Bafant, Z.P. (1982) "Mathematical models of non-linear behavior and fracture of concrete." in Nonlinear numerical analysis of reinforced concrete, ed. by l.E. Schwer, Proc., ASME Winter Annual Meeting, held in Phoenix, Arizona, pub!. by ASME, New York, pp.1-25.

Nonlinear NUDlerical Analysis of Reinforced Concrete

preSltnred ilt

THE WINTER ANNUAL MEETING OF THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS PHOENIX, ARIZONA NOVEMBER t4·t9, t982

sponsored bv

THE APPLIED MECHANICS DIVISION, ASME THE PRESSURE VESSELS AND PIPING DIVISION, ASME

LEONARD E. SCHWER SRI INTERNATIONAL

THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS

United Engineering Center 345 East 47th Street New York, N. Y. 10017

MATHEMATICAL MODELS OF NONLINEAR BEHAVIOR AND FRACTURE OF CONCRETE

ABSTRAcr

Z. P. Bai ... t. Prol_ of Ci"iI EngiNerift9 and Director Can., lor Concrete end ~te"als

TecllnOlo-;ical Institute NOItll_ltrn UniversitY

ell_ton, Illinois

A review of some non-classical mat~ematical formulations for the consti­tutive behavior and fracture of concret~ is presented. After pointin~ out certain lilllitations of the orthotropic IIIOdels, tne plastic-fracturing theory and the endochronic theory are discussed. Attention is then shifted on internal friction and dilatancy. particularly in relation to the fundamental stability postulates and work inequalities. Some possibilities of lIIodeling frictional phenomena due to oriented cracks are ouclined. and finally the lIIodeling of fracture by means of strain-softening constitutive relations is examined. Some typical cOlllparisona with test d.Ha are given.

INTRODucrION

Concrete. the IIIOst widely used material both in terms of tonnage and costs, is characterized by an extremely complicated microstructure which endows it with an equally complicated behavior. The microstructure consists of a random dense array of irre~llar large inclusions of a hard mineral, embedded within a softer and weaker matrix of lIIortar. which itself consist. of small hard in­clusions embedded within a still softer and weaker matrix of hardened portland cement paste. The bond between the.e inclusions (the pieces of aggregate and sand grains) and the matrix is relatively weak causing that microcracks begin to forlll at relatively small stres.e •• and tnat microcrack formation is the ~hief source of inelastic behavior, prevailing over plaatic deCorftations.

In the present lecture, an attempt is made to review and dummarize some mathematical models for the nonlinear inelastic behavior and fracture of concrete which were recently developed at Northwestern University. The literature on this subject has become va.t. and the reader must be warned that no attelllpt is being made for a cOlllplete and balanced presentation of the state-of-the-art. For this purpose. some other works have recently become 3vailable (1).

ORTHOTROPIC HODELS

It may per hap. be of intereat to begin by pointing out the 1imitationa of the orthotropic models --an approach which ha. been most popular and moat widely utilized in finite element progr.... In theae modela, one asaumea the incre­mental atreaa-atrain rel.tion to be line.r, ch.racterized by a t.ngential stiff­neaa or compliance .. trix th.t is of orthotroplc form, with zero coefficienta relating tbe normal strain incrementa to the shear streaa incrementa. Thia simplification has the adv.ntage that the variation of tangential elastic moduli or compliancea can be to some extent figured out by direct intuitive reaaoning. without the aid of more aophiaticated concepts such as inelaatic potentiala. flow rule and normality, stability postulates, etc.

An orthotropic form of the t.ngential .tiffnesa or compliance matrix ia obviously correct only.if the principal directions of streas and strain do not rotate. Problems arise in general loading situations in which the.e directiona rotate. The orthotroplc models can then be applied in two different waya:

1. Either the coordin.te .xea to which the .. terial orthotropic properti.a are referred are kept fixed with regard to the material during tha deformation process (.lthough .rbitrary coordinatea may of course be introduced before deformation begin.).

2. Or the axea of orthotropy are rotated during the deformation proceaa (fra. one loading atep to the next) 80 that they ramain parallel to the principal stresa directions at the belinning of each load atep.

Conaider tbe .econd method. which ia probably uaed moat often. The non­linearlty of the aaterial reaponae i. due to formation of certain defecta (e ••• , aicrocracka aa we conaider thea here, or alipa) by the previoua deforma­tion hiatory.

The rotation of the axe. of orthotropy .. ana that ve rotat~ the.c defect. agaluat the materlal. Thia ia of ~our.e phy.ically imposaible. and would be ac,eptable only if the defccta were either cauaed ao1ely by the current atrea. atate (and not by the previo~ deformational or if they had no orientad char­acter (e.I •• round void. rather than planar mlcrocracka). Such.n a •• umpclon ia obvlously unacceptable for &eomaterlala. tn continuum mechanic. of in­ela.tic behavlor it i. a lenerally accepted funda .. ntal fact that the material axe. muat remain fixed witb regard to the material after the deformation begina. The .ymmetry propertie •• auch aa 1.otropy, are characterized by the fact that any coordiDate axe. may be cbo.en before the deformation begina, and the obtained re.ponaea auat be equivalent. Thi. however does not permit rotating the coordiDate axe. after the material baa already been deformed.

Another probl .. with rotating the axea of material orthotropy appear. when we con.ider tbe principal direction. of atraln. Theae rotace alao. but ln ,eneral, accordin, to the orthotroplc .odel. a. well a. in reality, their rOLatioaa ara not tbe .... aa thoa. of the prlncipal Jirectiona of atre ••• Therefore, the prlncipel direction. of atr ••• and of .train cea.e to be parallel. Sbould thea the aaterial orthotropy axe. be oriented parallel to the princlpal .tre •• directlona or to the prlnclpal atrain directiona? Evidently, tbey cannot be parallel to both.

2

Conaider now the first method in which the material properties are de~ibed in terma of coordinate axes that are attached to the material aa it deforma, and consider the constitutive relations

dOij • DijkD (~) d~kD (1)

in which the tangential moduli tensor Di1km is a function of the stress tensor a.

For i.otropic (precis.ly, initially isotropic) .. terials, it must be possible to apply thes. relations for arbitrsrily chosen coordinate axes snd get equivalent results. Theref~re, thes. incremental constitutive relations must be form­invariant under any transformation on the coordinate axes. This form-invariance condition is .xpr •••• d .. thematically In the form (see e.g., Malvern (2 ).

(2)

in which a ij • cos(xi. xj > • the transfo~tion tensor containing the directional

cosine. of the rotated axes xi with respect to the original axes xj

• and ~. is

the str ••• ten.or in the rotat.d system of coordinate axes xi. obtained by the

ten.orial tranaformation of stress tensor ~ in the original axes Xi:

a~v • aug avh agh • It can b. demonstrated (1) that t~ foregoing condition is

always violated by the orthotropic incremantal .lastic models when they are applied using the first method. It can be also shovn (3) that the lack of objectivity due to the lack of tensorial invariance can ~ause serious dis­cr.pancie •• i •••• the pr.dictions obtained when working with different coordinate systems can gr.atly differ. sometimes as much as 50%. (Note that the fact that after coordinat. tran.formation the orthotropic matrix y~elds a ~~trix with non-zero coefficients relating normal ~trains and shear stresses is immaterial: this i. because the orthotropic symmetry prop.rties do not change with the coordinat. transformation and apply after transformation with respect to the rotated axes.) .

Th. usa of a relation between total scresses and total strains does not suffer with the limitations of the orthotropic models. In fact. differentia­tion. of such a .tress-strain relation yield. an incremental stiffness or com­pliance matrix which is. in general. fully populated. and is not of orthotropic form (with z.ros in the elements connecting .hear strains and normal stresses). This observation also shows that the assumption of irreversible defects and p.th-independ.nc. do.s not provide a justificstion for the orthotropic models, sinc. it lead. to the total deformation theory which is incompatible with these mod.ls.

The recent popularity of orthotropic stress-.train relations hss no doubt been caused by the recent .xagg.rated eaphasis on the cubic triaxial test •• as opposed to the classical cylindrical criaxial t.st.. In cubic specimens. the principal stress axea cannot b. made co rotate during the loading process. and by virtu. of symmetry the principal directions of stress and strain are forced to coincide. The cubic triaxial tasts have of course the advsntage that they can reveal the affect of the medium prinCipal atress. In many situations this aeem. to be however lesa important than the knowledge or the effect of rotation of principal streas directiona and the eff.ct of non-coincidence of the principal stress and strain directions (the lack of coaxiality).

To b. able to obs.rve the affect of rotating principal str.ss direction •• other type. of tests will be needed. One such type of teat is a classical teat in vhich a hollow cylind.r 1s .ubjected to axial load, lateral external and internal pressure, and torsion. In chi. test one can induce any combination of

3

0) Plastic b) Fraeturinq c) Plastic - Fracturinq

~

• Flg 1. Idealized Unloading Behavior Imagined

for Plaat1c-fraeturing Theory

- "-............. n....y -----."........,....--. -..

I I

I I I I I I , , , , , ,

\

, , , ,

, , , ,

, , ",ill ,

Fig. 4 Example of a Spring-Loaded Frictional .lock

(0 )

-cr

Cb)

..

a_.L_.S_ ", .

,~ ... s ... I' ..... '.',

l hi •• a.s .... , ••

cr,

Flg. 2

Failure en­"elopes ob­tained f rOfJ

plastic frac­turing theory. the load1ng surfaces of ..,hleh do not involve third invar1anta of IOtreaa and "tra1n

Flg. 3. Stalrea •• Loading Patb

-cr

Flg. S

Planic Strain I acrelDeat Rela­tive to Load­ing Surfae.

pr1Dcip.l stresses, and moreover one can make tb~ principal 8tr~ss directions have any.ngle w1th the spec1Dwn axis Qnd rot'I[l!. tdtht.·r cnntlnll,,"~ty 01" abruptly. dur1n& the luatilng. process. Mucb more attention shQuld be given to thili type of t.at in the furllfo

_. °

PLASTIC-FRACl'URING THEORY

Since the major source of inelastic behavior in concrete is mi~rocr3ck1ng rather than pl.stic slip, applications of the theory of pla~tlcity co concrete do DOt .pp.ar to be physically justified. Otber theories. wh1ch 1n ~c~ way take the phenomenon of .1eroeracking into account are th~refore appropriate. One auch theory 1& the plastic-fracturing theory. This theory represents an extenalO11 of cla •• ical incremental p13st1city (4) which adheres to the use of loading surface. and flow rule based on these surfaces. and introduces, 1n addition to pia. tic .train increments. the fracturing stre •• relaxation~ due to 1Ilicrofracturin& (Illicrocracking). The exten.10n to atodl.'l the incl .. stic phenomena dua to =teroeraeking appears to be essential for materials such 4S concrete 4H .. 11 •• rocka (5-6).

The nature of the tbeory .. y be illustrated by Fig. I. in which pl~~tieity 1s seen to b. char.e~riz.d by an elastic increment followed by a horizontal pla.tic scrain increment. A material which undergoes only micro(racturing and no plastic deformation. first studied oy Dougill (?~~ ). 1s shown in Fib. Ib where the elastic increment is followed by a vertical fracturing stress decre­ment. While pl •• ticity obviously does not allow strain-softening. i.e •• decline of str.s. at incre.aing strain. the fracturing theory do~.. To be able to distinguish strain softening fra. unloading, tn. unloading surfac~s for the fracturina theory muSt be considered as functions of strains rath~r than stre •••• , .. vaa first doae by Ooug111 (1=1 ). who also introduced the nor­malit, r~e in tbe strain space to obtain fracturing 9tre98 rela~tions. The plaatic-fT&cturing theory Is a comblna~ion of plastic and fracturing theories and is illustrated in Fig. Ie. where t:~e elastic increment is follo ... ed by a horizontal plastic strain increment 3n,l then by a vertical fracturing stre •• decrement. Obviously, the strain soft~ning 1. al.~ allowed 1n this theory. The pla.tic s-C'-ain increments and fracturL1& stress decrements are then obtained on the baai. of separate loading surface» in the .tre~s and stra1n spaccH (~.

The characterist1c property wh1ch allows us to distinguish betveen plastic an4 fractur1na phenomena i. the unloauing slope. Th~ plastic ph~nomcna do not lead to any chang. in the unloading stiffnesa. while the rracturing phenome~ are totally related to a cbange in the unloading stiffness. and in case of pure fractur1na behaVior they preserve total reversibilIty at complete unloading, as indicated by the fact that the unloading slope in Fig. lb shoots to the origin. In pl •• tic-fracturing .. terials (Fig. Ie) the unload1ng slope decreases but doe. not point to the origin. If the unloading slop~ 1s known for each point of the load1na diagram. it is possible to uniquel, scparate the plastic and fracturin& effects (.6.).

The pla.tie-fracturing theory le.J.ds Co th~ (ollowlng incrt!mt!'nt31 stress­Itra1n relacions ( 5):

do - )Kdc - 2K B du - 1 adK

5

(3)

(4)

in which

dK • RI(dy + o'd£). d~· H2 (C.kad£km + TK8'd£kk)

Here. C • elaatic .h •• r modulu •• K • el •• tlc bulk moGulu. (both moduli are variable); •• devi.tor of .tre •• t.neor 01j' e

1j • devi.tor of .crain teneor

£ij; ° • "~&D (hydro.t.tic) .tr •••• £ - me.n (volumetric) .traln· ckk/3;

(5)

~. K • parameter. for pla.tic and fracturinl deformation; T •• tre •• inteneity.

y •• train inteoaity • (eljeij/2)~; S'. a' • pla.tic and fr.cturinl intern.l

friction coeff1cieut.; B. a • pl •• tic and fracturinl dil.t.ncy f.ctor. aivina the r.tio of volumetric to d.vi'toric in.la.tic incr.ment.; and Kl • H2 •

hArdenina and .oft.nina .tlffn...... Th. inel •• tic r •• ponae i. here totally chAracceri&.d by .1x coefflcient., Hl , HZ' B', a', ~, o. the dependence of

wh1ch on tb. invar1&Dta of .tres. and .train _t b. deteralned frOAl ezper1meDt. V.rioue oth.r coneideration. .re Deeded for thi. purpo •• and .uit.ble functiooa for tbt •• coefficient. have been Identlfied, leadinl to • r~ther clo.e aaree­"Dt with a bro.d ranae of experi .. Dtal d.t •• v.ilable in the liter.ture (.ee llef. S ).

ID cooatraat to iDcrement.l pla.ticity. the con.titutive equation. of pl •• t1c fracturin I theory (Eq.. 3-5 ) involve not only terma which depend on .tre •• (the tera with 'ij in Eq.3 >, but al.o terma whlch depend on .tr.in •• uch

a. tbe tara iDYOlviDI '1j iD !q.3. The.e terma appe.r to be very helpful In

repre •• Dt1D& .. t.ri.l beh.vior iD .train-.oftenina realme., which i, due .imply to the f.ct that e lj incr ..... during .tr.in-.oftening while 'ij de-

cr...... The ••• tr.in-d.pendcDt terma ,ive r.th.r different l.teral .tra1na and thue .llow ua to model the larae volume dil.t.ncy during .train-.oftenina.

Another e.renti.l difference fro. pla.ticity i. that the .h.ar and bulk .la.tic moduli ar. variabl.. their decr.a.e i. tied to the arovth of fracturing parameter K, for which the follov1Da equation. vere obtained ~):

dC • ~ dX • - ~ ~ (6) zY 9 c

Rote that Eq. 3 applie. only to lo&diol; for unloading modified expreaaiooa .uat be introduc.d. In analogy to pl.aticlty. one .laht •• t dK • d~ • O. but th.D DO repre.eDtation of in.l •• tic behavior on unloadina. reloadlnl and cycllc loadlna would b. po •• ible. It la po •• ibl •• how.ver to formulate rule. which allow for DOD&erO ~ AD~ d~ durins unloadin& and cyclic loading (1). The •• rul •• conal.t in the eo-called juap-kine .. tic hard.nina. in which the ceDtar of the loadiDI .urfacs 1. juap.d to the l •• t extre .. atre •• or .tralo POiDt wheoaver loadin& 1& r.v.reed to ""lo.din, or vice veraa. Thr .. _a, 10ad1na-unloadlnc-reloadlna cri,.ria ar. needed for thi. purpo •• ~).

The aoat t.portan' adv.nta,. of the plaat1c-fr.cturing theory 1. tb. fact tbat the .tre •• -.tr.in relation.hip. (Eqa. 3-5) can be brought "(for loadina) to AD iDcr ... Dtally linear fora:

or (7)

6

1D ~ch Cijkm or £ represent the t.n.or.or matrix of the tangential moduli

vb1ch .r. functions of the invariaat. of the .'ress tensor ~ .nd .,rain tensor

~; .. e laf. 5. The use of ,angenti.l moduli Is the mos' .ffective approach in

.t.p-oby-s,ep fiDite element analy.is.

Observe that the matrix of tangential moduli i. non-.ymmetric. This 1s a consequence of internal friction and is inevit.ble for close represent.tion of experiment.l data on the mat.rial behavior. This i. cert.inly .n inconveni.nca for DUlllerical finite ele_nt .naly.ia. Kowever, it muat be _phash.d thet thi. tytle of non __ J1IIIMItry does not cause _tertal inst.bility .s long •• the inequality ,iven below in Eq. (12) 1. aatiaf1ed. It should be also noted that tangential moduli Cijkm have a generally anisotropic (non-isotropic and non-

orthotropic) fora. which Is • manifeatation of the .tre .... nd atrain-induced aninropy.

The stresa-straia r.lations in !q.. 3-5 vere derived by u.ing loadinl .urface. that do not involve the third invariant. of stre •• and .train. It i., however. intere.ting that the failure envelope. obtained fr~m these rel~eions (by recordinl the peaka of the stra •• -strain diagr.m. run at various fixed ratio. of of .tre •• cQlaPoaents) have a ehape thae is non-circular on the octahedr.l plane (w - projection). This is illustrated by the diagraa in Fig.(2) which va. con.tructed fraa the numerical value. of the coefficients of the pl •• tic-fracturinl tbeory a. liven 1n R.f. (~. This shove that the round.d trlanaular shap. of the octahedral .ection does not n.c ••• arily imply the in­fluence of the third .tre •• invariant. Thi •• h.pe c.n be equally vell explained' by the .imultaneoua influence of .tre •• and .tr.in on the failure Burfac •• nd the fact thet the atrains do not incre •• e proportionally to .tr ••• eB.

The plaaeic-fracturing theory has been ehoWD to be c.pable of representing a very broad range of inelastic ph.nomena. The.e includ.: Str.in-Boftening, inel •• tic dilat.ncy due to shear .nd internal friction, a. manifested by the gre.t .ffect of hydro.tatic .trength in tri.xial tests, the incre.s. of volume 1n pre-peak as vell a. po.t-p •• k deformation, l.teral strain., the incr •••• of appar.nt Poi •• on r.tio. hy.teretic loo~. during cyclic loading to small •• well a. high .trenlth l.vel., etc. (dat. from Refs. ·1-11 fitted 1n Ref. 1)·

ENDOCKIONIC THEORY

Another theory wbich ha. met with con.id.rable success 1n modeling the ob •• rved material behavior is the endochronic theory. Th. general framework of this theory v •• developed by V.lanis (1R). although an elementary prototype of chia approach hed been sugge.ted earlier (11). The central concdpt in this theory i. that of the intrinaic time, vhich i •• variable wh1ch depend. on the length of the path trac.d by the .tates of the material in the strain space. Th. original V.lini. definition of the intrinsic time, z, vhich vas us.d for concrete, i.:

dz • F1(z, ~, ~) dt, dt • #e1J

de1J

(8)

in vhich F ia a function of the stress .nd strain invariants and model. the hardeninl Ar .oftening of the material during the evolution of inelaatic atr.in. The in.lastic .train increment •• re as.umed to be proportional to dz and the constitutiv. relation of .ndochronic th.ory has the following form (26.27):

• d·1i ~ de1j 2G + 2G dz, de • da + dA

JK

7

(9)

H.r. an additional term, called inelastic dilatancy; d~, is introduced to model the In.laatlc dilatancy due to ahear. This term 1s al~o related to intrinsic time:

It can be ahoyo that the endochronic theory is a special case of visco­plaaticity in whlch the viscosity coefficients depend not only on atreas and atrain but alao on the atraln rate (~).

(10)

Th. chief advantage of the endochronlc theory, first recognized by Valania, 1. the fact tn.t It la capable of representing the unloading lrr evera1bll1ty, the aall.nt feature of inelastlc behavlor, without the use of any inequalitiea (unload!n& criteria). Thia makea the endochronlc theory extremely effective for cyclic loading. Furthermore, the fact that all 1nelastic strains are tied to one time-like variable dz makes it easy to control the st1ff~ning or soft­ening oi the material by changing the rate of growth of the intrinsic time. aue aap.ct which ia particularly eaaily modeled by the theory is the inelastic volume dilatancy dA.

Th. mo.t 8ignificant difference of the endochronic theory from cla88ical pla.ticity a. well aa pla8tlc-fracturlng theory 1a the fact that, even for loading.it cannot be reduced to an incrementally linear form given by £q. (7). N.v.rtb.l •••• In the vlcinity of any apecified loading direction the endochronic theory can be linearized. 1.e •• brought Into an incre~ntally linear form (JJD. Howaver. the tangential moduli Cijkm of this linearized form are not conatant

and depend on the choaen direction in the vicinity of which the behaVior i. linearized.

It .. y seem that the lack of incrementally linear formulation ~ould .cauae 81gn1f1cant difficulties in numerical computation. However, finite element programs utilizing endochronic theory have been written (e.g •• 19-23), m\d no particular conv.rgence difficulties have been encountered.

Endochronic theori •• have recently been criticized from the viewpoint of .. terlal atability and uniqueness of respons* (24-~. Subsequently, it was ahown. however. that ,the theory can b. either modified to satisfy these require­ments. for .xample by the use of jump-k1neaatic hardentng and by unload Ing- re­loadina criterla (18.27). or that the atrong uniquenesa requirements are them­aelve. in queation. For example. Rivlin (~ pointed out that vhen one considers a .taircaa. loading hiatory in the .train .pace (Fig. 3) and when one leta the 8i:e of the ataira .hrink to zero and their number goes to infinity, the limiting behavior does not approach that for the smooth loading path. It is, however. po.aible to formulate a refined definition of intrinsic time (intrin.ic tlme with finite reaolution) for which the limit coincidea (!), although at the same time the need for doing tbi8 may be questioned because tile stairea •• loading path mi,bt Dot cause the same type of daDage to the materlal .s doea the smooth loading path.

The comp.riaon. with available t.at data have been very 8ucceaaful. about equally good with the endocbronlc theory and the plastic-fracturing theory (11)· au. alabt thus wonder Vby two rather different tbeories allow representation of tbe .... phenomena. The answer ia that our inforaation on the .. terial behavior la far fro. complete and is inaufficient to completely deCine the mathematicsl formulation. Therefore, certain loglcal a •• uaptiona ~.t inevitabl, be used and the reaulting formulatlon alao de?enda on theae. It .ppear. that the =oat 8ignificant difference between Various th.oriea of inel.stic behavior ls ob­tained when a proportional loading i. followed by audden load incrementa to the

8

side of the previous loading path; e.g., when an incre •• e of normal stress ia followed by a sudden sbear stress increment. The different respon.es for such loading may be graphically illustrated in terma of the inelastic stiffness locua (18). It is unfortunate that for such loading path. measurements are most dilticult and the present experimental information is rather scant. How­ever, improvement of our knowledge for this type of loading is important be­cause the loading "to the side" is chAracteriatic of failures due to _terial instability.

INTEltNAL FRICTION ANt) DILATANCY

An important aspect of the physical .. chani.m of inelaatic responae of concrete ia, no doubt, the frictional slip of the surfaces of closed micro­cracke. Friction causes a major co.,lication in constitutive modeling since it leada to violation of the basic stability postulate, namely Drucker'. postulate (38,~, which s.=ves as the basis for the flow rule (normality rule). The postulate is given by the inequality

in which dW representa the second-order work diaaipated during a cycle of

applying and removing stre •• increments doij,and dti~l are the increments of

pla.tic strains. This postulate is known to repreaent a sufficient but not necessary condition for local stability of the _terial (27s, 28, ~. ~). It is important to realize thAt its violation does not necessarily i~ply in­stability. In fact, certain condition. under which this postulate may be violated yet stability is still guaranteed has been recently formulated (!) extending previous analysia by Kandel (~. The following more general in­equality which guarantees stability of the _terial has been found (~):

dW - X dW f > 0

(11)

(12)

in which X is a parameter which can have any value between 0 and 1. Drucker's postulate ia obtained for X • O. The quantity dW f represents a second-order energy expression defined as

* • 8' - 8 - * dWf 2C do(dt + 8 do)

which _y be called the frictionally blocked elastic energy. The materlal parameters in Eq. 11 are defined as

where

F(o, • t

T, yP ) • 0

(13)

(14)

(15)

representa the loading function of the _terial. The variable a represents the mean (hydrostatic) stress, t is the sfress intensity (square root of the devia­tor of the strain tensor t ij ), and yP is the length of the path traced ln the

space of plastic strains, which is used as a hardening parameter.

The meaning of the material parameters in Eq. 14 may be 11lu5trated taking recourse to Handell's example (~ (Fig. 4). A frictional block. resting on a rough horizontal surface 1s considered loaded by • vertical force simulating

9

~ !ti.;i·;: ~~h i:;:l~l .. b .. .:: ':.Pt"'! ~-t "~".f" h.:! hh .3 ~'t ..... " .. .8.:;" •• -<0

I ... ~ ~":~.,'11 .. ~ ....... "1· ~"" .. ~ ." ~ j. "!l .. -: ",l~-'::" ",.,.. ~ .... ~ . " ....... -" •• ' " !" " ...... ... .. ~ .. ~,. ......... ~"

.. 4"ll ~ -;:~.!IS 0 .11 ,,"0":" ""l~ z·..,: .. 'll"j , ... .., ~ . " , i·' .. .. .. " ....... .. ~: .. ::=:"1 ..., ... '" 0,--," •

.... ~.. • u

'I" .,"; . . .. .. " ,_1_ ... . "

, I" ....... 3 ";!';$ ~J'~ .. .. • .. ... ~ ... 11 "''$ .... ,d"l!" lId .... h .. j ~~ ...... -a.::: I "J;-!!."" ." "j" .-~.J"'!\ "'"I' :.: ..... ~.!" S'!""l, ", l~ , . .:12" .. ~ .. S'!l /d!. !l' .'jH,~ :S:;.¥ .. ;:Hh~ BU' iJ~.HH

11j:t j i,i.:: i1 ~.U~,:!S3 '" .... ",~~.::. ":: eU·al~~~';·!\ .. ::.--~"-".~ -::!.!d~, ... "~.'\h .... .. ~ u .. ~'" ~1""'o"1"" .. ... .... .. "'ftj1~·{(iha ~ ~ ... ., nu ,:d ij!51~~H~~j§ S~:: ::~!l!H"'! "-J"J .... 1'~ '1~\!'d.i:ll;: :I ...... !I ..... ~ ~:; g§l~:iIM~J~ . ::lliir!!.il~! "'~:.,!j·~~·r=·tl! m'!ll"!jli j ! .... j "'1' 1""j"" .:l~~ P.!l; !!~.,;~ ,.,h' 1'1,1' .... ·~~!::l~ :: I.i I!~~ ~ .... • ~'j~ ,.,';J"!: ~ p'l-~ih,~j ,"j ... ~Ii' .. ';'\1

!l ~ I 1.': "!:" 1: :.t-'J"'II""" 5Hi!'t1§ 5 U~

, '"j • B... • :: ~. R ~ ~ UJ:d: • "j!':'l:: ~ .. ~ ~ .. ~"

~i"H!l ~~ .. ,,; ~ ~j!H H~!31 ~3hH' j ,. =, J ~h~l'" ~1"t.l!I· , -, !!lPi.;

.... 1ff" "jhl , , . HUH !~! . .ig el.!l!

mm ~1~1l!',;

n:1:Il¥ ~no::_t

, , ". ,. 'l ... , .. ~ .!:, "0:" :::1 " .. ~~~ .. " ~" ~~"''''u··~I· U!~ ... ~.':l:~f~. ' .. ".,~.'l~ .I'''!" •

I~·~~u.';"·l j'" :H"!h oU

'\! .~~= ~ ... "~! .. ,, ,,~.. ... .~.l" 'I!!~"i·,!"l' ...... " ~~ ~ .:I ·PSBi·du~i;l· .. .,'\ ". I' Eil 1!!::l"l".tl~ .. ... .. .t • .i.~ .. ~1J''': ~ ~i;-,'l=! S ~ t.t U I'll i 0': =2J uu= -!: ::... I' ..... • .. ··· ... .."' ... .a .. .li;~ .. ut~ .p,:! ... :: ....... "~ !, .. w:'-'t

I l,,·£t,·!,l",'!l "', " ..... ~:: . . ..... '"'' .. .s.!: ~.q ,,! PH r=-S.:!

i '{II"!!!I!!'i!' .!' l". ,,11,1 pH. !;;1!;r.p .. s i '!ill]I'!!'!',j" ~ hI! ,'j*·<i'c ~ !iil;II;I!t~i'i1 i "'~!!i,!"'!i" !,~!" ':1', i .. i'l .33~, ... .':1I ... r ." .. ~$~S.f'ol~

;',.' 11 ;'f':~ B . !'I~:" .~~; . ":", ' "-!: -.~~. , ", J' :: u ~

.t"b .. !I" lH P;:;: ! t: 8~, "l'l !'"' , " i!~! t;:l , ' . ! • 'I ~ ~~ '.1 {;l! '" . J1l j,: i'i ,I! 'ili,)" Iil hi li'l i " ! ; i~ H, ili ! .".. .-iH: u:

b'.1 i3~

•

ill which

D·

-1 !: • D

(16)

ay..

(17) .

Here p and ~ represent the stiffn.ss and compliance matrices of the uncracked uurial.

To figure the change a in the foregoing matrices caused by progressive microcracking. it is helpiul to recsll first the stiffness matrix of a fully cracked material. i •••• an elaatic: material which is intersected by continuoualy distributed (ameared) cracka normal to axis z; see e.g •• Suidan and Schnobrich (lU). This atiffness matrix is known to have the form:

o

o (18)

ay.. o

This matrix is derived from the condition tbat tbe stress normal to the cracks ~ust be zero and that the material between the cracka baa the properties of an uncracked elastic .. terial. This l.at a •• umption is. of cours •• a simplifica­tion. since often the material between continuous cracks may be damaged by presence of diacontinuoull microcracka which formed before the continuous cracks were produced.

Comparing the matrices in Eq. 16 and 18. we aee that each element of the stiffne.s matrix ia affected by cracking. Thus. modeling of a continuous transition fromEq. 16 to Eq. 18 would not be easy since each term of the stiffness matrix would have to be considered a function of some parameter of cracking. It appears however (~ that the aituation becomes ~ch simpler 1f one us.s the compliance matrix C. In that case one needs to con~lder only one element of the compliance .atrix to change

11

Fl,. 6

Uniaxlal Stre •• -Straln Relatlon U.ed ln the Pre.ent Non-Llnear Fracture Hodel

c ......

2000.----------------, (e)

tor p .... ·43~0Ib.

1600 o

. 5 1200

::t -... .., c o ...

o

, , ..... .. _-0

0 0.4 0.8 1.2 1.6 ReI. Distance (x/o o)

, 02~~...-o---~~------...-o--------~ Q25~-----------------------------_, Nonlinear Theory-

fl.0

0.15 ......

III 0

fl.E

010

005

00

(0 )

, Po • 19,800 lb. , ,

T

~ i+R' H

1 Ie. A

W 't 0.2 0.4 0.6

Rei. Flaw Length (20/W)

( b)

0.20

.

lIiI.

" Linear Theory , Nau$ (1971) 0

" Po .1~.120 lb. o , ,

·-1

W/2

0 0 0.2 0.4 0.6 Rei. Flaw Len9th(20/W)

Fl,. 7. Comparison Wi~h Max1mum Load Data of Saus (1971) (after l&1&nt and Oh. 1981).

12

£Cu) •

.y..

in which u i. a cer~ain cracking parameter. There are two reasoc. Cor the effact of cracking on the compliance matrix to have this simple form:

1. If we ... ume all microcracks to be perf.ctly .traigh~ and normal to axis z. which is of course a simplification (in reality the microcrack orientation. exhibit a certain atati.tical di.tribution). then formation of the microcrack baa no .ffect on .tr ••••• a and a in the directions parallei to the crack x y plan... Therefore. appearance of =lcrocracks nermal to z .hould have only the affect of increa.ing tha .traln in the z directlon. which i. why. under this a •• umption. only the third diagonal term in Eq. 191hould be affected.

2. The .econd rea. on i. that. in the limit U • O. the inver.e of the complianca matrix in Eq. 19i1 identical to the .tiffne •• matr1x of the fully crackad material in Eq. 18.1 ••••

Dfr • tim £-1 (u) ~

(19)

(20)

Thi. relation (theorem) has been proven mathematically· (!Q) writing the general inver.a of Eq. 19 with the h.lp of partitioned matrice.. It is also physically obviOUS that the diagonal compliance term Cor a fully cracked material should approach inflnity becau.e the mat.riAl has no stiffness 1n the direction normal to the cr.cks and a flnita .tres. therefore produces an infinlte .train.

Th. us. of Eq. 191nstead of Eq. 18 in finite element progra~ing save. programmer'a .ffort •• ince only one element or the elaatic coepllance matrix needa to be changed. In practice one may replace e

33 by a large number

~ (say 10 ), and numerical matrix inver.ion then yields a matrix nearly exactly euqal to Eq. 16.

Th. ca ••• of an uncracked and a fully cracked materi.l are characterized. In ,arma of the cracking parameter, a. followa:

uncrackad: u •

fully cracked: u·· 0

Modeling of a progres.ive davelopment of microcracks now obviously require. prascribing the variation of cracking parameter U between these two limits

(21)

(Eq. 21). The vartation of thia parameter ~y be callbrate~ on the baai. of a uniaxial tenaile teat. From tasta of amall .a=p1es in an extremely stiff teat­ing machine. e.g •• Evana and Marathe, 1968 (!1), it 1s known that the ten.ile atreaa-.train diagr .. exhibita a gradual decrea.a of strain at increasing .train (atrain-softening), the aoftening branch being normally a rev times longer ~han the ri.ing branch. Although this tensile atresa-strain relation appears to be smoothly curved, 1t may be approximated by a bilinear atre~.-strain relation (Fig. 6). In such a casa the cracking parameter i. defined .a

13

-1

" t

-e33 ~ - e33 1:

0-1: . Z

t in whlch e33 i. the downward slope of .the softening portion of the ten.ile

(22)

atreaa-strain dlagram (neg.tive), .nd I: i. the t~n.ile strain 1n uni.xial ten. lIe te.ta .t whlch the stre •• i. reSuced to 0, i.e., continuous cr.cklfo~.

Eq. 22 fo~ the bilinear tenaile .tre •• -atr.in diagram may be easlly re­placed by • more ca.plic.ted equation correspondlng to • curved tenaile atre •• -.tr.1n di.aram. However, it .ppear. th.t the biline.r stre.a-atrain dt.,r •• is sufficient to obtain •• ati.factory agreement with existing test data On fracture. Horeover, the use of a straight-line str.in-softenins diagr.m .voids uncertainty with the point .t which unst.ble str.in localizatio" lead in, to fr.cture .ppe.ra.

The foregoinl stress-str.in ralatlon. involve only the normal stress .nd .train componenta. They Are obviously applic.ble only when the princlp.l .tr •• s directiona do DOt rotate durina the formation of fracture. Even theil, if the load1ng subsequent to fractur. formation produces .hear atr.ina on the crack planes, the atrea.-.train relation must be expanded by the inclusion of .hear terma (lead1nl to a 6 x 6 .tlffne.s or compliance matrix).

Another adjustment of the foregoing .tres.-strain relation i. appropri.te when there .re .ignlfic.nt coapre •• ive normal .tre •• es in the directions p.rallel to the crack pl.ne. In such • case it i. neces.ary to decre.ae the v.lue of the peak stre •• f~ as a function of Ox and or (lLQ).

Finally, note that then there is certainly. similarity between the pre.ent stre •• -atrain relation and the .tre •• -di.placement relation used in other vorks <ll-39).

An •••• nti.l a.pect in .adellng fr.cture by .tr.ln-.oftening stre •• -.train relations is to realize that the width of the crack band front mu.t be con­sidered .. a material property. Thera are very definite theoretical re.san. for chia.

•

1. If the cr.ck band width i. con.idered a. a variable, and the material 1. treated as a continuum, then an an.lysi. oC t~e .traln-localization inst~bility indicate. that the crar.k band would alvay. locali~. to a sharp front of v.niahing width. Thia situation i. however equlvalent to linear fracture mech.nie. aDd CaDDot de.cribe the test data available for rock and concrete.

2. It i. meaningle.. to consider the cr.ck band front to be 1 ••• than a few times the si~. of the heterogeneties in the material, such .. aralo .i:e 10 rock (or a"re,ate siz. 1n concrete). On such ... 11 dimensions one doe. DDt have • continuum, and the fore.oina con~iDuum .adel would be inapplicable.

3. finally, the matheaatical .adel must .ati.fy the conditions of objectivity: which include. the requireaent th.t the r.sulta of analysia auat be independent of analyat's choice cf tbe ... b (except for an inevitable aumerieal error which vaniabu .. tbe _ab is ref:1ncd). It has beell verified numerically that if the vidth of the filllte element over which the crack baed i. a.sumed to be distributed 1s varied, tbe fracture predictions vary too. The only way to achieve coasiateDt results is to assume that the width of the crack band at the fracture front i. fixed •

ief •• 40,41

14

Hodeling of fraccure by meana of seress-strain relations thus leads to a model characterized by three material parameters: the uniaxial tensile strength, f~. the fraccure energy Cf (defined as the energy consumed by crack formations

per unit area of the crack plane), and the width we of the crack band front. Since

(23)

where Wf is the work of the censile scress p.r unit YQl~,.~ual to th •• r •• t under the ten.il. acre •• -strain diagram (Fig. 6), the softening slope el3 is

not an independent parameter and Collows from the values of f~. Gf

• and wc'

The foregoing simple stress-strain relation far tensile strain-softening has been implemenced in a (inlte elemen~ program. for this purpose. the stress­atrain relation i. differentiaced and used in an incremental loading analysis.

FITTING OF TEST DATA ~.D VERIFICATION

Host of the important eest data from the literature (~~-i~) have been succes.fully fitted in report (30) with the present nonlinear fracture model. Some of the fita from Ref. (30)-;re shown In Figs. 7-10 bv th~ solid lines. The best pos.ible fits obtainable-with linear fracture mechanics are shown for com­pari.on in these figure. as the dashed line.. The fits were calculated by the finite element method u.ing square meshe.. The loading point was displaced in small step. and the r.sponse of the specimen was calculated in small loading Increment.. Tha .... stres.-strain relation, as indicated before. waa assumed to hold for all finite ele .. nts. However, only some elements entered non­linear behavior. A plane atress state was assumed for all calculations. The preceding stress-strain relations were differentiated and converted to an incremental form for the purpose of step-by-step landing analysis.

In fitting thes. t.st data, it was discovered that the optimum width on the crac~ band froat was for all eases between 2.5 and 4-times the ~~ximum aliregata a1ze and. furthermore, the crack band front width

w - 3d c a (24)

wa. nearly optimum for all calculations (da • maximuM aKgregat~ size). It was for this width we that the area under the stress-strain curve yielded the correct value of the fracture energy needed to obtain good fits of the test data. It thua appears that. at least for plain concretes, the width of the crack band front may be ea.ily predicted from the maximum aggregate size. However, we must caution that the foregoing simple relation might not hold for high .trength concrete., in which the fracture seems to be more brittle. the crack band more concentrated.

In view of Eq. 22, the fracture theory presented in this lecture is essentially a two-parameter theory, the two mater{al parameters to be determined by experiment being Cf and f~.

The aforementioned analyses of cest data involved the maximum load tests, as well a. tests of the resistance curves (R-curves), in which the apparent fracture energy is determined as a function of the crack extension {rom a notch.

15

09 Col ~.I ·977,11. CIII p .. "208111.

08 po.' I,so ,II. 08 p •• ' '.25'11. , POI' 4883 'II.

, Pu -6041111. 0.7 07 , , 06 ' 2 "11 , 05 05

• .-21n • 04 0 •• 41 IA. Q4

A .-IOIA.

4 6 • 10 12 16 20 4 6 8 10.2 •• 20

O. 0' Ie I p .. -7671a. Cd. ~ .. ' 803111.

oe , ~ ~ ·2S02111. 08 p •• ' 2410.11.

~ ,

p •• - 3836111. , p .. - 4016111.

07 0.7 , , - , u , , 2 a 06 0.6

":I.' u .. , , 0

, , .!a ~

.:{ , 05 , , , ,

I I , ,

c? , , , 04 ,

" 04 I I , , • ~

,. •• ./1 , .. 6 • 10 12 16 20 4 6 8 10 12

O. O. , •• ~ ... -13S •• 11. cr. Po. ·1163 lla. , ~~ .4.72111. po,· 34'0 .... ,

P".6953111. po. ·lI8.6 III. , , , , Z ~l 06

0'

Q4 Q4

4 6 • 10 .2 •• 20 4 6 8 .0 .2 •• 20

dIdo (lo~ scole)

Fig. 8. Comparison Wlth Maximum Load Data of Wabh (1972, for Six Different Concrete •• (after &alant and Oh. 1981).

16

1.0

,

..c:: -

"

o

" " ... o

... .....

(0)

... ... ... _-- Nonlinear Theory

-- - Linear Theory go 0.6 u ... en

a Shah and McGarry (1971)

g' 0.4 6 G)'n, Sfrenun, Arneun(l977)

..c:: co C Q ... en CO C

"0

"0 c: U

CD

...: G a:

1.4

1.2

1.0

; 0.6 CD

Q 0.4 a:

0.2

0.2

oL-_-L-_-'-__ '--_J-_-..L._~ o 0.2 0.4 0.6

Rei. Initial Nolch Depth' (oo/H)

,

-'-"- . -.-._. _.-- Nonlinear Theory --- Linear Theory _.- Model of Hillerboro, Mod~er, Pelersson0976)

a Tnr, of -"-

I) o~---~--~----J_--~----~--o ~ 10 I~ 20

ReI. Beam Dept h (H / wc )

Fig. 9. Comparison Vlth Maximum Load Data of Shah and McGarry (1971). Gj6rv et al. (1977) and Hl11erborg et a1. (197&) (after Balant and Dh. 1981).

17

4

'.1 • • •

3 .~-'N ;;, E ....

2 2 :I "'i: • ~ 18 ¥

-.5 ... .Ii -c «:»

-; ;to e .... Z 2 -C JIC

- NoNlneor The.,y

-- - Llneor Theory • Soll,8cwOll. ',arMi.l. 11979)

0 0 , 10 " 20 U

Rei. Clock Ex'en.ion (to I w~)

1.0--------------.., lbl 6 6 ____ .. ____ 6.

• 6

a • • • I

~~ - NonU,,_ Tl\eor,

- - - Lan..,. TlleCM'Y •• 6. WecllO,olana Mel $1\011 11980'

QZL-__ ~ __ ~ __ ~ ___ ~ __ ~ __ ~ o 4 • 12

R.I. Clock ExtenSion Ito IWe)

o.a

!~ - ~ - : -- - - ;n- • :

• • -0.,

~I 0 0.4

~ 0.2 - NonIInMt Tile.,.,

- -- L ..... ' T"-Y • ar ••• 1I.72)

0 0 , 10 " 20 U

Re .. Clock ES'enaion I to/w,)

F1,. 10. Compartsona W1th Resistance Curve Data of Sok. Baron and Fran\o1a (1979). Wecharatana and Shah (1980). and Brawn (1972). (~fter Ra!ant and Oh 1981).

18

The .ample. of te.t data available In the literature are sufficiently large for.a .tatiatical regression analy.i. of the errurs. Fig. 11 shova a resre •• ion analy.i, of the maximum load data for twenty-two different concretes (~. In th1A plot, the abscissa 1s X - Pt/Po and the ordinate 1s Y - Pm/PO'

in which Pm - me .. ured IUxi_ load P max' P t - theore tical va lUI! 0 r P max'

and Po - failure load. calculated according to the strength theory. If the theory were perfect. then the plot of Y vs. X would have to be a straight line o! slope 1.0, p .. sin, through the orlgin. Thus, the vertical deviation of the data point. from the regre •• ion line characterize the errors of the theory, and so do the deviationa of the resre •• ion line slope from 1.0 and of its intercept from the origin. The coefficient of variation, ~, of the vertical deviation from the regression line in Fig. llwas calculated to be

For the present fracture theory w - 0.066

For linear fracture theory ~ - 0.267 (25)

For .trength criterion w - 0.650

These results confira that the improvement achieved with the present nonlinear frscture theory i. quite .ignificant.

A .imilar .tatistical regression analysis hal been carried out (30) for the test data available in the literature on the I-curves. In this regression analysis. the fracture energy values were normalized with regard to the product f~ da • and the theoretical values of log (Gf/f~~a) were plotted against

the measured value. of this ratio. Again, if everything worked perfectly. this plot would have to be a straight line of slope 1.0 and intercept 0.0. The standard errors tor the vertical deviation, from this regression line have been calculated. for the set. of various test data available in the literature,

For the pr •• ent fracture theo~': •• 0.083 (26)

For linear frecture theory: •• 0.317

The value. of the fracture energy obtained for the uptlmum fits of various fracture data on concrete were further examined to see whether the fracture energy could be approximately predicted from elementary characteristics of concrete. The following approximate formula wa, found:

Gf

- (2.72 + 0.0214 f') f,2 d /E (27) t t •

in which f~ muae be in psi (psi· 6895 Pa). da • maximum aggregate size, E •

elaatic modulus. It must be emphasized, however, that Eq. 24 yields only the fracture energy value. for the present nonlinear theory, snd not the apparent fracture energy values determined according to linear fracture mechanics or those determined from oth~r theories.

Further interesting problems arise for large structures for which finite elementa whoae size i. 3-ti .. s the aggregate size would result in too many unknown •• In many such situations, e.g., concrete reactor vessels or dams. much larger finlte element. have to be used. In auch a eaae. it appeara that the parameter of overr1dlngimportanc. ia the velue of the fracture energy, and that the declining slope of the stress-strain diagram and the value of the atrength can be adjusted. for very large structures. so as to give the correct fracture energy value when a much larger finite element forms the crack band

19

o .. -• ..

1.80

I. 44

1.08

0.72

0.36

----(a) HONLINEAIl

THEORY

n • 68 i • 0.596 Y • 0.595 all • 0.l75

• + bl

a • -0.014 b· 1.023 ... 0.066

0.0 ~~----+-------+-------+-------+-----~

0.0

1..0

(1)>)

1.44 n i Y • II

1.0'

0.72

0.36

0.0

0.0

0.l6

LINEAA THEOIY

• 6. • 0.59' • 0.S95 • 0.43)

/ ",

0.36

",

"

0.72 1.08 1.44

. ",

/ ",

", ", ~

",

", ",

0.72 1.01

a • 0.093 b • 0.'44 to • 0.267

t.44

1.80

t.eo

F1&. 11. Plot of Measured va. Theoretical Value. of Maximum Load. From T45t. of Twenty-two Different Concretes e1ven 1n tne Literature (after s.tant and Oh. 1981).

20

front. For thia purpose. lf the structure la very large. one ~y consider a sudden atreaa drop after a certaln strength limit is reached. Occurance of the ~den atress crop must be determined either directly on~e basis of an -ru criterion f"L • ,:.ccure formulated for a blunt crack band. or in terms of an adjusted strengt •• limit. called the equlvalent strength. For details see laf. 30.

Further rather difficult question. arise when shear is superimposed on the crack planes. either during the progressive formation of microcrscks or after the full fracture forms. In the first case. one must take into account the fact that auperposition of the ahear cauaes a rotatlon of the princlpal stress direction. One can then no longer use the total stress-strain relatlons. a. ve outlined above; rather one must use a tenaorialy invariant incremental stress­straln relation. This problem has been discuased ln Ref. 59 and an invariant form of a atrain-softening stress-strain relation leading to a complete re­duction of streas to 0 has been indicated.

For the second case. namely when shear 18 superimposed on completely formed­continuous cracks. one needs to formulate a stiffnes8 matrix of a material taking into accoUQt the friction and dilatancy of these continuous cracks. Various forma of such matrices have been proposed. based either on the simple concept of friction or on more realistic. but also more complicated. stress-displacement relationa observed in tests of cracked concrete specimens; see Ref. 59-60.

CONCLUSION

The internal mechanism of the inelastlc behavior of concrete. characterized chiefly by progressive .. icrocraeking. crack friction and dilatancy lends us to formulate non-claasical mathematical models fur the constitutive behavior as well aa fracture. Due to the large size of the hetero~eneties in the micro­structure (the pieces of aggregate). the distinction between constitutive models and fracture models i. becoming blurred. Fracture a&p~cts. consisting in unstable strain localization. cannot be Cor~otten in the formulation of constitutive equations which involve strntn-soitening. vh!le fracture seems to be most reasonably approached by means of stress-strain relations for the fracture process zone.

Financial support under U. S. National Science Foundation Grant No. CEE-8oo9050 1. greatefully appreciated. Mary Hill Is thanked for her careful and prompt typing of the manuscript.

REFERENCES

1. ASCE Structural DiVision Committee on Finite Element Analysis of Rein­forced Concrete Structure. (chaired by A. Nilson). "Finite Element AnalysiS of Reinforced Concrete Structures," Chapt. 2. ASCE Special Publication New York 1982. '

2. Malvern. L. E •• "Introduction to the ~echanics of a Continuous Medium," Prentice-Hall, Englewood. N.J •• 1969.

3. Baiant, Z. P •• "Critique of Or;thotro ie ~odels and Triaxial Test1n of Concrete and Soil.... Structural Engineering Report No. 79-10 640c. Northwestern University. Evanston. Ill •• Oct. 1979; also 1n press. Journal of the Engineering Mechanic. Division, ASCE.

21

4. Fung. Y. C •• "Foundations of Solid ~chanica." (Chapter 6). Prentice Hall. Englewood Cliffa. N. J •• 197~.

5. Balant. Z. P •• and Killl. S. S •• "Nonlinear Creep of Concrete - Adapta­tion and Flow." J. of Eng. Mech. Div •• ASCE. 105:429-446. 1979. (Proc. Paper 14654).

6. Balant. Z. P •• "Work Ineq~11ties for Plaatic-Fracturing Materiala." Int. J. of Salida and Structures. 16:. 1980.

7. Dougill. J. W •• "On Stable Progreasively Fracturing SoUda, ZAMP (Zeitachrilt lUr Angewandte Mathematik und Physik), 1976. pp. 423-4~

8. Dougl1l, J. W •• "Some Remarka on Path Independence in the SIII811 in Plasticity." Quarterly of "'22lo Hath •• 32: 1975. pp. 233-243.

9. Aachl, H., Linse, D •• and Stoeckle S" "Strength and Stress-Strain Behavior of Conc:ete und.r Hultiaxisl Compre.aion and Tena10n Loading, 1976. T.chnical Report. Technical University. Munich, Cetlll8ny.

10. lallDer, C. C •• "Shearing Str.ngth of Concrete under High Tr1uial­Stresa-COlllputation of Mohr'a Envelope aa a Curve," Structural Reeearch Labora­tOry Report No. SP-23, 1949, Structural Research Laboratory, Denver, Colo.

11. Breal.r, B., and Pister, K. S •• "Strength of Concrete under COIIIbined Streaaes," Am. Concrete. lnat. J., 551, 1958. pp. 321-345.

12. Popovics. S" "A Numerical Approach to the Complte Streas-Strain Curvea of Concrete," Cement and Concrete Reaearch, 1973, pp. 583-599.

13. Kupf.r, H. B., Hiladorf. H. K., and RUsch. H •• "Behavior of Concrete under Biaxi&l Stresses," Am. Concrece Insc. J., 66, 1969, pp. 656-666.

14. Liu, T. C. Y., Nilson. A. H •• and Slate. F. 0 •• "Biaxial Stresa-Strain Relations for Concr.t •• " J. of Stl". Div •• ASCE. 98. 1972, pp. 1025-1034, (Proc. Paper 8905).

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