Lv hazanovich July

2 aira Tulubkov, May

3 Acknowldgmns I would lik o xnd my d graiud o my acadmic advisr Dr. Lv hazanovich for his guidanc and suor during my PhD sudis. My scial graiud gos o Dr. Sonia Mogilvskaya and Dr. Hnrik Solarski for hir classs, for srving in my commi and jus for warm frindshi, hl and advic during my sudy in gradua school. I am also graful o Dr. Prry Lo for srving in my PhD commi. I would lik o hank h fllowshi of gradua sudns of h Darmn of Civil Enginring for frindly amoshr, hling boh in hard sudis and in fun ims of daily lif. In aricular, I hank Elizava Gordliy and Ivan Gordli, for our communicaion and frindshi, which rsuld in my dcision o sudy in Univrsiy of Minnsoa. I rally arcia invaluabl conribuion d by Ivan Chikichv o my ovrall comur liracy. Many hanks go o naiv English sakrs in my rsarch grou, namly, yl E. Hogh, Mary ancura, Drk Tomkins, and Ria Ldrl for chcking my English in my hsis. Finally, I am vr graful o my family for hir lov and coninuous ncouragmn. Thy say always wih m a any sag in my lif, rgardlss of h disanc bwn us. i

4 Ddicaion This dissraion is ddicad o my hom counry and o my family. ii

5 Absrac Th focus of his rsarch is a micromchanical characrizaion of Porland cmn concr a arly ag lss han 8 days. Concr s viscolasic roris chang significanly a arly ag du o solidificaion of is rix comonn. Bažan s solidificaion hory modls concr as a rial solidifying in im. This aroach is gnralizd o a hr-dimnsional characrizaion of a comosi rial wih a solidifying rix and lasic inclusions. An ingral consiuiv rlaionshi was obaind using a gnralizd corrsondnc rincil and homognizaion chniqus for lasic comosi rials. In ligh of his aroach, ffciv cr roris of comosi shrical assmblag wih an aging rix ar obaind. In addiion, h lasic Hashin-Moniro modl is gnralizd o accoun for h ffc of h inrfacial ransiion zon roris on concr cr. An ffciv comuaional laform was dvlod o valua oraor xrssions in ordr o obain rlaxaion and cr funcions numrically. Through numrical xamls, i is shown ha riaxial gnralizaion of Bažan s solidificaion modl nabls robus and comuaionally fficin rdicion of cr dforions in Porland cmn concr. iii

6 Conns Lis of Tabls...v Lis of Figurs... vi. Prambl.... Prsn sa of knowldg Concr as a comosi rial Cr modls Objcivs and significanc of rsarch....4 Organizaion of Dissraion On-dimnsional solidificaion hory for homognous mdia Thr-dimnsional solidificaion hory for homognous mdia On-dimnsional solidificaion hory for wo-has comosi....4 Limiaions of xising solidificaion hory and ossibl aroachs o rsolv hm 4 3. Problm dscriion Prliminaris from lasic hory Gnralizd Eshlby s rincil Ur and lowr bounds of rlaxaion funcions Comosi wih shrical inclusions Bounds of shar rlaxaion funcions for vry sll and vry larg aricl concnraions Discussion Numrical valuaion of rlaxaion bounds Illusraiv xaml Inroducion Thr-dimnsional gnralizaion Numrical xaml for riaxial solidificaion modl Inroducion Hashin-Moniro s modl lasic cas Gnralizaion of Hashin s modl for aging viscolasic rials Zvin s mhod of iraiv funcions Cr modl for cmn as Snsiiviy analysis Comarison wih Bažan s B3 modl Conclucions Fuur work Bibliograhy Andix A iv

7 Lis of Tabls 5. Marial Characrisics of Shocr According o Hubr v

8 Lis of Figurs. Muliscal microsrucur of cmn-basd rials.. Couno and Hansn modls..4.3 Masurd lasic modulus of concr wih vs. Hashin-Shrikn bounds.5.4. Inrfacial ransiion zon ITZ. 5.5 Hashin-Moniro modl for hr hass Effc of rlaiv ITZ lasic modulus B3 cr rdicion bfor lf and afr righ modificaion Aging of cmn as as a growing of rix sklon owards h ors....9 Evoluion of Young s moduli.... Princil of homognizaion: diffrn lngh scals..3. Solidificaion modl undr uniaxial load...6. Uniaxial rhological modl Hrognous mdia and HEM wih h sam boundary condiions....4 Uniaxial modl, usd by Grangr and Bažan. 3. Hashin s comosi shrs assmblag.7 3. Hrognous mdia and HEM wih h sam boundary condiions Hrognous rial dividd ino comosi lmns Aroxiion of n-h comosi lmn by concnric shrs Coninuous variaion of on-aramric family of bulk rlaxaion funcions Coninuous variaion of on-aramric family of shar rlaxaion funcions Aging of concr as a growh of sklon oward h ors Gnralizd 3D modl and ffciv B3 modl Transvrs cr comlianc of h comosi Raio of ransvrs srain o longiudinal srain of h comosi Snsiiviy wih rsc o shar cofficin s Snsiiviy wih rsc o ITZ hicknss vi

9 5.3 Concr cr as rdicd by h roosd modl and h B3 modl for diffrn valus of w/c vii

10 Char Inroducion. Prambl An incrasing us of fas-rack consrucion has gnrad dnd for modls caabl of rdicing of concr bhavior for a wid rang of im of load alicaion ims from arly ag loading o long-rm bhavior. Du o his dnd and bcaus of consan modificaions in concr mix dsigns, mirical quaions which hav bn succssfully usd in h as canno m modrn rquirmns. This cras a nd for h dvlomn of nw modls, which can rovid accura rdicion of concr roris basd on a limid numbr of ss. Concr is a comosi rial consising of cmn as, fin and cours aggrgas, and air voids. Alhough concr can b considrd as homognous rials on a larg scal > - m, i y b inaroria o ignor is hrogniy on sllr scals On h cro-scal lvl - m, concr can b rgardd as a comosi ha consiss of minral coars aggrga mbddd in a rix of morar Grangr & Bažan 995. Th rix of morar islf can b considrd as a comosi on a mso-scal lvl ~ -3 m, consising of hardnd cmn as and fin aggrga sand. Th cmn as, in urn can b considrd on an vn sllr scal ~ -6 m as a comosi consising of hardnd cmn gl and ors filld by caillary war and air. This squnc of mbddd lvls of scaling can b coninud o mor rfind lvls of rial characrizaion, as shown in Fig...

11 Fig.. Muliscal microsrucur of cmn-basd rials Brnard al. 3. An imoran mchanical rory of concr is is im-dndn rlaionshi bwn alid srsss and xhibid srains. Undr consan loading, concr srains incras also known as h cr ffc, and h xhibid srains dnd no only on h duraion of h alid load, bu also on h im sinc h concr was cas of load alicaion. Th sudy rsnd hr will focus on modling of h cr bhavior for a wid rang of im of load alicaions and load duraions. Scial anion will b givn o rdicing cr ffcs a arly ags.

12 . Prsn sa of knowldg.. Concr as a comosi rial Th roris of a givn comosi rial dnd on h roris of is comonn rials. Howvr, whn hs comosis conain rials wih vasly diffrn roris, h modls mloyd o undrsand h comosi can bcom comlx. Dnding on h numbr of diffrn disinguishd hass wihin h comosi rial, modls of diffrn comlxiy migh b mloyd. Th simls comosi modls ar wo-comonn comosis in which wo lasic rials ar conncd as ihr aralll oigh or sris Russ lmns. Ths modls si h ur and lowr bounds for h ffciv Young s modulus for a wo-has comosi rial. Howvr, hos sis ar vry rough in ny cass. Lar by using variaional rincils, Hashin and Shrikn 963 narrowd h rang of ossibl valus of ovrall comosi bulk modulus and shar modulus μ, obaining h following bounds for saisically isoroic mdia:... 3c 3c c c c c... 6 c 6 c whr c and c ar volum fracions of hass; h subscris and corrsond o h roris of rix and aggrga inclusions hass, rscivly. I is assumd ha and. 3

13 Couno 964 considrd h cas whr an aggrga inclusion is lacd a h cnr of a rism of rix Fig.., whr boh h inclusion and rix shar h sam raio of high o ara in hir rsciv cross scions. By using a siml srngh of rials aroach, h Young s modulus E of h comosi is whr E E c c E E E and E ar h Young s moduli of comonns....3 m m a b Fig.. a Couno, and b Hansn modls. Hr m and sand for rix and aricl inclusion rscivly. Hansn 965 roosd a modl Fig.. ha considrs comosi lmn as a shrical inclusion locad a h cnr of a shrical rix. In his modl ce c E E c E c E E...4 A vry imoran xaml of a comosi civil nginring rial is Porland cmn concr PCC. I is naural o considr PCC as a wo-has comosi rial in which coars aggrgas inclusions ar disrsd in cmn as rix. Howvr, as was oind ou by Nilsn and Moniro 993, ny of h xrimnal rsuls Fig..3 obaind by Hirsch 96 viola h Hashin Srikn bounds, which ar valid for wo-has saisically isoroic rials of any has gomry. 4

14 Fig..3. Masurd lasic modulus of concr wih vs. Hashin-Shrikn bounds adad from Nilsn and Moniro 993. On on h xlanaions of his hnomnon is ha h wo-has modl dos no accoun for h ffc of h inrfacial ransiion zon ITZ, h rgion of lss dnsly ackd cmn as surrounding an aggrga aricl Fig..4. Fig..4. Inrfacial ransiion zon ITZ adad from Cwirzn & Pnala 5 5

15 Alhough his rgion accouns only for a sll fracion of h rix volum, i y significanly affc h siffnss and srngh of h comosi. Thrfor, for comosi rials lik PCC, h ITZ mus b akn ino considraion as a sara has from hos of h inclusions and surrounding rix. To dscrib lasic bhavior of concr, Hashin and Moniro roosd a slf-consisn hr-has modl of an lasic comosi. In his modl Fig..5, a shrical aggrga inclusion wih avrag radius of aggrga aricls is lacd a h cnr of a shr of ITZ, which is in urn locad a h cnr of shr of h rix. Figur.5 dnos h volum fracion of aggrga shr as c and volum fracion of ITZ and rix shrical layrs as c ITZ and c rscivly. I is noabl ha h Hashin-Moniro modl rovids an analyical form for h ffciv roris. A fini lmn soluion would rquir a sohisicad aroach for h numrical ingraion nar h ITZ du o rial disconinuiy L and Park 8. 6

16 Fig..5. Hashin-Moniro modl for hr hass. From h Hashin-Moniro modl, h ffciv bulk modulus xlicily as c c can b xrssd ITZ...5 3c * 3 4 whr,, ITZ ar bulk moduli of h aggrga, rix and ITZ rscivly; and ITZ ar shar moduli of h rix and ITZ rscivly, and * c / citz c ITZ...6 3cITZ / citz c 3 4 ITZ Th norlizd ffciv shar modulus soluion o quadraic quaion ITZ ITZ g * / is givn in h imlici form as a * * Ag Bg C...7 whr cofficins A, B, C dnd on volum fracions and rial characrisics of comonns. Th modl dscribs h shar and Young s moduli of ITZ as bing aroxily 5% of hos of h rix, whil according o h modl, h bulk 7

17 modulus of h ITZ is on h ordr of 7% of h bulk modulus of h rix Hashin and Moniro. A snsiiviy analysis Zhng al. 6, Li & Zhng 7 dmonsrad ha roris of ITZ significanly affc h lasic roris of concr. Figur.6 illusras h ffc of rlaiv ITZ lasic modulus E / E on h ITZ rlaiv concr lasic modulus E / E. Fig..6. Effc of rlaiv ITZ lasic modulus on rlaiv concr lasic modulus in body conaining ITZ wih hicknss 38 μm and aggrga aricls wih diamr 6 mm. A a is h aggrga volum fracion adad from Zhng al. 6. Sinc h ITZ significanly influncs h lasic roris, i is rasonabl o suggs ha ITZ affcs cr as wll. ITZ islf is nohing ls bu lss dnsly ackd cmn as wih similar cr roris. 8

18 .. Cr modls During h las wo dcads, significan advancs in h undrsanding of cr of concr hav bn achivd. Cr bhavior of concr was a subjc of numrous xrimnal and horical sudis. arious hnomnological modls hav bn roosd o dscrib concr cr, such as ACI 9 Mha & Moniro 6, h doubl owr law Zhng al. 6, h doubl owr logarihmic law Li & Zhng 7, CEB-FIP modl cod 99 and GL modl Cwirzn & Pnala 5. On of h mos rliabl and hysically sound is Bažan s B3 modl Bažan & Bawja 995. This modl rlas concr cr comlianc funcion wih concr mix roris: J,...8. q, q,q, q,3 ln[ ] q,4 ln whr rms q, k k,..., 4 dnd on war, cmn, and aggrga wigh conns. Howvr, on roblm wih h rviously mniond modls is ha hos modls ar buil mirically o dscrib h cr of ur concr. Thy do no accuraly rdic cr for loading ags of on day or lss L.Øsrgaard al ; D Ambrosia al. 4, and nd o b modifid o caur arly ag bhavior. To his nd, rsarchrs L.Øsrgaard al. roos modificaions o xising modls, such as h inclusion of an addiional rm in h B3 modl o br rdic cr bhavior a in vry arly ags Fig..7. 9

19 Fig..7. B3 cr rdicion bfor lf and afr righ modificaion adad from L.Øsrgaard al For h modling of h join forion of concr avmns, modls for cr in concr a arly ags ar vry imoran. Morovr, h availabl concr cr modls only dscrib mirically drmind rlaionshis bwn concr roris and concr comosiion. In h roosd sudy, aging is considrd as a filling of h ors in cmn as by hydraion roducs. On a micromchanical lvl, such an aroach rsums h ramn of h considrd body hr, a concr scimn as a growing body. This aroach has bn roosd by harlab 98 for on-has rials and is rcognizd in solidificaion hory Bažan & Prasannan Objcivs and significanc of rsarch Th aim of h sudy rsnd hr is o dvlo an analyical and comuaional basis for modling concr bhavior, scially a arly ags. For his sudy, concr is considrd on h croscoic scal as a comosi, which consiss of lasic aggrga

20 aricls surroundd by a viscolasic cmn as rix i.. h rix. Th novly of his aroach is in h considraion of h rix as a growing sklal srucur wih ors bing filld by roducs of hydraion Fig..8 Inhomognous isoroic agg ors or agg agg hydraion in or Fig..8. Aging of cmn as as a growing of rix sklon owards h ors agg dnos aggrga aricls. A arly ags, du o cmn hydraion, h concr s sklon grows owards h ors, filling hm u. This rocss causs changs in h croscoic roris of h rix during arly ags Fig..9. Ths roris ar h modulus of lasiciy E and h Poisson raio for lasic dforions. Th rial of h sklon is assumd a homognous, viscolasic and isoroic mdia. This characrizaion of h rix allows a rlaivly siml and maningful hysical xlanaion for h dndnc of h cr rsons on h momn of loading.

21 Fig..9. Evoluion of Young s moduli wih h dgr of hydraion for morars B35 and B6 adad from Boumiz al. 996 Among h chniqus mloyd for his sudy, i is worh highlighing Zvin s mhod of iraiv funcions Zvin 979. This mhod allows h calculaion of an ffciv cr comlianc oraor of h rix from h ffciv cr comlianc and vic vrsa. Anohr faur of his aroach is h ossibiliy of using h gnralizd corrsondnc rincil harlab 98; hanzanovich 8 o ransfr h viscolasic roblm of a growing body ino an lasic form in h ransformd doin. Anohr aim of rsn sudy is modling isoroic comosi rial as consising of hr hass, namly: h rix, inrfacial ransiion zon ITZ, and aggrga inclusions. Aggrga aricls ar surroundd by h ITZ, and hs ITZ-nclosd aggrga aricls ar mbddd in h largr cmn rix. Th aggrga aricls ar considrd as lasic shrical inclusions wih Young modulus E, Poisson raio,

22 volum fracion c and avrag radius a. Th rix on h croscoic lvl has bn characrizd as an viscolasic rial wih Poisson raio, volum fracion c and cr comlianc,, whr, ar h ag of concr and h momn J of loading rscivly. On h micromchanical lvl, h rix consiss of linar viscolasic sklon and ors insid h sklon. Fig... Princil of homognizaion: diffrn lngh scals adad from Sanahuja al. 7. Th aggrga aricls ar surroundd by ITZ wih hicknss i. Th ITZ also is rad as linar viscolasic rial wih Poisson raio ITZ, volum fracion ITZ, and cr comlianc,. In h currn sudy, h cr comlianc of ITZ is assumd J ITZ o b roorional o h rix cr comlianc as J ITZ, J,.3. whr. Th mirical analog of assumion.3. has bn obsrvd for h rlaionshi bwn h Young s moduli of ITZ and h cmn rix Hashin & Moniro. Th whol comosi is considrd as a linar viscolasic rial wih ffciv roris, namly Poisson raio and cr comlianc J,. 3

23 .4 Organizaion of Dissraion Th dissraion is organizd as follows. Char rovids a rviw of solidificaion hory, which is h hical basmn of aging viscolasic mdia. Consiuiv srss-srain rlaionshis for homognous and hrognous rials ar givn. Char 3 concnras on h micromchanics of aricula comosi rials, h comonns of which y xhibi cr roris; hs roris y dnd on h im of load alicaions. Aroria variaional horms ar rovn, and by viru of hs horms, viscolasic bounds for ffciv rlaxaion funcions ar obaind Char 4 rsns an invsigaion on gnralizaion of uniaxial modl o riaxial aging modl whn bulk modulus of h rix has is consan. Char 5 rsns a sudy on h ffc of inrfacial ransiion zons on h ovrall dforion rsons of h considrd comosi. Finally, Char 6 sumrizs h findings of h rsarch and oulins ossibl dircions for fuur work. 4

24 Char A aricula comosi, such as Porland cmn concr PCC, is dfind as a rial whr on has of discr inclusions is fully surroundd by anohr has of rix. Drmining h ffciv viscolasic roris of aging comosis using h roris of ach comonn is a common aroach whn characrizing comosi rials. This aroach has bn considrd for comosis such as olymrs Boyr al. 998; Fishr and Brinson ; Brinson and Gas 3, biological issus Gosn and Hashin 98; Laks al. 979, and concr Bažan and Bawja 995; Srcomb al. ; Grasly and Lang 7; Aly and Sanjayan 9. Th mhods usd o characriz comosi rials can b gnralizd o drmin h ffciv roris for comosis wih an arbirary gomric srucur, bu dail and accuracy of h modl ar los. To avoid his, his sudy focuss scifically on aricula comosis. Th aim of h sudy rsnd in his ar was o build a muli-axial gnralizaion of h uniaxial Grangr-Bažan modl o caur ffciv aging viscolasic roris of aricula comosis in hr-dimnsions. Th Mori- Tanaka schm from lasiciy hory was gnralizd for aging rials o drmin ffciv srain dforions. Th roosd modl is validad hrough comarison of h cr comlianc o ha rdicd Bažan s B3 modl using a numrical xaml ha is valid for boh modls. 5

25 . On-dimnsional solidificaion hory for homognous mdia Bažan s solidificaion hory 977 uss h growh of non-aging consiuns oward ors in h rial o xlain h im dndnc of ffciv cr roris. Sinc his modl is h saring oin for h subjc of his sudy, a brif ovrviw of his aroach is sumrizd. A mor daild xlanaion can b found in Bažan s ars Bažan 977; Carol and Bažan 993; Grangr and Bažan 995. In solidificaion hory, saisically homognous quasi-homognous rial is modld as a sklon solidifid r wih ors which fill u as h rial ags. As alid o Porland cmn concr PCC, h hydraion rocss is modld by ors filling u wih hydraion roducs Fig..8. Wihin linar cr hory, h srain rsons of a concr scimn o uniaxial srss,, is rlad in h following way: J, d.. whr J, dnos h srain a im causd by a consan uni srss, alid a im. Th rix rial can b considrd as an aging rial du o growh of h non-aging sklon as h ors fill in wih dosid hydraion roducs. Hydras from h war soluion rciia on h wall of h sklon o form nw layrs microlmn of solidifid r and bgin o bar a load. In h mos siml form Bažan 977, h volum of hydrad cmn as is assumd o b growing by dosiion of hydras, as shown in Fig... Bažan 977 suggsd ha h 6

26 croscoic srain is qual o h avrag srain in all microlmns of h solidifid r. Fig... Solidificaion modl undr uniaxial load adod from Bažan 977. In his cas, h croscoic srss,, is h sum of all srsss in h microlmns. Srss,, in h solidifid r micro srss a im varis wih locaion. Th mos g suiabl aramr wih which o characriz h locaion of h srss, according o h modl shown in Fig.., is h volum fracion,, of h hydrad as. Thus,, dnos microsrss a im,, du o h volum of solidifid rial,, wihin h currn microlmn locaion. Bažan s modl assums aralll couling of solidifying microlmns, d, whr dosi layrs sar baring load immdialy afr solidificaion. Thrfor, h oal croscoic srss a im is: g, d.. 7 g

27 Now considr h srain a h locaion whr solidificaion occurs a im ', rcding h currn im,. Th rial of h sklon bwn ors is assumd o b homognous, viscolasic, and isoroic, wih a micro-comlianc funcion, and Young s modulus as a consan, E /. Thus, h srain-srss rlaionshis wihin h micro-lmns ar as follows: g g ', ' d..3 Bcaus hydras rciia a a srss-fr sa, ', ', Bažan 977 obaind h following rlaionshis from.. and..3: 8 g d..4 Howvr, hos facors ar no rvailing in h hardning sag bcaus hy would conradic h assumd linariy of h srss-srain rlaionshi..4, which was xrimnally confirmd for h srvic srss rang of concr. In microrsrss-solidificaion hory Bažan al. 997, h uniaxial Hook s law.. is xndd o h muliaxial cr law as follows: ε J, Gdσ..5 whr σ and ε ar srss and srain nsors, rscivly. Th 3 3 rix G dnds only on Poisson s raio, which dos no dnd on im. Howvr, quaion..5 dos no work for young concr, whr Poisson s raio varis significanly wih im. I should b nod ha hr ar various facors ohr han hydraion causing aging of h rial, such as h growh of microcracks, microcrack haling, and bonding or

28 dbonding wihin h solid sklon Bažan and Hu 999. A good ovrviw of ohr hysical mchanisms causing aging is givn in h work of Bažan al. 4. Howvr, i has bn xrimnally confirmd ha hos mchanisms ar no rvaln in h hardning sag for h srvic srss rang of concr; his is foruna bcaus ohrwis hy would conradic h linariy of h srss-srain rlaionshi..5. On ohr aging mchanism which dos no inrfr wih h aformniond linariy is h rlaxaion of localizd high srss aks in h microsrucur of cmn gl, dscribd by h microrsrss-solidificaion hory Bažan al., 997. Ulm and Coussy 996 sudid h kinics of h hydraion racion and is voluion wihin h hrmodynamic framwork of young concr. Grassly and Lung 7 combind his hrmodynamic framwork wih a im-shif modling aroach for cmn as gl o mor accuraly rdic h aging viscolasic bhavior of Porland cmn as. Ricaud and Masson 9 considrd aging as h rsul of h imdndn orosiy of h solidifying comonn of h comosi. In hir work, im varying orosiy affcd h viscous roris of h rix comonn, whras is lasic roris wr consan in im. Howvr, Bažan s solidificaion hory srvs wll as boh a saring oin and as a bnchrk hory for his sudy.. Thr-dimnsional solidificaion hory for homognous mdia Dforion roris of isoroic rials can b dscribd in a nnr similar o lasic hory by using wo comlianc funcions o characriz muually rndicular srains. For ur concr, whr Poisson s raio, ν, is assumd o b consan, a 9

29 uniaxial comlianc funcion J,, mulilid by and 3, can b usd o drmin boh h volumric and dviaoric comlianc funcions Bažan, 988. L, dno a laral srain rsons, ransvrs o h dircion of uniaxial J consan srss, alid a im τ. Funcions J, and J, y srv as anohr alrnaiv air of comlianc funcions o dscrib h rial rsons in h muliaxial modl. In h microrsrss-solidificaion hory Bažan al. 997, h uniaxial Hook s law..4 is xndd o h muliaxial cr law as follows: ε J, Gdσ.. whr σ and ε ar srss and srain nsors, rscivly. Marix G dnds only on Poisson s raio,,which dos no dnd on im. Howvr, quaion.. dos no work for young concr, whr Poisson s raio varis significanly wih im. In gnral, hr is no closd form soluion for drmining comlianc funcions, J, and J,, sinc h aging law,, and h microcomlianc funcion, ar arbirarily chosn. Nvrhlss, for scial cas whn raio bwn ras of muually ransvrs srains J, and, dnds only on currn im J, / J,.. and dos no rval viscolasic naur, harlab 98 obaind a soluion for h laral comlianc funcion in h ingral form: J J ' ', ' d E ' g '..3

30 .3 On-dimnsional solidificaion hory for wo-has comosi Concr can b considrd as a comosi rial, in which a rix has modls h cmn as and inclusions modl coars aggrgas. Th rix has, xhibis cr roris, whil h inclusions ar lasic. A homognous rial ha has h sam croscoic rsons o h sam alid boundary condiions s figur.3 will b rfrrd o as h homognous quivaln mdium HEM. Th volum chang of h HEM will b dnod as an ffciv volum chang, and h microcomlianc funcion of h HEM as an ffciv microcomlianc funcion. Th consiuiv rlaionshis..4 hold for h HEM wih and. From hr, h ffciv Young s modulus and viscolasic oraor of ffciv cr comlianc can b drmind as: E / and J d, rscivly. Th rlaionshi bwn h comosi roris and roris of is comonns vary from on modl o anohr. Among h simls, is h on-dimnsional comosi modl Fig..4 roosd by Grangr and Bažan 995. This modl combins aralll and sris rix cr of cmn as in a orous solidifying rial wih volum chang. Th o lmn of h sris couling rrsns layrs of rix rial wihou inclusions. This o lmn orion of h modl is s qual o -ξ. Th boom lmn of h sris couling rrsns layrs of rix rial wih inclusions. This boom lmn orion consiss of rix rial and inclusions

31 could in aralll, and is s qual o ξ. Th aggrga inclusions ar assumd o b a is ximum comacnss and balanc. Th aramr c /, is a cross-scion aramr, whr c is h volum fracion of aggrga aricls. a b, homognizd rix inclusions Fig..3 a Hrognous mdia and b HEM wih h sam boundary condiions. Similar o..4, srain and srss in h rix ar rlad by: J.3. whr J d microcomlianc krnl of h rix, '. is h viscolasic cr oraor wih a

32 E ', rix E ', rix E, aricl -χ χ Fig..4 Uniaxial modl, usd by Grangr and Bažan 995. Solidificaion hory Bažan 977 says ha an lasic Young s modulus, E of h solidifid rix grows roorionally o h volum chang,, of h rciiad hydras: E /.3. E From his, an ffciv Young s modulus in h Grangr-Bažan modl in Fig..4 can b dfind as: whr 3 E E E E.3.3 E is Young s modulus of h aggrga inclusions. Th ffciv volum chang for h ovrall comosi rial can now b found using E / E. Th ffciv cr comlianc funcion, in his uniaxial modl is drmind as follows Grangr and Bažan 995: J E R, J, J, E E E E R,.3.4

33 whr R, and J, ar rlaxaion and cr comlianc funcions of h rix, rscivly..4 Limiaions of xising solidificaion hory and ossibl aroachs o rsolv hm Paricula comosi rial rsns addiional challngs which do now allow for a sraighforward gnralizaion of soluion..3 for ffciv laral srain,. In hrognous mdium, h rlaion bwn srain ras J, and, is mor comlx han h raio dscribd in..3, and ha raio canno lay h rol of Poisson s raio anymor. This horical obsacl forcs a rconsidraion of h dscriion in rms of rlaxaion funcions, analogous o shar and bulk moduli in h lasic hory. In a non-aging viscolasic roblm, krnls of oraors dnd only on a im shif, whr is ag of rial and is loading im. Thrfor, h corrsondnc rincil is alicabl, and i allows rformulaion of h original roblm as an lasic roblm in h Lalac ransform variabls doin Eshlby 957, Chrisnsn 969, Paulino and Jin. Onc ha quasi-lasic roblm is formulad, on y dal wih ingral oraors as wih lasic consans, and aly any homognizaion schm known from lasic hory o drmin ffciv rial roris. In cas of h aging roblm, krnls of oraors dnd on ag of rial and loading im saraly so Lalac ransform is no longr alicabl. Morovr, i canno b assumd ha oraors ar commua wih ach ohr. In h ligh of hs horical obsacls, i was rasonabl o consrain h considraion o h class of oraors which 4 J J

34 will commua wih ach ohr, y will broadly rrsn characrisic roris of xising aricula comosis. On of siml modls fulfilling his rquirmn is roosd in h rsn sudy. In h roosd modl, h bulk modulus of solidifying rial is assumd o b consan in im. This assumion allows h alicaion of any avraging mhod usd in lasic hory o drmin ffciv aging viscolasic roris. In Scion 4, h Mori-Tanaka schm is adad for drminaion of ffciv viscolasic oraors using hs concs. 5

35 Char 3 Bounds on rlaxaion funcions for aging comosis Drminaion of dforion roris of comosi rials from roris of hir comonns is an imoran roblm. This ar concnras on h micromchanics of aricula comosi rials of which h comonns y xhibi cr roris and hs roris y dnd on h im of load alicaions. Eshlby s horms for lasic rials and Hashin s aroach for homognizaion of lasic roris was gnralizd o obain h bounds for h rlaxaion funcions. For Hashin s assmblag, furhr simlificaions of h bounds ar dvlod. An aroach for drminaion of ffciv shar rlaxaion funcion ha can b inrrd as a gnralizaion of h Mori- Tanaka schm is also roosd. 3. Problm dscriion Drminaion of dforion roris of comosi rials such as concr Bažan and Bawja 995; Srcomb al. ; Grasly and Lang 7; Aly and Sanjayan 9, olymrs Boyr al. 998; Fishr and Brinson, Brinson and Gas 3, and biological issus Gosn and Hashin 98; Laks al. 979 from roris of hir comonns has bcom an imoran roblm. Thr ar diffrn 6

36 ffciv mdium concs of lasic comosi rial; comarison bwn hm can b found in Wall 997. Th acivly usd avraging horis ar h gnralizd slfconsisn schm Bnvnis 8, Halin-Tsai modl, nrgy aroachs Hashin 96; Hashin and Shrikn 963; Wall 997, diffrnial ffciv mdium hory and fini lmn modls Garboczi and Brryn. In ny cass i is also imoran o know h horical bounds in ordr o s xrimnal rsuls and various hurisic horis, as wll as for srucural oimizaion of comosis Gibiansky and Milon 993. In addiion, on of h bounds ofn rovids a rlaivly accura si Hashin and Shrikn 963 of h rial roris vn whn h rcirocal bound divrgs from i Torquao and Lado 99. I is difficul o giv clos bounds of ffciv roris for comosis wih arbirary gomrical srucur. So far, h bs achivmn in ha dircion is h Hashin- Shrikn bounds Hashin and Shrikn 963. Howvr, som classs of comosis can b sudid mor horoughly. On racically imoran class is aricula comosis. A aricula comosi is dfind as a rial whr on has is rrsnd as discr inclusions fully surroundd by anohr has which will b rfrrd o as a rix. For som has gomris nrgy mhods y rovid xac bounds. This was dmonsrad for ffciv bulk modulus in Hashin s assmblag also known as comosi shrs assmblag Hashin 96. Hashin s assmblag fills h whol sac of h comosi wih inclusions surroundd by shrical shlls of h rix so ha h raio of h our shll radius o h radius of h inclusion is consan s Fig

37 - Fig. 3. Hashin s comosi shrs assmblag. Th bounds of h ffciv bulk modulus, c c, for h Hashin s assmblag ar: c c whr and ar h bulk moduli of h rix and inclusion hass, is h shar modulus of h rix, and c is h volum fracion of h aricl inclusions. I can b shown ha ur and lowr bounds in 3.. coincid. For h shar modulus, Hashin has also drivd xac bounds for Hashin s assmblag whn h volum fracion of aricls c is clos o zro or. This ar concnras on h micromchanics of comosi rials of which h comonns y xhibi cr roris, i.. h srss-srain rlaionshi of h rix has can b xrssd by h following form: kk kk and of h inclusions has: kk kk

38 whr and ar comonns of srss and srain nsors in h rix and h aricula hass ar dnod by surscris and, rscivly. d is h ronckr dla: is im. Th oraors,,, if i j,, if i j and f f, d, f f, d 3..4 f f f, d, f, d 3..5 ar hrdiary ingral oraors wih h bulk, and shar, rlaxaion funcions characrizing h rix roris. Analogously,, and, ar bulk and shar rlaxaion funcions of h inclusion has. Th consiuiv quaion for h homognous quivaln mdium HEM can b wrin as whr HEM HEM HEM HEM kk kk HEM HEM and ar comonns of h srss and srain nsors in a homognous quivaln mdium HEM, Fig. 3. and h ovrall ffciv bulk and shar modulus oraors of h comosi rial ar: f f, d, f f, d 3..7 whr, and, ar h ffciv bulk and shar rlaxaion funcions, rscivly. 9

39 , U i, S homognizd,, Fig. 3. Hrognous mdia on h lf and HEM wih h sam boundary condiions on h righ. Pas rsarch in his ara inly concnrad on homognizaion of viscolasic comosis, which is a scial cas of his gnral roblm. Proris of viscolasic rials ar dscribd wih Equaions 3.. and 3..3, bu all krnls of ingral oraors 3..4 and 3..5 dnd only on a im shif.g., 3. Eshlby 957 was on of h firs o show ha h hory of lasic comosis can b xndd o viscolasic comosis. Th corrsondnc rincil sas ha if h boundary consrains ar sarabl funcions of sac and im, hn subjcing all quaions of h boundary valu roblm BP o an ingral ransform for xaml, h Lalac ransform lads o an associad lasic roblm Chrisnsn 969, Paulino and Jin, Mukhrj and Paulino 3. rix inclusions Wang and Wng 99 xndd h Eshlby-Mori-Tanaka mhod ino h Lalac doin for wo ys of viscolasic rials: a ransvrsly isoroic on wih shroidal inclusions and an isoroic on wih randomly orind inclusions.

40 In gnral, whn using variaional bounding for viscolasic mdia i is hr no uniqu osiiv dfini forms, which ar comlly analogous o lasic srain nrgy Brur 975. As i was mniond by Chrisnsn 968, hr ar svral nrgy funcionals ha could srv for dfiniion of ffciv roris. Chrisnsn 969 sablishd minimum horms for som nrgy funcional in h Fourir doin. By us of hs horms, h drivd bounds of ffciv comlx moduli for viscolasic aricula comosis. Lar Hashin 97 dfind ffciv comlx moduli by h corrsondnc rincil for Hashin s assmblag and for gnral incomrssibl-rigid and incomrssibl-orous viscolasic comosis. Alrnaivly, Gibiansky and Milon 993 inroducd a quadraic quasi-convx funcional dfind ovr h sac of rlvan sas associad wih diffrnial rlaions bwn srains and srsss. Onc such a quasi-convx inqualiy is formulad, i can b usd o driv a variaional inqualiy analogous o a variaional inqualiy for an lasic cas. Th rsul of h gnralizaion of h Hashin-Shrikn mhod givs a sysm of four inqualiis, which characriz h ur and lowr bounds of h ral and iginary ars of h ransformd ffciv comlx moduli. Using h corrsondnc rincil, Hashin 965 has gnralizd som of his xac rsuls for Hashin s assmblag from lasic hory o h viscolasic cas. H obaind h following bounds for h Lalac ransform of h bulk rlaxaion funcions: k low s k s k c4 s 3k s k s k s3k s 4 s 4c s k s s k s

41 whr k k u s k s s d c k s k s 3k s 4 and k s s4 s 3k s 3c k s k s d s s 3..9 ar Lalac ransforms of bulk rlaxaion funcions of aricl and rix rscivly; s and s Lalac ransforms of shar rlaxaion funcions; k low s and k u s of Hashin s lowr and ur bound for h ffciv bulk modulus. ar similarly ar Lalac analogs Th soluions can b alid only o viscolasic roblms wih imshif invarian krnls which is a srious consrain whn modling aging comosis hazanovich 8. Rcnly, hr was inrs in homognizaion of comosis wih im-shif non-invarian rlaxaion funcions of comonns. Ricaud and Masson 9 considrd a scific cas of such rials and alid h Mori-Tanaka schm a ach im incrmn o drmin ffciv rlaxaion funcions. Howvr, no rsarch has bn don o drmin h bounds of such rlaxaion funcions. This ar aims o fill his ga. 3. Prliminaris from lasic hory Hashin 96 has roosd a hory of quivalncy bwn a aricula comosi rial and a homognous solid basd on qualiy of srain nrgy sord in boh rials subjcd o h sam boundary condiions. Basd on his cririon, h drivd h lowr and ur bounds for h bulk and shar moduli of h quivaln homognous lasic rial. Sinc Hashin s aroach will b gnralizd in his ar, his scion rovids a brif ovrviw of ha aroach. A daild xlanaion can b found in h ar of Hashin 96. 3

42 Considr a hrognous aricula comosi body wih volum and boundary surfac S in which h disribuion of inclusion aricls is saisically isoroic. Consiuiv rlaionshis dscrib lasic bhavior of h comosi if h rlaxaion funcions in h oraors ar consans and for rix bulk and shar oraors, rscivly, and and for bulk and shar oraors for inclusions, rscivly. Th ffciv bulk and shar moduli of a homognous quivaln mdium HEM ar dfind by h consiuiv rlaionshis 3..6 wih I, I, I is an idniy oraor so ha h lasic nrgy of h HEM Fig. 3.b is qual o h lasic nrgy of h hrognous comosi Fig. 3.a, i.. whr and d d 3.. HEM HEM ar srss and srain filds, rscivly, in h comosi rial; HEM and HEM ar h srss and srain filds, rscivly, in h homognous quivaln mdium. Considr a boundary valu roblm wih h rscribd dislacmns U i x a h surfac S of h comosi rial. I is assumd ha no body forcs ac on h rial. Dfin any dislacmn fild saisfying h boundary condiion as an admissibl dislacmn fild. To consruc admissibl filds, h comosi body can b subdividd ino N lmns, wih ach lmn occuying volum surroundd by a rix Fig. 3.4a. n and conaining an inclusion 33

43 i n c l u s i o n s S Fig. 3.3 Hrognous rial dividd ino comosi lmns. Prscrib dislacmns U n i x a h inrnal boundaris S n Fig. 3.4a such ha Combining h soluions for dislacmn n U i x U i x for x Sn S. u ~ i, srss ~ and srain ~ filds of h boundary valu roblms for ach lmn rsuls in admissibl dislacmn, srain and srss filds for h nir comosi body vn hough srsss y viola quilibrium quaions on h inrnal boundaris bwn comosi lmns. rix rix S n inclusion n S n S n b n inclusion n a n S n a Fig. 3.4 Aroxiion of n-h comosi lmn on h lf by concnric shrs on h righ. b 34

44 According o Eshlby s rincil, for any s of rscribd dislacmns on h inrnal boundaris, h following qualiy holds: whr N ~ ~ ~ d d ~ d 3.. n is h volum of n-h inclusion and n n and ar soluion srss and srain filds, rscivly, for h sam roblm if hr ar no inclusions in h rix. From h rincil of minimum srain nrgy Hashin 96, Chrisnsn 969, h soluion srain fild ha is associad wih h soluion dislacmn fild minimizs h srain nrgy funcional d ~ ~ d 3..3 If h rscribd dislacmns roduc admissibl homognous srain, hn from Eshlby s rincil 3.. and minimum horm 3..3, h following inqualiy givs h ur bound for h ovrall bulk modulus : N n ~ d Similarly, alicaion of Eshlby s rincil for h body wih rscribd hydrosaic rssur on h boundary n kk N ~ n 3..4 ~ ~ d d ~ d 3..5 and h minimum nrgy horm 3..3 lads o h following lowr bound for h bulk modulus: n N ~ kk n n d

45 Boh bounds hav bn drivd by Hashin 96 for aricula rials wih an arbirary gomry. For shrical inclusions in Hashin s assmblag, hs bounds wr furhr simlifid rsuling in bounds 3.. for h bulk modulus. In h nx scions, Hashin s aroach will b gnralizd o obain bounds for rlaxaion funcions of aging comosis. 3.3 Gnralizd Eshlby s rincil As i was shown in Scion 3., drivaion of Hashin s bounds for lasic comosis uilizs Eshlby s horms. In his scion, Eshlby s rincil will b gnralizd for a scial class of aging aricula comosis xhibiing cr roris characrizd by consiuiv rlaionshis bu wih h rix rlaxaion funcions rlad by a dimnsionlss osiiv ingrabl funcion R, C, [, as follows: * * *, R, 3.3.a *, R, 3.3.b whr and ar rial characrisic funcions. In addiion, R, dfins h oraor R as follows: whr h h Rh R, d is an arbirary wll-bhavd funcion. Considr a comosi body wih a singl inclusion occuying volum. I is assumd ha no body forcs ac on h rial. Th boundary valu roblm for his body is comrisd of h srss-srain rlaionshis 3.. and 3..3 for ach has as 36

46 wll as h following quilibrium quaion, dislacmn-srain rlaionshi, and boundary condiions whr 3.3., j ui, j u j, i n n, u u, S j x j x S is surfac of h inclusion and i i i x i x U x u U xs n T x S j i S S U S T T. Th soluion srss and srain filds ar dnod as and, rscivly, for h roblm whn and. Thorm 3. For a comosi body wih a singl inclusion and rscribd boundary dislacmns, i.. S U S and S, h following qualiy holds: T 37 R R R d R d d whr and ar h soluion srss and srain filds, rscivly. Proof: Following harlab 968, inroduc ransformd variabls U i RU i, ui Ru i, R, R,, Mulilicaion of boh sids of quaions 3.3.3, 3.3.5, and 3.3.4b by oraor R from h lf and us of h ransformd variabls in hs quaions as wll as in quaions 3.., 3..3, 3.3. and 3.3.4a lads o h following sysm of quaions: kk kk , j 3.3. ui, j u j, i 3.3. ui U i xs 3.3. n n, u i ui, j j x x x x x S 3.3.3

47 whr * * I and I for h rix and R and R for h inclusion. For ach im momn h sysm of quaions dscribs a boundary valu roblm for a comosi rial wih an lasic rix. Thrfor, according o Eshlby s horm, h ransformd srsss and srains should saisfy h following condiion: d d Subsiuion of quaions ino his qualiy lads o rlaionshi In a similar nnr, Eshlby s horm for rscribd boundary racions can b gnralizd. Thorm 3.. For a comosi body wih a singl inclusion and rscribd boundary racions h following qualiy holds: R R R d R d d Proof: Mulilicaion of boh sids of quaions and 3.3.4b by oraor R from h lf and us of h ransformd variabls in hs quaions as wll as in quaions 3.., 3..3, 3.3., 3.3.4a lads o h sysm of quaions consising of quaions , and h ransformd boundary condiion for racions d n j T i x ST whr T i T. A ach im momn, h sysm of quaions , i 3.3.3, and dscribs a boundary valu roblm for a comosi rial wih 38

48 an lasic rix and rscribd surfac racions. Thrfor, h ransformd srsss and srains saisfy condiion 3..5 which rovs h validiy of quaion Ur and lowr bounds of rlaxaion funcions Dfin ffciv roris 3..7 in consiuiv rlaionshis 3..6 for h HEM Fig. 3.4b such ha h following qualiy holds: whr and R d R d 3.4. HEM HEM ar soluion srsss and srains, rscivly, in h comosi body. Th objciv is o obain analogs of Hashin s bounds for ffciv oraors 3..7 rovidd ha inforion of h cr roris of h rix and inclusions ar known. Considr a boundary valu roblm wih h rscribd dislacmns x, a h surfac S of h comosi rial. I is assumd ha no body forcs ac on h rial. Dfin any dislacmn fild ui ~ saisfying h boundary condiion a any im as an admissibl dislacmn fild. Similarly o h lasic cas dscribd abov in Scion 3., h admissibl fild ui ~ and h associad srss ~ and srain ~ filds can b consrucd by subdivision of h comosi body ino N comosi lmns Fig According o Thorm 3.: 39 U i N ~ R R ~ ~ R ~ d R d d 3.4. n whr and ar soluion srss and srain filds, rscivly, for h sam roblm if hr ar no inclusions in h rix. I can b asily shown h soluion srain and srss filds and, rscivly, should saisfy h inqualiy: n

49 Combining on gs R d ~ R ~ d N ~ R R ~ HEM HEM R d R d d n Inqualiy rmis formulaion of h ur limis for h rlaxaion funcions of h comosi rial. Thorm 3.. Considr a solid body occuying h volum and consising of rix and inclusion hass. Thn h bulk rlaxaion funcion of h homognous quivaln mdium,,, saisfis h following condiion: n N,, ~ d kk n n whr ~ kk is h soluion srain wihin inclusions for h rscribd boundary dislacmns alid a im ha would rsul in h uniform srain fild if h rial is homognous,i.., d of h sol rix has. Proof: L h boundary dislacmns U i x, b Ui x, xi H whr dos no dnd on coordinas x i. Thn srains in h HEM and in h rix wihou inclusions ar HEM H 3 Insring hs srains ino and alying 3.., 3..3, 3.3.a rsuls in:, R,, R, R, ~ N n n kk d Dividing boh sids of his quaion by R, lads o h inqualiy Thorm 3.. Considr a solid body occuying h volum and consising of rix and inclusion hass. Thn h shar rlaxaion funcion of h homognous quivaln mdium,,, saisfis h following condiion:,, d N ~ n whr ~ is h soluion srain wihin inclusions for h rscribd boundary dislacmns alid a im ha would rsul in h dviaoric srain fild if h rial is homognous,i.., d of h sol rix has. n

50 Proof: L h comosi body b subjcd o h following boundary dislacmns a im : u x H, u x H, u 3, x S Th rsuling srains in h HEM and in h rix wihou inclusions ar dviaoric, i..: HEM HEM H / and h rining comonns of HEM and ar qual o zro. Inqualiy 3.4.4, quaions 3.., 3..6 and 3.4.9, and h symmry roris of h srain nsor, ~ ~, lad o h following inqualiy: ji, R, d N ~ R,, R ~ n n, R, d d 3.4. Sinc h rlaxaion funcion R, and srain do no dnd on coordinas and accouning for 3.3.b N ~ R ~, R,, R, R, Using rlaions 3..3 and 3.3.b and dividing boh ars by R, lads o h inqualiy Inqualiis and consiu gnralizaions of h Hashin ur bounds for rlaxaion funcions of aging comosis. In a similar nnr lowr bounds can b drivd. Considr a boundary valu roblm wih h rscribd racions x, a h surfac S of h comosi rial. Dfin any srss fild ~ saisfying h boundary and quilibrium condiions a any im as an admissibl srss fild. Similarly o h lasic 4 n n T i d

51 cas dscribd abov in Scion 3., h admissibl fild ~ and h associad dislacmn ui ~ and srain ~ filds can b consrucd by subdivision of h comosi body ino N comosi lmns Fig. 3.3 wih boundary racions T i x, on h inrnal boundaris. Combining h soluions for dislacmn u ~ i, srss ~ and srain ~ filds of h boundary valu roblms solvd a any im for ach lmn rsuls in admissibl dislacmn, srss, and srain filds for h nir comosi body vn hough h dislacmn fild u ~ i y b disconinuous a h inrnal boundaris bwn comosi lmns. From quaions 3.3.4, 3.4., and i follows ha for h roblm wih rscribd surfac racions h following inqualiy is saisfid: N R ~ ~ R HEM HEM R d R d d 3.4. n n This rmis formulaion of h following horm: Thorm 3.3. Thn h bulk rlaxaion funcion of h homognous quivaln mdium,,, saisfis h following condiion: N, I I Ln, 3.4. n whr L n is h oraor rlaing srsss in h n h inclusion wih h rscribd boundary racions if h body is subjcd o h hydrosaic rssur alid o h boundary S a im. Proof: L h comosi body b subjcd o h uniform hydrosaic srss. Thn h boundary racions ar: T i n i

52 whr n i ar h comonns of h uni ouward norl o h boundary S. Sinc boundary racions ar urly volumric, hn srsss in h HEM and h rix wihou inclusions ar as follows: HEM Us of h consiuiv rlaionshis. and.6 and quaion rmis rwriing inqualiy 3.4. in h following form: R N R ~ kk ~ kk R d R d n n d From 3.a: R * Alying quaion firs o and hn o ~ ransforms inqualiy o: R d R d R ~ d Sinc h srss fild dos no dnd on coordinas, ingran in h lf hand sid of can b akn ou of h ingraion sign. Dividing boh sids of by and alying oraor R from h lf rsuls in h following inqualiy: N n n kk N ~ kk d n n If Ln is h oraor rlaing srsss in h n h inclusion wih h rscribd boundary racions whn h body is subjcd o h hydrosaic rssur alid o h boundary S a im : L n ~ kk d hn inqualiy y b rwrin as 3.4. whr oraor low is dfind as: n low kk 43

53 N low [ Ln] 3.4. n Slcing H,, subsiuing i ino inqualiy 3.4., and alying oraor o boh sids from h lf rsuls in h inqualiy low Thorm 3.4. Th shar rlaxaion funcion of h homognous quivaln mdium,,, saisfis h following condiion: N I M n,, I 3.4. n whr M n is h oraor rlaing srsss in h n h inclusion wih h rscribd boundary racions if h body is subjcd o h dviaoric srsss alid o h boundary S a im. Proof: L h comosi body b subjcd o h dviaoric boundary racions T, T, T n n whr h shar srss gniud dos no dnd on coordinas. Thn srsss in h HEM and in h rix wihou inclusions ar as follows: and h rining comonns of 3 HEM HEM HEM and ar qual o zro. Inqualiy 3.4., quaions. and 3.4.4, and h symmry roris of h srss nsor, ~ ~, lad o h following inqualiy: ji N R ~ ~ R R d R d n From 3.b: R, * Subsiuion of quaion 4.6 ino 4.5 givs: n ~ ~ R * R d R d R ~ N n n d d

54 Sinc srss dos no dnd on sac variabls, h ingran in h lf hand sid can b akn ou of h ingraion sign. Dividing boh sids by and alying h oraor R from h lf lads o: 45 N d n n If M n is h oraor rlaing srsss in h n h inclusion wih h gniud of racions alid a im M n ~ d hn inqualiy y b rwrin as: whr h lowr bound oraor is dfind as: low n low N M n low n Choosing H,, insring i ino 3.4.9, alying oraor boh sids from h lf, and aking oraor Comosi wih shrical inclusions ~ o low ou of h bracks lads o inqualiy Hashin 96 has valuad ingrals in inqualiis 3..4 and 3..6 for h assmblag wih shrical inclusions which rsuld in bounds 3.. for h lasic bulk modulus of h homognous quivaln mdium. In his scion, h gnral bounds and 3.4. will b valuad for his assmblag o obain h bounds of h bulk rlaxaion funcion. Considr a shrical comosi lmn Figur 4b wih h radii of h inclusion and h rix shll of a and b, rscivly. Plac h origin of h shrical coordinas r,, a h shr s cnr, whr r is h disanc from h origin. If h comosi

55 shr is subjcd o h rscribd dislacmns 3.4.6, hn according o h gnral soluion in h lasic comosi shr Lov 97, a any im h radial dislacmn insid h rix shll can b wrin as: u r r, ra r B 3.5. whr A and B ar h cofficins drmind from h boundary condiions. In h inclusion shr h condiion of fini dislacmn a r imlis ha B. Thrfor, u r, ra 3.5. r Th srain and srss nsors hav no dviaor ars du o radial symmry and can b wrin as: u r r, r rr r, ur r, r, r, r, r, ur r, r, r From consiuiv rlaionshis and 3.5. h radial srss comonn in ach has is qual o: 3 3 A, rr 3 A 4r B rr R-wriing condiion for radial dislacmns in h shrical coordinas lads o: u r b, b / whr H. A h inrfac bwn h rix and inclusion h following condiions should b saisfid: rr ur a, ur rr a, rr a, a,

56 Subsiuion of quaions 3.5., 3.5. and ino quaions and lads o h following sysm of linar oraor quaions for funcions B : 3 aa A A A, A, and a B aa a B 3 A b b B b / Th sysm of ingral quaions can b solvd by h mhod of Gaussian liminaion. From quaion 3.5.7: Subsiuion of quaion 3.5. ino lads o: 3 A Rgrouing rms in 3.5. : 3 A A a B a B 3 A a B and alying h oraor rsuls in: A A a 3 4 B o boh sids of quaion 3.5. from h lf a 3 Subsiuion of ino lads o: B 3 4 B /3 3 3 b a 3 4 3I b Finally, subsiuion of ino and 3.5. and h volum fracion c a 3 / b 3 rsuls in h soluion for h rining unknown funcions A and A : A 3 A T 3 whr h oraor T is dfind as follows: T 3cI 3 4 3I 3 4 3cI

57 Subsiuion of h funcion A ino quaions 3.5. and lads o h following soluion for h srains in h inclusion has: T N From Thorm 3. and accouning for c n rsuls in h following inqualiy: n u,,, c TH whr u, is h ur bound for h bulk rlaxaion funcion of h homognous quivaln mdium. I should b nod ha h ordr of oraors in quaions and is imoran. For xaml, h oraor T is no ncssarily qual o h oraor T. To obain h lowr bound for h bulk rlaxaion funcion, considr h boundary valu roblm for h shrical comosi lmn wih h boundary condiions This lads o h following sysm of linar ingral quaions for cofficins A, A, and B : A a a B A a 3 3 A 4 a B 3 A A 4b B /3 Solving sysm by Gaussian liminaion in a nnr similar o h rocss dscribd for h roblm wih h rscribd boundary dislacmns rsuls in h following soluion for h cofficin A : whr S is h oraor dfind as follows: 48 A S /

58 S 3 4 3I 3 4 4c Subsiuion of h funcion A ino quaions 3.5., and.3 lads o h following soluion for h srsss in h inclusions has: r, S If an assmblag consiss of N h shrical comosi lmns hn for an n-h lmn h oraor L n, dfind by quaion 3.4.9, can b r-wrin as follows: n Ln S Us of Thorm 3.3 and quaion 3.5. and accouning for inqualiy: c N n n rsuls in h low,, I c S, 3.5. whr low, is h lowr bound for h bulk rlaxaion funcion of h homognous quivaln mdium. Th ur and lowr bounds and 3.5., rscivly, rovid bounds for h bulk cr roris of aging aricula comosis wih h gomry dscribd by Hashin s assmblag. For lasic comosis wih h Hashin assmblag, h ur and lowr bounds for h bulk modulus coincid. I can b shown ha h bulk rlaxaion bounds coincid as wll s Andix A, i..: I c S,, c TH Th ur and lowr bounds and 3.5. ar valid only for Hashin s assmblag. Howvr h chniqu rsnd abov can b also alid o mor comlx comosi srucurs whn combind wih analyical chniqus Mogilvskaya al. 49

59 3.6 Bounds of shar rlaxaion funcions for vry sll and vry larg aricl concnraions Hashin 96 showd ha h ur and lowr bounds of h ffciv shar modulus for Hashin s assmblag coincid whn aricl volum fracion, c, is vry sll: 5 u low c O c Analogously, in h cas of larg aricl concnraion, c, h bounds coincid as wll: u low c O c For an arbirary aricl volum fracion c, Hashin suggsd an aroxi formula for h ffciv shar modulus which lis bwn h ur and lowr bounds: 5 c c In his scion, xrssions 3.6., 3.6. and will b gnralizd for h shar rlaxaion funcion. Considr a shrical comosi lmn Figur 4b wih h radii a and b of h inclusion and rix shll, rscivly. Plac h origin of h Carsian coordinas a h shr s cnr and dno h disanc from h origin o h osiion x, x, as r. x3 If h comosi shr is subjcd o h rscribd boundary dislacmns 3.4.8, hn according o h gnral soluion for h lasic comosi shr Hashin 96, 5