A plastic-fracture model for concrete

61 I Sdad~S7mctumsVol 18 No 3.p~ 181-197.1%? Prmcd mGna1 Bnlm

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A PLASTIC-FRACTURE MODEL FOR CONCRETE S. S. HSIEH,E. C. TINGand W. F. CHEN School of Ctvtl Engmeering, Purdue Utuverstty,

(Recewed 20

Aped

1981;

III nviscd

West Lafayette, fon

3 Augwsl

IN

47907, U S.A

1981)

AMraet-The paper summanxes recent efforts m formulatmg an elastic-plastic-fracture model for the fimte-element analysis of concrete structures. Based on the geometrical considerations, a four-parameter fracture (or ywidmg) cnterion was proposed whrch embraces some of the snnpler existing models fsotroptc elastic and amsotroptc elastic behaviors were proposed for the Initial loading and the post-failure behavrors A plastic model displaying mixed hardening was proposed to describe matertal behavrors between the uuttal yieldmg and the fracture failure Incremental stress-strain relattonships were denved based on the assocrated flow rule and ZtegJer’s kmemattc ha&rung rule Three diierent types of fadure modes were considered A stmpk crushing coefiicient was defined based on a dual criterron to identify the crushmg type, the crnckingtypeandthemtxedtypeoffaihtre Matertalparametersrequiredforcachelemeatof theplastic-fracture model were determined. An important feature of the paper is that matrix formulations for all the constttutrve equations were derived and are available for finite-element tmpkmentattons NOTATION experimentally determined material constants abbreviated functtons defined in eans (24) and (25) parameters in eqn (11) and eqn (12j elastic stiffness tensor tangent stifFtress matrix of plasttc concrete stiffness matrix of fractured concrete Young’s modulus loading functton failure function ultimate strength of concrete under tuuaxral compression, unaxral tenston. btaxud equal compresston and triaxial compression wtth confinement pressure f;O respecttvely mittal drscontmuous strength of concrete under uniaxial compresston confinement pressure for maxtal compression shear modulus work hardenmg modulus work hardenmg modulus asso&ted with tsotropic expanston first stress invariant identity mamx second deviatroic stress invariant bulk modulus parameter of tsotroptc hardenmg effects maximum pnnctpal devtatoric stress devtatortc stress tensor coordinate transformation matrices for stress and stram components abbrevtated functton in eqn (18) crushing coeflictent translation component for untaxral compressive stress condttton translation components of the center of ytelding surface reduction factor abbreviated functions m eqn (18) Kronecker delta prmcipal strains equtvalent plasttc stram due to tsotropic hardenmg equivalent plastic stram Increment elasttc stram Increment, plasttc stram Increment, ~sotrop~c hardenmg plastic stram mcrement, and kmemattc hardening pfasttc stram mcrement, respectrvely posittve scalar functton m the normality condition Potsson’s ratto Haigh-Westgarrd coordmate system prmcqral stresses uruaxial comprescne stress maxrmum prmctpal value of gq reduced stress.and deviatoric stress components maximum pflncIpll stress and &WI mcremental stress and strain tensors mcremental stress and strain matrrces for the post-fadure regton released stress matrtces due to crackmg and crushrng, respectively tsotrop~c strain hardenmg rate functron abbrevtated functtons m eqn (18) angle that describes the crack plane dtrectton 181 SS Vol IS, No 3-A

S S HSIEHet al

182

INTRODUCTION

With the rapid development of computer-aided design of structures, considerable research has been focused on the modeling of progressive failures exhibited in the concrete structures subjected to complex loading conditions. Different approaches for the mathematical modeling, such as material-nonlinear elasticity{l-31, rate-mdependent and rate-dependent plasticity theories[54] have been proposed in recent years A literature review including cntiques of the various modeling techniques was given by Chen and Ting[9]. Concrete behaviors are exceedingly complicated. A constitutive model which embraces concrete characteristics for all types of loading and environmental conditions, and yet IS sticiently simple for present design applications, seems beyond reach. However, a reasonably complete constitutive model for short-time loadings shouid include some of the principal features of cracking behaviors: (i) The brittle cracking in tension, (ii) the ductile failure in compression, and (iii) the post-failure stress-redistribution due to local cracking. Focused on these three primary phenomena, we summarize in this paper our recent effort in formulating an elastic-plastic-fracture model for concrete structural analysis. The present consideration IS limited to plain concrete behaviors subjected to tn-axial short-time loading conditions. The model assumes a linear or nonlinear elastic stress-strain relationship until the combined state of stress reaches an initial yielding surface. The initial yielding criterion is assumed to have the same geometrical shape in the stress space as the failure criterion. A four-parameter failure criterion is also proposed to define the ultimate state of stress. Between the initial yielding state and the failure state, an incremental stress and strain relationship is assumed to define the plastic behaviors. The plastic relations are based on a mixed-hardeningmodel and the classical associated flow rule. Fundamental concepts and the verification of the four-parameter failure criterion with experimental data are discussed. For the post-failure models, the concrete behaviors are defined by three different types of failure modes, namely, cracking, crushing, and a mixed mode. A crushing coefficient based on a dual criterion is proposed to identify each of the failure modes. Procedures have also been developed to handle the stress-redistribution for the fractured concrete. The procedures are tailored for the finite-element analysis of concrrete structures. For the fractured concrete stress-strain relationship, an anisotropic elasttc model 1s proposed. Different stages of the proposed elastic-plasttc-fracture model mentioned above can be illustrated schematically in a typical uniaxial stress-strain curve for plain concrete shown in Fii. 1. A FOUR-PARAMETER FAILURE CRITERION

A four-parameter failure criterion is proposed to define the ultimate state of concrete behavior. The same form of formulation and the same proportionalities among the material parameters are also adopted to describe the initial yielding. This can be geometrically inter-

timate Stress _______----Stress

Release

Post

Failure

Fig 1 An ldeahzed typIcal umaxial stress-stram curve for phn concrete

A plsstic-frocaae modelforcmcrete

183

in

prcted as the initial yielding surfacetbe stress space bas the same shape as tbc fracture

surface. Only the Iimiting values arc proportionally smalkr. Concrate behaviors arc assumed to be elastic for stresses within the initial yielding surface. For stresses fallii between tbcse two sllrfaces, concrete bcllaves plastically. Beyond the failure surface, an anisotropic CIastic post-fracture behavior is assumed. Chamctuistics ofa faiht surface It is generally acceptable that the macroscopic fracture behavior can be &SSWWIto be isotropic. Tbis implies that 8 failure function can be written in terms of the principal stresses (s, g2, u3) or in the ~~es~~d coordinate s~~rn~l~] (the stress Kit space)

f(P,r,@=O

(1)

where

e-cos-‘aA p(saoo, 2 fl’ S,

= the rn~~

pan

deviatoric stress

It = the lht stress invariant =Ul+U3+@3, J2 = t&c sccmd

deviatoric stress invariant

=gu, - u3’

+ ([I)

- a3y

+ (a3 - a,)7.

A geometrical aeon of the coordhate system is shown in Fii. 2. The eqlicit fona of the failure function is defmed by the experimental data. Uniaxial and biaxial tests of plain comrcte arc welldocumented in the literature. To name a few, reports by Kupfer and Hikdorf(ll] and Tasuji d uf.[I2] nearly cover the full area of the biaxial stress section.

Fig. 2 Gunusl failuresurfam in psiiqd

stress space.

184

S S HSIEH et al

However, for the triaxial state, only the test data for limited ranges are available, and generally have wider scattering among different testing apparatus. Among them, we mention the results by Mills and Zimmerman[l3] Launay and Gachon[14] and Gerstle ef al.[U]. The aforementioned experimental data also forms the basis of the failure function proposed herein. The available data clearly indicates that the failure surface plotted in the principal stress space, Fig. 2, should form a cone shape with smooth curved meridians and non-circular convex sections in the deviatoric stress plane. Since the hydrostatic pressure alone will not cause failure, the failure cone should have an open end in the negative hydrostatic axis. In addition, the intermediate principal stress should be accounted for. This indicates that the deviatoric cross-section of the failure cone approaches to a circular shape as the hydrostatic pressure increases. It proportronally shrinks into a triangular shape as the hydrostatic pressure decreases. A four-parametercriterion

The present criterion IS motivated by the geometrical requirements of the failure surface cross-section in the deviatoric plane mentioned above. Observe in Fig. 3 that for a constant value k, r cos tl = k represents an equilateral triangle and r = k is a circle on the deviatoric plane with I@( r&l”. Hence, given two positive constants a and j? with a + p = 1, a combined equation r(a cos 0 + /3)= k yields a smooth function between 101I 60”on the deviatoric plane and bounded by the two extremes of equilateral triangular and circular shapes (a = 0 or @= 0). Recall the convex meridians shown in the experimental data. This indicates that for a constant value of & r should be a nonlinear function of p. Hence, p and r”?terms are added and the resulting form is f(p,rA=a~+t

acos0+fl)r+Cp-l=O.

(2)

Equation (2) may be written in terms of the stress invariants defined in eqn (1):

where the parameters are nondimensionalized by using the unlaxral compressrve strength of concrete f:. In eqn (3), (Tois the maximum principal stress with a positive stress value representmg a tensile stress. I, is the first stress invariant, and Jz the second deviatoric stress invariant. It is interesting to note that although the basic form is originated by the geometrical consideration in the stress space, the resulting functional form appears to be a linear combination of three well-known failure criterta, namely, the von Mises. the Drucker-Prager, and the Rankine’s criteria. In some aspects, eqns (2) and (3) resemble the forms proposed recently by Ottosen[l6], Hansson and Schimmelpfennig[l7], Willam and Warnke[8], and Nagamatsu and Sato[l8].

b) =K

Rg 3 Geometry on the dewatorlc plane

A plasts-fracture model for concrete

185

This is not surprising, since all the above forms are also based on the similar set of geometrtcal considerations. However, the present form appears to be simpler for the determination of material parameters and possesses some convenience in numerical calculation. Equations (2)and (3) also bear certain similarities to the forms suggested by Argyris et al. [19], Mills and Zimmerman[l3] and Hannant and Frederick[20]. However, the present form has the advantage of satisfying the convexity requirements for all type of stresses. Verifications of the present failure criterion by plotting the function in the octahedral shear and normal stress plane, the biaxial stress plane, and the deviatoric stress plane have shown good agreement with the reported test data. For comparison, the material constants A, B, C, D in eqn (3) were evaluated based on four basic stress conditions: the simple tension (0, = f;, a2 = a3 = O), the simple compression (u,, a2 = 0, a3 = - f:), the biaxial compression (a, = 0, uz = a3 = -&), and the confined triaxial compression (a, = a2 = - fk, a3 = - fi, with f:, > &,>. The stress values were assumed to be f: = 0.1f:, f;K = 1.15f:, fAc= 0.8f: and fl = 4.2ff. This gives A = 2.0108, B = 0.9714, C = 9.1412,and D = 0.2312.Figures 4-6 show the comparisons of the prediction and various reported data. We have also applied the failure criterion to study an elastic-fracture analysis of a concrete splitting test for the purpose of illustrating its tractability in numerical calculations[213. Criterionfor initialyielding It is convenient to assume a criterion for initial yielding to have the same functional form as the failure criterion. In the present model, the material constants are also proposed to remain the same, except that the nondimensional constant, the compressive strength f:, is replaced by a different value fc = 0.3 -0.6 fb The exact value of fc can be taken from the uniaxial compressive stress-strain curve of the specific concrete used. ELASTIC

AND PLASTIC

REGIONS

The initial yielding criterion and the failure criterion define the limits of the elastic region and the plastic region. Within the elastic region, concrete can be assumed to be an homogeneous, isotropic, linear elastic material from the macroscopic point of view. The constitutive relation is defined by a stress-strain relationship with two elastic constants, the modulus of elasticity E and Poisson’s ratio v, or alternatively, the bulk modulus K and shear modulus G. For finite-element applications, the matrix form of constitutive relations can be found in a standard textbook, e.g. Ref. [22]. For a state of stress beyond the initial yielding, irreversible

Dota taken from

0

Reference

16

Reference

13

Reference

1

(DetermIne matertal

constants) ?

Fig

4

F;ulure cnterlon m octahedral shear and normal stress plane

186

S.S.HSEH etd

-

Failure Critertan

o--- Data taken from reference

14

hg. 5. h~Iurc

cntmon

tn devhkmc

stress plane

..-04

-----0

Foilurr Criterion Kupfw et al (11) Tasujl et al (12) Data taken for dcrrminhq matwlol caWmtr

.,-OS

-I0

Fw 6. F&EC cntcrb IR b&id prihpd stressplane.

deformations become @Scant. It is couvenieut to follow the classical plasticity theory[lQ] to an incremental form of stress-strain relatioaship. The total strain increment is taken to be the sum of the elastic iucreuient and the plastic increment

use

dq = dcj + de& Adopting the lbw de with a plastic potential function F and

A

plastic-fracture model for concrete

187

we may assume that the plastic potential coincides with the subsequent loading function which

has a similar form as the initial yielding function or the failure function. In eqn (5), the associated Bow rule, Us is the total stress tensor and dA a positive parameter to be determined. To include concrete behaviors under the local unloading and the cyclic loading conditions, the kinematics of the loading surface is assumed to follow a mixed hardening mode1[23,24].The model allows the loading surface to translate and simultaneously to have isotropic expansion. The specific form of F is taken to be (Fig. 7) F(&,,,7) = A

-& +BV.& +Cc?,+DI;+ +,,‘)

= 0,

where the tensor a,, characterizes the translation of the center of the loading surface, r(eP’)an isotropic hardening function, 4’ the equivalent plastic strain due to isotropic hardening. 4, B, C and D are material constants given in the failure criterion. The stress invariants are I, = ti,,, & = l/2 $,&, where 4 = ui, - ati $ = Sb -ail + 1138~~ S,, is the deviatoric stress tensor and 6, the maximum principal value of the stress tensor c?~ Using the condition of consistency, i.e. dF = 0, and noting that

we

have

Equation (7) may be used to evaluate the flow rule parameter dA. To do so, we consider the plastic strain increment. It is convenient to write dr$ = dr$’+ dr$ =Md+(l

-M)dcj;,

(8)

Initial Yielding Surface

Fatlure

Fig. 7. Schemnt~cof failure, mitd ylehiing and subsequent loads

surface.

Swface

188

S S HSIEH ef al

and introduce a constant parameter M to define the isotropic hardening effect. The remaining plastic strain increment de;’ is then due to the kinematic hardening. The strain increment de; is related to the increment d7 of the isotropic hardening function. Define the equivalent plastic strain increment to be (9) where the definition can be applied also to the components de: and dt-:’ which are denoted by de; and dcPk,respectively. In view of the definition of the isotropic hardening function I, we have dr = ti dq’ = Mfi deP,

(10)

where g is the slope of the 7 - e,,‘curve. To determine the tensor increment da,,, Ziegler’s kinematic hardening rule is assumed[25], da,, = d&

(11)

The parameter dk is further assumed to be a function of the kinematic strain increment m the form dp = c de; = c( 1 - M) de

(12)

Using the tlow rule, eqn (5), to replace de,, by the corresponding stress components, d7 and dru,, can be solved explicitly in terms of the unknown parameter dA. Substituting the resulting forms into eqn (7) and noting that the elastic strain increment is governed by a linear elastic behavior, do;, = &, deh

(13) (13)

where C, is the isotropic stiffness tensor whose components can be expressed in terms of elastic moduli. The parameter dr\ can be solved explicitly in term of the stress eomponents. If dA is again substituted back into eqn (13), a total stress-strain incremental relationship for the plastic region is obtained:

where

aF aF aF aF G““=ba,,aa,=~~’ Note that eqn (14) is based on the mixed hardenmg model. A weighting coetictent M is included to allow the freedom of selecting di#erent proportions of isotropic and kinematic effects in the mixed model. M can also be a negative value, so that isotropic softening can be considered. The advantages of using the concept of mixed hardening have been demonstrated by Axelsson and !Samuelsson[23]in describing the loading cycles of metals. They showed daterent degrees of Bauschinger effect can be considered. The mixed hardening gives much better curve-fitting results than either the isotropic or kinematic hardening model. In their consideration, M is arbitrarily set at 0.15 of 0.20. Equation (14) also contains two material constants fi and c. They can be related to the test results of a sample subjected to simple compressive loading. For the uniaxial compressive

A plastx-fracture

model

for concrete

189

stress condition, the only non-vanjshing stress component is u33= - &, and according to Ziegler’s hardening rule, the only corresponding non-vanishing component of a,i is a33= - 2. Hence, from the stress-plastic strain plot, we may evaluate the tangent modulus H, i.e dd= HdeP

(15)

The degenerated form of the loading function given in eqn (6) becomes

(16)

fi-4’7.

Using eqns (10)-(12)and eqn (16),we find

Since M is arbitrary, this implies fi=Hand

c=$.

(17)

To implement the incrementa stress-strain relationship for finite-elementanalysis, it is convenient to write eqn (14) in an explicit matrix form. After carrying out the algebraic manipulation in substituting the loading function into eqn (14),we have {da}= [P’l&l,

(18)

where

For i= 1,2,3, E(l-

v)

c~=(l+v)(I-2V)-W

E2-__e6_A2 ( I-2v+1+v

For i=4,5,6,

,,,L_qA-)2. Z(l+v)

Fori,j=1,2,3andifj,

For i= 1,2,3 and j=4, $6

W l-tv

) *

190

S S Hsmetaf.

Fori,j=4,5,6andi#j,

It should be noted that since the loading function beoolnes singular at +, = ti2> & or B= 60”, for this specific state of stress the limrting values of f12and p3 should be implemented in the numerical caicW.ions where

A note of further interest is that since the failure criterion (and the loading fun&on) is a generalization of several other theories of simpler form, it is particularly convenient from the program averment point of view. For example, by stressing material consents A and D and so appropriate material v&es for B and C, the amputer program can be adopted for the Drucker-Prager’s plasticity modeII261.By the choice of parameter M = 0,1, or other values, we may select the hardening model to be kinematic, rsotropic, or of the mixed type. Thus, the program developed based on the present model in fact embraces some simpler modeling techniques. FAILURE MODE CRITERION

Concrete fails or fractures in extremely complex modes. Aggregatetypes, mixed design, and loadii conditions among many other parameters all play important roles in the cause of failure It wouki be difhcult to classify and de&m precisely the faihrre modes. However, m a general sense, the mode of failure may be categorixed into three types, namely, the cracking, crushing and a mixture of cracking and crushing. ~cumented test results for tensio~te~sion or

A plastic-fracturemodel for concrete

191

tension-compression biaxial conditions show the cause of fracture is primarily a brittle splitting in the plane normal to the maximum tensile strain direction (e.g. [12, 13, 271).For the triaxial compression tests, depending on the magnitude of confinement pressure, it seems that all the three types of mode are possible. When the ~~~rnent pressure is much lower than the axial compression, rough crack surfaces can be formed in the direction normal to the maximum tensile strain, possibly due to the connection of numerous microcracks. For nearly uniform hydrostatic condition, crushing failure is more common, possibly due to the rupture of mortar in the concrete.

In view of the failure modes due to various types of loading conditions, a crushing coefficient a is proposed to identify the mode being either a pure cracking, a pure crushing, or a mixture of the above. The coefficient can also be used to estimate the proportions of cracking effect or the crushing effect in a mixed type failure, This is particularly convenient when the post-failure behaviors of the fractured concrete are considered. The concept of crushing coeffcient is based on the consideration of a dual criterion in defining the pure cracking zone and the pure crushing zone in the overall spectrum of failure mode. SpecifIcally,the pure cracking zone is assumed to satisfy the maximum tensile stress condition

Written in terms of the stress invariants, we have

It may be shownthat the upperlimit of the pure cracking con&ion satisfies fhe t&axial and the biaxial compression faihrre test data, see Fii. 8. For the pure crushing zone, it seems reasonable

to assume that ah three principal strain components are all comwessive strains, so that the crack mecw can not be developed in the light that no tensile strain could appear in any direction. This imp& that the rnax~~ principal strain is non-positive

Fig 8 Fadwe zones in octahedralshear and normalstress plane

S S. HSEHet al

192 Using the

Hooke’s law, the same condition becomes @I -

v(q

+

u3)

<

0

with

al

>

Q

>

03.

Rewrite the inequality in terms of the stress invariants, tt becomes

Corn~~

eqns (19)and (20)a crushing coefficient a is defined as I,

a = - 21/‘3t/Jzcos 6’

/t?Js 60”.

Bl)

The failure modes are then iden~ed as (i) Pure cracking, (Y< 1, (ii) Pure crushing, a > H,

(2%

If Poisson’s ratio is taken to be Y= 0.2, we have a fs LOand 2.0 as the boundary values separating the three d&rent faihEe zones. Note that in obtair@g the simple crush& ooe&ient Q Hooke’s law of elasticity was employed to obtain the stress criterion. Strictly speaking, this is inconsistent in an elasticplastic-fracture model; Hook.& law may not apply ~rnrn~~t~y before ~~~ However, judging by the complex nature of concrete failwe and the simplicity in the appfioation of the crush& coefficient, the elasticity assumption may represent an acceptable ~~o~~on. For more a&urate descriptions, the original dual criterion, i.e. cl > 0 and e1< 0, may also be used. POST-FAILURE

BEHAVIOR

OF FRACTURED

CONCRETE

To complete the constitutive model, we also need to define the post-failurebehaviorsfor each of the failure modes identified by the crushing coefficient.For the pure crushing zone, the crushed concrete can be viewed to behave like a grannuhn material under the confinement of neighboring materials. Material stiffness in compression or shear, although reduced, should still exist. However, for simplicity, we may neglect the residual stiffness and the residual strength of a crushed concrete element in the analysis. Thus, the post-faihrre behavior becomes perfectly deformable. For a concrete element subjected to pure cracking, the post-failure behavior is assumed to be anisotropie elastic that the element have lost its rigidity in the cracked planes, see Pig. 9. An extensive discussion of the kinematics of a cracked concrete element was reported by Chen and Suzuki[28]. Within the mixed failure zone, the value of the crushing coefecient is between 1.0 and 2.0 for v = 0.2, for example. If the crushing coe&$ent is adopted as a measure of the degree of crushing in this partially cracking and partially crushing concrete element, then we may view that the post-failure behavior is also a linear inte~lat~n of the perfectly deformable behavior and the anisotropic elastic behavior. Hence, it is proposed that the concrete element wiit.lose its rigidity in the cracked plane according to the ~imum tensile strain direction and the anisotropic stifllnessof the fractured element will also be proportionahy reduced according to the magnitude of a. Note that for Y= 0.2, mixed failure lies between a = 1.0 and IY= 2.0. Thus, the values of ~1behind the decimal point represents the ~rcen~~es of crushing and also the percentages of sti&tess reduction. Finite-elmrent implementation To formulate the anisotropic elastic behaviors in matrix form for ~nite~lement apphcation,

A plastic-fracturemodelfor concrete

(a) Stress ~~fi~ti~

Just Before Crocks Are Formed

193

fbf Stress DI~I~I~ Just After Crocks Are Formed

Fig 9 Pattern of cracks and stress datrtbutmn m a crackedconcrete

we define the pre-failure principal axes be (x’, y’, z’) with x’ denoting the maximum tensile strain direction. Then, the incremen~l stress-strain relation for the post-failure behavior has the form

in which

is the released stress components due to cracking and {Q} is that due to crushing. [T,] and [7”] are the transformation matrices for stress and strain components between the principal coordinates and the original reference frame. @ is a reduction factor related to the crushing coe~ciesta.Fora
C**= C33) = (2 - /3)E(l - V)/(l+ VMl- 2Y), ~u=l;n=(2-B)Evl(l+yXl-2v), c&+=

‘

(2 - #?)E12(1 f 4,

and all other components vanishes. It IS expedient to list the matrix equations for some special cases: (a) ?%eu~isymm@t~cal pr~b~ern. Written in polar coordinates, we have

(6) The plane strain problem.

194

S. S HSSEHet al

The abbreviated functtons are cos23, WB

=

sm2I#J 2 sin * cos 4

where 4 is the direction of the cracked plane, see Fig. 9, [II is the identity matnx. (c) Tllccplane sttess p&~feru. Special attention should be given to the plane stress condition. For tens~o~tens~n and tensio~ompress~n conditions, the rn~rn~ tensile stress is in the plane, and the concrete always fails in the crt&r.ing mode. However, for the compresstoncompression case, normal stress fll = 0 represents the maximum stress. This indicates that the crushing parameter a = 1.0 for all stresses. Hence, for plane stress condition the mixed zone and crushing zone are collapsed into a singular point. Physicatly, it can be interpreted that baaed on the zero-thickness ass~p~ of plane stress, the element thiikness simpl,ydoes not permit cracks to generate, and pure crushing would rest& due to the sim~tan~us compressions in the plane. The corresponding matrix formulations are: for tension-tension and tension-compression cases,

APPLICATIONFOROTHERMATERIALS The present model is proposed for concrete materials. However, the model can alao be extended to represent properties of other ~~~ materials of similar nature. As mermtiontd prev~~~y, by inserting approp~~ mate&i con&u& (see TabIe 1) the mode1 can be degenerated to simpler forms, such as the von Mises criterion which has been used for metals. It can alsa be reduced to the Drunker-Prager criterion which has been used as a simptified version for the Mohr-Coulomb model for rocks and soils. The proposed four-parameter failure criterion and the associated constitutive equations arc thus the generalization of some models used for a wide variety of materials. Due to the fkxiity of having four material ~ons~ts, material properties can be simulated more accurately. For example, the Drucker-Prager yield Table I Mammal consiants matcbcd with other cr~tcm

A p&k-fracture modeifor concrete

195

function, depicted in the three dimensional principal stress space, can be represented by a cone shape with the circular base located on the deviatotic stress plane, while the Coulomb criterion is a pyramid shape with an irregular hexagonal base. Thus discrepancies are expected when the ~cker-Par model is used to simulate the behaviors of a typical Co~orn~t~ material such as soils. Efforts have been made to minimize the discrepancies by relating the DruckerPrager constants a and K to the Coulomb coefficients c and (d by using various matching approaches. However, successful matchings are limited to certain special cases. Using the four-parameter model proposed in the study the material constants A, B, C, D can be used to match CouJomb’s mate& properties 4 and c as shown in TabIe 1. With two additional cons~ts they seem to match very well on the de~atori~ plane. The d~eren~ is less than 4% (see Fig. IO).Plotted both criteria on the biaxial stress plane, without further adjustment in the material constants, the maximum di5erence is found to be less than 15%in the compressioncompression region and 4% in the tension-compression and the tension-tension regions as shown in Fig. 11. SUMMARY AND CONCLUSIONS

A plastic-fracture constitutive model for concrete structural analysis has been developed, with the primary objective being that the material model has the capability of describing the essential features of concrete behaviors, and is yet sufficientlysimple for which the model can easily be implemented for finite-element analysis. Based on the geometrical considerations in the stress space, a fo~~meter faihtre (y~di~ criterion is proposed. Parameter determination for the criterion has shown to be simple. In addition, the criterion proves to be a linear combination of three simple models, namely, von Mises, Drucker-Prager and Rankine models. Hence, a plastic-fracture theory based on the present failure (yielding) criterionin fact embraces some simpkr theories. This is particuhu~ytractable in the program development and in the selection of modchng techniques often required in the material chara&&ation. Resting on the similar ~~~, a plastic model ~~~~ mixed hardening e&t has been proposed. With the sekction of a constant parameter, the hardening rule has the choice among the isotropic type, the k&m&c type, and the mixed type. To identify the faihtre mode, a simple crushing coefficient based on a dual c&ion is proposed to subdivide the spectrum of faihue into the crushing type, the cra&ing and a mixture of cracking and crushing. For each type of the faihrre modes, the post-failure anisotropic elastic behavior has been de&ted. The crushing coe5cient is also used to estimate the amount of crushing in the mixed mode of faihue, and to determine the loss of material rigidity after fracture.

Modet

ss

1%

HSIEH

Coulomb Four-Parameter

Model

et al Model

I

f cc

Fig. It Compartsonof Mohr-Couiombmodeland the four-parametermodelm bmxudstress plane,

Since one of the considerations in the development is that the material model developed should be convenient for numerical calculations, matrix equations for the constitutive relation-

ships at all stages of the loading and unloading process discussed in the above have been explicitly formuiated. This includes the consideration of adopting a stress-re~t~bution process commonly used in ~nite~~e~nt anaiysis to handie the progressive failures. In conclusion, we wish to emphasize the generality and versatility of the present plasticfracture model, the simpkity in using the crushing coefficientto hatile various types of failure mode, and the tractability in implementing the model for concrete structural analysis. REFERENCES L. Cedobn,Y. R J Crutzen and S. D Poh, Tnaxud stress-stramreiattonshtpfor concrete i J&gngMe&. I)ss AXE ldS(EM3).243-439,Proc Paper 12969(June 1977) T C Y LM A H Ndson and F 0 Slate, Btaxlatstress-stram relations for concrete,I Strucr.Mu ASCE,98(ST5), 1025-1034Proc Paper 8905(May 1972) R Palamswamyand S. P. Shah, Fracture and stress-stram relahonshIpof concrete under triaxlat compresston J Stnret Div. ASCE, lO@@TS), 901-916 Proc Paper tOS47(May 1974) Z P Btint and P D Bhat, E~~mc theory of l~~asttc~tyand farlureof concrete. f Engng Me& L)ru.ASCE l~EM4), 70t-722 Proc Paper 12368(Aug. 1976) 2 P Baiant and C L Shieh, Endochromcmodelfor no&near trlaxtaibehaviorof concrete Nuclear Eagng &slgn 47.305-31s(1978) A C T Chen aud W F Chen, Conshtutlve relations for concrete I Engng Me&, L&U. ASCE 141(EM4),465-M Proc Paper 11529(Aug 1975) A C. T Chen and W. F Chen, Constltuttveequatmnsand punch-mdentattonof concrete 1 Engng Mch Div AXE, lQl(EM), 889-9%.Proc Paper 11809 (Dee 1975) K J Wdfamand E P Wamke,Cons~~tlve model for the tnaxtai behaviorof concrete. ~n~rn~~o~ Austin of Bruigeand StructuralEngmeersSeminaronConcrete StructuresSubjectedto Tnaxud Stresses,Paper IIX-I,Bergamo, Italy, 17-19May, 1974 9 W. F. Chen and E C. Tmg, Constnuuve modelsfor concrete structures J Engag Me&. au ASCE l%(EMi), l-19. Proc Paper 15177(Feb 1980) A Mendelson,Plastcciry 77reoryand Applicattons McMdlan,New York(1968) 1: H Kupfer and H K. H&do& Behavtorof concrete under bmxudstresses ACI I 64@),656-666(Aug. 1969). t2: M. E Tasup, F 0 State and A H Nilson, Sttess-strain response and fracture of concrete in biaxial loadmg ACJJ 75(7),306-312(July 1978) 13 L. L Mtlisand R M. Zimmerman,Compressivestrengthof ptatnconcrete under mu&tax&loadutgcondmons ACJ3. *lo), 802-807(ocl 1970) 14 P. Launay and H. Gachon, Strain and ulttmatestrengthof concrete under tnaxlai stress SpecialPI&L,SP-34,AC11, 269-282(1970) 15 K H. Gerstle et ni, Strengthof concrete under multlaxlalstress states Speed Pub1 SP55-5,103-131ACI, Dougius !&Henry Intematmnal Symposiumon Concreteand Concrtte Structures(1978) 16 N S Ottosen,A farlurecnterion for concrete i EngngMe& T#a ASCEl~EM4). 527-SProc Paper 131Ii (Aug 1977l

A plastic-fracturemodelfor concrete

197

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25. H. Ziegkr. A mod&&ion of Prager’sItardeomgrule Quult Appl. Mu&m&s 17,55-65(1959). 26. D. C. Druckerand W. Pnaer, Soil mechanicsand plasticityanalysis of hmit design, Aupl. - Cbrurt. _ . . Muthemut~cs 1((z), 157-165(1952) 27. J. N carin0 and F. 0 Slate, Idmibng tensde stram cnterton for failure of concrete, ACI 3 73(3),M-165 (1976). 28 W F Cbcn and H Swdu, Constitutive models for concrete, ASCE Annual Conventtooand Exposition,Chrcago, Illinois, October IGO, 1978,Preprmt 3431,51-B