corso di perfezionamento per le costruzioni in cemento ... · delle strutture in cemento armato...
TRANSCRIPT
Batant, Z.P. (1978). "Inelasticity and failure of concrete: A survey of recent progress,· Proc. of Seminar on "Analisi delle Strutture in Cemento Armato Mediante iI Metodo degli Elementi Finiti,· held to commemorate 50th anniversary of School
IH£USflcrn AND fAIlURE Of COIClfTE: I SUHm or mE NT raoGRESS
by Z. P. BAfAIIT
POLlTECNICO DI MILANO
CORSO DI PERFEZIONAMENTO PER LE COSTRUZIONI IN
CEMENTO ARMATO • FRATELLI PESENTI· Cinquanrenario della iscicuzione 1927/28 - 1977/78
ANALISI DELLE STRUTTURE IN CEMENTO ARMATO MEDIANTE IL METODO
DEGLI ELEMENTI FINITI
seminario 20 - 23 giugno 1978
Prof. Zden~k P BAZANT Norchwcscern Uni\'ersicy, Evanston, Illinois
Prof. William C. SCHNOBRICH Universicy of Illinois, Urban;l.,llIinois
Prof. Alex C. SCORDHIS Uni"ersicy of California, Berkeley. Cal iforni<l
Z. P. Baaant. "Inela.ticity and Failure of Concrete. A survey of Recent Progre ..... Reprinted from . Proceedina. of Seminar on "Analiai delle Strutture in cea:nto Armato Mediante il Metodo degl! Element! Finit! (ADelySis of Reinforced Concrete Structures by Finite Element Method). held in ca..emoration of 50th Anniver.ary of School of Reinforced C Politecnico di Mil DO Hil oncrete. pp. 5-59. a. an. Italy. JUne 1978.
ltelZa .econda ",.t<t dlZ Pllbb7'fZiD 19Z8 inizim>anD aZ PoZit." .. iCD di NiZa7ID le p"':- Z .. a,;""i d .. Z COl"eo di p"l'r .. aw-..to pel' Z. C J.t",_i""i ill C_to ~to. ""ZlIto " patl'OCinato dazZa !~Zia P"aftti """ 10 aaopo di """tl'ibkiNl a .fOlWrU'tl dllgZi "i"f/"pa6l'i ape"iaZiuati i .. tlCtt" Z. appliaaai""i dlZ ..-..to • • • p.<:iau. .. t" ... zz. CDatl'lCa~i i .. ,,_to _to-. Ri"o..,.. CD.l i .. qk./tto <DIftO aaaad-nco il tri_t"NU"iD di _ato iatitIC
ziOPW!. Za ""i doria i l'iPOl'tata ... Z ""Z_ fIV (1977) d"l1ti StlCdi • R.,.di"""ti del C01'IlO.
Pl'a Z. "",Zu i .. iaiat':". po .. ibili pel" """ degrra .. "tebNsiDM di """ data "oei iIopo .. ta1Ite .i I .... Zta QlCelZa pill CIO""';~t. alto a<:opo istitICaionaZ" dll eo ... o. Nlaz,:uaMc i .. lII:VIillN liNN. pttbbZicuz lOt SIlIrril'I<Zl"io ehl d./tel"iua 10 "/ttato d.ZZ 'Ql't,," ""lto """li.i d.n" stl'lCttlCl". in e ....... to <U'fll2to ",.dianu it ... todo degZi .t_ti .'i .. iti. TaZ ... toda. di.eNti .. _*, 10 atl"kttleJoa i" IDI inai_ di el-..ti, pe,..,tt. di adDtea..e pel" cia.CIDID di qkll8ti ta dAraeris"'''' a>laliticuz pill appJ'Ol'riata. C;W.to I pal'tit:Oz.u-..ta ... tant<t MU. atl'lCttIC ... i .. ..-to _to. pIlN:hi """ a.,.t. di u... ... """to. in 1DI' .... icuz analiai, aia di .. ~ti diu.,. •• qkQJl
to a tipoZogia st"'ttleJoaZ •• aia del e"""" .. _to oZtl'llJ!lOdQ COIIpZa .. o deZ cal ..... t"" .. o di ...... to. _tt<tria~to da lego. ccatitICtilJD """ Zi".a.... d".fo,.,.,.z~i di tipo viaClOao " r""lO'GaiDne.
IZ SIlIrrinal"io • • tato id.az-nta divieo i .. Ilk. pal'ti. 14 prillca ri-I!' .. ~" ! ·.'!'l'siai"",. d"n" l'i .... ,...,.. ... tati"" aZta .'o.-tasiDne dli Z.~ eoetit .. tivi da iIIt~ neZta anali.i. c....,o ill ..... Z. poa.ibilitd analit""'" dlt _todD i .. OfIIIlltto .op~ di f1PaII lI.M{!Q Z4 t1DN1.t: ... :uz -1.1 ooat.riale daZ "....to di vi.ta .fi.iC!O. 14 .....,..aa ripo,.ta _ d .... ri.i"". delta _todDlcgia di analiei • dille .... appliaaai""i ai print:ipali tipi di .t,...ttICN ddta irl(l.l1""ria <:ivit., ta qwat .... ct. in .vid ...... I. OOI'II!izioni "",: dI"" .oddi.!al'Il it ..ad.Z 10 cmIllitit:o ar!iMhi poaaa riprodwl"N ,:z """""PtaIIIeIoto NaZ. delZe 8tl'lCttICN. A tau propo.ito .... to ,;z delle ... di l'i"fll'Q&iaN i proo' ... ol"i za .... k P. Bazant. I>'iZli(llll St:lmobricIJ • At" C. Scal'ddi •• .. "" SOJlO staH alii pioni.ri in qkIl8to c....,o di rit:.roca. pe .. 1 ·",t ... iaPfO t:01f cui hanno at:t:llttato di til ...... m."sto s...nJIIJrio, • PIll' 10 fIIJJli.N CIOat ad.,....tIl atlo .pirito i .. .folWltlto ... con cwi Jranno p"'pal'ato iZ t.sto d.ll. Zo,"", les~i. RingraaiD aMlt. iZ oriD t:aZZabol'ato,... irl(l. twigi CedoZi ... pe" atIIl ... avoZto cod .rlicat:_te it Za1IOJ'O di aool'di_"to dn diTJel'si i .. t.,..,.,,".
A1I<>M i" ",,".ta occo ....... ri"""'l10 alta latriglia P ..... ti • • PIll' ... a aZta Soci.td Ita~ti di /lel'gCllllO, Z ... p,.. .. ""'" delta grotit .. di ... pe .. it ..... "Aatiawoo """ cwi Ita da • .,." .... oue""to iZ Co ... o di P."'ezio_"to. "h. Ita ,...0 po •• ibi Z. aneh. OIIe.ta iniziatiTJa.
MiZano. 20 giugnD 1918
n prof"MlI' ZUrtI1c P. BdaIIt ha ric_to it dipZ- di dottorato daltd ~ d6lla S4Ji_ C._Z- lid II1J • di_ doc:a..t. al Poliutmic:o di Proga ... 1 1117. Dope _P riaopcrto i~i di w.,.,aIlW7lto al C.E.B.'/.P. di Parigi, aZZ'/h..,;lJ8raitd di '/Orollto'. aZZ'lIIIivaNitd di Catifo .... ia a Bft,.~e roil, .!aX 1.S~ .fI.i i""fl'lOl praaecl lG 'ort'-,u,.. uIIiI18Pait¥. dava ... 1 II'1J di_ prof .. .., ... oPdinario. E' _1 ... " di AI'fI_ llatio .. at LaboNtorr. Gb1c Ridge .rational i4b0l'lZto", • SaI'fIetIt aIIIf LIDIdJ, E1tgp •• IltJ PNaUdMto td ccarrl.nou ABa.."., ... lla Pl'Op1"Utd d6i "",t.rUlli • fa part. d6Zla diN.iorN aditoriala d6i ,w",tdi ASCE-"" • C_t aIIIf COOlON" R._h. n prof'''or IltJaaott l'icauatta ta -'aglia RILDt 1161 1176, ....... oitato pal' "brillaltti .lIilkppi ",zta ~ dai ""'t'rUlH, ",Ita t.~oa d6i f.-i iii_i • • ",ltd Uoria datta .tabilitd. fortdarldtJ ~ a~"-taZ • • fo_t<zaitww taDric:a" • .. , 117' riclHtu daZtd ASCE H pNlll'io II. L. likbal' ~r td ric:aroa ",Ita illfl'fllWN oillil •• • aundo oitato ~r "ric:aroa "'fIH 'ff.tti d6llG lIi_ntd, i ... taaeioitd • eo>tt ..... to di wrriditd ... 1 Cldlc:a.tnaao. """"",rt..wllto IIorI UVaN • dipartt/6rlta daZ t.,., ... Zla atl'kttkN, Uoria d61td atabiUtd e d61td /Nttk:J'Q" • ... 1 1177 it premio '1.r. Lill par it .:."j"to _to pre""""1'8 .. 0. par ta -"" "Vi..,oaitd • ritiro ... i ,...ci di eo>tt • .n-to di Hattori _laGri". EgU ha appAa ric:atlkto td pN.tigiQaa berea di atktHo ~ .... par td ric:aroa ~I' l'/DIJIO ~ao Zi'l'-'II.
....no. E' co ..... t ... t. di "",:t • • O<riaed di propttaaiorN ~I' ! '_lui di ael'l<ttkN a".ci4ti ill _to _to. E' """"'ro d6ltd ~aait1lV ASCE"., ~I' i ... todi IIIIZt_tici • d6U. atJIIWIi .. iorli ASCE-S'fD ~I' la .tl'l<ttkN .".ci4li, lG propttuiolla d6Ua atl'l<ttlJie a ,...t:io di a..llto _to a t '_lui ~I' d_ti finiH d6U. atl'l<ttkN ill _to _to. ,*,lta part. d61td actillitd di rioaroa 1161 prof."ol' S4JltIIobri4h • • ta;;a d6dieata al-10 mlkppO di ... todi <l>l:2litiai ~I' 10 atkdio d6ltd ri.poata .taticG • di-wa d6U •• tl'kttkN. Si • OOCt<pato ill IIIodo partioolaN ddl'_li.i ~I' at_ei !, .. iei di atl'kttkN ~ a a Q.aio, i .. .:_to ~to, ~I' la qk4ti Ita .IIiZkppOto .. todi di oalc:oZ<! capaci di t_N _to di l.pi 00-
atitktilli IIorI li .. eari .• f •• lkrdliolla ora .... iual'aa~t. adottati. R • .: ... -t_ta .i • OOCt<pato d6lla Nnau..aa aU. aaiorli .i.acM ili parati di "",,_to • torri tH Nfrig'1'GIIioIIa •. ad iparboloida. 2'ra la "",h. C1p8N
di .:..i • • tato _t ... ta pel' td propttaaiorN IIG I'ioordato it "~i .... aIIIf lfatatori...," 1161 Gaorga JoIill,- Colt., •• • at 1I?? Ita ri_to it p"'",u, A. Vorl IlIII/Iboldt ~I' la S4Ji ......
Prof. Atn C. SooPd.li,
II prof .. ..". Ala C. 5;Jorclalia ~tt. it dip'- di IaoIIelor of ~ daZZ'lhtiueNitd di CaZiforrria ",t ISH' ad it dipZ_ di Muter of ~ dat Nu--.a.tta I .. aeitku of r.clwloloflll ... t ISH'. 'dto .u .. o _ ~ ad i",8f"tZN pN610 l'lhtiuerlitd di CGlifomia, daI18 _ c:a.n_u .fi_ prof ... ON orcli>tarLo. E' _z...t. di ftue .... ,'Iocie-td d .. prog.ttaaiolla ad ag ....... fIOlI8,.,..,ti.,. par t '_Hai di "I'kttkN a p.84io • ponti i .. ~to CZl'IIIlto. £' pN.w...t. d6ttd ~ •• io ... ASCE ~r l • • tl'kttkN ill ~to ~to • IIIIIIbro dell. ~a.iorIi: ASCr-ACI • lASS ~r i fII.'oi i .. "_lito _to, ASC£-S!'D ~r t 'analiai ddla at"'ttkN i .. _to _to _ iZ .. todD dagli .l_ti filliti, ACI ~r i ponti ill "...... to _to. t.a ric:a_ avotta dal prof ... or Sc:orclaU.a • • uta .... N rivolta allo .tkdio taoric:o • a~ri ..... tal. 11611 • • tl'kttkN iJt _to __ to, ad i .. particot<zre d6i fIk.d • d61la tralli da poilU a __ • rill daflU --a: '60 ha ric:oIIoaoil<to l. poa.ibiZitd d6t .. todo II6f1Zi .. *-.. ti, ill uMlOiaaiorN _ to mlkppO d6i ordcoZatori .lattrootiai, ~P 1'_lin d6Zla .tl'l<ttkN a gwoio ad • atato tN i prUrri ad appZic:aN i 1 .... -todo d.,U .:-ti filliH alto .tkdio d.U. UI'kttkN ill _to _to. Pra 1 • ..,Zt. O~N di c:ki • atato """",z...t. ~r td prog.ttaaiorN IIG ricordata ta copartkra a parabot0id6 iperboH.:o dellG St. 1trrrt '. Cathedral SaIl F-raNriaco. Nel lI76 Ita ric'''''to dall' ASCE it premio l*Ii .. aiff ~r td -.oria "AIlaU.i '"'" till'ON di gk'c:i di C~"""to _to di fo- qk41.ia-8i", e ",I 1978 • • tato eletto ... ftlbro d.ttd Ac:caderwia Naiorrat. di IIIfIegneria degli StaH lIIIiti.
Table of Co.ce.u
I.eroductio.
Pall I
II£USnCITf AD FlILII£ IF C.ICIfT[: A sOlm If I£C£IT rlDCIESS
bJ z. P. JAfAIIT
Prof ••• or of Civil l",i •• eri.1 Iortbwe.t.ra Vaiver.itJ
lv_to •• l11iaob 60201. USA
CIW'TIIl Al DlClDllllTALLt LDIIAI TIIIOaIlS FOR ftlAXlAL I!IIAYIdl
Al.l Slapl. Ceaeralizetioa. 'of El.sticitJ Al.2 lacr_.tal P1.acicieJ
A1.3 Pl •• t~c-rr.cturi", Hat.ri.l vith rrictioa ... Dil.t.aeJ
ClIAPTEJ AZ DlClEllEllTALLt IIOIILDIIAI TBIOIIIS FOR ftlAXlAL IIIL\9Ita
AZ.l ladochroaic Theory .ad Vi.copl •• ticity AZ.2 lefi_nta aDd C"""ariaoa. of ladochro.ic aad
I.cre .... tal Plastic Poraalatioa. AZ.3 CJclic Loadinl by Indocbro.ic Theory
6
1
7 10 13
19
19
CIIIIPTEIl Al CRACXI!IC. STAllLln AIID STlUctUlAL AlW.1SIS 31
Al.l Ta •• ile Cracki", aDd Net-Rei.forced Concrete 31 Al.2 Str.in-Localiz.tioa InseabilitJ Due to Strain-Softeninl 40 Al.3 St.p-by-Step "'-rical Allorit'" 43
CIW'TIIl M CUEP. SIIRINItIIGE AIID UFICTS or II)ISTUU AIID TlMPPATIIU 45
A4.1 Ther.04~c. of Yiscoela.tic SolidifJinl Haterial A4.2 Co •• tituti ... Iqustioas A4.3 Moisture Tr_fer ... Ther.al Iffecta
Concludi", R_rka
IIcknowledpenu
Referencea
45 47 S2
54
54
55
AO.l
Dlt'IaIUCTICII
I!IIerl •• ce of powerful COllpUtat~ Mtboda. wda .. the f1a1Ce ele
_t _thod •• ..u. lt todaJ po •• lbl. for the .truct...,.l aaalyat ~ eoa
.1d~r rather COIIpl1cated _chaD1cal propertUa of _terl.... llbile c-tJ
Jura qo the lalavledl. of .. terial bebaYtAor .... bead of the cOllpUtat1oa
al capab1l1Uaa. raeently lt he. falt. bebiJld. Thl8 18 .apac1a1lJ true
of coac:r.t •• a .. t.rial vith eztr-tJ cOllpl1catiad. _ cou.ld -J ..... ., ••
propertia.. ViCh the powerful fialt. el_t cocIa. av.Uable. 1M 1DIl.ed
c:ould tod.J aaal.,.e CODcr.te .tructuraa _re accuratelJ 1f _re realistic
~ela of .. urial hebaYlor _ra availabl. ~.... Ia fact. thia ia aot
oalJ • po.a1bllity. but • c ...... llial Ileed ... fer" •• certaill ~ara type.
of atructur ••• e.I •• the coaer.t. reactor vuael. or oc ..... tr""turaa
.ra coacaraed. It v11l be eheracterl8tic for __ tiM to c...a thet ia ill-
ftstipUll1 t" ... tbod. of atructural aaalpl8. 1 •••• the ftllite al_t
... tbod ..... v11l ...... ~ paJ • ar-t dMl of attellt10a ~ the ~la of .. -
terial bebaY1Ilr CD lie ueed ... the affaeUv ..... tat1oa of t .. eOllpUtat~
u.nt ... to thaaa ~ ••
S1p1f1callt adVUlC .. ill the ~aliq of the 1Ilala.Uc behaVtAor of COll
er., ....... be_ acldeY&d duriaC the leat decada. n. purpoea of th1a work
ia to attapt eo 11v •• ""tv"J of the recent devalopaanta •• enlllS as •
lIactar-d ... ~ of • aerie. of tve1v. lectur .. daliy..,ed .t the
Pol1tecaJco di II1laao .. part of _ iat_ioaal .-1aar _ratiq tlla
SOt. _ber.., of tile ekeduata Scllool for la1aforced CeDent, Structur ..
(Poodaz1oaa Prata1li .. _ti). 110 c.la1aa for cOllplat .... of thie """.J
.ra •• t; • cartaill biu ..,he.1z1ll&· the COIluibutloll. aad •• t the writer'a
hoM iAatitut10a 1a 1MY1tabl .... paciallJ aiae. -J of the resulta sur
v.,ed ...... _t pt beeo publiahed ill periodical ..... al8t &till oalJ ill
report fora. The ruder .hould judi. the foll~", taze ill thie lilht.
Dophaa1a v11l lie placed OIl the f.....s-tal. of t .. coa.u.euUve equ-
. ti •• for -t1llaar tr1aJlial lIaIIavtAor. The coyer.l. of creep .... _lature
affects v1ll be diaproporU_talJ unov ... v11l he focoaed oalJ oa
aelected fev vary recent deva1opooallt. He ...... _preb .... iv. surveJ of
thi. subject ... publiahed aot too loa& ,,0 [39].
Al.I
CllAPTD Al
IJICIPIIIItALLY LDJEAa TII!OllI!S FOJ. 'BLUIAL BEHAVIOR
AI.l Siaple C •• r.UaaUCDa of Ua.ticity
The uniaxial .trsa.~.train dialr .. of concrats ia ... ily daac:ribed by
• stapl. for.ula. sucb a. tho.s due to Sa ..... Sarlin. 'opovic ••• te. llhars
the eoapla"ity e_. fro. 18 tbe triaxial nature of tbe r.sponsa and tbe
f.ct tbet the ra.poo .. _st be DOdel ... without viol.ti1l, b.de prineipl ••
of cont1.a_ •• chanic. [1). whieb i1lclud. tbe priocipl. of objeetivity ... d
tbe ... oei.t ... cODditiCD. of inv.riaoc. duriq • routioo of coordinata
ax... Many of tbe ree ... tly propo.ed aodel. for DODlin .. r tria"W beha"ior
of concr.ta do DOt .. ti.fy tbe.a iov.rianca coaditioo. UDd.r Isner.l lO8d
inl' Con •• quently. tb ••• .od.1& .ra of liaitad u .. fu11l." .nd wiU DOt be
deseribed herein.
lecolni&iq tbet cl ••• ical pl •• ticity ••• de"eloped for .. t.ls. ia
in.pplic.bl. to concreta. Coon .nd !vana (2) .doptad tba f.r aora laoer.l
fr.-work of bypoela8ticity. in which tbe incr_ts of .ues. tensor '1 •
["ij) .ad atrain tensor! • [Cij ) .re ........ to ba related liD&arly .ad tba
proportlonality coefficients .ra ......... to ba function. of" .nd C but lj ij
DOt of thelr hl8tory •• ad not of their rat ••• Thu.
1.a vhlch c lj .... t ... sor of tUleot .Ulfna .. DOduli .Dd Dij
.... t .... or of
tan,ent cOIIPlisnc •• (fluibllitie.). Lover c.s. 1.Un subleripu rafer to
carte.l.n coordin.t •• Xi (1 • 1.2.3) and rapetltion of • subleript laplia.
...... tlon. OIIly ... U (linearized) Itr.i1ll ar. consldared in tbie vork.
Assualnl that the raaponsa dlalr ... ara aaooth .t l ... t up to the peak
point (stren,tb). vblcb i. true of concreta. tbe pe.k point of tba respon.a
d1.sr .. ls obt.lned by sattinl da ij • 0; Eq. (Al.l) thea foraz • ba.o,aoaous
linear equ.tion .y.ta •• nd a 1Ion-zaro solution d'lj is poSlible only if tbe
deter&1111nt of tbe (6"6) .. trix of C1jka v.nlsbe •• vhicb y1elds tbe strenlth
criterlon (2). To •• tisfy tbe condition of isotropy. C (or D ) _at 1j'" ij'"
Al.2 8
be an tsotropic teasor polYJlO1lial 1.a a 1j .nd • ij' the I_r.l foraa of which
have beea det ... ined by True .... U. .01JDomals of d.lre. two ..... hilher
involve too ..... y COll.tuU •• ad so Coon .nd EYu. (2) lia1t'" .tt.nUon to
a linear t.n.or pol)1Io .. ial. which cont.ins only .evea .. t.rial constuts.
two of which are relat ... to 1'DUDI'. lIO<Iulu. ! ud rois_·. ratio y in
.tre .. -free .t.te. Thi. fo..-ul.tion I.v. • re •• on.bl. deac:riptioD of tbe
unlaxisl aad biaxi.l r •• pon.. dl.lr ... up to .bout 7S1 of tb. peak point
(.tr&lO,tb). I_er. the f.Uur. condltion r .. ultinl fr_ thl8 fo..ul.tiOD
v •• very poor and. e.pecislly. the .tr.ins that corresponded to the peak
po1.at v.re infinite. Th •• e .hortc.,.in,. could undoubt.dly b •• Ueviat ...
by ulinl hilh.r-order t .... or polJDoaisls. but tb ... involve too .. ny t ....
aad too .. ny COD.t.nts whlch .r. not pr.ctic.lly .. Da, •• ble. Ther.for ••
phydcal conc.pta .r. ne ...... to ",ide tbe SelectiOD of the functiOGaI d.pen-
denc. of C1j ... or Dij ... upoD '? .nd!. I1Icr_ntal pl.aticity. which can b.
r ... rd ...... epeeisl e ••• of hypoel •• ticity. off ... the 1Iee .... ry phydc.l
concepu.
All iIIportant property reflected in Eq. Al.l 18 the stre •• -indue ...
aniaotropy (or .tr.in-iaduced .nl.otropy. or • coablDatioo of both). Tb.
t .... ent 1IO<Iu11 tensor Cij ... haa an .nisotropic fol'll unl ... the str ... i.
&ero or the stu .. t ... sor is isotropic (hydrost.tie pr ........ ). lad ..... if
va .... it that the t .... ent lIO<Iull d.pend on .tre ... &ad if the .tr ... tauor
is DOt isotropic. it would be eotirely unr .. sooabl. to • priori ........ n
anisotropic fo .. of Cij.... This is coofir.ed by tbe f.ct that DOn-isotropic
.tre ••• t.te •••• 1 •• the uniaxial ca.pr ••• i01l or t .... ioo. praduc. mcro
cr.ck. that ara DOt oriented randoaly but exhibit • c.rt.in prev.leot ori
.nt.Uon ~.pe"'inl on the .tr.... Th. property of .tr .. s-i1lduc'" a1lbo
tropy 18 characteristic .f all lIO<Iel. for tb. behavior of coocnte. esp.
eially tb. locr ... nlal pla.tic and fracturi ... .odel ••• vall .a tbe .ndo
ebroaie theory •
V.rious siaplified f .... of tbe hypo.la.tic for.ul.ti01l ~. been .p-
pearl ... in the Uteratur •• ad are widely u .... in finite el_t prolr ... .
10 oa. fo ... tbe ("'6) taDl.1It coap11anc., .. trix R ia upr ... ed in t .... of
I and y i1l tb. __ (botropic) fot'll .. for the str.ss-fr.e stat •• and tb.
eanleot 80dulus i. varied as a function of the .axt.ua principal .trai1l.
Alt.rnativ.ly. ~ i •• xpr ...... in e.l'II. of bulk lIO<Iulus l .nd sbear DOdulu. C
in tbe _ fo ... s for the stress-free st.te .nd G .nd l .re varied ..
Al.3
functiOG. of the .tr •••• nd str.iD iDvariant.. Th •• e foraulatiOG •• of
cours •• do Dot r.flect the .tr ... -induced anbotrop,. Another .odel.
called "orthotropic." assu ... the tanaeDt JIOdul1 .. trix , to b. of the
.... fo~ a. for an orthatropic .. terial aDd dependinl OG the principal
9
• tr ...... straiD •• and th.ir ratioa. This JIOdel cannot reflect. h ..... ver.
a leDeral .tr ... -induc.d aDi.otropy which 1& alwa,. produced if tha in-
cr .... of .tre •• coapDDent. or .train coapoD.nt. i. DOD-proportional. i ••••
wheD the diraction. of priDcipal .tr ..... or .tr.in. rot.te iD the COur ••
of loadinl. Th. orthatrop1c fo~atloD. bava b.en aad. to repr ••• Dt UD1-
uial and bluial .tr ... -atralD curu .. for proportlonal 10alUna up to the
p .. 1t str ... point. but they da DOt .... to be c.pabl. of r.pra .. Dtinl a
broader •• t of t •• t d.t. ( •• p.cl.11y on volva. ch.nl •• nd l~ter.l .stralna).
Another ,.DaraliaaUa of .la.ticit,. ori,iull, iDtroduced b, ...... It,
and N.dai to d •• cribe pla.ticit, of aetala. 1& the defo .... UOD th.or, (or
total-.trdD theory) (3-5.11. lD which a unique relatlOD 1& •• ......r to nbt
b.tween the total .tr ..... aDd the total str.tD.; for an 1&otro,ic .. t.rial:
'lj • ZC(!.!)a1j ............................................. (Al.Z)
lin • 3r.(~.f>&n ............................................. (Al.3)
wh.re .1j' .1j • d.v1&tor. of 0lj and 'lj I C end r. ar. DOV the .ac.Dt .haer
and bullt .odull. Thb fo .... latiOD ba. the .dvaDt ... that ODS do •• DDt D.ed
to worry about the .tr ••• -lnduced ani.otropy. for if the .tre •••• and
.trains are referred to the lDitial .tres.-fre •• tat. which i. i.otropic.
the stress-strain r.latlon aust alao be of 1.otroplc fora. So. the .ecaDt
DOduli C and K .. , b. a •• uaed to d.pend oal, aD the iDvariant. of str ••• aDd
.traill. Such a forallaUon has b.en develop.d for concret •• for e ... .,le. by
leupfer aDd Cantle [61 and by louovo. and N.-.D (71. and it has be.D
.hown to fit rea.onably well proportional biaalal aDd triuid teau. but
IIOt the straill-aoftaniDI branch •• aDd the p.alt loads. The .. hi.f shortcoaiDI
of the defonoatioD th.ory 1& tbat strains do Dot d.pend all the str •••
and strain patha prior to the current .tat •• which is not true. lIev.rthel ....
theories of this type leDerally 11v. a better prediction of the .tlffna ••
of the .... terial in the dlractioD tanl.nt to the current lo.,diDI surfac. than
do.. iDcremental pla.tlcit, with the nor .. llty rule (whlch w.,. for .. tal.
pointed out by Bud1aDski [II). Differentiating Eq. (ALl)
Al.4 10
d.ij
• 2C d. ij + 2.ij (::ka d°ka + :~ka dC"") •••.••••••••••••• (Al •• )
da • 11 dc + l< (!!- do +!!- de"') •••••.••••.•••••• (Al.S) kit kit Itlt a0ka .. h ..
we ••• that the r.latioo b.twe.D d.lj
and d.lj
b llD .. r but DOt laotropic •
So. the defo~tloD theor, .shlbit. a .ort of .te ••• -induced aDl.otrop, •
Al.2 Incr.aeDtal Pla.tlclty
Incr_tal plasticity results alsci in a linear relatlon batweeD Itrea.
and strain 1ncr ... nt., sa_ a. the theori •• diScu •• ed .11 far. IIowaver. the
difficult qu •• tion of .tre.s-inducecl aDiaotropy b DOt attaclted in a dlrect.
ph._nolollcal ....... r, but phya1cal concapt. are ..,loyad to a •• war thb
qua.tiOD indirectly. a. a b,-product. The •• phydcal conc.pt. c.nt.r 0. tha
DOtiDD of yl.ld surfac. (yleld fuactloD) '(Oij) • O. deflned a. the .avalop.
IIf .U .Ut •• that C&D b. r.ached fr_ the currnt .tr •••• tat. ala.tic.U,.
ObYlously this prop.rty should in ,eneral d.pend OD the curr.Dt state. and
ao ehe ,1.ld aurhc. 1& len.rall, .ariabl. in the loadl .. proc •••• which 1.
called hard.DiDa. (Thl. f.atur. 1. D.,l.cted in ideal pla.ticlt,.) the
axbtanc. of a ""nant ,i.ld .urfac. 1& an a.su.,Uoa. dDC. for concr.u
practic.ll, DO adjac .. t atat. can be reachad b, pur.l, al •• tic d.foraatlOD
(u 1& actuaUy reflected in the .... ochronic tbaor,); thb b to a 1 .....
eataat true .... of _tala (9.10).
the cha .... of tha loadiDl surfac. in ehe peac ... of loadlq _, b.
charace.r1&ed by hard.oiDI par ... t ... ~. wlUo, tb. currant loo.i ... sur
f.c. a.
'(Olj'~) ·0 ................................................ (Al.6)
Choo.lq (ar/a~)d~ > 0 for loadl ... (1D.1a.tlc: .uainlq). va obtaiD
............................................... (Al,.7)
a. tbe loHinl crit.rion (or the c:ondlt~ for inelastlc: acralDlq). Reoc ••
d'ij -.at b. a function of thb ... pr ... lon. A.euat ... a lin.a1' d.pendence
of laala.tic .trains dCij UpOD dOlj v. further obtain
.u.5 11
del'; - 'iJ (ir/aa ... )da... • •••••••••••••••••••••••••••••••••••• (Al.e)
where 'iJ are eoaa proportlODallty coostADts.
»ote that It ~d b. 1acornct to 1Dtrocluce a lNdla, ultuioD .. p
ar.tely for 1adlyldual stula co.po .... t •• "Yilla. e., •• that Cu loads if
dC11 > 0 aDd c22 ualo"'. lf deZ2 < 0; tha fal1aci 10 thia approach ls that
vpoD rotatl00 of coord1Date. the .taoa of 1adiytdual .trala lacr .... t. ..,.
bacaae dlffer.at. "k1aa lt a.po.albl. to .. ka the loa41 .. crlt.rlOD lavari.at.
All •• s.tial fvrther coac.,t ia the IInckar'a at.blUty poatvlate (11,12.3-5), which require. tbat
•••• •••••••• •• •••••••• .•.••. ••.••••• . •• (Al.9>
1h1a coadlUoa .... raot ••• that the work 40Ge 00 tha t ial d 'ft_ - er ut_ a c,cl.
of ."lylq aad r_iq 401j be _PUy.. (Th18 ia oo1y tM _004-
order part of the work. aiDc. tbe flr.t-ordar work ~ ° dc 1 ." lj lj' caac. • wlth
the werk ot applied l0.4s aceocdiq to the priDeipl. of virtual work.)
UDd.c e vld. rao,e of .1tuetioaa. the aoo-o.,atlyeae •• of jV2
1. equivalent
to a .. terial atabUlt)' c0041tioa. lID_ar. th18 1& aot tR. vbea hictlOD
ia lDYolv .... AI hal been a"- by .1. lCeadel (l3] &ad C. llaier (14]. Ia
fact. a .1apl • ...,1. of a .priq-loaded fr1c:t1oa block llhic:h 'dolat .. 10-
etlul1ty (Al.') yet 1& .tebl. iD the tllpuaov .en.e Caa he ltv... It be.
beaD • .urpdae when - ._pl ..... dl&Cover'" (lS] for aa oppoUt •• 1tua
tioo; aa.ely, a ca.a Vbea the .. tacial .. tl.fl.a Drucker'. po.tul.t. tAl.,) ),et is UDstable la Ltapuaov ....... (Thls .. y be ob •• rved la ... terial vIIlcb
.... DO elastlc cupon ••• 1.a •• 11 lnela.Uc elso for UDlo..sla,. e. la tcu.
of coacr.te.) COIIHqu.otl,. the COftoactlon of Drucker'. pollulete vith .tabillty i. aot UD.ablluou.. levecth.le.. viol.tiDO of II L_' • ruc_r • postulate ..... ld b. acla1tted oal, vIIan th.re 11 a cl.ar ph,. leal ru.oo for 40iDi eo.
auch •• the effact of iDtamal fricUoa. whlch 1& abMot 10 utala but !aportaGt for cOIIcr.t. aad .011 ••
AdopU ... Drucker'. postulate (Al.,). c .... riq lt to laaquaUty tAl. 7).
..... ootiDl thet both 1DaquallUaa .. at oold for uy do... lt toU ..... tbet
ar del'; - a0
1j 411 ... ............... •••• ................ ...... (.u.10)
Al.6 12
vbare the proportiaaality coa.taot 4~ la. b, ca.parlaoo vith 14. (Al.e)
4" _ !(!L do l h 30.. .....
(.u .11)
vbar. h 1& a par_tar d.,. .... iD' oa alj aDd e1j • 14. (.\l.lO) 1& the ... 11-
kaova nor.ality rule. which tr .... 40u.ly .!.pltflea tlla treaa.eDt of lael •• -
tic ph_. Accord1Da to th. previou. 41acuaatoo. "-'ter, tlla .... of
noraUty nal. 1& DOt a oece •• it,. la fact ve could a .. iD ,1ve .,...plu of
v.rt ..... frictional _tarials vIIare Yiolatloo of the DOral1ty rule ... t be
... pect.... I., .. _ -J cooddar • p11e of &haec. <. book) aul>Ject ... to
&llear. Accord1Da to tlla aor_Ut, rule. tlla p11e _uld bave to 4U.te upoo
.hear1Al. but it doe. DOt. Or ooe _y cooaidar tlla altd1Da of two plate.
vitia tooth'" ~f.cea; if tlla ... rfece teeth .r •• teep _ ..... , the dlletaocy
( __ t .pan) safore... by the altdiq .. y exc.... tllat 1adlc:ated by tbe
aonlllUty nala.
IIotiD, that d01j ia iDyolv.d la 14. (Al.ll) liDurly. it 1. clear that
the .tr ••• -.tr.l0 rel.tl00. of iDer .... tal plasticlt, .. y .lvay. b. put 10
hJpoala.Ue fon dClj - Dlj .. (~.!)411m. Thb fon _y funh.r b. 1Dvart ...
to yl.U th. ta .. _Ual .duU C1j .. <!!,!). Althou,h the 1Dv.rsiOll of •
(6-6) .. uia la .. _al lud. toproh1hiU".ly c .. las up&" •• a101la •• rathar
atapl. eapre .. l00 fO&" tbe laverse .. trtz of CiJ
... 1& po.sible h.r •• a. va.
ficat recop1&ed bJ 'ralar [16). All _lOIOUS iDune4 _trill ..... iDtro
dueed "y T_a ..... zt_k1ev1cz 1'5.66] whea they fo~lat.d the fiait. d.
.. at .. thod for iDer .... t.l pl •• ticlty •
The ,leld fu.ctiao for concr.t .... t r.flact the iot.rael frletloo.
Th •• 1Ioplaat fon 18 thea the loadl ..... dace of Drucker-Pr.,er type [64):
F(OI, .... ) - T + 1(0) - II - 0 •••••••••••••••••••••• ,...... (.U.12)
- 1 " whare a - 0",,/3 - ...... tr .... ~ • (~ ....... ) - aue •• lnteDlit,.
Th. iDu_t.l atr ... -Itr.ln r.latioo. be.ed 00 thl. loedl .... urfec. Call
r.pr._t r .. aoaabl, _11 1I01l .. ial. biul.l ... d tr1allial au ... -Itraln dia-ar.... ao-v.c. lataral strel0 aad vol.- dllat.acy data ........ p.ciaUy
tbe .tra1o-.ofteoiq branch ••• c ... not ba duecibed .eU, aot ."eo vith .or.
c..,Uc:ated fora. of th. loadiDl f."Cion. Th. probl. 11 DOt vlth tbe
.ha,. of the loa4iDi aurfac.; .b ..... oalol the von K1a .. -typ. d.viatori"
At.7 1)
loadtol ... dace, baaeel OD T, .,cI totracluctal IIOra cOllplu for .. tbat tavolva
13
, the thir4 iovariaDt of erre .. , brtala oaly ataor 1IIprov_ota .... ich are
tacODcluaive ta view of the acarcity aacI Kat tel' of teet data GO tr 1A>rial
behavior of 0 ... aacI the _ type. of coac:r.C.. The probles 1e rath.r vith
the iaher_t UaitatiOD. of the tacr_tal-pleetic fo~l1 .. a. it. UD-
... itabUlty to "preaeot .tra1D-8OfteaiDi' rue 11llitatiOD ceo b • .ur_teel by a _ coac.pt, the theory of fracturtaa .. terial.
At.) Plaatic-Fr.cturtaa Mat.rial vith FrictlOD eacI Dllatancy
'laeric ell, do .. DOt lead to .ofte.1D1 of the .. t.rial. The ... 1,
phyeical MCballi_ whlch ceo uplaia atrata-8Oft.ataa 18 the aJ.crocracktaa.
1111e _a re.11 ... b, Oo ... Ul (17], tIbo proceecleel to foc.&lat. a theory for
a puraly fracturtal .. t.rl8l (00 plaaUdty) which 18 c.pl.tal, aoal0lou.
to iocr_tal pl .. Uc1£y. B. beai.a by po.tulattaa the ul8t_. of a
fraetur1fti surface
(41.1)
.... ich ... velopa .U atat.a tbat caD be raacba4 vithout furtb.r fracturtaa.
ChoodDi (a./a .... )..... to be oepUv. vhaa fracturlDi occur. ( .... ere the
fraccuriDi ,.r ... cer.), the co.lcioa of fracturtaa ia obc.1DecI aa
itt rc-cle .. ! 0 ••••••••••••••••••••••••••••••••••••••••••••• .. (Al.14)
Sle~e tbe .. tria of tbe purely fract~r1q .. t.rial la perf.ctly elaa
tiC, the rel.tl00 0lj - CijkaCka .ust bold, lac11e.t1nl tbat upon ualoacliq
the .. terl.l alwaya retur •• to the 101tLal str.in-fre. at.te. 'y cl11fer
.nUaUDI .... th.a bave
h cl01j - Cij .. clellS - 401j .... ............................. (Al.l5)
fr: 401j - -clC1j .. c.. ..................................... (Al.16)
~.re OOijfr are ~~Ll.d fractur1al .tres. lncr ... ata (actually clecr ... nt.).
Iy cOllparisoa of lq. (At.16) vlth Eq. (Al.l4) ve acquir.
At.1 14
fr (a. 1 .*' lj - -&lj ae .. cle..,
(41.17)
whera 11j ua _ pl:oportiooalit, c .... "., ••
To pia a as.pt., ...... eabl. fom of the theoc, ... further .eed .
vork ta.qualit, aoa1op. to Drucker', poatoleta. 1111e 18 ,rovlcl.cI by the
U',..ah1a'a pMtulat. (ct. laf. 11):
4112 -l*'i/r 4eij !.0 .~ ••• : ••••••••••••••••••••••••••••• (41.11)
where 4112 repr ..... te the eecoacl_r4er c:o-.l_ury lIOR claDe 011 the .. t.
rial cI..rtal the a"licati ....... r_al of the I£rato 1acr_U cI'lj'
C.,.rt_ vith 1oequalit, (A1.::.~) aocI fether COIIparl_ vith 14. (Al.17)
til ... ,ial.
........................................ (At.19)
2 clot -. : .. ·b ...................................... (1.1.20)
ta _1°11 vith ptaaUelt,. .... (Al.tt) r._u the oorallty rule •
S1a1larl, to the Drucker-haler ,taU fUIICtioa to pla.tidt" the
fracturtaa eurface .. , be totrocluc" a.
.(I£1j' ... ) - ; + k(t) - 81 • 0 •••••••••••••••••••••••••••• (Al.21)
where , - "kIt"' ; - (l.lj .1j)" - .nat. tat .... tt,. Appl,tal 14', (At.l9)
..... (41.20), 1£ follow. tbat (19]
d. ij - 2G '.13 - 4a1/1'
• d·1/r
- ·lj T ...... ...... ..... (At.22)
'11 _ 2Uc _ 411fr , 'lIfr .. j e'I< ....... .... .. .... .... ... (Al.23)
clot - 1 .(4; + .. 'cit) 11 41< !. 0; alaa cIoc - 0 (At.24)
10 whicb D' - .k(t)/4e. FUll aor .. 11t" .a 10cIlcated by Eq. (1.1.19), 18
obtata .. for a - .. '; INt tbie r:.atrictl00 1. deliberately ralased, for: tha
41.' 1S
.... raa_ ••• tho .. reprdilll iIlteraal frictlOD.
With r.p" to plasticit,. there Is. h_v.r. OIl. "o".pl"uou. dUfar
."". ill tbat the .la.tic properti ••• C1j .... r. DOt COll.t.llt but v.rlabl ••
lIpu,Ul (17) baa explor .. th. COll ....... C •• of the DOrMl1t, nal. (£4. Al.n)
for the dearad.tlGD of Mterlal coo.ta",. c1j .. (141. Al.16) .... for the ad-
8laaible .he,. of the loa.ina .urf.c.. It .ppear" thet • fracturlll, .la.
tic .. t.riel vbicb exhibit. full path-ind.peade"c. i ••• scrib .. b, a fr.c
turina aurf.c. thet 1a lill .. r ill t 1j" Thi. corr •• ,... •• (17] to a .. terlal
thet con.bt. of • r .... oa .,.t_ of boad ... trai,ht Ubers. which 1a DOt •
reaU.tic: aadel fot cOller.te. Thar.for •• vb_ aadeU,.. the .ctud behavior
of co""rat •• OD. CaDDOt follow the COlle.pt. of fr.cturiq aurfac •• ,.tb
1adepend .. ca aad 1l_11t, rule coapl.taly. .
Whil. tb •• tr •• ~-lnduc" ... i.otropy of the relation b.tveea the total
.tr.l8 .... .tr... l8cr_t. 1 ....... tiel. th.ir el •• tic part ... , .pproxl-
.. taly be related b, • t .... e8t _dull tenoor that Is isotropic (e&e.pt for
pural, t .... U. cracUna. for which the a1crocr.cb .r. hl,bl, ori.llt").
Thl. "el .. tic" 1.otropy he. 18 'f.ct b ............ 18 £4 •• (Al.22) .... (Al.2l);
thu ••
....................... (Al.2S)
Squarl8l .... ~ of tb. first r.laUOD fUrDl.b •• the approlliaaUoa (19).
dl __ 2.!!! 9 t •••••••••••••••••••••••••••••••• (Al.26)
froa which the fracturh, .ilatucy factor .... y be expr ..... a. U9].
ftdlt .. - 4y iii: ................................................... (At.27)
This rel.tion .11ows OD. to ut11iz. a th.or.tical rd.UODship b.twe.n I: ....
, fDr ." elastic .. teriel contal"l", • r.ndoa arr., of crKk. of varyina
.. ea. Such a .. hUonahlp ba. recently b •• " .erlv" by 1u.1aD.U and
O·Conn.ll 120) vain, Hill' •• alf-c:ondatant .. thad for c:oapodt... 11118
.avdopaent .llow. u. to theor.Uc.ll, det.r.ine th •. f""cUOD (19J:
~d' f~) o ................................................. (AI.21)
Al.lO 16
vbere C. - v.l ... of C 18 the .tr ••• -free .t.t.. the .1latucy f."tor t ....
follow. froa £4. (Al.21) •• nd deady lt doea DOt olley BOraUty ex".pt by
Kd .... t. £4. (Al.21) pv •• rea ... llAble ear .... llt vltb t •• t •• ta for ,,-
pre .. l ...... ft ... lna of _"rete (19].
the ...,baIll_ of 1Ilal.stic: .eforat18ll of _cr.t. c .. 1au of both
.ic:rocrackl8l .nd pl.atic: alip. the f_r pr ... U •• t low bydro.t.U"
pr ....... aad 18 tb. 'l.ter .t .... of the ua1ulal "c.pr •• s1 ... t •• t. The
latter doainat ••• t blab bydro.t.tic: pr ... ur •• aad 1a .1 ... laport.llt ill th.
early at.I" of tb. ""i.xia! "c.pr •• "OD t •• t. All lapr ••• iOll of tb. nature
of 18.la.Uc .craill My be pin" fro. tb. alo,.. of th. lIIlloadl8l bru"b .. ;
•• lonl as tbey .ra roulhly par.Ual to the l8ltlal alo,. .t the Unt
loadilli. fr."curl8l ".nnot c .... Utut •• aipUic:.llt ,.rt of 18.laatic
.tr.in. aad _1, vb.n the unloadina slop. bac_. auc:b 1 .... t.ep thell the
i8itiel loadl,.. alope ••• is "heract.rlsti" of tb •• t.t ••• t tb. f.11l8l
br ... "b of tb. _iex1al coapre •• lGD .ielr_. the fr."turina behavior b.co ...
laportant.
,,",cor.I"lly •• raaU.tic: aadal for c:oa"rU. an caab1aa both th. pl •• -
tl" aad fracturlna .tralll.. ..c.u.. the pl •• tic etr ••• -.tr.ill ral.tl ....
.xpr ... the .t •••• l""r ..... t •• tb. _ of th. al •• tic aad l_lutic .t •• ir.
1Ilc:r_u. vbU. ill the f.Kturl,.. •• el It Is tb. eluU" .nd 1aalastic:
.cr ••••• r.ther thell .tralll. vbic:b .r. _ •• It Is _ca ... r, to Urst ill-
wart the iIl"r ... 8t.l plutlc .tre •• -.tr.iIl r.l.tlO1l. b .... Oil I •• (At.12). After doina this. tbe pl •• tic .... fr.cturina .tr ••• iDe ..... t. My b •• uper-
iapoa". This procedure l •• d. to the foUowina ralaUollS (19).
.·lj - 2G .·lj - 2C ·ij ~ - ·lJ T ....................... (Al.~')
• 0 - little - 21:8 ... - ; ..... (41.l0) .
ill vb1cb
G .... e .. + i I.'.e .... ... - n(G + h + U.') if ... !. 0; .h .... - 0 ••••.•.• (Al.ll)
... - t (.y + .. ·.e) U d .. !. 0; .h. doc - 0 (Al.32)
whU. th. v.rieUGIl of G .... I: ie el.teraiDed froa ! •• (Al.26). Th. coafU
ci ... t. h ••• t. ". ca. ca' are certain fUDction. of .tre ••• nd .train l"v.r-
A1.11 17
iaDU (a. well a. ehe char.ct.riatic. of cOIICr.C., auch •• &ereo,th f;).
Vlth the balp of v.riou. iDtuitlv. coo.ider.tlODs tbe prop.r fora of ch •••
functiOD. hea b_ d.duc.d aDd MCerial par_tera have be.D Uueified by
fittiD, v.rloua t.at data available 10 the llteratur. (19). A v.ry ,oad a,ra_t wicb t •• t d.ta, coapuabl. to that prevloualy achieved with the
aDdocbrODic theory (aDd • ...,Ufiad iD ri,a. AZ.2-6 iD the .equal), has
b ••• tt.iDed for t •• t. under ... Dotonic.lly 10crea81D, load (19). JIovaver,
the b .... vlor .t unloadi", .nd cJClic loadl", la not pr.dicted by 14 •• (Al.29)-(Al.32), eod furthar d.v.lo,..Dt. would be Dec •••• ry for thl.
purpo •••
AZ.l 19
IJICl£ItD/TALLt IIONLUIEAa TaEOIl!S fOil nUlm llIIAVlOIl
AZ.l EDdochroolc Theory ad Vi.copla.ticity
EDdochroolc theory 1& • noval fora of COD.Ututi". r.lat1oGa for t18&
iDd.pendeDt •• vall •• t18&-d.peDdaa.t loal •• tic behavlor. Th. chaory 1&
characterlzed by the u •• of reduced or locr1Daic C18&, • DOD-dacrea.1D,
acal.r v.rlable who.. 10cr ... ot. d.p.nd 00 atr.lo locr..-ota .a veil ••
actu.l t18& locu_ta. Th .... t laport.ot dlUar_c. witb r .... ct to 10-
creaeatal pl.stlclty and .11 oth.r cheorl .. dlacuaaed ao far 1. that the
r.tatioo batvaaa. do 1J aDd dC1J
1. not lloear, 1 ••• , aodochrooJc cheory 1&
DOt •• ,acl.1 ca.a of h,poala.t1clty. Tba aDdocbrOllic cbaDry for COKut.
will b •• ,.,ldoad wlth laa. det.ll thaD tha precedlo, 00. becaua. a tborou,h
expl .... Uoo la avall.bl. 10 a recant paper (U) aDd raport (22J.
EDdocbroo1c theor, 1& but approached aa a a,.cW _ of v1&co,taa
ticit), 11S]. To elucidata 1t, cOll.ider tb. cla •• lcal vlaeoplaatiC cooaU
tuU" ralaUOD:
(.\2.1)
Vb.re .uperlapo.ed dot. refar to d.rlvatlv .. witb re.pect to tt.a t, r(!)
1. a 10adlD, aurface and lit M)' b. ra .. rdad a. a vlae •• lt, par ... tar.
Schaper)' [231 (1969) ,ropoaed a ,aoaraliz.tioo .uch that the .. laeo81t,
••• ocl.ted wlth tha 10ala.tlc .tralo rata 1. al.o d.p.Dd .. t OD ClJ~
(AZ.2)
If t'" inele.tlc .tral0 davalop. ,r_dually, a. 10 cODcr.ta, fuoct1oG .a(i) M, ba ... pacted to ba cootlouou. and _tb, .... SO lt .at allow raylor
.eria .... paosion [21,15):
_" • t, ... _. '" ...... ~"' .. , ..... no ... , .......... " .. • 1<,. ," , .... ,,_. ................. , '" ....... , .... ..., - .. _ ... ' (0) +, _., ... , ...... "OJ' ......... . "') ..... ",._, ... _ ... ,.,. ' ...... tho TO'''''' .-... " " ..... ... - to tI .... oaI "" ...... __ • ,_'" ' ..... to 'J, ... " -, ................ -........... - .. ,-,,_ ..... .. ......... " ... , ....... '"'''''' .I&nt.; .. .
,Ia \ ... · .... "1' • .,. •• '! ,_.
.... ,~, ........ o. _" ... " ~ of '00_ •••.•. {'iJ ''')~' .. ... _ ... 0.- ........... ,_ ....... ,'- ........ , ...... <mi.
.... ., •••• .,_ ...... -1/ ..... W.')'" _ ...... " ... '" ....... . ..". . . "1 'J'" .. ' 'W
.. -/~)·.~·. OJ 'j
.. ;,.~ .. '"
., • -1,<.1" .,!] ....
"""'. 'I"'" ................ , ........ o1oa' ..... _.
,12.>j
. ...•.. W.')
............ "'" _, ...... f , .... __ .. .. ........... ... ..... ..
' ........ ., .... , ....... , ........ . ...................... ..,. .. '<.I". "'" ._ .... -., .. .. , ... '''''' too .,... .............. , ......... .
.. 1_ 0' .......... "_",,, "' •• " .. , ............ f ......... <1, •
...... , .. '''' i" .... <j ............ _._ ..... ". ......... ", ... ,.,
''''!.o.~ .... _ .. .,. •.•..
" •• {!.,.<>< ..... , ... ] W.>J
..... ........ -- ........ _ ...... f._ , ..... ' .. n ..... .. , .. , '''', 'u .. _0< ... _'" .. f~., ..... f !. , .... ' ............ 0..,. ... "'.'" ., .................. , .. _1_ ...... " ol ...... " .... ".
.......... .. , ... _ ...... of ... ,.. .............. ,,_, .01 ..... ''''''
.... "' .... , ......... """ '''p. 'oo 'hi. '''_ ..... 1_ ..... _ ....
·.' " ''''' ... """ _ ....... ' ..... " • _,t ....... ..
.. ••• '(!,~.<')~ .......................................... )
.. .... " .... _ ...... ". <all" .-... Iao In) ..... _ ... _ ...1.1. _ .. , ..... 01, , ........ " .. _ ... ,_ .. .,....,.... l .. j ___ '"'' t. _''' ......... M __ .. _ .. , to, _ ... ...
_, ..... _<1 .. , """"." .,.,« , ......... _ .... _ ... .. _.10 ... " .. "" ....... G .... , ... _ ............ ~ ....... d_
.. , ... of , .. __ .. f.-.." ....... 11 ... , .. , ...... , .... ...
... .... " ,lao ....... ,.., .. , to , .. ,_ ' .... , ,. ...... ml .....
..... ••••• , .... , ... , .. ", ...... , .............. , ... <t. W.O) .. --, ..... ---'''' ......... , ............ -. ....... '"'' .............. ,_ t .... tA>'». no ....... JAr ..... ~. __
,....<- •• , ........ 102.') _ fin ..................... " ..... _
• .... .,. ...... lo ("') ........ " ....... -,. ........... _ ...
_'--_. vi ......... 1 ...... 1 ................... _
... _ .... U. lo, .......... "'" 1-"" , ...... " ......... "' .. " of ..... , .... , .......... _. of ...... , .......... ,,", ... , .. ..
.... __ .... _ ........ a,,". ",.'. {o ••• , 'ill, ".""" ..... ..
........... ' .f ..... H) •
... _ ... Iaob" ......... _to .. eo! , .. " ...... _ ..
• ...... .,. ...... _ ......... 1 ...... f_ ......... _ .... , ... '" -............... ...." --.... -'" ,.. ....... -.. - .. _" .... "" ............ ~u.._ ... ......., I ... ' .... "' .... ,.. .. .
'" _ .. , .. _,ook ,.-....." ..... "'""""', ...... to .......... 1 ..... Ial ............. _ ..... n.o .. _ .... f_ ................. ..
_"" tOT _ ..... ",. InJ ....... " ..... _"", .. , • ...,.,,1< • .... ...,. ....... _ .............. U' ... , .. , .......... _ ....... on _. _ ....................... {o«_OJ.
n.. .... _ .... _ ......... f ... "''--q -..... • ....u .... ....................... _ .. _ .......... _ .... n.. ...... .
...' .................... lI ........ 1.0 ........ 1 " .............. 10-._.0 ... _ .......... 00 , ... _ ... _ ............. _ •
,. ................ _ ...... ~ _r •• J. _ , .. " ... , d ........... .
,. ............. "'! ......... "'. """, ...... " .. u ............... , ..
U.4 22
dope .ce.p.r t ..... the 10adiDi alope •• 0 chI irrevenibUiCy. chI .. li ... t
feacur. of io.le.cic behavior. 1. builc 111 • priori. withouc aoy lIIequal
tU •••
'ia. Al.l SupupodUOD of !laatic aDd lDala.eic Ser... Iocr.-eDc. ac Load1o& ad UDload1o& 111 OrdiDal"J IDdocbrODlc Theol"J.
The aer. facc chac aDdochronic eheol"J var'" ." ... vicboue Ml ... 1D1 cri
teria le .... the theol"J arue "erlOtU1t, ill soda!1o& ehe re.pODIa co r .... ced
.... c,clic loada. ....."er. _ UIlIOUaface0l"J pr .. 1cttaea ara abuiDed for
cyclic load. of ... 11 aap1ttude luperiaposed on a lerle .tatic load, aDd ic
appeara tbat for the .. attuaUonl c.rtaill uDload1o&-raload1o& criteria 1hau1d
b. iotroduced (15]. Th. abllDce of such crlceria 10 chI ord1Dary "'0-
cbroo1c sodala 1. DOt the b.lic feature of theorJ; rather, 1C 11 che iDcre
aeDta1 DODliDaarlcy.
The 10crillilc tisa te a coo" ..... t "artabla for d.lCrtb1o& cbe hard .. -
10& or loften1o& of tha _tarlal. TM ... , be acc ..... U.h .. by •• CUDI
ruDctt. fen) t. called hardenlo, fUllCclOD aDd ic _ae prtaarily delCrtba
chI face ehae che h,.cer.al. loop. are ,en1o& oan_r a. c,cl1c 10ad1o&
procr...... SiIIC. 111 c,c11c load1o& ( .. y ba COR""red a •• par_tar for
the o ..... er of cycl •• , " .. y b. cOD.idered to b. a f .... cU ... of ~ or of '"
.... tch 1 •• qulval.oe. Th1l fuocttoo _.t ,rov vith " .... the .tapl •• t po.-.tbl. cboice 1. f(,,) - 1 + '1'" a. flrae lIItroduced by Valaola for .. tal.
(24]. A .oaevhat _r. c_,lea for. 11 .... ded co sodel tbe r •• poo .. for a
,reat o .. b.r of c,cle., aDd a .liaht d.p .... uc: .... the nre.a .Dd stulll
tavniaou _ae be iotroduc" to soda! c,clic behavtor 10 the stra10 sofe
eD1D, uDI.. FlmcUon '(!,!1) _.t reflaet tbe aofeeo1D1 of CODcret. at
u.s
hl,h .tr.lII. ..pact.lly the str.to-softea1a& raD,.. The .0fteotDi t. cau •• d
hy akrocraclr.1o& ""ich is l""....,..saot of vol_uic co.pre.aloo .... "paDd.
_lDly on devi.tor deforseU.... So, _at ioer .... with J,(!) •• Dd thi.
toer .... _.t fad. a. c ... ftoio, hydroet.ttc pre .. ur. i. introduc". Md1o&
"artau ••• c ...... ry effacc •• a ... it.bl. for. of fUllcC10n , .... ,tv .. 111 (21).
Another •• _tial propercy of concr.t. i. the inal •• Uc dUU.DCY. 1.
It te cau." ., stcrocr.clr.1D& ..... ich r.sults cht.fly frOll d."iator .tr.iDa.
So. it .... po.wleee. 111 (2l] chac
dA • '1 (l.!.!)dt •••••••••• •••••• •••••••••••• ••••••••••••••• (Al.10)
""'cUoa '1 reflaeu the facc chit dl distal •• a vich lIIcr~a.ta, coofta1o&
'r ..... r.. tacr ..... vtth J2 (!). e.paciall, 111 U. atralll-lIOfc ... to, rllll, ••
... dacr ..... if lares dileCaDc, hal dr ... , b_ produc .. ; _ (21). 11 •• -
tic sodull Pauld .leo be c .. l1der"to d.,.. .. 1 beeau .. cbe, is ,_rd
an .ffacc" b, poro.ity cbe ..... (21]. To d.aerib. chI lnele.tic behavior
aC ll1,h hydro.ueic co.pr •• l1oci ..... ich te cauMd by pore coll.p .....
clodo, of void •• 011. _.c lIIcrocluc ..... iti ... 1 "ol_tric 1Da1ascic .tr.lII
ocIa'/l1I. ....... de' 11 ~a!ac" to dC' (Eq. (Al.I» [25). P'urtl:er.::re, it 11
proper co dtec1D&u1lh _char 10.le.Uc 901_ c ..... , •• d1', .... ich r'pr"'oc.
ch. Ca.pac:UOD _ ... b, .ddlUou1 pore colle,.. cuc 11 1Dduc ... c cbe
be,_l01 of dear .UatolDi UIlder hiah coef1ll1 .. pre .... ra (25).
' .... 11110& .11 c.rse, .... choodDi the .... ochrOlllc 10adtDl fUilCUOD ,
co he .f Druclr.er-Pr ... r cype (vUh voo Hi ... d."i.tor parc), che .u ... -Itr.la r.leU •• of endocbraaic cheery .. , II. win... •• (25]:
dalJ ~ d.1l
• 2C + 2C de •••••••••••• •••• •••••• •••••••••• •••••• (Al.l1)
de _ ~ + dA + ..l!.. dz' + dA' lit lit
(Al.U)
A"aU.ble tut data ..... r to be lII ... ff1ci.t to decid ..... ther a _re c_
plea fuocUoa r br1o&. aboue aDy isproy_e or DOt.
Usio, tbe .... ochroolc fo~lattOD a. Just outU .... , ao UIlprecadeoted
_ce .. h .. b •• o achi."ed 1n aDdeUD, tha telt data r.ported 111 the litera
t"re [25,26). Th. fo~letioa hal bee ... h_ (21,U) to .odd qu1t. cloaely
; .;
i
~.,.
F "
•• 111 lu •••• C.
-. I '...ell, •• .. ,-..... , "'-'U. Hll ....... ftJKft,
..r i -. a
-J
1_ '.' . ..., .. . --I ...... .....: 'NOr, (lin. J.OOO ........... ..u.e)
.,."' ;,;eM, eo ~ .. . .. ' ..... -.....
Itr.M. 'I
F11. AZ.2 -.•. Endochron1c ~ory Fita (aftar (25]) of Un18x1a1
Teat Data of Popov1~a (58] (1973). Roan.atad. Hanaon and McHenry [55] (l955) and 8in181 Teat Data of Kupfer. Hi1adorf and ROach [56] (1969).
'Ua.R r.
-lIlr------------,,------,
-It
-II
: .. " 5 .. : ! ."
............ -l ......... act .... ,
1 .. " ...... C_", •. .... --...... .. , ..... 'r"~'.-"'
...... -_ ... . . ,.,.--... . .
!. .....,., .. .... ~
~ -............ K.~
".'.- -11IIt ...
~~.o .,,_ ~ __ ..... ..... .... ...... .. ... "r •. fI. I,
i
Mill1. 1.' .. :: " 5 •• ..,
-EndDcNOft&C ' .... r'
.......... -....... . $tr.I"'.
• • • • •
• •••
, h • L--_ a, •• ••.• ." .•.•
5tr •• " I, ." •
Pil. AZ.3 Endochronic Theory Pita (after (25) of Triaxial Teat Data of .. 1 .. r(52) (1949), H111a [57) (1967) and Robba (54) (19'4).
~ ...
N ..
~
N ...
---~r"---------' ·"r'--------------------------.., "
. .
_, ...... JOftIC 'MOl,
t .•
..
.,
•
" .... "' ... '" CIW'C,eM. 1 ...... "''"'' 'UH •
.. ·r.' •
Gr_, _,1'71 ·e·· ,. ... f ..... _1000 •• ,· ...... ,.....,
= " .-~ ..
·11
.... onl. Lan ••• 5'_1. 117' '.' •• '00"'1
--EncIOCft,onac , ...... ,
v .; .;.----:rt
'. S I'
t ...
i
--------." "
...., •• Ha.1teN'. 1biMft. 1_ t, . 4110 .. "
-E.-cMona.c 'fteD"
L., •• al Su.," e,Y. • .... --- ';i.'-- -.. ,
'.&.AI S"nlt " 1ft S S\r ... 41.. ".1 .... VOl~'nCl Itt .... I"
'i •• A2.4 Indochronie Theory Pita (after (25) of 'rri.d.l T .. t Dat. of Gra." ."t! ~an.on (53) (1973). Biaxi.l .nd Tri.xi.l r.ilure Z"velop •• fro. Propor
tional Loadi". T.at. of Aachl, Lin ••• ad Scllell (51) (1976), and Volu ..
Chanlle Dat. of Kupfer, Hlladorf ead RUach [56) (1969).
I • ..,. ....... c -... . 1---:=' Aii" , ",-. -~-," I I
I " I I I ,..J.,l,,'. ~ , I : r..
, 'I' , , " ' , , u I 'I I I ,
.:t. 'I I , , -( ...... -..c f.-, " I I, ' \ \
• ~ 'I ' \ \ I II I \ IA I II I \ \
i,·t~·,·Y fJ -~~KIf " ! \\ I, \ \ I \I \ \ \ , 1\ \ '\
IJ , \...', ' , \ ,
" :::>~ ',..... \ ' ........ C'fQ.l "1 I 111 11 U
__ .--.-.~ ..... ~ _____ -..It.._ ~~ IJ IA U U 1.1 U
t,/.,
'* II w ... c ..... c" -,- -"" ~ICIIO"'.''''''''''' ... ,
ri,. A2.5 Indochronic Theory rit. (.fc.r [251) of Vo1u.. Cha",e Det. fro. Trl.xia1
Te.t. of Gre." .nd ~'D'" ['3) (1'73) .... of Later.l-co-Axia1 Str.1D .. tlo fro. CycliC Un1 •• 1.l T •• t. 01 Sh.h and Ch.ndr. (S9) (1970).
:. :" ..
N
'"
t ...
..
A2.l0 28
the uniaxial stress-str.in diagr .... including the at rain-softening tail.
biaxial and triaxi.l .tress-str.in d1.gr .... l.teral atrains .nd volu..
chanlea in .11 th ••• tests. failure envelopes for biaxial .nd triaxial pro
portion.l loadiola. shear-caapression failure envelopes. uniaxisl cyclic
loading teats .t ... 11 .s well •• hilh str.in. ioclud1og lateral etraina
• nd a hiah ouaber of cyclea. Eza.ple. of fita of sa.e b.sic teat data
[51-59) ara liven in Fiaa. A2.2-5. Further..,r •• with the inc:ludon of the
te~ dt2 in Eq. (A2.6). the r.te aenaitivity of uniaxial coapre.aloo t.ata.
the nonlinear creep isoehrones. and the phenoa ... on of cyclic creep ~.
been also ..,deled .. tiaf.ctorily (21). The aaterial par ... tera hava been
exprea.ed a. a funct10n of concrete .treDlth. fc'.
A2.2 Reftn .. tmU and Co!parllon. of Eodochroolc .. d locr_otal Plaatic Por.ulaUon.
As has beeD aade clear 10 Sec. 2.3. the fracturlog aaterial theory
yielda lael •• tic .traina which ar. particularl, sultable to d.acrlb. the
strain-aofteniog b.havlor. This is becau.e theae ter .. depend on strain
cij .... ich 1Dc:r ...... 1001 the .0ft..,log hraoch. rather thao "ij' which
decreaa .. elonl the .ofteoiog br.nch. Guided by thia ldea. It has b.en
tried to introduc. into the endochroa1r *".1 additlonal in.la.tic: .traios
doc • ·ij i •
fr z dc • leaeloc •••••••••••••.•.••••.••••••• (A2 .13)
with d~ - Fo(~.!!.t>d~ ••••.••••• ••••• •••••••••••• •••.• ••••..•••• (A2.l4)
whlch .re ."alo,ou. to tho._ in !q •• (Al.22) .nd (At.23). but sr. related
to path l.ogeb t. This refin_t allowed a1apl1f1c.UOCI of the function.
that define the r .... 10108 inelastic t .... [261. as well as *"e110g of the
decrea.e of the unload in, .lope ••• tr.l0 grove ioto the falli08 branch.
ADother 10lical refin ... ot of endochroo1c theory conslats lD iotro
duciog a stress-induced anisotrop, of coefficienta 'ljk. in Eq. (A2.6) de
flolDI the latrlaaic ttae (15). One poa.lble fo .. 1.
dt • f!t(d_ijdelj + P181jdelkdeik) .......................... (A2 .15)
A2.11 29
lIovever. the preaently .v.ilable test data ar. Dot sufficient for d.t.r
mata, coefficient PI'
A further lea..-.11 •• tion. which ,l1d prove to be u.eful lor wideniog
the hy.t.re.la loop. .t unload-reload cycl... i. an introduction of k1oe
aaUc hardeniog into th_ andochrooic theory. S1all.rly •• a pluticit,.
tile eDdochronic loading function i .......... d to be centered in tha .tr ...
.pace at poiot ea1j which IIOveS. i. ••• F • r{olj - eaij
). Thia COllC.pt live.
e.I •• endochronic: aelalUc .train 1acr_ta of the fora
1 d.1j • 2C<.ij - eaij)d, •••••••• , ••••••••••••••••••••••••••• W.16)
The rule for the v.ri.Uon of eaij ia the ceDtra! ..... tion which is hard to
elacid .... the beals of .v.il.ble t •• t data. • ... rthale ••• OD. cyp. of U ... ·
aatic hardeninl prov •• to b. appropriate for ualoacIiog and cyclic loadinl.
OIl puraly theor.tical IrOUDd.. Thla will be d1acua .... latIN".
The .v.ilabl. t •• t data have b.... fitted al8D.t ..... 11' well b, an
SDdo~hrooic for.ulatioo and b, a pla.tic-fracturiog for.ul.tioa. What i.
cben the diff.renc. b.eween th •• e CWO for.ulatioo.? Sioe. the .. t .... tical
for.. ar .... It. diff.reDt. where 1. tha dlfferenc ... olf •• ted? The •• qu •• -
tion. can be ao_red b, po.tuiatiog th. cooc:.pt of "in.lastic .tiff ••••
locu." [IS'. Thi. locus i. defin.d •• th. locus of th. tip. of all .train
1acr_t,dCiJ
(vactora) which liva tile _ aal.Uud. of inalastic .traio
incr_nt • I dC"I. {The .... ltud. aa, be •• Uoed, a.l. a. It I. (. • )" • , • -1j-lj - l"'lth of vactor 'lJ ta the dz-clt..oslooal str.ta .pac •• ) Th. eli.ceoc.
.f th. inela.Uc .Uffoe •• locua frCla tha 101tial .tr ••• paint 18 ,ropor
tlonal to in.lastic .tlffn ••• ~ulu ••
U.inl the tan,ent .Uffo ....... ult. 0". caD show that for eb. plutic
a. wall •• pla.t1c-fracturlog fot8Ulatlon •• tha toela.tic .tlff •••• locus iD
.. , two-diMosiooal sect10ll of the .1Jr-cl1aeDa1_1 .traiD .pac. is a scraigh
lloe (F1g. A2.6). If the nor_lity rule is saU.Uacl. this lin. is p ... Uel
to the ta .... ot of the loadinl surfac. r in the 1n1t1al nate (or tha frac
turinl ."rface). Otherwise It is 1oclloed.
ror tha endochronic fot8Ulatlons (of ordinar, t,p •• with dt •
" ('t d.1jd.lj ) ). lt can be .hown that the inal~atlc .tiffo ... locus 1. a
circl. about th. inltial .tate po14t iD any two-eftaeodonal croa ••• ctioa of
the deviatoric strain .pac. (whUa in tha y - t pl .. e It h a •• t of two
Al.12
1
Su, ~deu InelastJc.~'1' e22 stiffness r_ I de I Deus ~ileij
II "".,... A __ de" ..rA~- ~
1 F 511' ell, I $", ell
5". e"
T
522. e22
Fil. Al.6 Inela.tic SCiffn.a. Locua for Pla.ticity (top),
Ordinary Endocbronic Theory (siddle), Endochronic
Theory with Stress-Induced Anisotropy (bottoe left)
and Taasential Linearization (botta. right).
30 A2.13 31
parallel lines); see Fia. A2.6.
If the .... terial parameters are fitted to predominantly proportional
loading data, tbe circular locua and tbe atraisht line locu. will touch at
a point which lies in the direction normal to the current loadins ourface
(straight-ahead direction). The difference between the two inela.tic
stiffness loci beco .... s larger as the strain direction deviateo frOll the
straight-ahead direction, and it becomes lars. at when the .train direction
i. tanaent to the current loading surface (.trainins-to-the-.ide). In that
case, the inelastic strain dt~j for the endochronic theory 18 as oisnificant
as in the straisht-ahead direction, whereas in the plastiC and fracturinl
theories dt~j 1. zero, i.e., the respons. for stralnina-to-the-slde 1.
purely elastic.
The latter feature J. definitely DOt correct. ThJa has beeD realized
by researchers 1D pl •• ticity of _tala, aDd to rated), the situ.clOD they
Mve introduced the so-called vertex-harde!'l1ng) in .... ~,i.:~ u::.e aliSU.a.c& lne
formation of a local vertez on the loading surface at the current state
point. One particularly stAple and versatile for. of vertex hardeniDI hall
been introduced by Rudnicki and lice (27 J aDd applied to rock.. By virtue
of the vertex, one can achieve that the response for lo.diDI to-tbe-side
becoaes .lso inel.stic. as test data aa well •• aicro.echanic5 .odels 5ul
,eat.
For plasticity with verta hardenlna. the inel.stic stiffness locua
•• suaes either a vertex shape or a curved shape. Thus, the trend towar4
vertu-hardeninl modela brinSI plaatic ity closer to endochronic theory, 10
that both exhibit inelastic stuin for the loadin& to-the-aide. Neverthe
leiis. there i_ Itil1 one noteworthy difference. In endoc.hronic theory, the
&training-to-the-lide produces an Inelast lc strain locreaeo[ .. hose vector
has the straight-ahead (nor .. l) direction, while io all vertex twlrdenln&
waodels (e.g., [27]), this straining produces inelastic strain increments
whose vector deviates frOID the straight-ahead directioc.
The differences 1n the response to-the-side have little effect on the
lita of .ost test data available. This is because well controlled test.
perfor8ed thus far are either proportional or almost proportional loadin&
tests. It might see8 that the triaxial tests, in which all three principal
stresses are first raised simultaneously and then they are varied individu
Ally. are an exception. ~ot so, ho.ever, because the initial hydrostat1c
Al.14 32
~ What ODe cblef-10 ... 101 pr .... ce ..... orieDt'" u.ale 10 tbe a1cronruc .ure.
1y needs are tesu where ODe lOa be. flr.t ... e or ~ .ue ...... 0 •• to pro
duc. orleDted 4._le. and tbeo ".rle. the other atre •• e •• creat1o& iD the
d..,iator .tr.b .pece • petb that baa • aharp corner. Sucb te.t •• re. of
cour ••• diff1cult to c.rry oat 18 a val1-cODtrolled _er. The loeb.tic .trab for loa4ina to-tbe-d4. ha. tbe areate.t iapor
t.oc. for pre4ict1oos of .. terial 1oatab1l1ty. sucb ••• tr.b localizatloo.
a. bas beeIl ~ __ b, ..... o1cki and a1.c:e 127J. The lar,er tb. bel.at1c
• tr.b for loadina to-tbe-a14e. tha lover the crit1c.l atre •• for aucb 1a-
• tabiUti... t.,- tbb 11,bt. plutieity vlthout ".rtea bardeob, b the
leut ute ........ tioo that caD be ..... if the test det. on atra1a1a1 petha
vltb • coro.r .r. l..::kina. Tbu •• the u •• of ..... ochr.1c tbeorie ••• _11
•• v.rt.a-bardeo1na tbeorie. 10 fblt. el .... t proar'" 1. b the bt.reat
of uf.ty of 1n.tabllity predlctioaa (1SJ.
It ia iot.r •• Una to Db...,.,. 10 thb CODtext that for loadtna to-the
.14e the d.for.atloo theory of pla.ticlty (the variable seC.Dt -.sull theor,
for c ... cr.t.) liv ••• 10 ....... that is soft.r thao elaaUc. llence. tb1.
t...., b DOr. correct tllan 1nc:re.eotal plaat1cit, b tbb r ...... t. The
IUcr_hao1c ..... 1. for plastic behavlor all lad1c.t. that thare shoa14
be 1aelaatie .trab for loa41aa-to-the-.14a (1.10).
The concert of ioala.t1c .tiff.... locus Ul_t.ate. tha Mtbod of
l1near1&atloo of eJIdochron1c tbeory (15). L1nearizaUoo require. that the
circular 1_ be r.placed 'Y .. taD,_t .tr.iabt l10a (ria. &2.6). Tb1a
taD be 4 •• caly 11 the dlrectioD blj of "actor d£ij ie .pproat.at.1y t.ovD. and tb. liaeariz.UOIl vUl .. acc.pt.ble caly for atr.10ina 4!recti .... that
are clo .. to 4£lj' It can be ebovo that tb. U .... riz.tioo of the eo4ocbroo
ie lor..lattoa .. , lIa achl..,ed II, replacina dC vitll the azpr ... 100 (15.221:
tiber.. -~ ....................... (Al.17) 1j mt .. b ..
Alternativel,. an .... ivaleat 11neariaaUOIl can II. canled out dtrectly b
the .tiffoea. coeffie1eot. of . Uolt. el_ts (lOJ. Th. 11oeariz.tiOll 1 .. 4.
to • DOr •• ff.cti". oUMr1tal et.p-lly-step .1,ot1tba for .tructural anal,.is. Vitb the sub.titution of Eq. (Al.17). the for-.l.tl00 bac_ 14entical
to • pl •• tie-frKtur1n& fOrallat101l. an4 lt 18 .1ao po •• ibl. to azpr ... the
Al.ll II
teosor of taoleotW _dull C1j... Ob.erve. bowYer. thar tht. ter. 1a
41ffereot for each .tralolnl 41rection b1j
• Thls ls rather uopleaNQt
vbao tbe 10"'101 petb exbiblts •• barp coroer ( •• 1. u .... lly tbe c ... for
_terial 1oatabl11ty) becau.e .... 1004 e.tt.at. of b1j
can be -.s. and .aD, po.sible 41rect1oos b1j .. at be trie4. 501vina tba si&.ovalu. probl. of
10atabl1lty for ODe cho .... dir.ctton bij 11v •• DO iDfo.,..tloo on tb •• itu.
tioo for otb.r po •• lbl. bij • a fact whlch .. at be k.pt 10 olnd 10 1ot.rpre
tina fl01t •• l--.ot claculation. for -t.riala whoa. 1Dala.t1c atUfo ••• locus 18 oot atral,bt •
Th. circular .hape of the 1oel •• Uc .Uffo ••• locus 10 ..... ochron1c
tbeory cao be DOtb1na nor. thao an approllt.aUon. 111 ,.oeral 00 .... t
expect the 4btanc .. to tb. 1oelaat1c stUfo ... locu. 1a tl!e bacitvu4 direc
tion and the uo,.oUal directloo to be 41ffereat fro. that io the • tra1 Iht
ahead dir.ctloo. ... eodochrODie theory vb1c:b allow ..... to control the ..
two properties -y be obtalned by r.pl.c1o& 4t vith Eq. CAl.lS) or with
vb.r. dt' • "" 4.1jd.1j .......... (AZ.lI)
r •• pectiv.l,. The •• two azpr ... 100 •• r ...... 1 •• of po.aible Ura .. 10-
duced anisotropy of teuor Plj .. dafloi ... ehe btriD.lc t1De. The corra.-
pondiol 10el.stic stlffo ••• locus is .0 ecc:entr1c clrcl. or an .111p •• (IS)
(FlI. Al.6). The available t.st det •• "-wer, ... to II. iaauffieleot for
1deotif,101 COD.t.Ot. PI .od Po 10 E.a. GAl.16) end (AZ.17).
The concept of lotrlodc -tiae baa racentl, b .... ,.Deralized 10 • 411-
ferent "00.10 b, Val.ola (21). eoo.1der1n, the uoiu1al ca ... b. propo."
that 4t - 14t - k d£.,1 wher. 4,., • da/E._ and a~ that for k • 1 en
..... ochroo1c .. urlal v1th thla iotrln.ic t1De daUoitioo bec_ ..... 1v.lant
to aD dUUc-perfectl, plaseic .. t.Uel. 10 thet ca .. the theor, .1.0 ut
hU .. the oo.,..11t, rule .nd Drucker'. po.tulate. Bov.y.r. the c ... k • 1
1. DOt of pr.ctic.l tnter •• t. whil. for k < 1 Druck.r'. postul.te 1. DOt
."Usf1ed ("nd 10 the ca .. of cycl1e 10adlna dbcus •• d 10 the nut .ection.
the reloAdlOI slope h 8tUl predlcted _Uar than the unloadbl alDpe).
A2.16 34
A2.) Cyclic Lo.cUo, by !lldochroolc theory
The prop.rties of eodochroolc forwul.tioos io cyclic lo.dio. have
recently heen criticized by Sandler [29J. who cl.1aed th.t ouaerical ioeta
bilities would tesult fro. inevitable o~rical errors iD dJUaaic fiDir.
el...ot .nalysis. Yet. the endocbzooic forwulatio"s outl1oed h.ve already
been trled iD dJUaaic fiDlte el ... ot analy.is. an4 the iost.bilitie. ex
pected by Saadler dicl oot _terWize.
the quaaHOD is .s follows. Coulcler the cue of e .. ll cyclic
nus. a supers.posed OIl st.tic stresa 170' 1.e •• 17.- 170 + • ato wt. It
can be found that accordio. to the eodochroolc theory the re.poose c(t) _y
deviste withiD sufficieot tWe h ... the static val,.e .0 by aDY ........ t DO
utter "- ... ll is .pl1tu.de s. Tbh ... os that the andochrooic tbeory
cloes DOt parutee st.bility io the LiapUDOV aense. ""ich 1& the .. sence of
Sandler's critici... Bovever. this fact per .. cunot be objectionable be
c.u.e io classical viacopl •• ticity, a phy.ically well founded theory. tbe
situatioo is prechely the _ .. e. Moreover, the endochrool" formulatiDo do ••
sati.fy • weaker cooditioo of st.bility, whlch .. y be .t.tecl .s
11111 I,(t) - .01 - 0 ... 0
for aDy liveD t (A2 .19)
Thi. condition repre.ent ••••• otially • continuity cooelition, a re.uir ... ot
that the probl. be "property poseel."
the l.ek of atability io the Li.punov sena. ia co""aeted to tba fact
that eadoelltooic theori .... y violate Drucker'. po.tulate. 10 parUeul.r,
for a ... 11 uo10acl-reloael cye1e of .tre •• the .eeooel-order work fiaurinl io
the poatul.te can b. Delativ. aceorcli", to the ordinary endochrooic for.u1.
tlons. this i. b.eause io eodoc!lzooic theory the .10p. .t the atart of re-
10adlo1 c.o b. 1 ••• at •• p tham the previous un10.dinl alop., whicb i. DO"'~
ly DOt ob.erved in t •• ts. let, •• f.r a. the ... 0 alop. of the unload1", .~
r.loadl", branche. ~a cODcazmad, th1a behavior i. correct becau.e otherwi ••
the pbeoo-.non of cyclic .train acc ..... l.tlon (cyclic ere.p) could oat be
.o&.le4. Mev.rth.l •••• Drucker'. postulate should mot be viol.ted unles.
there 1 •• 0" clear physlcal reason for dolnl ao, such .a the friction.l
effects, .od for uoia1ial cyel1", lID such reasoo is •• en.
It. w.y to .edlfy the endochronlc theory so •• to satisfy the Drucker'.
A2.l7 3S
postul.te io cyclic loadina of arbitrarily ... 11 a.,litude. yet allow
representin. the cyclic creep. has been found [tSJ. One _at u.e k1De
_t1e hardeoina such that the eeoter "ij of the lo.dlna .urface j...,. to
the extr ... stress pOint ""enever loadiol revert. to ualoadina or ualoadlaa
reverts to r810adiDl Ouap-ldn_tic hardenlna). 70r ",,10ad1ll1, de _at be
repl.eed by cud~. and for nloacliDl. by crele. It CAo be shova that
Drucker's postulate Is satisfied for the ualoael-reloacl cycle if Cu
< cr
~ 2cu ' liviol response with AW > 0 a. shovn 111 711. Al.7 [IS).
Pic, Al.7 Byateresi. Loop of liIIall .-pUt""., S&tiafYlaa Drucker'. Po.tul.te and eivinl Cycllc 5tr.io Acc-.l.Uom (after [15]).
A. for the un!o.dinl .ad re10adinl criteria, tbe alan of the iocr ... nt
of total worlt, V. 171jd'ij' is a .uit.ble indicator (15J. It. refiDed ....so
chrontc theory which e.o alw.y ... ti.fy Drucker'. poatulate haa b.en fo..u
lated. It allowed aD ezc.l1.ot r.pr .. aot.tion of a .... r"". teat clat. for concrete (.s we1l a. aand).
Al.l 17
CUCItDIC, STAIn-In AIID snuctUlAL AlW.1515
A).l reo.ile Cr.cki .. aa4 .eC-"1Dforcad CoDerece
reosl1. cracklol i. perba,. be.t creaC" .. par.cely froe ocher iDel.s
Uc bahavior. ODe _y to approach tbe probl .. 1& to .pars.po .. tba .tuia
4". to tensUe cracks .. poe the .tr.iD of cbe CODcrece betwe .. cbe cr.cke,
IIhJcb c.n b. c.le"l.tad .ccordiDl co the andochrODlc tbeDr, (or pl.stlcit,)
(30). Thia .Dpro.cb, .lbeit pl .... ible. 1& DOt entird, .. tbfactor, .. it
do .. not aaswer tbe probl .. of lud ... l and partial tendle cr.ckina aod of
partial sbe.r cr.nsfer d .. e to .ur ... te iDcedock •
.. ,re.eat.UOG of Iradual t •• U. crackina 1& eapacially UportaDt for
4yoaaic fiDiC. el_ac aoalya1& of coacret. UDder blast load.. If ODe
••• ..aes 10 aucb an anal,ais tbe tens11a cracks co fora 8Uddenly •• vaY. of
at •• p froat is .. lttad .t .acb crack for .. tion, Which .. y c..... tha adjaceat
fidit •• l_au co cr.ck aDd allit further vay.a ca .. d,.. tba atruct .. re to
41s1ntelrate iD a cbaia r .. ctloo. SQch babayior i. .pur1~ aa4 doea DOC
cec"r 10 r.aUc, bee ..... cracke fora araduaU, aa4 CODcr.t. ublbUa cen.11.
aoreenlal. Th ... ,rop.rU ..... c be r.flected In a propar .. th_tlc.l
DOdd. A relaUydy .t.pl. trut_t of t .... U. soft_ina 10. lar ...... 11c·
It flnlce .l ... at .oaly.i. of a coacr.t. rUctor y •••• l v •• 1atrod .. c" in
laf. 1l11. A .ore .o,hi.ticat" .odel .. y b •• chi .. ed by iatrod .. clq teaaor a
lj Which deUna. tbe UIlcrackad .rea fracUon for Y.rioua directioa.. Th.
.Uffoa .... tr1ce. for CODer.t. f .. lly cracked in OD., tvo Or tbr.e direc
CloDS .. , tb. b. linea.rt, ca.llined accord1,.. to t •• or a1j
(491. Another
.. ,. to tr.at tbe ,robl_ is to introd .. c. DouI11l-t,.. fract .. ri,.. .. t.rial
' .... ry. Endocbronic ch.ory ca.biaad vicb ceo.il. cracki ... baa b.en ..... ...it.
auccescfully to pradict cb. r •• poase of b.... co .. isaie-C". c,clic loedinl
112]. ~ exaaple of the aareaunt vlth t.st data 1. liyea 10 Fil. Al.l.
rbe theor, allow" 8Od.lial r.aUstically tb. effect of .• cirr .. p. OD duc
Ulit,.
Al.2 38
The probl .. of teasile er.ckiDl i. of particul.r 1Aport.oce to COD
crete tbat is re1Dforcad by • steal bar aet Whicb at.bilize. tbe cr.cke.
ril. Al.l rita of P.rk, IaDt sad s..paon'. r.at Data [60] (1972) OIl C,clSc a..peas. of ..... by IIldcebroaic Theory IIlbaDc" by eraclL1Dl 1111 .. (after lalaat aad lbat [13]).
Thb ,robl_ bas 1a tbe ,.st ...... .tudied 18 t • .,.. of 1111it dedp (.·1·,
Jef. ll). Sa which tbe ... albl. fdcC10a OD the cracke bas b •• a.llect".
lac_elJ, l18lt d.alp 1a _lela frletlOD GO tba !track 1& DOt a.llected baa
......... lored [l4J sllll. curiously. it a,pea .. " tbat a beavier ratber tbaa
1ilbter re1Dforc_t 1& r .... ir .. vhe fr1ctlOD OD tile craeke is ace_t"
for. Thu •• a oell.et of frletloo 1& DOt be ... Oft tbe dole of safety. Thi.
i. becau.e tb. t .... U. forc •• 1a yieldlol reloforc_t _.t not oaly bal
aace the app11ed load .... t ... t alao provide tb • ., ........ dve force aero ••
the crack tbet 1& aead" to davelop fricUoa. This force is alvay. ioduced
bJ dUatloa oa the cracke .. pOll .Uppi,.. of roulh, iaterlocked crack surface ••
(0)
Cb)
2.0
1.0
o Fli. Al.2
1.0 20
a = 30° m=0.5
~et ' • a .... nfarced Slab Vith Friction a th and Corre.ponelin, Lt.it De n. erack. (.) Friction Coefficientl k. (;!~c!~v:!~~:'i!b~ :o~ ~~rlou, eailtinl th~ory tapli •• k. -.) (After Bat ~~. TsubaU r 14)). ant .....
Al.4 40
DestlD dlalr_s for thl8 type of aptia ... liait dea1ID h.ve beeD developed
(34] (Fil. Al.8), .Dd tbe r •• ultinl .nvelap. of safe delilDa i. liven by
«n _nO) - a (n _DO»)( (D _onO) - a (D -1I0)J • (a (1- .)ain 211]2 (#.l.l) "" Iyy yy 1ss 2
in which a l • (1- sin a)/(I+.in a), 82· l/(i+ ain ,), 8· "'Ile of
) • a friction on the crack (probably 8 • 1.4 to 1.7 , Ds· N;tNI' n • N-'N1 , • • Y Y
•• N2/NI ; N,,' Ny - yield farcu iD the st.el rdnfarc_nt in arthalonal
directianl " and y; III' N2 • principal iDternal fare •• d .. e to applied laed,
II •• nlle of N1 with rupeet to .,,11 ". CeOlUtr1caUy, Iq. (Al.l) repre
.eats • hyp.rbol. in the (D", ny) d •• ian plan., aDd it il DOtevortby that
ita ..... ar. inclined wbereas for aero friction con.idered thus far itl
.... 1 are ortholonal.
Al.2 Str.in-Lacalization Instability 0 ... to Strain-SafteniDi
It 11 w.ll known that, except at very rapid dynaaic loadiq, a conti ....
u ... caD .xhibit no .train .ofteniq bee ..... it 18 unltable. Yet, Itr.iD
aafteninl c.n b. ob •• rved exper1aelltally even und.r Itatic lo.diq, aDd all
our previoualy di.c .. ssed aod.ls .xhibit ItraiD-softeniDl •• an esa.nti.l
property. Th. reason is th.t, .lthoulh va have DO other choice but to uae
cantinu ... concepts of .tres. and Itr.in, the .. terlal is actually nonboao
leneous, prevent ina localization of Itrain into reaians tbat .re tao ... 11.
If ane assuaes that the width 'a of • strain-loc.llzatian band cannot be
narrower thaa the siz. of allrepte, d, .. terial Inst.bility is obtained at
a certaiD nan-zero, finite value of the downward .lap. It of the Itrain
lofteninl dlalr .. (lSI, dependinl on the .i •• of the Ipec1aeD relative to
a&lrel" •• ize, the .tlffness of the t.stinl .. cblne, .Dd possible par.tlel
eLa.tic ••• traintl. This .. y b. ao.t .laply deanastrated ia a .. niaxial
aodel which lives a rather siDple condition for stability (15,171:
1 1-
a
1 AI
+ 1 (C u+ C ) a • p
(#.l.2)
Al.S 41
where I .. b the lIIlloadiDl .odulu. (IU > 0), C. - .tiff ..... (.priq coutant)
of the te.ti", .. chiD., Cp
- .tiff ..... of eYaDtual .l •• tic rod. par.llel to
the 'peeS-, A - cro ... acU .. ar .. of apecS-, , - iU leaath, '. - Dd,
Il • 2. TrOD chS. fo ..... 1a we Clll calculat. the critle.l value of talll'llt
_ul ... It' frOD which the failur. poiDt OIl the .tr ... -atreiD diaar_ Clll
the be loc.teel. Por. t"ical .tr .. a-atr.iD diaar_ the r .... lu ar. uiD Pil. A3.3 wh.r. ductilit, i. under.tood •• the r.tio of the .cr.ill .t
i ... tabllit, to the .treiD at peak .tr ....
A .1II11ar .:Iaople f ....... l. CIIl he d.ri" ... for c .. n.tur. loc.U&atiota iD
c .. creu fr .... which ubibit soft"'111& (35). Yariou ... pacta of such
fr .... C.1l elso b. deac:r1I1" "-111& .oft_1II& pl •• Uc hilll", ..... .dopUIlI
thi •• pproach pea.tratiq .nal,al. of .tabUit, hes ...... ~rri'" out b,
""in at al. (0'1., Iof. 36).
A deteraiai.tic .pprooch to loc.licati.. iD.tellilit, ca .. 0Dl, depict
the dependance of ductilit, ... i ... Dd .tiff .. o •• of support. (.tor"
anuu). IIowevor. in. further i .. "O.ti"UCID (37) ill whleh • .od.l c .... ia
UDI of •• yat_ of parellel atr.iD-.oftaDiq el_u who.o proporU ••• ro
r ...... ' distribut" ovor the .,st_ .... cODalder", it .... fOUDd that tho
loc.liz.tiCID iDstebllit, duo to .tr.iD-softeaiq c .... lso doac:rill. the
effect of alze ..... support .Uff .... upoD the atreaath Cpeak stn •• ) of •
cClDcreta .peeS-. nia ia II_sa iD tho por.llal .y"_ tho 10caUzatiOD
iDatabilitio. occur Iraduall, ..... ",uaDtiall, •• tho .tr.iD ia 1Dc:r ... ".
thb .taU.tical .ffect GIl .traaath b f ___ tall, diffor_t trOll that
_dod lIy the _b.t liDk iD • chaiD, .. u .... 0 f.r.
A atudy of the .ffect of Jllrellel rostralat provid .. II, atool uiD
forc_nt upoD the .tr.iD-localiz.ti.. inat.llilit, wUl II. crucial for tho
theory of deforaatiaa ..... crackiDl in .ot-roinforc" CODcr.to.
The q .... ti .. of iDatellllity due to atr.iD-aoft_iq 11 'D Sapon.at
out.t ..... iD, queati.. iD the fiDit. el ..... t .... l,.i. of concroto .truct .. ro ••
Special st.biUt, checb .oed to. be d.".lo,... for theao proar-. no
• tructurol t.aleat atiffa.a ... tria •• a usually sat up, i. iac.pobl. of
reveallal .11 iIlat.blUU •• , espeeiall, tho .. which iwolv. "atr.iDiDI to
tbe-.ide" dbc ...... befor., bocauae the.oatir. struct .. ral .Uff ....... tria
1& dlfferent if • differnt atrain iDcr_at direction ia the .tr.in apace
is caaa1dered for onl 11_t. thla _a. that on. vould have to aet .. p •
.. ... " J
: :; &
...
1.1
1.2
1.1
I.' I.'
..,
1.2
1.1
Al.6
:,1 c,- 'iT
c -!!!.. • u c.-
C,-I.' "-II.a 1.0
ri,. Al.3
, , 10 • • I , to 20
L .. ,le 114 (Iill .rr.eI)
Ductll!t, of Concrete •• fllDcUon of SpocS- Size, Hac:h1n •
Stiffnl" C. - c. A lu'l .... Coupl .. 'arellal lar SUff .... s
Cp - cp A ' .. /' C.fter lallllt ad P .... la [37]).
.\3.7 43
.iff.rent .tructur.l .tiftDe .... tria for each po •• ibl •• tr.iD iDcr ... ot for
each fiDit •• l_t. o .... ioual' • ~oh1hit1". t&alr.. 'or thb "'._ •• illpl.
ph,.ic.l coacept •• 8UCh •• the prope .. tioa of aoft.a1D& atraiD-localiaatioo
bud ..... t be ..... loped.
T •• il. craclr.1l11 i ••• pacial. lWt1D& eas. of atr.ill-aoftao1D&. The
curreotl, us" .. thod of .... 1'.. cracka withJa tbe fiDit •• 1"'lIt auff.r.
trOll. wk of COIIverl.llc. vhaIl the dce of tbe f:1A1t. al_t 18 r .. uc ..
to c.ro. Coo.ider the .l .... t •• t the "lUllt tip of ..... r •• crack b ... .
The .cr ... "I iD the el_t .h ... of the blaot crack b ... tip i. u .... U,
t.ken •• Ch. coodltlon of cr.ckl... JIlt this .tr ••• could have .ny value
(.vao for a f1xacl load) d.p.O<Iill, OIl the dc. of the fiDlt. al ... IlU •• "'"
theoretically t ...... to Willit, a. thla .iae .helolt. to aen. Thu •• the
aoal,., could .. Ita the crack b."'" ,rope .. t •• t ... arbltraril, _11 loa ••
ju.t b, .. kina the f101t •• l .... t • ..all eIIOUlh. thi. 1a a fua. ... lltall,
ObJeetioaabl. prop.rty ual.a. the f10ite al_t .1&. 1a alr.ad, tb •• ioi
DUll alze that ia ."iaaibla 10 r.l.tlO11 to tb. a&&r .. at. aia. a"'" tb. at .. l
.. .II a1&.. OD. va, to r_. thie v1olattOll of the pdac1,le of obJectivit,
i. to u •• fractur • .adlaica criteria for the crack baDd ,ropa.-Uoa.
.\3.3 St.p=br-Ste, IUller1cal AllOrlthaa
~ .iSCu.a1011 could h •• evoted to lbb topic. wt VII _at COllt .. t
ourealv •• with ootlo, thet ao .ffeetl ... laorll~ for atatic .oalyai. With
ltar.tiOll. 111 .. ch at., has b ... livWl ill hf. 21 .... verlfl .. by UaLta
ale'""t .... ly ... iD Raf. 22. So.a illprov_u 10 tha d,orith. vere .. de
'y ~rao.eD. Aro ......... "rlao [3IJ. La.,.. fiDlt. el_t proara •• hay.
by _ beell .ev.loped ill aev.ral laboratorl.a; ...... 1 •• aat. [30] where
a careful ItU.7 of the .ffeetly_e .. of yarlou. dloritt.. va ..... aD. furth.r illprov ... llu for atructural aaal,a1. b .... Oft .... ochrooic theor7
... r •• ropo ....
.\4.1 45
CIlAftU #0.4
ClUP. SHIUJlUGE All\) EffECTS or 1I)1STUU AIID TlKPWTUU
".1 Theraodya!!ics of Vi.co.lastic 5oli.lf7iD. Mat.rial
Creep of COllcret. i. a fa.c1aatlll, .ubject of lr-.-"GUS braadth. 10
....ich cootilluua ..ehaoic •• c_t ph7aic •• phyaical chea1atr,. therllO
.,aaaic •• pro ... bi11.t1c theor7 .... ouaerlcal .tructural ... 1781. loter.eet.
SillCa.n e"tao.lve lurve7 of thb ... bject va. pubU.W Dot roo lona .10
(1915 1)9». the rev1_ of the cr .. p theory will b. focu ••• 001, 011 v.ry
rec .. t d.velo,...t ••
.\ rtaor0U8 trur_t of cn., r .... 1r •• for.alaciDl the r.lev.llt coo
" ..... theraodyDalc.. Tbe _t t.portaDt vork 10 thi. r .. peet has Mea
thn of ArlYris. Piat.r, S%1Iaat and 101111 .. l40j. who Ulad the iDt.mal
.arl.bla foraal1 ..... expr..... the co.pleaeetary fr.a eD.fl7 a. • fWftc
u_ of oot 0017 the Kr ........ 1oteroal yartUl.a wt elao the _iatllra
conteDt aDd t .... r.tll'.. FIIrtber an_loaa of thi ...... 11 ... t vork .. n ••
earrl" out to accouot ill a proper .. _r for the effact of a,i •• u. Co
aoU.1UcaUoo or hydrecioo of c_t pa.t.. Ju.t l1ke 111 ch.raodyuaJc.
0' chealc.1 r.actloo •• a propar .p,roach r.~ir.a that va i.entify aod 10-
troduc. tiDe-lnv.rtaot .. tarlal cOllpOoeat. of filled properti •• (a., •• COII
.laat .l •• tlc .odulul EO. conltallt yiaeo.ity ~'. eo. treat the var1.tl00 of
th~ overall .. t.rlal proper tie. ( •• , •• .odlllul E(t) of concr.te) •• a COQ
•• ,uellca of volu .. Irovth or bon. 111, (pol,..r1 .. t100 of caeent ,.1); 1 ••••
I(t) • E'v(t) vltar. vet) • voluu of c .... , ,.1 part1clpat1o& ill r •• l1tlna
,~. loa.. The ca.pl ... atary fr.e eo.r,y D i. tball obt.lne •••• ti .. -hl.
lDr, iDtelral 10vo1v1o, volu.. vet) .... lt cell be expre .... a. a fllnct100
.f Cllrr.,ac .tat. ollly 11 y • cODltaDt. III conlequaoc., .trall1. C.DDOt b.
•• ,1ved •• £ • aa/.o but 00& ... c uee
(A4.1)
u •• eu.ph. we aa1 ,1ve tbe c...,l ... lItar7 frea eoern· for liDe.., 1,1111
creep 10verll&4 by the Maxwell challl IIOd.l:
.\4.2 46
t dO' (T) • • HI ~Jo,. + I V',. ........................ . ,,0 II "
i - £0 (,\4.2)
ia whicb ~ - iaterDal varl.~le. • .tr.ia lu tbe d.ahpot of II-th Kax».ll II
... it. 0 _ •• _lated .tre ... y (t) - •• _lated _1_ cr_h. tile dlffer-II " ••
_tlal ..... tloo. of tbe Kax»ell cbelA ...... 1 are obt.ia'" •• £ - aa/3o vUh
,,- I"O'" .... G - aUlae " "
................................................. (,\4.3)
vbile tbe .toslpat1oe 1Dequal1ty 1 ••• ,l.fied by .. tt1o&
.................................................................. .. (M.4)
iD .... lch qO _ iDcrlAaic t1ae-coD.taDt ,,{scGoIty Gf tbe ,,-th Maxvall uoH.
" AD 1oportallt fact 1. tbet fGr ... c1A& ucerlal. tbe .lasUc .cre .. -
.tr.iD ralat10e •• , be "'itt..... .." - Io,,(t)dc(t) - I(t)dc(e). n. r.l.t1oo 0 • Io,,(t)c • I(t)c 10 iDadot •• lbl. bec.u.e It would "iDlat. tho
dl .. 1paUOG lA ..... Uty fGe tho .tralA an.rIY challi. c.u ... by "01_ Irovth.
lOwev.r. fGr • d~aolidlfY1A& .. t.rial (debydr.tiD. CODcr.t •• bo". lOO·C).
tho r.l.tiOD 0 - ID,,(t)£ 10 tbe coreoct ODO.
'10 obow tbet tla1a 10 1DIl." 00. va _y expr ••• the otor'" lA.uutaa-
oou.ly e.c09.rabl. (r .... r.ibl.) ... r., U of tbe .. t.elal aove,.ed b, 0 •
Io,,(t)£; w. be". U • ,,2/2ID,,(t) .... 0'
• do • U • " 10,,(t) + Dcb•
D __ ~ i1£L. ..................... ('4.~) cb 2if ,,(t)2
wheee DCh • ch .. ic.l dl •• lpat1oe of .tor .. alastic .D.rlY due to roaov.l
(dia.GlutiOll) of .cr ..... _uer. Tbe .ecoad 1 ... of th.,.....yDAa1ca r.-
quir •• DOt OG1.y tbet tbe frictS-l (,,1acou.) dl .. ipatiOG D be _-... pt1ve.
hut alao tbet Deh!. O. a_. tbe ralat10e " - I(t)l 10 po .. i1>l. GIlly if
• c 0 Gr i c 0 froe wb1cb It fGll .... further that do • I(t)d£(t) io tho
~y po .. l1>-;11:y if i!. o. 'ftIe. loot cClllClu.10D .100 follow. fr_ • sicrG
otructural cOD.~deratloD ....... 00 tb. f.ct that Dowl, .01idlf1.d .. tt.r
vb_ it 10 add.d within the slcro.tructur. to tho load.d F.re ou.t b. iD1-
d.lly iD aD uoau .. o" .ut.. lD .. u, work. udUI rbeolollc.l .... ele fGr
.. lAC llAoar ere., the forelO1DI lA.qualiti •• have beea vl01.t ... .
.\4.3 47
.4.2 Con.titutiv. Egu.tilln.
bt.-type .... ela .re requir.d not oaly for fora.laU .... the therwo
'JD&aic. of cre.p hut .lao. aa4 "laly. for flAlt •• 1e.eDt cr •• p .naly.i.
.f otructur... tIli. i. to .~oid the prohibit Iv. .tor... ...d co.,utlDi tt..
• ....... po •• d by tbe 'irect u •• of the cr •• p l.v 10 the fOnl Gf • her ... ltary
iDt.cr.l. til ... lAt.,ral-type l.va can. fortunat.ly. b •• pproxt..t •• by tho
rat.-typ. creep lev •• clo •• ly a. d •• ir.'. end .ff.cti". co.,ut.r .1,Grithaa
for this purpo .. hev. b .... 'ev.1Gped 141,39). 0... such al.Grit" for .,iD.
cr •• p of CODcr.t. he. b .... lAcGrporat'" lA lar,. c08lpUtu cod. SAt I~O).
10 cla.olcal vl.coela.ticity. tho ~ll sod "lviD cheln ...... 1. ar •
"- to' be equivalent. tIlia I. true for .,lA, .. t.rlab GIlly If aepU".
opriD. 80CIuU aDd D.pth. "lscG.IU ... r. .llowd. which -ta. however.
ther.oclyaaaic.lly 1Aedai •• 1bl.. If thea. are required to b. po.ltl".. It
ceo be oh ...... that the llaxvell chelA i. _r. c .... ral thaD tho .. hiD ChaiD.
... hlth 1& • rat!: ... surpr 151:>& ruult. :; ... ly, th. lattu la capabl. of
repr ..... Un' only coover, ... t creep curv .. for dlff.reat load in •••••• vlaU.
the for.er caD r.preullt both dlv.r,_t .... c_HI_' cr •• p curv ••• a prop
erty .... lch I. 'yplcal Gf cr • ., te.t d.t. for coacr.te. It C.D be alao .hGWD
thae dlv.r._t creep curv •• do DOt vlolat .... y thanlodyoulc r •• trlcU •••
Thar.for •• GDly llaxvell chaiD .hould be u.ed 10 the fiDlt. el ... DI .DAlyoio
or .10. aD .rbltr.ry UDju.tifl.d It-lt.tl. i. iaplied. (As a con •• qu.nc.
of ebh f.ct. the Ma ..... 11 ChelD C.D .i" •• upoo ."lic.tiOD of the .upu
P001UOD priDcipl •• __ otODlc creep rocov.ry curv .. without viGlaUDI
.. , th.,.,dya_lc reatrlctiODo. whll0 tho "1,,1. chata ca. ,lald oilly __
coatc r.cover, curv.o.)
n •• fore_Uoo'" find in,. ,r •• eat a further or ..... t .,a100t the
.... of tho ao-c.l1ed "l.pro"ed Diochin •• r .. thod" I. ,r.cUcal cod. r.c_
MadaUoo.. AdoptlA, the fGnlUl.U_ h_ the eu..o DDI ea. •• thb fo,..
I.U- has just b .... lAcorpor.ted lAto the C.E.I. lIodal Cod. (197'). It
.... arpad lA its favor that It all .... 81.,ler .tructural _lyot. (.0_
Ually .ccordlDI to' Dbchllll.r'. cl ••• ical .. thod). _41 that the .. pu.
eloa of d.layed .lalttc .nd flow t.roo 10 cre.p t •• ppaa11o& to 1000 eolt
MUI. lIovover. this oep.rati_ 10 DOt Ju.tlfied b, JODI-tt.. creep data
141). aDd tb •• Ir ... eat vlth t •• t data 1. f.r inf.rlor to other .val1.ble
,uetical _ela •• uch •• the IraGSOD·. product type ..,del (adopted 1>y ACI
M.4 48
eo .. lttee 209) aad the refined product-type .adel based on the double power
law. Moreover. a Dew .. thod for creep calculatiODs in desian office. baa
been developed (Trest _thod or the a,e-adjusted effect ive _dulua _thod).
allovinl an even si.pler calculatioo tbao the DischiD&er-type fo...ulatioos.
aad for any for. of the creep function [42]. The t.proved DischiD&er-
... thod corresponds to a Burlers rheolosical 8Odel. whicb is a special case
of Kelvin chain. AccordinS to tbe precedinl discussion, this ..,.sel is in
hereotly incapable of representinl d{versent creep curves. aDd this is part
of tbe reason vhy the t.proved Dischinler fo...ulation i. incapable of a
sati.factory description of 10ns-ti8e creep data. Another reason ia that
this for.ulation t-plies the shape of the creep curve a. well as the a.iol
effect to be defined in teru of one aDd the ..... fuoctioo, whlch does Dot
correapond to tha real it,. The double power law. alonl with the practical _del for shriDi<ale and
dryiDS creep (43), have been recent 1, developed into a full practical pre
diction _del. Iy co.puter anal,sis of practicall, all teat data available
in the liCerature. for.olaa to prediCt the .. terial puneten fro. con
crete atreRlth aad caapD.itioo baYe been developed (44).
"icramechaoic, ~els of the solidification proces. can ba IUcca •• full,
utilized for derivinl a proper for. of creep fuoction of cOGerete (45]. Ooe
such ~el (45). 10 which la,ers of un.tressed .. terial are belnl deposited
upoo loaded c ..,.epinl sol ill, is sbOYD in Fli. 1.4.1.
bte-type ~dela for creep ara alao needed to tor .. late the effect of
variable h~idity on creep (in addition to the effect on shrinkala). A
..,del ba.ed 00 tbe the..o.tynaaics of .ultipha.e equilibriua betweeo adsorbed
vater io the .tcropores of c.-ot ,el, capillary vater aDd vapor. aDd 00 the
hypothesi. of coupled diffusion of vater and 801i1ls witbio tne .tcropore.,
baa beeo developed and.bowo to live r ... onable .. re .. eot with te.t data[3~.
... ed 00 aD idealized co~cept of creep .achaoi... .s well as the addi
tivit, properties of linear creep. a stochestic process 80del for CODcrete
creep has beeo deduced receotly [46]. The basiC creep is treated as aD ad
dltive OOD-.tatiooar, process w1~h independent lOcr .. eots and le.ds to •
DOo-atatiooar, local ..... proce.s [46J. This ~el caD be utilized for
eatrapolatiol ahort-tiae creep test data and for deteraioins co~.!UeQC.
It-its of a liven probability cut-off. An euaple of thirty .. trapolation
A4.S
Fis. A4.1 Hodel for Creep of Sol1d1fyinl Materid (after (45)).
trials a!.aulated on ""!Iput.r 1a .ho .... io ril. A4.2, io co.p.riloo with
observed creep. The a!.aulatioos are built up forvard .a weU aa backward
fro. the l .. t .... ured value.
A leoerd coostitutive equatioo for nonllnear creep of concrete. which
is aportaot at hllh .ere .. levels as vell as certaio strab histories at
low stress, suat ",,,o.paas botb th. lio.ar creep fonoulaUDD and the fo ..... -
lation for t18e-independeot DOo1inear triaxial behavi~r. For this reasoo.
it has been atte.pted to extend tbe endochronic theory to ooDlinnr creep.
This required usioS a differeot lntrinsic tae for each unit of a Maxwell
chaio [47], lout otherwise very few additional par ... te,,; aee Fi,. A4.l.
The endochrooic theory autouttcaUy yielda not only creep iDcr ••• e at hiah
atreal but also the Itlffeninl due to 10nl-tiDe C ... pr.sstoo of s=all value.
A .tapler 80del which i. applicable for noolioear un1&aial creep caD
be obtained by leoerall.iol the iotelral-type creep law based on the prlO
"lple of superpositloo. The prevloul works (especially io au •• ian litera
ture) atte"'pted to leneralize tbe intellral ~xpre.s1 ... for total .traio t (t).
Hovever, any physically conceivable .echanls. of nonllnur creep increase
Wit be expected to be weak. 1n .emory. and so • saple additioD of a visco-
iii Q. .. .. Q.
ID '2
c:
~ ~ I-\1\
\1\ ;:) 0 l.J <= :! <= :! ~ + Q. !<J ~ ~
-CO
-:::;
L', I
.. I ;
A4.6
_ ..... ,' .. 1_ , ...... ; ..... :='
• -.... to.'" r.a.z.l ... t-'
•
,-t. TIM£ FROM I..OADING IN DAYS
fi,. A4.2 St.u!atiDQ of Stocha.tic Proces. for CoDcrete Creep (after ~1Dlar et a1. 146J>, eo.pare4 with Uru' Teat Data 162] (1969) for Dvorahak Du.
50
10
-.-
z , ;'10-'
A4.7
........
• .,'tI' ,- ..... LMUt(·,._ ._-.,-~ ... .,-
• fl,. A4.3 Eodoch:onlc Theory flts of ~llQ.ar Creep Deta by
Ka.111an (61J (1959) (after (47J).04 by KUsch (63)
51
A4.8
pla.tic flow t.ra ae ... to be quite .ppropriate. Thus. a ~el based OD
tbe equatloo
. .im.t~~ . e(t) - I(t) + 1 at I+i("t') + 1(0 - Clh(t) ••••••••••••
• (A4.S)
has bean tried (19J aDd .ppe.red to .ive surpri.1D.ly .ood •• r .... llt vith
t •• t d.t.. a.r. J(t.t·) - 1in .. r creep fUDctiOD - linear atr.in .t tt.e t
c.u.ed by .ue.. 0 - 1 .pp11ed .t t'. The iacr ... e of cre.p .t hip .tre ••
18 aodeled by Che la.t cera. fa which .. - state ".riable that corresponds
to kiD_tic harcl.Dina iD pla.Ucity ...... t(t) clec.Yina functillll .biaa tha
a.e .ffect. The iDtear.1 t.ra ..... 1bit •• DODli .... rity thet i. oppo.it. to
that at hi,h .tre •• , o ... ly •• cI.cr .... rather t .... incr .... of cr •• p.clue
to .raclus1 .dapt.UOD iDto 101l.-tt.e co.pre .. illll of _U _pitud.. The
.rovth of .tau uriebl •••• Dd • 18 clefiDed iD t.ras of .1IIp1. D0811_
flr.C-orcl.r cliff.rantiel r.l.tiODs (19). Acc.l.r.tiOll of .,iaa c.u .... by
.. st.iDed lov coapr ... illll i •• lao accOUDted for iD &V.luetin. J(t.t') for
Eq. (A •• 5). The con •• quaDt iDterpla, of cre.p sofcanina clue to the last t.ra
in Eq. <A •• 5) (hl.b-.tr •• s DODlinearity) .Dd of cr •• p stiff.Dina due to tbe
-..ory t.ra. i. r •• pon.ibl. for the coapltcated r •• pon •• ob.ervecI iD t •• t ••
"'iChout botb of cbe .. DODliaeariU •• it i. po .. ible to fit ODly .. lacted
t •• t det ••
M.3 Moistur. Tr ... t.r &lUI 1'herasl Uhcu
Ca1cul.tloo of aoistue. conteDt .nd tap.r.tur. is incIbp ..... ble for.
ratioDal .pproach to cr •• p and .hriDtaa.. Diffu.ion theory i •• pplicabl.
for this purpo ... but tb. probl_ is _liDear b.c.us. it v •• foune! that
peraeabiUt, •• vell as clUfu.bit, cI.cr ..... bout tvuty t1ae ••• pore
relati". hualdlt, dro,. froe 1001 to 60%, d. (n).
At hllb teaperatur •• (.bov. lOO·C), prediction of pore v.ter pre • .ure
i. of .cut. concera ( ••••• for sspl0.iv. spallin •• nd nuclear .ccident aDal
yd.). Rather iDt.r.nina phen .. eaa .r. founcl to occur in tbb ranle (48).
The dependence of psraub11ity on pore pr.ssure se... ~o vanish .nd petllU
b11icy ..... ibit •• n upvarcl juap .bout 20o-Ciae •• s lOO'C is p.ssed. Thi.
1.4.9 S3
vas cb1efly deduced f rOD tb. dryiD. t •• t. .t
Phydcal ."pl.uUon of thi h 1I0rtbv .. tem Vul".,..tcy (481 • • p en_on can b •• tv b
cheIl8·. in the n.cks ill aol.t .f. - Y poarul.tiD. C.rt.ill ut. _raU""
preaaure .Dd ao!atur. cOlltent _ b L_ pa....... The relat10ll of pore 111 y • ..... ed l1li theraod--<-c.p ary "C.r. pro"ided thet ,-.0 proparu •• of
d OIIe po.tulat •• • (.) < __ ue to d.hydr.uOIl of C_t "·.D _r .... ot pore .pac
pa.c ••• Dd Cb) u 1nc • able to free (capill.ry) vat.r cau.ed b r •••• of pore .pace ."ail-u.a y pra.aura iacr.. "The
.ource due to liberaUon of ch_tcaU " ••• cl18tributed cons1der.d ill the .... con y bald .. t.r hydrate vat.r 8U.t be
senaUoa ",uUion Dad di •• rea .nd .orption isoth.rae sh • uciaa the per"'bility
vith .".il.bl ..... ur ... t OVD 1a Pta. ".3 ... ti.factory •• r .... Dt
L • 1
D • l1li "-ted Cllllcr be at. • beaD .chi."ed (481.
• . :-r::::r.;: ~ .. . . ........... , ..... .... ..
'.1 .......... ..: .. .. - ..
.............. --... , .. UJ
Hoi.ture 'er. •• bllit cI Char.cteristic of C y aD Sorption laotheraa C.tter (481). ODerete .t IItlh T"perature
..
A4.10 S4
CONCUJD llIC IIlltAIlKS Aa is clear frOll the preced1l1, .urv.y. a rea11.tic and &ecurate de.-
criptioD of the 1I1el •• tic properti •• of CODcret. nece •• it.ted theories that cut .cro •• "0, diacipliae.. Thu •• .o.t .i~1ficant future .dvance • .u.t be ezpeeted frOll an iDterdiacipltDary approacb.
no. i ..... U.bl. tbeor.tical ... tur. of v.riou. qu •• UOIl. often reapt. ODe to eRP,. 111 reaearch for the aake of tbeor,. To ait1&.te this wid.spread .. lais •• it i •• ther.for •• of pera.ount 1JIportance to keep io a1ad that .U tbeoretical developeellu .ust he checked .,.iD.t esper wental dat •• aDd not just a fev cOllveni_t1y .. leeted fllYClr.b1. d.t •• but .U re1ev.nt data av.il.b1e in the 1it.ratur.. Vitb the advent of cOllput.r opt1aiz.tion technique. for dat. fittiDl. this ta.k ha •• fortunat.1,. been ,reat1, reli.v.d of the t.diou.n ••• it ODC. 1I19Olved. no. pool of te.t data av.ilabl. in tb. 11teratur. is v •• t and .u.t b. tapped. even tbou'" .. ny tuta b ... not be.n .s w.ll doc_ted aDd •• clo •• l, coatrolled •• OD. ailbt have d .. ired.
ACIQIO\ILIDQGHtS Mo.t of tbe researcb covered iD thi. surv., has been carried out .t IIort ...... t.rn UniY.raity with th. support of U.S. latiooal Sci.nc. Foundation (uDd.r Cr.nts CIt-26030. DIG 75-14848. DIG 75-14848 A01 aDd DIG 77-0676~~. P.rt of tb. re ... rcb surv.,ed ... also supported b, U. S. Ioer., Ie.ear and Developll8Dt AdaiDietr.tlon under contr.ct with UniOll Carbid. Corp. throulh oak W,. Jatiooa1Lahor.tor,. aDd anoth.r pert of tbe vork wa. carri.d out ""U. tb. writ.r va. vi.itina aci.ntiat .t Swedi.b Ce .... t aDd Concr.t. Institute (CIl) .t lD,al In.Utut. of TechnololY. Sweden. no. writer i. thankful to Pe1ict.nico di Milano for provid1l11 the .t18Ulu. for thie vork •• nd particul.rly to Or. LuiSi Cedo1iD of Pol1tecnico
for .. n, h.1pful sulle.tiOll ••
M.ll
UFERENCES
1. Malv.rn. L. I .• "Iotroductlon to the. Mechanic. of a Continuou. Kedi ...... Prentice-llall. InSlevood Clifb. N. J. 1969. 2. Coon. K. D •• and Evan •• R. J .• "Incr_ntal Constitutive Law. and noe1r As.oci.ted 'ailure Criteria with Application to Pl. in COncr.t .... International Journ.l of Solid. and Structure •• Vol. 8. 1972. pp. 1169-1183.
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V. C. SCIINOBRICH
PUT ][
Film (l(IIEIT 1I1L1SIS Of IEllrORCEI COIICRETE mUCTUI(S
by
A. C. SCORDELIS
Professor of Civil Enaineering University of Illinois Urbana. Illinois 61801
Professor of Civil Enaineering University of California Berkeley. California 94720
Teble of Contents
CHAPTER B1 FINITE ELL~~, MODELING OF REI~rORCED CONCRETE
STRllCTURES
by A.C.Scordelis
CHAPTER 112 THE SOLL,IO/I OF THE liO!'iLlliEAR EQUATIONS
by V.C.Schnobrich
CHAPTER Bl FINITE ELEMENT STUDY OF REINFORCED CONCRETE
IIU.'1S VITH DIACO)lAL TENSION CRACItS
by •• C.Scorde1is
CHAPTER B4 SLABS PP.aIlLOtS
by V.C.Schnobrich
CHAPTER C1 P .... 'iEL A!ID lIALL PROBLEMS
by W.C.Schnobrich
CHAPTER C2 ANALYSIS OF SHELLS
by V.C.Schnobrich
CHAPTER Cl NOHL lNEAR .... 'iAL YS IS OF REI!>'FORCED .... 'fD
PRESTRESSED CONCRnE IIU.'tS AND FRA.~S by A.C.Scordelis
ClIAPtER C4 .... 'IIALYSIS OF CURVED SEGl!Elo,ALLY ERECTED
PRESTRESSED CONCRETE BRIDGES
by A.C.Scorde1is
61
115
129
155
177
211
245
295
61
C I NQUANTE!'lAR I 0 ~El CORSO D I PERFEZ I ONA~IENTO PER lE
COSTRutlONI IN CEMENTO ARMATO "FRATElll PESENTI" (1928-1918)
ANAL lSI DEllE STRUTTURE IN CEME!'lTO ARIIATO
IlEDIA~TE IL METenO CEGll ElEME~TI F'~ITI
• SE'lIHAIlIO 20-23 GIUG)iO 19i8 ..
... rtedi ~O Giu9no IOrS:
9.00 -
9,30 -
11,30 -
14.30 -
16,30 -
9.30
11,;)()
13,00
10,00
18,00
Prolusion. del Cirettore del Corso (Pr-of. A. D.i Pol i)
I ncr ... nta I Pla.ticity (Prof. z..p. Bazant)
PI •• tic F'racturing .. teri.r Theory (Prof. Z.P. a.:ant)
Endochron ic Theory (Prof. Z.P. aa:ant)
C4MI~i.on 0' Enclochronic and Incr ... ntal Plastic F'oreulrtions (Prof. !.P. 9.%.n~)
Meccoledl :1 iiuono 10~q:
9,00 - 10,30 Tensil. Cr~in9 and ~.t-R.inforc.d Concrete (Prof. I.P. hzant)
11,00 - 12,3,0 Cr •• p Meehani .. , Th.,...dyn .. ics and C~ns~itutiy. Equat io". for Cr •• (Prof. I.P. Se:e"t)
. 14,30 - 16,00 Rat •• Typ. Cr •• p Fo~l.tioft. (Prof. Z.P. S.zantl
16,30 - 18,00 PrllC~ icel C .... A"el )'Sis (Prof. :.P. 9u ..... )
9,00 - 1~,3~
11.00 - 1:,3J
14,3C - 16,CC
16,30 - 15,~C
Finit. Ele.ent Mod. I ing of 'teinforeeo ":onc:rete Struc~ur.s (Fr-of •. ".~. Scord.1 isi
Th. Solution of ~h. ~~nl in •• r E~u.tior.s (Prof. *.;:. Schnobrich)
Fin i te E I_ent St:udy of R. in forced :o"ncrete Be __ with Oiagon.' Tension Crack_ (ProF. A.:. Scordel is)
Slabs Probl ... (Pr-of. w.e. Schnobrich)
Y,n.rdl ~i ';iugno lO~S:
9,00 _ 10,.30 Panei and Wall Probl ••• (Prof. W.C. Schnobrich)
11,~ - 12,30
14,30 - 16, CO
16,30 - IS,~
Anal~'-is of Sh.lls (Prof. w.e. Schnobrich)
~onl inear Analysis of Reinforced And P~s~~essed Concre1;e Sea.s and Fr ••• (Pr:.f • ..\.:. Sco,.d~1 is)
Analysis :.f ::Jrved Se~enc.3T Iy 'Erected Pres'Ct"'''essed Concrete Sridges (:'I"'oi. ~.~. 5c:ordel isi