Fracture Mechanics of Concrete Sweden

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DIVISION OF BUILDING MATERIALSLUND INSTITUTE OF TE HNOLOGY

FR CTURE MECH NICS OF CONCRETENordie seminar held at Division of BuildingMaterials November 6 1986

REPORT TVBM 3025LUND SWEDEN 1986


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CODEN LUTVDG/ TVBM-3025)/1-155/ 1986)

FR CTURE MECH NICS OF CONCRETENordie seminar held at Division of BuildingMaterials November 6 1986

SSN 348 -791


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CONTENTpage

Pre face 2Lis t of par t i c ipan ts 3Fin i te element analys i s of mixed mode f rac ture 5using a discre te crack approachu. Ohlsson, K Gyll tof t Sweden)Smeared crack analysis of concrete using a nonlinear 31f rac ture mode lo. Dahlblom, N. Saaby Ottosen Sweden)A descr ip t ion of concrete and FRC mater ia l s by means of 47composite mater ia l theory and continuum damage mechanics

H. Stang Denmark)Fracture mechanics and dimensional analysis for 69concre te s t ructuresN. A. Harder penmark)Time-dependent f rac ture of concrete, an experimental 79and numerical approachE. Aassved Hansen, M Modeer Norway)Applica t ion of f rac ture mechanics in computer 89ca lcula t ions of minimum reinforcement in concretes t ruc tu res subjected to res t ra ined shrinkage

M Grzybowski Sweden)Modelling of hook anchors 105L. Elfgren, U. Ohlsson Sweden)Modelling of mult iple crack formation in reinforced 119concrete s t ructures Analysis of beam with- longi tudinal reinforcementB. Bergan Norway)Pred ic t ion of shear force and shear displacement for 137construct ion j o in t s and cracks in concreteJ Norberg Sweden)l ee-abras ion of concrete 151M P. Lanu Finland)


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Pre faceFracture mechanics of concrete has aroused much in te res t in manyunivers i t ies within the Nordic countr ies. Therefore a need was fe l tto arrange a seminar in order to exchange information on the workwhich i s going on. The seminar was arranged as a mini-seminar i ea one day seminar with par t ic ipa t ion mainly re s t r ic ted to act iveresearchers in the f ie ldAt the seminar a number of papers were presented and discussed. These papers are presented in th i s report . The papers are presented inan informal way. o reviewing has been performed and no attempt hasbeen made to unify the presentat ion or typing. Some papers may appear more or l e ss unchanged in other publicat ions.I would l ike to thank the par t ic ipants of the seminar for the i rcontr ibut ions to the discussions a t the seminar and for the i r contr ibut ions to th i s publ icat ion.Lund March 1987Arne Hil lerborg


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LIST OF PARTICIPANTSFrom DenmarkHarder N. A.Holkmann OlsenNicholausStang HenrikFrom FinlandLanu Mat t i

From Norway

3

Ins t . f . Bygningsteknik ADC Sohngrdsholmsvej 57 K 9000 Aalborg.ABK DTH Bygning 118 DK-2800 Lyngby.

Technical Research Centre of FinlandKemis t in t ie 3 SF-02150 Espoo

Aasved Hansen Einar SINTEF-FCB/Inst . f . be tongkons t r . NTHN 7034 Trondheim-NTH.Bergan BjrnFrom SwedenG y l l t o f t Kent

Daerga Per Anders

Elfgren Lennar tOhlsson Dlf

Sta tens Provningsans ta l t Byggnadsteknik Box857 S-501 15 Bors.Kons t ruk t ions tekn ik Tekn. Hgsk. i LuleS-951 87 Lule.

Grzybowski Miroslaw Ins t . f . brobyggnad KTH S-lOD 44 StockholmNorberg JanDahlblom Ola Byggnadsmekanik LTH Box 118 S-221 00 LundGustafsson Per JohanPe te rsson HansSaaby-Ot tosen Nie lsHassanzadeh Manoucehr Byggnadsmater ial LTH Box 118 S-221 00 LundHil le rborg ArneZhou Fanping


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FINITE ELEMENT N LYSIS OF MIXED MO E FR CTURE USING A DISCRETECR CK PPRO CHy

Ulf Ohlsson Kent GylltoftDivision of Structural Engineering, University of Lule, S-951 87,Sweden

PREFACE

This report presents an experimental and numerical study of thefracture of a notched unreinforced concrete beam. The work has beencarried out at the Division of Structural Engineering LuleUniversity of Technology under the direction of Professor LennartElfgren. The tests were planned by Kent Gylltoft in 1984 when he wasan assistant professor in Lule. The tests were carried out by LeifHedlund and Ulf Ohlsson. Ulf Ohlsson also carried out the numericalanalysis and wrote this report. The work has been supported by TheSwedish Council for Building Research BFR 830900-3 .


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1 INTRODUCTIONA crack in a structure can propagate in different ways To facili tatethe analysis three basic mades have been named see Fig. 1.1. A lot ofwork has been carried out to study mode I, the opening mode see forexample [1J, [2J, [3]. The other two modes sliding and tearing have sofar only been studied in a few tests.

Fig. 1 1 Basic modes of crack extension:I, opening mode; II, sliding mode; III, tearing mode [3]

In this paper a study of a mixed mode fracture is presented. Experi-mental and analytical results are compared for a beam according toFig. 1.2. Here both mode I and II occur.The i rs t experiments on the beam in Fig. 1 2 were carried out byArrea and Ingraffea in 1982 [4]. They subjected the beams to eyelicloading. Af ter each loading cycle the crack growth was inspected. Oneconclusion Arrea and Ingraffea drew from their tests was that thefracture mechanism was a mode I opening tensile) fracture.Another type of beam that has been tested is the double notched beamn Fig. 1.3. Four different sizes of the beam were tested by Bazant

and Pfeiffer in 1985 [5J. Compared to the beam in Fig. 1 2 the loadingpoints were here located more elose to the notch. The fracture mechan-ism was said to be a mode II sliding shear) fracture.


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7

0.1 3P p

96 2.0 3_ ---------..--........---.

Fig. 1.2 Tested beam of Arrea and Ingraffea

d o 38/16 152 305t:: 38 mm

p

2 b1d

Fig. 1.3 Tested beam of Bazant

305

d


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Several investigators have modelled the beams in Figs. 1.2 and 1.3 withthe finite element method using both discrete and smeared fracture models.

Arrea and Ingraffea 1982 [4]; used a discrete fracture model basedon linear fracture mechanics with different mixed-mode fractureinit iat ion theories. Crack trajectories observed in the tests werereproduced with the f inite element analysis.Glemberg 1984 [6] modelled the beam in Fig. 1.2 using a smearedfracture model. The calculated load-displacement curves are in agreement with the experimentally obtained curves but the calculated cracktrajectory does not match the laboratory tests.Rots 1985 [7] modell ed the beam in Fig. 1.3 with a smeared fracturemodel. He succeeded in reproducing the experimental crack trajectories.Oldenburg 1985 [8J also used a smeared fracture model for the beamin Fig. 1.2. But his model also included the possibility of plasticdeformations in compression. He succeeded in matching the calculatedload-displacement curves and the crack trajectories with the experimental results.De Borst 1986 [9] analyzedthe beam in Fig. 1.2 using a smearedfracture model. The load-increment is determined from the conditian ofconstant increase of Crack Mouth Sliding Displacement CMSD at everystep. He obtained a very good correlation between the experimental andthe calculated load-displacement curves but the crack trajectoriesdid not match the experimental ones.

2. EXPERIMENTAL STUDY2.1 Test specimenFour concrete beams were cast for the experiments. They were cast inplywood forms and in two different batches. Steel plates thickness3 mm were fixed in the bottom of the forms to make the notches. Sixconcrete cubes 150 x 15 x 15 mm3 were also cast in each batch.


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9

[mm]

Fig. 2.1 Test Specimen

to,- = ~ ~ - - ~ ~ = - ~ ~ 5 2

Af ter the casting the beams were covered with plastic film and keptwet. The forms were removed af ter three day s and the beams were thenmoved and cured in water until the day of testing.The concrete mix had the following proportions: cement 10.6 , fineaggregate 41.9 , coarse aggregate erushed stone, maximum size 16 mm)39.9 and water 7.6 . The cement used was Cementa Std Slite. The wa-ter/cement ratio was 0.72.

Results from the cub e tests are summarized l.n Table 1. The meanvalues of the tensile and compression strengths in batch 2 weref = 3.0 MPa and f = 35.4 MPa.et ccTable 2.1 Cube tests

Cube No. Age Density Splitting Compressionstrength, strength,Q f ct f3 ccdays kg/m MPa MPax 1000

Batch 2 68 2.39 3.0 33.92 68 2.39 3. 1 35.43 68 2.38 3.2 36.14 68 2.39 3. 1 35.25 68 2.38 2.8 35.76 68 2.40 2.9 36.0


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2.2 Test set-up

A servohydraulic testing equipment with a capacity of 270 kN was usedin the tests. The test set-up is shown in Fig. 2.2 and 2.3. The de-fleetian of the beam a t the loadpoint r was measured with two LinearVariable Differential Transformers, LVDT attached to a steel frame.The Crack Mouth Displacement CMD was measured with an extensiometer,see Fig. 2.4.

STEEL BE MHEP toO

Fig. 2.2 Test set-up

Flg. 2.3 Test set-up

LO D CE..LLLVDT

c- EXTENSIONEiER


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Fig. 2.4 ExtensiometerThe CMD measured in the tests includes the crack opening as weIl asthe crack sliding. The ratio between the Crack Mouth Displacement,CMD and the Crack Mouth Slidi ng Displacement, CMSD is 1.3 at peak-load according to the FEM-calculations.

The load was measured with a load cell with a range of 500 kN Theloading of the beam was strain-controlled with the extensiometer atthe crack t ip as the feedback source to the hydraulic cylinder. Theloading velocity was equal to an increase in CMD of 0.08 ~ m / sTest data was collected on two xy-recorders plotting load versus CMDand load versus deflection.2.3 Test resultsTest results from two beam tests are presented in this report. The twobeams were east in the second batch. Test results from the beams castin the i rs t batch are not presented due to difficulties in controllingthe load during the tests.Fig. 2.5 shows the load versus CMD and Fig. 2.6 the load versus deformation curves of the tested beams. Only one load-deflection curveis shown, due to a transducer fault.


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...ZY-

C l


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The experiments were run in a very stable manner and could be controlieduntil the dead-weight of the beam broke t into pieces.

The crack trajectories are presented in Fig. 2.7. They are curved andin accordance with the tests of Ingraffea [4].

BEAH 1(S \DE. 1 J ~ DBEA 1 :2SIDE. 1 AwD

2

2

/

Fig. 2.7 Experimental crack trajectories


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Figs. 2.8 2.11 show the tested beams af ter failure.

Fig. 2.8 eam 1 af ter testing

Flg. 2 9 eam 2 af ter testing


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Fig 2 10 Crack surface of be m

Fig 2 11 Crack surface of be m 2


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3 NUMERIC L STUDY3 1 Finite element modelThe adopted material model is based on nonlinear fracture mechanicsThe element mesh is shown in Fig 3.1.

a)

B76

Lb)

Fig. 3 1 al Finite element model of the beamb Linkage elements


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Plane linear elastic three- and four-node elements together withtwo-node linkage elements were used.

A fracture zone with the same trajectory as the crack in the tests,was modelled with the linkage elements.The linkage element can be visualized like in Fig. 3.2. t has nonlinear material properties. The 1-direction is normal to and the 2-direction, the shear-direction, is parallel to the fracture zone. Thematerial properties are different in the two directions. Fig. 3.3describes the constitutive relations for the linkage element. There isalso an interaction between the stresses in the 1- and 2-direction, sothat when the peak stress is reached in the tension-direction, thefracture stressline 2 in the shear-direction is lowered Fig. 3.4.

2

Fig. 3.2 Linkage element [3

Fig. 3.3 constitutive relations for the fracture zone


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A

18

8S

Fig. 3.4 Lowering of the fracture stressline 2 in the shear-direction[3]

The linkage element has been developed by Kent Gylltoft and is furtherdescribed in [3]. The linkage element is implemented in a specialversion of the f inite element program FEMP [10].The fOllowing material parameters were used:E = 30 GPaf t 3.0 MPaf s = 3.0 MPaf s 5.0 MPa upper part of the beam where high compressive

stresses are presentEU 20Gf 150 N mThe width of the fracture zone was 10-5 m The loading of the beam wasperformed by stepwise moving the point C in Fig. 3.1 downwards.


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l

1.6E 02

1.2

0.8

C 0.4...J

19

3.2 Results from the f inite element calculationsThe numerically obtained curve of load versus CMSD is shown in Fig.3.5. The solution is stable up to peak load but the softening part isunstable and gives numerical problems.

0.4 0.8 1.2 1.6E 01CR CK MOUTH SL OING DISPL CEMENT CMSD (mm)

Fig. 3.5 Load versus CMSD curve from the finite element analysisThe curve of load versus CMD from the FEM calculation is in Fig. 3.6compared to the test results from beam 1. There is a good correlationin the prepeak part of the curves and the peakload differs only 15 \but the model gives a too ductile post-peak behaviour.


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o

1 6E 02

1 2

0 8

C 0 4J

20

FE ANALYSrS

EXPERIMENT BEAMl

0 4 0 8 1 2 1 6 2.0 2 4 2.8E 0lCRACK MOUTH DISPLACEMENT MD mm)

Fig 3 6 Load versus CMD curve from the finite element analysisWe shall now look at the stresses in the fracture zone at differentload leveIs The location of the linkage elements 1.5 shown in Fig 3 1band in Fig 3 7 the tensile stresses in the elements 1 to 5 are shownas a function of the CMSD.


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roCl'UUUJo::U

zUJUJ-1UJ

4. OE+06

3. O

2 0

1 0

21

2 3 4

1. O 2.0 3.0 4.0 5.0 6.0 7. OE 02CR CK MOUTH SUDING DISPLACEMENT (mm)

Fig 3 7 Tensile stresses in elements 1 to 5

The figure shows that the crack in the FEM model developes as a ten-sile crack Af ter peak load for MSD = 0 07 mm element number 4 hasreached the maximum stress 3 MPa. The stress in element 1 has de-creased to 0 8 MPa.

The elements 6 8 are subjected to large compressive stresses togetherwith shear stresses The shear strength is therefore raised to 5 MPafor the three elements Fig 3 6 shows the shear stresses in the ele-ments. No fracture occurs in these elements during the calculation


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4. OE 06

3 O

' 'e: 2. Oflfl

UJo:::fflf:zUJ:::E:UJ 1. O--1UJ

7

6

8

1 0 2.0 3.0 4.0 5.0 6.0 7.0E-02CRACK MOUTH SLIDING DISPLACEMENT mm)

Fig. 3.8 Shear stresses in element 6 to 8Plots of deflections and principal stresses at three different loadlevels are shown in Appendix 1.

4. nJSCUSSrON

The numerical study shows that the fracture of the beam can bemodelled as atens i le fracture. The model gives a good prediction ofthe peak load and the crack growth at different load levels can bestudied.

When the peak load is passed the model gives a too ductile solutionand i t soon runs into numerical problems. What can then be improved,to get a better result?A finer element mesh will probably give smoother load-deflectioncurves. The crack is now modelled with only 8 linkage elements.


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The chances of getting a more stable post peak curve had probably beengreater if the loading was performed by giving desplacement control tothe nodes on the top of the concrete beam instead of a prescribeddisplacement at. t.he loading point of the steel beam. The steel beamcannot be modelled infinitely st i f f and if the steel beam is weakcompared to t.he concrete beam the solution af ter peakload will neverconverge.Fig. 4.1 shows the experimental load deflection curve on the secondbeam at the loadpoint C in Fig. 4.2. The deflection was measured withtwo LVDT:s attached to a steel frame placed on top of the beam. Theload deflection curve therefore shows the deflection related to a linebet.ween point A and B in Fig. 4.2.

1.6 ~ - - ~ - - - r - - ~ - - - - r - - - ~ - - - r - - ~ - - - - ~ - - ~ - -E+02

1.2 l ~ .

O.B

2or l 0.4


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Fig. 4.2 Definition of deflection

What is then the real deflection of the beam? If we look at thenumerical study of the beam and calculate the downward movement ofpoint A and B at the peakload of beam 2/ point A has moved downwards adistance of 0 026 mm and point B a distance of 0 006 mm When the loadaf ter peakload decreases again point A and B will start moving up-wards The real load-displacement curve will then be like in Fig. 4.3.

1.6 r---r--- - - - - . . - - - r - - - r--- - - - - . - - - r - - - r---E+02

1.2

0.8

zc' : 0.4'...J

1 0 2 o 3 o 4. DE DlDEFLECTIO mm)

Fig. 4.3 Modified load deflection curve


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The post peak part is now much steeper. his shows that even i theloading was performed by giving displacement controI to the nodes onthe concrete beam i t would be diff icult or impossible to get theright p o ~ t p e a k solution.


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APPENDIX

This appendix shows plats of the displacements and the principalstresses at three different load levels in the F E ~ c a l c u l a t i o n .

Fig A 1 1 Displacements P=72 kN

L ~ L ____________________ ~Fig A 1 2 Displacements P= 32 kN

Fig A 1 3 Displacements P= 37 kN CAFTER PEAKLOAO


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L, \

~ \

\

~ \

//

f

ffI//

II//

27, \ -/ / \ - , ,. I I \ ,

I f l \ \ ,f f f I I \ \ \f f I f I \ \

f I f I II I

I

I

j 1 ~II ~ \ \ ,

I i If \ 1 \ \ 'I f f I I \;>I

f

ffII

/

II

I f / I \I f I f If f I

f;>I

f >


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P=72 kN

Fig A 1 7 Principal stress@s

P= 32 kN

Fig A 1 8 Principal stresses

P= 37 kN CAFTER PEAKLOAO

Fig A 1 9 Principalstresses


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REFERENCES

[1] Wittman, F.H. Ed. 1985. Fracture mechanics of concreteElsevier Netherlands.

[2] Wittman, F.H. Ed. 1985. Proc. of International Conference onFracture Mechanics of Concrete Elsevier Science Publ.Amsterdam, Netherlands.

[3] Gylltoft K. 1983. Fracture Mechanics models for fatigue inconcrete structures. Doctorai Thesis 1983:25D, Division ofStructural Engineering Lule University of Technology, Lule.

[4] Arrea M and Ingraffea A.R. 1982. Mixed-mode crack propagationin mortar and concrete. Report No. 81-13, Department ofStructural Engineering Cornell University New York.

[5] Bazant Z.P. Pfeiffer P.A. Shear fracture of concrete.Materials and structures Vol. 19, No 110, pp 111-121.

[6] Glemberg, R. 1984. Dynamic analysis of concrete structures.Publication 84:1 Department of Structural Mechanics, ChalmersUniversity of Technology, Gteborg.

[7] Rots J.G. 1985. s t r a i n s o f t e n i n ~ analysis of concrete fracturespecimens. Proc. Int. Conf. on Fracture Mechanics of ConcreteEd. F.H. Wittman), Elsevier Science Publ. Amsterdam,

Netherlands.

[8] Oldenburg, M 1985. Finite element analys is of tensilefracturing structures. Licentiate Thesis 1985:010L, Division ofComputer Aided Analysis and Design, Lule University of Technology, Lule.

[9J De Borst R 1986. Non-linear analysis of frictional materials.Dissertation Delft University of Technology, Delft Netherlands.

[10J Nilsson L. and Oldenburg, M 1983. FEMP - An interactivegraphic finite element program for small and large computersystems. Technical Report 1983:07T, Lule University of Technology, Lule.


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SMEARED CRACK ANALYSIS OF CONCRETE USING A NONLINEAR FRACfUREMODELbyOla Dahlblom and Niels Saabye OttosenDivis ion of St ruc tu ra l Mechanics Lund Ins t i tu te of TechnologyBox 118 S-221 00 LUND SwedenSUMMARY

The smeared crack approach i s adopted and to circumventthe problem of lack of object iv i ty th is smeared approach i sbased on a re in terpre ta t ion of the f i c t i t ious crack model ofHil le rborg e t a l . This re in terpre ta t ion i s based on thein t roduc t ion of the essent ia l quant i ty the soca l led equivalentlength which is a purely geometr ica l quant i ty determineden t i re ly by the s ize and form of the element region ofin te res t . Following the same concept t turns out to beposs ible to obta in an object ive descr ip t ion of the shears t i f fness along the crack p l n e ~ which depends on the s ize andform of the element region in quest ion and the elongat ionacross the i n f i n i t e ly th in crack zone.

The theory is implemented in a f in i t e element program andre su l t s obtained for the response of a concrete tensionspecimen support the appealing fea tures of the proposedapproach.INTRODUCfION

Modelling of cracks in concrete canaccomplished by two dif ferent approaches:smeared approach.in p r inc ip l e bea discre te and a

The d i sc re t e approach Ngo and Scorde l i s [ l J and Nilson[2J i s physica l ly appealing as i t re f lec t s the loca l izedna ture of cracking but i t s numerical implementation i shampered by the need for l e t t i ng the cracks fol low the elementboundar ies which in turn requires the in t roduc t ion ofaddi t ional nodal points . Al ternat ively the cracks can beallowed to develop in arbi t rary direct ions and therebyin te r sec t ing the elements but th is approach requires aredef in i t ion of the or iginal mesh arrangement. Obviously t h i sredef in i t ion complicates the approach s igni f icant ly even thughautomatic procedures for the r edef in i t ion of the mesh have beendeveloped Ingref fea and Saouma [3J .


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Computationally, the smeared approach is much simpler. aspossible s tra in and displacement discontinuit ies introduced bythe cracking are ignored and attention is given only to thechanging abi l i ty to t ransfer st resses across the crack plane.Consequently, only the const i tut ive model expressed in terms ofstresses and strains needs to be modified by the appearance ofcracks. This modification is performed in a smeared manner overthe element region of in teres t and i t therefore leads to a veryconvenient numerical scheme, original ly proposed by Rashid [4J.

However, this original smeared concept exhibi ts anessential shortcoming, as demonstrated by Bazant and Cedolin[5J. For a completely br i t t l e behaviour together with a simplestrength cri ter ion to detect the in i t ia t ion and propagation ofcracks, the loading that results in propagation of cracks in astructure loaded in tension, depends entirely on the mesh size.This socalled lack of object ivi ty also implies that the totalenergy dissipated by the c racks approaches zero, when theelements become inf inite ly small.

Significant improvements of the modelling of cracks wereprovided by the crack band model of Bazant and h [6J and bythe f ic t i t ious crack model of Hillerborg e t al . [7J. Whereasthe crack band model describes the bi l inear behaviour of a.crack by three parameters ( tensi le strength U t fracture energyG , and size of process zone w a l l considered to be materialc cparameters, the f ic t i t ious crack model is a two-parameter model(at,Gc )

Moreover, whereas the crack band model of Bazant and h[6J is essentially a smeared approach. the f ic t i t ious crackmodel of Hillerborg et a l . [7J is original ly a discreteapproach. Nilsson and Oldenburg [8,9J pioneered the idea ofmaking a smeared version of the f ic t i t ious crack model without,however, being able to obtain entirely satisfactory results ,Oldenburg [9J. Also the la ter composite fracture model ofWillam [lOJ bears similari t ies with this work. Pursuing thisconcept and following the l ines of [ l l J the f ic t i t ious crackmodel has been reformulated in [12J so that i t can be appliedin a smeared manner. This reinterpretation of the f ic t i t iouscrack modelleaves i ts predictions ent i re ly unchanged, but inorder to obtain a smeared interpretation i t becomes necessaryto introduce a socalled equivalent length. This implies thatalso the smeared version of the f ic t i t ious crack model becomesa three-parameter mode l , but i t turns out that the equivalentlength is a purely geometrical quantity determined ent i re ly bythe form and size of the element region of interest . whereasthe corresponding quanti ty in the crack band model of Bazantand h [6J, the size of the process zone w , is claimed to be acmaterial property.

In the present paper. the smeared version of the


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f ic t i t ious crack model [12J is presented. Finally some resultsfor tens i le t es t specimens are presented.FICTITIOUS CRACK MODEL

The f ic t i t ious crack model of Hillerborg e t a l . [7J takesas i t s basis the experimentally observed fact that cracking isa discrete phenomenon and that cracking is not completelybr i t t l e but rather is characterized by a softening effectcaused by cohesive stresses in the microcracked region. For aconcrete bar loaded in tension into i t s post-peak region thef ic t i t ious crack model describes cracking as e las t ic unIoadingalong the ent i re length of the bar and an addit ional elongationoccurring in an inf ini te ly thin cracked zone. I t is of interes tthat the f ic t i t ious crack model does not rely upon a . re la t ionbetween s tresses and strains as the behaviour of the inf ini te lythin cracked zone is described by a const i tu t ive relat ionexpressed in terms of stress and elongation . This descriptionwas original ly suggested using experimental evidence only buti t is of interes t that i ts acceptance can be based on purelythermodynamical arguments [11J.

Following the discussion above., the f ic t i t ious crackmodeldescribes the post-peak behaviour of the uniform andhomogeneous concrete tension bar of length L as i l lustrated inFig. l , where for simplici ty a l inear relat ion between s t ressand crack elongation w is adopted.

LRegion

2

Region 2}

Fig. 1 - Fict i t iouscrack modelof Hi l lerborge t a l . [7J

The resulting elongation u for the entire bar becomes

where E = Young s modulus , t = tensi le strength and N is theslope of the a-w curve. Tension and elongation are consideredposi t ive quant i t ies . The total energy dissipated Dt , when thestress eventual ly has dropped to zero is re la ted to the area


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under the a-w curve of Fig. 1, i .e .Le. 2)

where A is the cross sectional area of the bar and G denotescthe cr i t ical fracture energy. Observing that o = a lN, cf.Fig. l we obtain from Eq. 2

Moreover, defining the quantity A by2G EcA = 2a t

we obtainEN = -

3)

4)

5)I t should be observed that A the socalled characterist iclength, is a material parameter having the dimension of length.DEFINITION OF EQUIVALENT LENGTH

The previous equations can be reinterpreted in the spir i tof a smeared crack approach by introducing an equivalent s tra in, uniformly distributed on the bar length, Le. c = u/L.From Eqs. 1 and 5 we obtain

6)The tangential st i ffness modulus, ~ then becomes

da EET = A1 - [As the bar is assumedelongation, ET must beres tr ic t ion

to be subjected tonegative, providing

7)

an increasingthe following


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L < A (8)Equations 6 and 7 clear ly show tha t the post-peakbehaviour not only depends on the material parameters E and Abut also becomes size-dependent through the appearance of theterm L. This effect i s i llus t ra ted in Fig. 2 and in accordancewith experimental ev dence, i t appears that the longer the bar,the more br i t t l e the response. However i t is of importance toreal ize, that irrespective of the length L the descript ion ofthe post-peak behaviour given through Eq. 6, implies that thecorrect total energy = AG will always be dissipated when the. c

s t ress eventual ly has dropped to zero.

1\J I \ ,J I \ ,J I \J I \ Al L =3I \ 'J I \ =/ =1=0.5 \

I \2 3

ELat

Fig. 2 - Behaviour of bar for different AIL-ratiosStar t ing from the original discrete approach of the

f ic t i t ious crack model, we have arr ived a t an al ternat ivedescription given by Eqs. 6 and 7, which provides exactly thesame predict ions. I t appears, however, that Eqs. 6 and 7 yielda const i tut ive relat ion expressed in terms of s t ress ands t ra in This means that Eqs. 6 and 7 can be used in a smearedcrack approach. I t is important, however, that to achieve thisformulation, we have introduced a length L into theconst i tut ive relation.

For the onedimensional case considered, this length L issimply the length of the bar considered. If more onedimensionalf ini te elements were used to model the bar, we must for theelement, which exhibits cracking, use the const i tutive relat ionEq. 6 where L now should be the length of tha t element.Consequently, we are led to introduce a socalled equivalentlength, L , into the const i tutive relat ion Eq. 6. For theeqonedimensional elements considered, this equivalent length issimply the length of the cracked element normal to the crackplane. This observation suggests the following def ini t ion ofthe equivalent length applicable to al l elements: theequivalent length, L , of a f ini te element i s the maximumeqlength of the element region of in teres t in the directionnormal to crack plane. This def ini t ion is i l lus tra ted in Fig. 3for the 3-node tr iangle and the 8-node isoparametric elementwith 2x2 Gauss point integrat ion.


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Fig. 3 - Definit ion of equivalent length LeqNote that for the 3-node t r iangle, the element region ofin terest is obviously the total element region, whereas for the8-node element, the element region of in terest is the regionrelated to the Gauss point in question. The above def ini t ion ofequivalent length applies for three-dimensional elements asweIl.

SMEARED CONSTITUTIVE EQUATIONS - PLANE STRESSHaving established the definit ion of equivalent length forarbi t rary f ini te elements wi th arbi t rary crack direct ions, weshall now generalize the smeared consti tut ive relat ionsdiscussed above.For simplicity we shall assume l inear e las t ic behaviourand eraeking under plane s t ress c ond i tions (consideration to

plast ic effects and other s t ress s ta tes is s t ra ight forward).Assume that a crack plane develops normal to the x-axis.In accordance with the experimental ev dence that nonecking effects are related to the c r c ~ development, thedisplacements in the inf ini te ly thin eraeked zone are assumedto be purely el ongat i ona i . e . no Poisson effect is present.Therefore, the s tra in in the y-direct ion is determined by theusual Hooke s law. In the x-direction, results in a s tra inx

component given by Eq. 6, whereas provides a s tra inycomponent in the x-direct ion equa l to the usual elas t iccontribution. The above observations lead to

E l O A l= A v c r o txl v - l-AlL O eqL v c A vy eq eq y l v - Y L Oxy l v A/L xy eqO O eqGE ef (9)


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From Eq. 9, we have the relat ion T = G ,where G f =y e y eeffect ive shear modulus. The determination of Gef is a subjectof much controvercy. see for instance [13J. I t s inclusionmakes i t possible to simulate shear s t i f fness along the crackplane, i . e . aggregate inter iock. Usually, one writes Gef = aGwhere G = e las t ic shear modulus and a = shear retention factor

O ~ a ~ l ) . Of ten, the a-factor is simply given a constant value,typically a = 0.1-0.4. and many numerical experiments have beenperformed in order to evaluate the effect of differenta-values.In real i ty the a-value must be a function of crack width,relat ive displacement tangent ial to the crack plane and of thenature of the crack surface. Also, the a-value is expected todecrease with increasing crack width so that for small widths,

a is close to uni t y and for large widths, a is close to zero.Moreover, jus t l ike the tangential slope of the softeningbranch, ET the a ~ f a c t o r must also depend on the s ize of thef ini te element in question in order that the total shearbehaviour of a concrete specimen does not depend on the numberof elements.

The object ives above can be achieved in a manne rcompletely analogous with the derivation of Eq. 6 by assumingthe relat ive displacement tangential to the crack plane, w .srelated to the inf ini te ly thin cracked zone to be determined by

ww = TS G ys 10)where the constant G = s l ip modulus. In the f ini te elementsoutside the infini tely thin cracked zone, the usual e las t icshear relat ion T = is assumed to hold. The assumptionsy yabove imply that

G f =e 1 G 11)G wG Ls eqwhere w as before denotes the crack elongation normal to thecraek plane and where L is the equivalent length of theeqf ini te element region in question. The simple relat ion, Eq. lO,provides elose agreement with the experimental results ofPaulayand Loeber [14J when G = 3.8 MPas

Eq. 9 is wri t ten in a total form, but for numeriealpurposes the eorresponding ineremental formulation might beadvantageous. Noting that Gef is not a eons tant , but depends on


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w which is given by w = ox-Ot) IN cf. Fig. 1, we obtain from.Eqs. 9 and 11 af ter some algebra

[WX E 1 O dE.2 A xdo l-v - l-AlL O dE.y eq 2 2 eq 2 ydT G fA Y G f AV l-v -AlLxy e xy e xy eqG d'YG L E GL E E ef xys eq s eq12)

1t is of interest that whereas Eq. 9 is symmetric, Eq. 12 isnot, Moreover Eq. 12 clearly shows that the incremental shearstress is coupled not only to the incremental shear strain, butalso to the incremental normal strains.FINITE ELEMENT RESULTS

The theory proposed above has been implemented into afini te element program and some resul ts for the behaviour ofthe concrete tension specimen shown in Fig. 4 will be givenbelow. Due to symmetry. only one quarter of the specimen willbe considered. The dimensions of the specimen are taken fromthe experimental investigation of Petersson [15J.al y bl

Sissipation

oM

u50

Fig. 4 - a) concrete specimen of Petersson [15J, dimensionsare in mm A = cross sectional area; b) calculatedtotal force - total elongation diagram shown inprinciple

The f ini te element solutionapproach and the following scheme follows a tangentialmaterial parameters l lrepresentative for concrete were adopted: E == 0.2, 0t = 3.3 MPa G =130 N/m G = 3.8 MPa.c s

42.1-10 MPa

By prescribing equal, increasing displacements along theupper and lower surface of the specimen, the specimen is loadedinto the post-peak region unti l i t has disintegrated


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completely. In r ea l i ty , cf . Petersson [15J, t h i s d is in tegra t ioncorresponds to the development of a s ingle crack having thearea A as shown in Fig. 4a) .The purpose of the ca lcula t ions i s not to compare r esu l t s with

e ~ e r i m e n t a l evidence, but ra ther to document tha t thesuggested smeared approach i s objec t ive in the sense tha t fordecreasing f i n i t e element s ize the to ta l energy di s s ipa t ed dueto eraeking approaches the cor rec t value. By def in i t ion , thecor rec t t o t a l diss ipa ted energy i s AG, where G i s thec cprescr ibed f r ac tu re energy = 130 N/m. From the calculat ionr esu l t s , we can construct the to ta l fo rce- to ta l e longa t iondiagram as shown in pr inc ip le in Fig. 4b . The ca lcula teddiss ipated energy i s obtained from the area under t h i s curveand by dividing by the cross sec t iona l area A we can determinethe calculated f rac ture energy G l For decreasing s izes ofc ,ca .the f i n i t e elements , G l should converge towards the exactc ,cavalue G ,cobjec t ive .

i f the suggested smeared approach should be

Using a 3-node t r iangular element and a 4-nodeisoparametr ic element, the f i n i t e element mesh of dif ferentmodels i s shown in Fig. 5. The crack pa t te rn , when the specimenhas .d is in tegrated completely, i s a lso shown.

Mesh A Mesh B Mesh C Mesh D

Mesh E Mesh F Mesh G Mesh H

Fig. 5 Fin i t e element meshes and crack pat ternsBecause the s t r es ses and s t r a ins in the constant s t ra in

element are evaluated here simply a t the cent roid of theelements, the ca lcula ted crack d i rec t ion might d i f fe r from thet rue hor izonta l direct ion. This implies tha t shear ing along the


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crack plane will contr ibute to the load bearing capacity. I t i sof in teres t that even for the tension specimen considered thismobilization of the shear s t i f fness might be quiteconsiderable.

To i l lustra te this mesh C of Fig. 5 was used incombination with the usual simplified assumption for the shears t i f fness along the crack plane i e. Gef = aG where a isconsidered to be a constant. For typical a-values = 0.01 0.1and 0.4 the obtained force-elongation diagram is shown in Fig.6. I t appears that even for this simple tension specimen is anaccurate description of the varia t ion of the shear modulusGef very essent ial . Consequently in a l l the remainingcalculations the consistent approach proposed by Eq. 11 wasapplied.

ForcelNl

lsplacement Im

Fig. 6 - Force-displacement curves for mesh C of Fig. 5 withGef = aG where a is considered to be a constantThe resul t ing force-displacement curvesshown in Fig. 5 are shown in Fig. 7 and

improvement compared with Fig. 6 i s obvious.forthe the meshess ignif i can

The result ing calculated fracture energy G l is shownc cain Fig. 8 as function of the number of degrees of freedom. I tappears that the calculated fracture energy converges veryclosely to the exact value G documenting that the adoptedcsmeared approach is objective. Moreover we observe a dramaticimprovement of the resul ts as compared with the correspondingresul ts of Oldenburg [9J.


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Force {NI Force (N)_lO

\ .50

Dlsp ocemen1 Im) Dlsplocem nt Iml.00

Fig. 7 - Force-displacement curves for the meshes shown inFig. 5

Gc,caIIN/m)

160

150

O

Fig. 8 - Calculated fracture energy forthe meshes ofFig. 5 as functionof number ofdegrees of freedomDOF)

The resul ts shown above cannot be claimed to be ofcomplete generali ty as the cracks mainly are paral le l to oneof the element boundaries. In a general f ini t elementanalys i s , cracks will have arbitrary direct ions with theelement boundaries.To investigate this aspect the highly distorted meshes

shown in Fig. 9 were adopted. The crack pattern when thespecimen has disintegrated completely is also shown.


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Mesh Mesh J Mesh K Mesh L

Mesh M Mesh N Mesh O Mesh P

Fig. 9 Highly distorted f in i te element meshes and crackpatternsThe result ing force-displacement curves are shown in Fig.10. I t is apparent that due to the low order of the t riangularelement the distorted meshes have. dif f icu l t ies with reflecting

the correct deformation mode. Even though the response shown inFig. 10 is inferior compared with the resul ts shown in Fig. 7i t is of interest that a smooth convergence occurs. Incalculat ing the fracture energies of Fig. 11 the ta i l of theforce-displacement curves have been replaced by straight l inesas s h o ~ T in Fig. 10.

orceiNIl

1.50

Fig. 10 Force-displacement curves for the distorted meshesof Fig. 9


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Gc col lNlml

DO

L

150

100

0.02 Q Ol 006 (NOOFl

Fig. 11 - Calculated fracture energies for the distorted meshesof Fig. 9 as function of number of degrees of freedom(DOF)

The obtained resul ts demonstrate that the presently usedsmeared approach is object ive. For distorted meshes oftr iangular elements the estimation of the fracture energy ishowever. a bi t high. This is a resul t of the inabi l i ty of theelement type to mode l the inhomogeneous s t ra in dist r ibut ioncorrectly. when highly distorted meshes are used. I t may beconcluded that elements of low order is unfavourable inconnection with smeared crack analysis . This is also supportedby resul ts obtained with eight-node isoparametric elements.

ON LUSIONSThe f ic t i t ious crack model of Hillerborg e t a l [7J isoriginal ly formulated as a discre te model. In recogni t ion ofthe essent ia i numerical drawbacks related to such an approach.we have made a re interpretat ion of the f ic t i t ious crack modelin the sp i r i t of a smeared approach. The key to thisre interpretat ion is the introduction of the socalled equivalentlength being a purely geometrical quantity determined ent i rely

by the size and form of the element region of interes t . Thein t roduct ionof the equivalent length implies that irrespectiveof the size of the f ini te element. the dissipated fractureenergy remains. in pr inc iple . the same. Following the sameconcept. i t turns out to be possible to obtain an object ivedescription of the shear st i f fness along the crack plane. whichdepends on the size and form of the element region in questionthrough the equivalent length and which also depends on theelongation across the inf ini te ly thin crack zone.

Through f ini te element calculations of a tension concretespecimen. the theory was demonstrated to be objective even forhighly distorted meshes . However. the resul ts obtained wi thdistorted element meshes indicate that a low order of theelement shape functions is unfavourable in connection withsmeared crack analysis.


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REFERENCES[ lJ Ngo, D., and Scordelis, A.C., Finite Element Analysis ofReinforced Concrete Beams , Journal of the American Concrete Insti tute, Vol. 64, No. 3, March 1967, 152-163.[2J Nilson, A.H.. Nonlinear Analysis of Reinforced Concreteby the Finite Element Method , Journal of the AmericanConcrete Insti tute, Vol. 65, No. 9, Sept. 1968, 757-766.[3J Ingreffea, A.R., and Saouma, V., Numerical Modelling ofDiscrete Crack Propagation in Reinforced and Plain Con

cre te , in Fracture Mechanics of Concrete: StructuralApplication and Numerical Calculation. Ed. by G.C. Sihand A. DiTommaso, Martinus Nijhoff Publ., Dordrecht, 1985,171-225.

[4J Rashid, Y.R., Ultimate Strength Analysis of ReinforcedConcrete Pressure Vessels , Nuclear Engineering andDesign, Vol. 7, No. 4, April 1968, 334-344.[5J Bazant, Z.P., and Cedolin, L., Blunt Crack Band Propagat ion in Finite Element Analysis , Journal of the Engineering Mechanics Division, ASCE Vol. 105, No. EM2 April1979, 297-315.[6J Bazant, Z.P., and Oh, B.H., Crack Band Theory forFracture of Concre te , Materials and Structures, RILEMVol. 16, 1983, 155-177.[7J Hi l lerborg , A. Modeer, M., and Petersson, P-E., Analysisof Crack Formation and Crack Growth in Concrete by Meansof Fracture Mechanics and Finite Elements , Cement andConcrete Research, Vol. 6, 1976, 773-782.[8J Nilsson, L., and Oldenburg, M., Nonlinear Wave Propagat ion in Plastic Fracturing Materials - A ConstitutiveModelling and Finite Element Analysis , IUTAM Symposium:Nonlinear Deformation Waves, Tallin 1982, Springer,Berlin, 1982, 209-217.[9J Oldenburg, M., Finite Element Analysis of TensileFracturing Structures , Licentiate Thesis 1985: OlOL,Div. of Comp. Aided Anal. and Des., Lule University,Sweden, 1985.[lOJ Willam, K.J., Experimental and Computational Aspects ofConcrete Fracture , Computer Aided Analysis and Design ofConcrete Structures, Pineridge Press, Swansea, UK 1984,33-70.[l1J Ottosen, N. S., Thermodynamical Consequences of StrainSoftening in Tension , to be published in Journal ofEngineering Mechanics, ASCE 1986.


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[12J Ottosen, N.S. and Dahlblom O.: Smeared crack analysisusing a nonlinear fracture model for concrete, NumericalMethods for Non-Linear Problems, Vol. 3, Edited by C. Taylor, D.R.J. Owen E. Hinton and F.B. Damjanic, PineridgePress, pp. 363-376, 1986.[13J Finite Element Analysis of Reinforced Concrete, ASCE

New York 1982.[14J Paulay, T., and Loeber, P.J . , Shear Transfer by AggregateInteriock Shear in Reinforced Concrete, Vol. l SpecialPubl. SP-42 ACI Detroit, Michigan, 1974.[15J Petersson, P-E., Crack Growth and Development of FractureZones in Plain Concrete and Similar Materials , ReportTVBM-1006 Div. of Building Materials, Lund Inst i tute ofTechnology, Sweden 1981.


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A DESCRIPTION OF CONCRETE AND.FRC MATERIALS BY MEANSOF COMPOSITE MATERIAL THEORY AND CONTINUUM DAMAGEMECHANICS

y

H. StangDepartment of St ruc tu ra l Engineer ingTechnical Univer s i ty of Denmark

Paper presented a t ordiskt MiniseminariumBetongens Brot tmekanik

LTH Lund den 6. november 1986.


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INTRODUCTIONIn te rms o f c o n s t i t u t i v e equat ions mater ia l s such as rock con-

-c re te and o t h e r cement based composite mater ia l s a re ch a rac t e r ized by t he f ac t t h a t they ex h ib i t a soca l l ed s t r a in sof t en ingbehavior pr io r to fa i lu re .Though t he re is still a grea t deal of di scuss ion on the v a l i d i -ty of s t r e s s / s t r a i n curves which inc lude s t r a i n sof t en ing seee .g . [1 ] it i s genera l ly agreed on t ha t f a i l u re of rock and cement based composite mater ia l s i s always governed by t he nuc lea t ion and growth of microvoids or microcracks [2 ] t i s a lso agreedon t h a t it i s t h i s darnage evolu t ion t ha t i s re spons ib le fo r t hen o n l in ea r i t y of t he s t r e s s s t r a in curves fo r the above mentionedmate r i a l s . Darnage evo lu t ion i s a lso re spons ib le for the development of a so -ca l l ed process zone ahead of t ens ion cracks in rockand concre te .In the ca l cu l a t i o n of e .g . concre te s t ruc tu re s loaded in t ens ionor bending darnage evolu t ion has been accounted fo r in a nurnberof d i f fe ren t ways. As two extremes one might m n ~ i o n the f r ac tu remechanical approach where t he process zone has been inc luded as af i c t i t i o u s crack [3] [6 ] and t he continuurn approach, see e .g .[7] [11] where t he darnage evolu t ion i s inc luded v ia a s t r e s s s t r a in curve with s t r a i n sof t en ing .Fur the r informat ion on the t rea t rnen t of s t r a i n sof t en ing and crackdevelopment can be found via the d e ta i l ed bib l iography in [9 ] .Though the co r rec t approach to damage model l ing in concre te s t r u c t u re s i s not gennera l ly agreed on, it i s ev iden t t h a t a continuurnapproach i s very su i t ab l e in descr ib ing darnage evolu t ion in f ib rere in forced cement based) mate r i a l s s ince we are dea l ing he re witha high ly homogenized kind of damage, and it i s poss ib le to def ineand exper imental ly exarnine a weI l def ined s e t of damage modes, e .g .matr ix c r ack ing f i b r e crack ing f ib re /mat r ix debonding, and so on[ 1 6] In t h i s paper t he continuum approach to darnage model l ing w i l l bei n v es t i g a t ed . We wi l l not dea l with t he s t ru c t u r a l l e v e l but witht he microscopic l eve l . Thus to be p rec i s e we wi l l i n v es t i g a t e t hep o s s i b i l i t y of der iv ing phenomenological c o n s t i t u t i v e equat ionswhich can be used in a continuurn approach to e .g . concre te . The


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too ls t h a t w i l l be used are the so-ca l led continuum damage mechani c s theory CDM), see e .g . [12] - [16] , and composite mate r i a l theory CMT), see e .g . [17] - [19] . CDM-theories cons t i tu te a group oft h eo r i e s which der ives loca l cons t i tu t ive equat ions in the co n t i nuum mechanics sense from t h ree d e f i n i t i o n s or l awsjassumpt ions :

1) A d e f i n i t i o n of i n t e rna l v ar i ab l e s which desc r ibe the damagemodes in quest ion.

2) Cons t i tu t ive equat ions involving the inf luence of the i n t e r na l damage v ar i ab l e s on t he overa l l mater ia l behavior .

3) Evolut ion equat ions which descr ibe the damage evolut ion asfunct ion of the appl ied load.

By applying composite mate r i a l theory the f i r s t and t he second problem of t he CDM-approach has been so lved in one s tep , as we s h a l lsee in the fo l lowing. Furthermore, due to the r a th e r genera l CMT swhich a re av a i l ab l e , t i s poss ib le to a) model damage with a 3Dmodel, b) model the e f fec t of an i so t rop ic damage, and c) model t hee f f e c t of damage geometry e .g . spheres contra microcracks , givena damage evolu t ion law) .Fin a l ly , t should be mentioned t h a t the r e s u l t s from 3D-compositemate r i a l theory can be reduced to 2D or 1D models, as t w i l l beshown in the case of the der iva t ion of the co n s t i t u t i v e equat ionfo r a unid i rec t iona l f ib re re in forced composite mate r i a l .

COMPOSITE MATERIAL THEORY FOR MATERIALS WITH VOIDS OR CRACKSF i r s t we w i l l summarize some of t he bas ic r e s u l t s of e l a s t i c CMTThese r e s u l t s a re g e n e ra l , i . e . they apply to any composite mate r i a l whether t conta ins i n f i n i t e l y s t i f f i nc lus ions or voids oreraeks . The r e s u l t s a re due mainly to H i l l [20] .The s t a r t i n g poin t for a 3D-composite mater ia l ana lys i s i s usua l lya so -ca l l ed r ep resen ta t i v e volume element , RVE, with t o t a l volumeV inc luding voids) and sur face orV) (which does not inc lude i n t e r na l su r faces on voids and/or eraeks) . On the s t ru c t u r a l l ev e l aRVEcan be thought of as a mater ia l po in t , however, on t he microscopicl eve l the RVE i s an element conta in ing many inhomogenit ies.


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In t he f o l lowing it i s assumed t h a t a l l phases matrix and in c lus ions a r e l i n ea r e l a s t i c and smal l s t r a i n e l a s t i c i t y i s usedth roughout the an a ly s i s .In order to i n v es t ig a t e t he behavior o f the RV two kinds of boundary condi t ions a r e presc r ibed . E i the r

o . . n. = n.1J J 1J J on V) , 1 )where o . . i s the microscopic s t r e s s f i e l d , n. i s the outward1J 1u n i t normal to V) and Lij i s a s t r e s s t ensor which i s con-stant:. in space .Or

on V) , 2)

where u. a r e t he microscopic d isp lacements , E . . i s a s t r a i n t e n -1 1J .sor cons tan t in s p c ~ and x i are the coord ina tes .The t ensor s 1J and E1J a r e c a l l e d t he macroscop ica l ly homo-geneous s t r e s s and s t r a i n , r e spe c t i ve l y , s inc e they represen t t hehomogeneous f i e l d s which would be i n t roduced if eg. 1) and 2)were pr e s c r ibe d on a homogeneous volume e lement .Denot ing by an overbar the volume average of t he s t r e s s and s t r a i nf i e l d :

-o . .1J

e . 1J =

it fo l lows

o . . =1Jand

e 1J =

V

o . . dV ,1J

f e 1J dVV ,from t he divergens

L1J given 1 ) ,

E . .1J given 2)

3 }

4)

theorern t h a t

5 )

6 )


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Furthermore, it fol lows from the pr inc ip l e of v i r tua l work t h a tthe volume average of s t r a i n energy densi ty , W, i s given by

w= er i j E i j dVgiven the boundary condi t ions 1) or (2) .Next we def ine the f ree energy dens i ty fo r the RVE as

= W er . . n . dA,~ J

7 )

8 )

where o V) denotes the p a r t of V) where the s t r e s s i s pre -pscr ibed . It i s read i ly shown, using the divergens theorem oncemore, t ha t

1 - - 1) ,2 CJij Eij given 9)1 - - given 2) 1 O2 er . . E~ ~

The s t i f fn e s s and compliance of the composite mat e r i a l a r e ingenera l defined by

- i j = Li j k l Ek l 11)given 1) or 2) 12)

where 1= Mi j k l

We w i l l not go i n to d e t a i l s about the determinat ion o f Li j k l orMi j k l j u s t mention t ha t there ex is t s a whole c la s s o f so lu t ionswhich are based on eg. 5) and 6) and on the solu t ion for ane l l i p so i d a l inc lus ion embedded in an i n f in i t e matr ix . Non-d i lu teconcen t ra t ions of e l l i p s o id a l voids or cracks can e .g . be d ea l twi th using Levin s method, [21] and [22] .The so lu t ion fo r the compliance, obta ined by using L ev in s method,can in general be wri t ten as

Mi j k l = ~ k l + S Hi jmn ~ ~ n k l 1 3)


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where supe rsc r ip t m re fe rs to the matr ix mater ia l , and B i s as ca l a r which descr ibes the crack densi ty or the volume concen t ra-t i o n of the voids , and where Hi j k l i s a four th order t ensor . Wewi l l consider three cases here :

i) Aligned, penny-shaped cracks: Here B i s given by3 = 471 33 n a

where n i s the number of cracks per volume anda i s the crack radius . The t ensor Hi j k l i st ransverse ly i so t rop ic . The e x p l i c i t express ioni s given in Appendix 1.

i i ) Randomly d i s t r i b u t ed penny-shaped cracks: Here 3i s aga in given by eg. 14), but Hi j k l i s now i so -t rop ic . An express ion i s given in Appendix 2.

i i i ) Spher ica l vo ids : Here 3 i s given bys =

where c denotes the volume concentrat ion of thevoids . The H-tensor i s again i so t rop ic and an express ion i s given in Appendix 3.

14)

15)

Note, f ina l ly , t ha t with the composite mater ia l compliance givenby eg. 13) the corresponding s t i f fn e s s i s given by

1 6)

with1Ri j k l = I i j k l + S Hi j k l ' ( 1 7 )

where I i j k l denotes the four th order i den t i ty t ensor .A complete ly d i f f e r e n t approach to the determinat ion of compositemater ia l s t i f fn e s s and compliance i s the. composite mater ia l elementapproach. In t h i s approach the s t i f fness jcompl iance of a s ing leinhomogenity surrounded by a su i tab le amount of matr ix mater ia l i sdetermined by sub jec t ing t h i s mater ia l element to appropria te bound-


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ary condi t ions . Fi n a l l y , the s t i ffn es s / co mp l i an ce o f t he composi temater ia l element i s equated with the s t i f fn es s / co mp l i an ce of thecomposi te mat e r i a l as a whole.Using t h i s approach the 1D ax ia l compliance of a f i b r e r e in fo rcedcomposi te mat e r i a l conta in ing u n i d i r ec t i o n a l i n f i n i t e l y long f ib re sand t ransverse mat r ix cracks along with f i b re / ma t r i x debonding hasbeen determined. The so lu t ion can be w r i t t en as (E, M, and cr denot ing sca r Ia r s ) :

s == M a ( 18

with-bR., bErn d IE f + ~ + t ( 19

where b denotes matr ix th ickness , tfibre th ickness , R. l eng thof non-debonded i n t e r face , and d l eng th o f debonded i n t e r face . Seea l so f ig . 1. Ef and Em denote Young s modulus fo r the f ib re s andthe matr ix r e s p ec t i v e l y , and n denotes the number o f t ransverseera eks per l eng th of the f ib re . Given n and d where

d < 2n (20)we have

(21 )

and AR. i s determined according to

(22)

Here \ rn and \ f denotes Poisson s r a t i o fo r the matr ix and thef ib re s , r e s p ec t i v e l y . The r a t i o b / t descr ibes the volume concen t r a t i o n of the f ib re s s ince

tb t == c

f (23)

Once M has been determined, the s t i f fn e s s o f the cornposite mat e r i a l can be determined from


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1L = 24)MThe so lu t ion 19) has been derived by means of a shear lag typeof ana lys is . The d e r i v a t i o n i s out l ined in Appendix 4.Note, f i n a l l y t h a t eg. 18) i s the one-dimensional eguivalent toeg. 12) . t i s c lea r t h a t there e x i s t one-dimensional eguivalentsto egs . 1) - 11) as weIl .

b. .E 2t

b

L ~ A /

Fig. 1. The composite mat e r i a l element represen t ing the f ib rere in fo rced mat e r i a l with matrix cracks and f ib re /ma t r ixdebonding.

THE DM PPRO CH

In the previous sec t ion some express ions for the de terminat ion o fthe s t i f fn e s s and compliance of d i f f e r en t mater ia l s with voids o rcracks are given, along with an express ion for the f ree energydens i ty r e la ted to the volume e lement under cons idera t ion . Theexpress ion for the f ree energy dens i ty depends on the type o fboundary condi t ions prescr ibed to the RVE.In terms of darnage desc r ip t ion the D - mater ia l with voids o rcracks are descr ibed by a non-dimensional darnage parameter Swhich can be s p l i t i n to two non-dimensional darnage parameters1 2 In the case of penny-shaped cracks we would ge t


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25 )

and in t he case o f s p h e r i ca l voids we would g e t

fa 1 a 2 < 1a 1 a 21 - a 1 a 2 , 26 )

where in both casesa 1 = k)3 n , 27)

41T ~ ) 32 = 3 28 )where k i s a c h a ra c t e r i s t i c l eng th and a denotes t h e r a d i u s ofe i t h e r crack or vo id .The damage of the f i b r e r e in fo rced ma te r i a l was desc r ibed by twodamage paramete r s , n, number of c r acks pe r l en g t h , and d , l eng thof t he debonded f i b re jma t r ix zone. DimensionIess vers ions o f t heseparameters a re e a s i l y in t roduced by w ri t i n g

da 1 = t 'a = n t .2

See eg . 1 9 .Assuming t h a t a 1thermodynamic fo rce

A =y

andAY

29 )30 )

a 2 a re independent i n t e rn a I v a r i ab l e s , acan be def ined as

y = 1 2. 31)This thermodynamic fo rce can be i n t e rp re t ed as a D energy r e l ea s er a t e .With the 3D composite mate r i a l theory the damage fo rce can bew r i t t e n as

AY1= 2

given t he boundary condi t ion 1) , o r as

y = 1, 2 32 )


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y = 1, 2 33)given the boundary condi t ion 2) .Simi lar express ions can be given fo r Al in the case o f the onedimensional descr ip t ion of the f i b r e r e in fo rced composi te mat e r i a l :

AYwhen a

Al

when a

1= 2macroscopic

= _ 1 E2 aL2 aaymacroscopic

34)

s t r e s s I: i s p resc r ib ed , o r

35 )

s t r a i n E i s prescr ibed .The l a s t s t ep in t h e CDM-approach i s to def ine a damage evo lu t ionlaw. We chose here the s imples t poss ib le , the s o ca l l ed b r i t t l e law.For a given . admissible) damage s t a t e , al admiss ib le s t r a i n o rs t r e s s s t a t e s are def ined by a co n s t an t pos i t ive sca l a r func t ionf , such t h a t

f lo. < 1.y = (36)Fur thermore , we assume t h a t damage growth follows an evo lu t ion lawwhich can be wri t t en as

day = (O, O)

day

day

= O, O)

df= o dAy

when f (Al ) < 1f A y = 1when

af dE ij~

f Ay = 1when

af dE . .~

(37a)

37b); O a f dL < Or a I: ij ~

37c)> O df dL > Or L ~

depending on t h e boundary condi t ions . In eq. 37c) o i s given by


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38a)

o = ~ ~ ) - 1 _ af )d a dE ij~ (38b)depending on the p rescr ibed boundary condi t ion . S t r i c t l y speakingegs . (37) and (38) apply to the 3D case . In our D case thel a s t term on t h e r i g h t hand s i d e i s s i mp l i f i ed s ince the ind icesi j van ish .We a re now, i n p r inc ip l e , ready to mode l any s t r e s s - s t r a i n h i s s to ry and i n v es t i g a t e the in f luence of darnage descr ip t ion andevo lu t ion laws. We can e .g . s t a r t wi th some i n i t i a l darnage s t a t e

1 , 2 (which might be (0, O)}, and. then s t r a i n the RVE. Atsome p o in t in the load hi s to ry we w i l l reach f = 1 and thedarnage 'evolu t ion w i l l s t a r t . Addi t iona l s t r a i n i n g w i l l r e s u l t indamage evo lu t ion such t h a t f 5 1 accord ing to eg. (38b). Thed i r e c t i o n ( in the a 1 a 2 - plane) in which the darnage t akes placedepends on the i n i t i a l darnage s t a t e and the f - func t ion accord ingto eg. 3 7c) .Resul t s a re most e a s i l y obta ined by using a nurnerical scheme whichmodels the load h i s t o ry by means of incrementa l s teps . Note int h i s connec t ion t h a t if the darnage evo lu t ion r e s u l t s in a descending s t r e s s - s t r a i r i curve, the n the load h i s t o ry must be desc r ibedin terms o f s t r a i n and not s t r e s s , in o th e r words o must bepos i t ive .

SOME EXAMPLESTo i l l u s t r a t e the method descr ibed above, we have p resc r ib ed au n i ax i a l t e n s i l e s t r a i n f i e l d to aRVE conta in ing spher ica l voids ,randomly d i s t r i b u t e d , penny-shaped cracks , o r penny-shaped cracksa l igned so t h a t t he crack surfaces a re perpend icu lar to the s t r a i nf i e l d .As f - fu n c t i o n we have chosen a ' super e l l ipso id

(39)


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where Ac r i t A cr i t1 ' 2 (Ac r i t Acr i ta re cons tan t s . Writ ing 1 2 as(40)

we can say t ha t Ac r i t descr ibes the ove ra l l c r i t i c a l energyl eve l and g (or g) descr ibes the di f fe rence in the energy l eve l sr e l a t ed to the two damage modes. Following t h i s l i n e of th inking,we can th ink of the f - func t ion as a descr ip t ion of the i n t e r ac t i o no f the damage modes.In f ig . 2 to f ig . 4 we have out l ined the in f luence of the i n i t i a ldamage vec to r given the damage desc r ip t ion and the f - func t ion;fur thermore , the in f luence of the f-funct iorr given the damage de-s c r ip t i o n and the i n i t i a l damage and, f ina l ly , the in f luence oft he damage descr ip t ion given the f - func t ion and the i n i t i a l da-mage.

S I G ~ A IJE m Aor CDM-ca l. Al i S ned l enny shaped cr-acks2.0EOOOO1. I ~ O O /,. 0.:05,0, 5 ....... m ~ ~ ~ ~ r ~ T I O O 25

v LORD .H I STORY:: / 0(=(0.1 1 : lINIAXIAL STRAIN

1.0EOOOO~ L :/ / [ ~

1 > = 2 0.2)1/ : ~ > ~ ~. . . . . . . . . . : ..... ... , :,.. ..... .

l /1 :, . l / . .1

. . ,, , . . . . . . . , , , , , . , , , . . . . . . . , . . . . . , . . . . .; / /,r1/ /.OEOOOO r,-. Aor.OEOOOO 1 . OEOOOO 2,OEOOOO 3.0EOOOO 4.0EOOOO 5.0EOOOO

Fig. 2. CDM - ca lcu la t ion of ma t e r i a l con ta in ing a l igned penny-shaped cracks . Effec t of i n i t i a l darnage.


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:31 GI1A CD 1-c a.J Em Ao r . Aligned penny shaped c r c k ~1.6EOOOO_ T - - - - - - - - - - - ~ - - - - - - - - - - ~ - - - - - - - - - - - - ~ - - - - - - - - - - ~ - - - - - - - - - - _ r

ALL CURI.jES:

1.2EOOOO

4 . OE -001

:::::::---- . . POl SSOH' S RATI o 0. 25\ -... . . INITIAL 0


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In f ig . 5 we have shown some exper imental r e s u l t s for s t r a i nso f ten ing in t ens ion by Balavadze shown in [24J) and Hughes andChapman [25] . We can see t ha t some of the theo re t i ca l curves r e -semble the exper imenta l ones a g r e a t dea l however, t he va l i d i t yof such a comparison depends heav i ly on the degree of homo-gen i ty of the s t r e s s / s t r a i n f i e l d s i n the t e s t specimens.

r MPa]2 . S ~ - - ~ ~ - - - - - - - T - - - - - - ~ - - - - - - ~ - - - - - - ~ - - - - ~ - - - - - - ~ - - - - - - ~

2 . 0 r - - - - - - + - - ~ = - r - - - - - _ + - - - - - - + _ ~ ~ _ + - - - - - - ~ - - - - ~ - - - - ~BALAVADZE

1 . S I - - - - - j r - + - - ; f - - f - r _ - - - - - \ r l - - - - - - + _ - - . . - _ + - ~ - - ~ - - - - ~ - - - _ _ _ 1

1 . 0 1 - - - - , 1 - . Y - 4 - - - - - + - - - - - - + - - - - - ~ ; : : : - - - - ~ - - _ _ l I F ' ....=_ __t ___;

L ____ ______ ______ ______ ______ ____ _____________ E0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8

Fig . 5. Exper imenta l r e s u l t s for concre te in tens ion .

We have a l so shown some t h e o r e t i c a l s t r e s s / s t r a i n curves for thef ib re re inforced composi te m a t e r i a l see f ig . 6. In t h i s case wehave used( A 2f = +Ac r i t1with

( A 2Ac r i t2

thus we have e l imina ted the f i b r e debonding damage mode.

41 )

(42)


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The mate r ia l parameters used in f ig . 6 cor respond to a polypropylene f ib re re inforced cement matr ix . Some exper imenta l r e su l t sfor such a mate r ia l cons i s t ing of i n f i n i t e l y long uniax ia l ppf ib res in a cement matr ix sub jec ted to un iax ia l tens ion i s a l soshown in f ig . 6 and we observe a nice agreement . Note in t h i sconnect ion t h a t the only curve f i t t i ng parameters t h a t we usedare Ac r i t and the i n i t i a l damage s t a t e and note a lso t h a t a l lthe t h eo r e t i c a l curves have been determined using the same valuefo r Ac r i t .Fina l ly it should be noted t h a t the experiments c lea r ly showedt h a t the damage in the t e s t specimens was homogeneous. t was notposs ib le to de tec t any v i s i b l e cracks even a t 1.5 s t r a i n thusit i s reasonable to assurne t h a t t h e non l inear ef fec t s are due tomicrocrack nuclea t ion as assumed in t h e theore t i ca l model.

SI GMA CrvPAJ I D t 1- c l CIJ 1at i cln o f PP FH C m.ater i .all1.8E0001

1.5E0001

1.2E0001

9. O OOOO

6. O OOOO

3 0EOOOO

OEOOOOOEOOOO

' + + . , .. , ,. . . . , . . . . . , . . . . . . . , , ... , , , . . . . . , , , . . . . . . . , ..... . . . ,

. . c . c c c cc cc

. c f =4.3

ALL PO I NTS: EXPER H1Et'lTAL RESUL TSALL CURVES: THEORY WITH Ef =8 GPa Em=32 GPa Acrit=.3 MPa

2. OE 003 4 . OE -003 6 . 0 r 0 0 3

Fig. 6. FRC mate r ia l CDM curves and exper imenta l re su l t s .

REFERENCES

[1] H.E. Read and G.A. Hegemier. St ra in softening of rock,so i l and concre te A rev iew a r t i c l e . Mech. of Mater .

~ 271-294, (1984) .


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[2] S. Ziegeldorf . Phenomenological aspec ts of the f rac tu reof concrete . In Frac tu re Mechanics of Concrete , ed. F.H.Wittmann. Elsev ie r Science Publ i shers Amsterdam, pp.31-41, 1983).

[3] A. Hil le rborg M. Modeer, and P.-E . Pe tersson . Analysis ofcrack format ion and crack growth by means o f f rac turemechanics and f i n i t e e lements . Cement and Concrete Research,... 773-781, 1976).

[4] A. Hil lerborg . Analysis o f f rac tu re by means of the f i c t i t i ous crack model, par t i cu la r ly for f ib re re in forced conc re te . The In t . J . Cement Comp., 177-184, 1980).

[5] P.-E. Petersson. Crack growth and development of f r ac t u rezones in pla in concrete and s imi l a r m a te r i a ls . Divis ion ofBui lding Mater ia l s Lund I n s t i t u t e o f Technology. ReportTVBM-1 OO6. 1 981 ) .

[6] P.J . Gustafsson. Frac tu re mechanics s tudies of non-yieldingmater ia l s l i ke concre te : Modelling o f Tens i le Frac tu re andApplied Strength Analysis . D ivis ion of Bui lding Mater ia l sLund Ins t i t u t e o f Technology. Report TVBM-1007. 1985).

[7] Z.P. B a ~ a n t and B.H. Oh. Rock f rac tu re via s t ra in - so f t en ingf i n i t e elements. J . Eng. Mech., ll Q 101.5-1035. 1984).

[ 8 ]

[ 9 ]

[ 1O]

vZ.P. Bazant and B.H. Oh. Microplane model for progress ivef rac tu re of concre te and rock. J . Eng. Mech., 111, 559-582.1985) .

vZ.P. Bazant . Mechanics o f d i s t r i b u t ed cracking. Appl. Mech.Rev. , ~ 6 7 5- 705. 1 9 8 6) .

L.S. Cost in . A microcrak model fo r the deformat ion andf a i l u re of b r i t t l e rock. J . Geophys. Res . ~ 9485-

1983) .

[11] L.S. Cost in . Damage mechanics in the p o s t - f a i l u re regime.Mech. of Mater . i 149-160. 1985).


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[ 2] D. Kra jc inov ic . Cons t i tu t ive Equat ions fo r DamagingMater ia l s . J . Appl. Mech., 50, 335-360. 1983) .

[ 13] D. Kraj c inov ic . Continuum Damage l-1echanics. Pppl . Hech. Rev. ,2J... 1-6. 1984) .

[14] D. Krajcinovic . Continuous Damage Mechanics Revis i ted :Basic Concepts and Def in i t ions . J . Appl. Mech., 52, 829-834. 1985) .

[15] J . J . Marigo. Modell ing of b r i t t l e and fa t igue damage fo re l a s t i c mate r ia l by growth of microvoids . Eng. Frac . Mech.,

~ 861-875. 1985) .

[16] F. Sidor f f . Damage mechanics and i t s app l i ca t ion to compos i t e mate r ia l s . In Mechanical charac te r iza t ion of loadbear ing f ib re composi te l amina tes eds. A.H. Cardon andG. Verchery. Elsev ie r Applied Science Publ i shers . pp. 21-35 1985) .

[17] Z. Hashin. Analysis of Composite Mater ia l s . - A Survey.J . Appl. M e c h . ~ 481-505. 1983)

[18] L.J . Walpole. E las t i c behavior of composi te mate r ia l s :Theore t ica l foundat ions. Adv. Appl . Mech., 21, 169-242.

1981).

[19] R.M. Chr i s t ensen . Mechanics of Composite Mater ia l s . AWiley- Inte rsc iente Publ ica t ion John Wiley and o n s ~1979) .

[20] R. H i l l . E l a s t i c prope r t i e s of re inforced so l ids : Sometheore t i ca l p r inc ip les . J . Mech. Phys. Sol ids 11 357-372. 1963) .

[21] V.M. Levin. Determinat ion of ef fec t ive e l a s t i c moduli ofcomposi te mate r i a l s . Dokl. Akad. Nauk SSSR, 220, 1042-1045, 1975). Sov. Phys. Dokl . , 20. 147-148, 1975).


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[22J H. Stang. St reng th of composi te mater i a l s with smallcracks in the mat r ix . Accepted fo r pub l i ca t ion in In t .J . Sol ids S t ruc . (1986) .

[23] R. Hi l l . Continuum micro-mechanics of e la s top las t i cpo lycrys t a l s . J . Mech. Phys. Sol ids , 1l 89-101. (1965) .

[24J J . Kasperk iewicz. Frac ture and crack propagat ion energyin p l a in eonere te . Heron, ~ 5.,..14. (1986) .

[25J B.P. Hughes and G.P. Chapman. The complete s t r e s s - s t r a incurve for concrete in d i r e c t tens ion . RILEM Bul let in , . No.l..Q, 95-97 . (1966).

APPENDIX 1In t he case o f un id i rec t i ona l penny-shaped cracks t he H-tensor i st r ansver se ly i so t rop ic and given by the fo l lowing express ion:

4 r 1 - vm 2 2 1 - vm 4 1 - vm) vm 5 ]HiJ kl = - L . DiJ kl + D i J k l + m D i J k lT 1 _ 2vm (2 _ vm) 1 - 2 v .(A. 1 1 )

where vm denotes Pois son s r a t i o for the matr ix m ate r i a l and45where Dijkl Dijkl and D i j k l are t h ree o f t he s ix t r ansver se -ly i so t r o p i c elementary t enso r s given by Walpole, see e .g . [18] .2 4 5(Denoted Ei j k l , Ei j k l , and Ei j k l by Walpole) .

APPENDIX 2In the case of randomly d i s t r i b u t e d penny-shaped cracks the H-tensori s i so t rop ic and given by the fo l lowing express ion:

1 _ vm )2)1 - 2vm

8J i j k l + 15TI

(A.2.1)where vm denotes Pois son s r a t i o fo r the matr ix m ate r i a l and


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where J i j k i and Ki j k l are the two i so t rop ic elementary tensorsgiven by Hi l l [23] .

APPENDIX 3In the case of i s o t r o p i ca l l y dispersed spher ica l voids the H-tensori s i so t rop ic and given by:

where

3= 2i s Poisson s r a t i o fo r the matrix mate r ia l and

(A.3 .1 )

J i j k land Ki j k l are the i so t rop ic elementary tensors mentioned inAppendix 2 .

APPENDIX 4F i r s t the compliance of the s ing le composite mater ia l element shownin f ig . 7 w i l l be determined .The behavior of the element i s descr ibed by the fo l lowing equat ions:

p ix - O2 - T = , A. 4 1p f ix O2 - T = , A.4.2)p

Emm

2b = E: x (A .4 .3 )p f Ef f2 t = E: x A.4.4)i m fT = tp u - u , A.4.S)m mE = UX ,x (A .4 .6 )f fE = UY ,x

A.4 .7)

where


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pf tf ax dy2 = A.4 .8 )obpm f x dy2 = A.4.9)t

and where supe rsc r ip t s m, f , and i r e f e r to the matr ix , the f i b r eand the in te r face , respec t ive ly .The fac to r i s determined by assuming a 2. order var ia t ion inthe y-d i rec t ion in the ax ia l displacements u, and by ident i fy ingum and u f wi th the fol lowing mean values of the displacementf i e ld :

t= 1 Ju dy ,t o

b= i ; f u dy .

t

Furthermore , the boundary condi t ionsT = O are fu l f i l l ed .y=b+tWe ge t

b Gf t Gm

Ty=O = O, Ty=t i= T

fu , and using the boundary condi t ions

A.4 .10)

A. 4 11 )

and

A.4.12)

Now we canO, p f U,) =pf = P we

so lve forp , um O)ge t

= O, uf O) = O, along wiht the condi t ion

t anh Ax)E f 2 t A l Em b+ EIDx 2b -E-=f-t-+-E-m-b

where A s given in eg. 22).Thus the complince, c fo r the s ing le element i s given by

c CQ,

A.4. 13)

A.4.14)


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The compliance , C fo r the mult ic racked element shown in f ig 1can then be wri t t en as

c = N j c .Q,) lE t R = d2n A.4.1S)where N i s t h e t o t a l nurnber of cracks elements) and where n = N/LThus t he s t i f fne s s of the mult ic racked composi te mate r i a l elementi s given by

1L = Mwhere

M=2(b+t)n-jc(R,)++ l .E t .Q.= d2nThe so lu t i on i s va l id fo r

1d < 2n

fb+-

2ttbl

/1.=0

iliil lillil III I llillll I III II i li II I I lil i lli II I I llllil :1111iilll llll l lil I111::I"I I iili2 l ~ t

Fig. 7. The bas ic composite mate r i a l element.

A.4.16)

A.4 . 1 7)

A.4.18)


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FR CTURE MECH NICS ND DIMENSION L N LYSIS FOR CONCRETESTRUCTURESN A HarderInstitute o Building Technology and Structural EngineeringUniversity o alborgDenmark

SUMMARYt is shown how some results obtained by dimensional analysis can be used to check a

new fracture mechanical theory the fictitious crack model and to determine an optimal linear a w relation to be applied in this theory.

INTRODUCTIONt is weIl known that a small-scale model of a concrete beam is comparatively stronger

than a similar full-scale bearn. In [2] Hillerborg, using a new fracture mechanical modelthe fictitious crack mode , has shown in a qualitative way, why it must be so. In thepresent paper the same aspect is discussed in a more quantitative way applying dimensional analysis.

Among other things, the role of the brittleness modulus is explained in a new way.Le. one of the modellaws from dimensional analysis says that the model must have thesame brittleness modulus as the prototype. As this, however, cannot be achieved, it isshown how to find a correction factor to the modellaw giving the ultimate load of theprototype determined from a small-scale model test. t last it is shown how the resultfrom this dimensional analysis can be used to determine an optimal linear a w relationto be used in the fictitious crack method.

FRACTURE-MECHANICAL RELATIONSIn the present paper use will be made of some relations from linear fracture mechanics[6] summarized as follows.

t f t t t t t t tr lO2aFigure 1


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The simplest problem dealt with in linear fracture mechanics is that illustrated infigure l showing a crack of length 2a in a medium under uniform tension 00

The condition for the crack to be stable isy EG ca


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Figure 2.where s a typical length. All sirnilar elements have the same critical brittleness m-odulusBe which can be determined by putting the ultimate load at yielding equal to the ultimateload at crack instability [6].

THE FICTITIOUS CRACK MODELIn this model it is assumed that yielding does not take place beyond a very narrow zonecalled the fracture zone or the fictitious crack. In the fracture zone the constitutive relation between the tensile stress and the opening w of the fictitious crack is defined bya relation as shown in figure 2, [2].Hence, the area under the a w curve is the fracture energy GF. In a way this providesa connection between the linear fracture mechanics and the fictitious crack model.

DIMENSIONAL ANALYSISConsider a linear elastic body with or without initial cracks and under a process of deformation. The loads consist of constant loads p, q, r multiplied by a load factor Iranging from zero to IU at ultimate load. IU Can be written as a function

Taking force F) and length L) as fundamental dimensions, the dimensions of theparameters being the arguments of the function are as follows:

p a concentrated force: Fq a line force: F . L 1r a surface force: F L-2U a prescribed displacement : L

a typicallength: La length of a crack at the surface of the body or hal f the length of an internal

crack:Le characteristic length of the material: L

v Poissons ratio: Dimensionlessau ultimate stress: F L-2


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I t is assumed that the material is linear elastic up to the ultimate stress au i.e. yieldingdoes not occur. But when a tensile stress at a point reaches au then a new crack startsat this point. Taking p and Qas fundamental parameters, the following complete set ofdimensioniess parameters called 7T-products [l] is obtained.

1rr =Y (A)T = q-Q p-1q B)Tr = r Q2 p-1 (C)T =UQ-1u (D)T =a Q-1a (E)T =Q .Q-l2c c (F)7TE = E Q2 p-1 (G)v = v (H)

T a .Q2 p 1au u (l)

The set of 7T-products is said to be complete, because the parameters are independent,i.e. no single 7T-product can be expressed by a product of the other 7T-products. For theunknown parameter Yu the following 7T-product can be formed

(K)

If all the 7T-products in the model index M) are taken as being equal to those of theprototype (index P) the following MODEL LAWS for complete similarity are obtained.

Yp :: lYMqp Qp -1 PpqM = CQM ) PM

ap QpaM = QMQcP _ QpQcM QM

Ca)

(b)

Cc)

(d)

(e)

(f)


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YuP=1YuM

73

g)

h)

i)

k)

The equations a) to i) determine the dimensions, the loads, and material parameters. in the modeLAnd from YuM the unknown YuP is determined from equation Ck).

In. practice, however, it is not possible to satisfy all equations a) to i) at the sametime. f he model is made of the same material as the prototype, then it follows fromg) that

Pp Qp 2=PM Q 5)

and all modellaws can be satisfied except f) which says

6)i.e. the brittleness modulus shall be the same in the model as in the prototype. This,however, cannot be achieved, becausewhen using the same material in the model andthe prototye one has

instead of, from 6),

Now, using k), YuP can be written

where the correction factor g can be found by calculations using the fictitious crackmodel on the following two models:


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Modellis identical with the physical model and for this model

Q 1) = PcM cwhich means

and all modellaws except f) are satisfied.Model2is a fictitious model being the same as model l except for ~ l l being determined fromf):

Q=Qc pwhich means

As all modellaws for model 2 are satisfied, it follows from k) thatY(2)

V ,(2) - uM - g 0 VuP - uM m YuM - luY(2)uMg= i )YuM

YuM

Defining the scale ratio o byp:=Q

the following relations are obtained

Q 1) =cM cP

(7)


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B 2) - BM P

75

When calculating Y ~ and Y ~ ~ using the fictitious crack method, the only differencebetween model l and model 2 is that w 2) =ex-l w(l) which follows from ~ ~ =ex-l G ~ ~ .Le. the a w curve for model 2 is affine with that of model 1.

In order to sketch the function g = g ex), a new parameter, THE BRITTLENESS INDEXexc is defined as follows

8)

Hence,

i.e. model 2 is ductile for exc < l and brittle for l < xc Furthermore,

i.e. model l is ductile for exc < X and brittle for X < exc Here, ductile and brittle donot mean anything more than

ductile: B


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vlEG (2) 'E l G(l)0 2) = M v J X M =_l 0 1) _ lcM eva; ~ .fii CM -..;ex; Ou

Now, consider the two cases Xc < l and l < cxc' separately.Ctc < 1:Model 2 is ductile and assuming cx> l , modellis duetile. Hence

(2)'YuM Oug= I)= -=1'YuM Ou

1 < cxcModel2 is brittle. For Xc < x modell is ductile and (lO) gives(2) (2)'YuM cM 1g= =


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From figure 3 it is seen that the function g = g(a) in the modellaw I uP = gl uMis completely determined by the value of the brittleness index ac.

Although ac is the critical value of a stress a which is often not actually existingit is believed that the result obtained above from (9) and (lO) borrowed from the lin-ear fracture mechanics, can be used in the following way: g =g(a), see figure 3, canbe determined by calculations using the fictitious crack method. Only a few calcula-tions are necessary because if g = l for the first trial, then a c < l and g = l for alll < a. Furthermore, if two calculations give the same result g =gc < l, then fromfigure 3.

laclg

ya1g gcFc

and also the critical brittleness modulus Bc is determined by

AN OPTIMAL LINEAR a-w RELATION

11 )

n order to make a check of the theory g can be calculated for different values of a inthe interval l < a a c . Then, provided it is p e r ~ s s i b e to use 9) and 10) from linearfracture mechanics, the result shou d be g = l/Va, especially for a = a c one should ob-tain g = l ly a c and the theory can be checked without making any experiments. Notrelying on (9) and C O the theory can only be checked by making experiments.

If, as an approximation, the a-w relation is taken to be linear, this relation is determinedby the two parameters au and GF or au and Qc.au can be determined by a cylinder splitting test and GF by a 3 point bending test asproposed in [4]. As there are some problems, however, regarding this test [5] it is proposedthat the test is made on two similar models with different values of a. The largest modelthen could be looked upon as the prototype (a = l) andI uCa=l)

gl = I Ca> l)uobtained from the tests could be compared with

I ~ i 1g = mI uM

obtained from calculations.Then GF can be adjusted so that the difference between gl and g2 is minimized.

n this way no use is made of linear fracture mechanics, only results from dimen-sional analysis are used to determine GF .


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8

REFERENCES [1] Langhaar, H.L., Dimensional Analysis and Theory ofMo dels John Wiley Sons,

1951.[2] Hillerborg, Ao A Model ofFracture A nalysis Division o Building Materials, The

Lund Institute o Technology. Report TVBM-3005, 1978.[3] Hillerborg, A. Analysis of one single crack. Chapter 4.1 in Wittman, F.H. (ed.),

Fracture Mechanics of Concrete Elsevire, Amsterdam, 1983.[4] RILEM Draft Recommendation. Determination o he Fracture Energy o Mortar

and Concrete by Means of Three-point Bend Test on Notched Beams. MaterialsStructure, Vol. 18, No. 106, 1985.

[5] Hillerborg, A., Results o Three Comparative Test Series for Determining the Fracture Energy Gp ofConcrete. Material Structure, Vol. 18, No. 107, 1985.

[6] Harder, N.A. Notes on Fracture Mechanics. Report: R8612 (in Danish). Insti tuteo Building Technology and Structural Engineering, AUC, Aalborg, Denmark, 1986.


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TIME-DEPENDENT FRACTURE OF CONCRETEAN EXPERIMENTAL AND NUMERICAL APPROACH

Einar Aassved Hansen Norwegian Ins t i t u t e of Technology Divisiono f Concrete St ruc tu re s and SINTEF Cement and Concrete ResearchI n s t i t u t e N-7034 Trondheim-NTH

Matz Modeer Norwegian Contractors Hol te t 45 N-1320 STABEKKand Norwegian Ins t i t u t e . of Technology Divison of ConcreteSt ruc tu res N-7034 Trondheim-NTH

ABSTRACT

The necess i y o f t ime-dependent f r ac tu re mechanics analyses ofconcre te f a i l u r e i s pointed out . A proposed numerical methods imula t ing a v i sco-e las t i c f r ac tu re behaviour i s summarized to ge ther with fu ture plans . The aim and con ten t o f an experimentalprogram regarding the t ime-dependent behaviour of concre te aregiven.

INTRODUCTION

Appl icat ion of f rac ture mechanics to f a i l u r e analyses o f concre tehas l ed to a be t t e r unders tanding of the concrete fa i lu remechanisms. Ma

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