Materials and Structures, 1994, 27, 563-571

Size effects on tensile fracture properties: a unifiedexplanation based on disorder and fractality ofconcrete microstructureALBERTO CARPINTERI, GIUSEPPE FERROPolitecnico di Torino, Dip. di Ingegneria Struttura/e, 10129 Torino, Ita/y

Tests were carried out using three independent jacks orthogonally disposed, making it possib/eto app/y a purely tensile Jorce, so that the secondary fiexural stresses, if kept under control,constitute a degree of error comparable with the values allowed [or norma I testing apparatus.The method enables a stress versus stra in curve to be plotted with the descending (soJtening)branch up to the point where the cross-section of the tensile specimen breaks away. Theprincipa/ purpose is to avoid any spurious effect that might pro vide a fallacious explanation ofthe recurring size effects on apparent tensile strength and fictitious fracture energy. Once thesecondary effects have been excluded, only the disorder and fractalit y of the concretemicrostructure remain to explain such Jundamental trends. In the case of tensile strength, thedimensional decrement represents self-similar weakening of the materia I ligament, due topores, voids, deJects, cracks, aggregates, inclusions, etc. Analogously, in the case oJ fractureenergy, the dimensional increment represents self-similar tortuosity of the fracture surface, aswell as self-simitar overlapping and distribution of microcracks in the direction orthoqonal tothat of the [orminq macrocrack.

1. INTRODUCTION

Robert L'Hermite, in his well known book of 1955 [lJ,considered the diffìculties arising when a pure andobjective tensile test on concrete is to be performed.Quoting from his work,

'Pour connaitre la résistance du béton en traction, ilreste donc l'essai direct, très peu employé à cause desa difficulté de réalisation. En admettant que lecentrage de l'effort soit fait avec toute la rigueurdésirable, certains auteurs craignent que toute fissu-ration conduisant à une dissymétrie de la sectionentraine la production d'un moment de flexion quiaccélère la rupture. Ceci pourrait ètre la cause d'unediminution de la résistance apparente en traction.'

The study of the tensile properties of heterogeneousmaterials (concrete, mortar, rocks, etc.) has involvednumerous researchers in recent years. Different test tech-niques were developed and several theories were for-mulated in support of the results obtained. The substantialdifficulties that the tension test had to overcome are 1,to obtain a stable and complete post-peak softeningresponse, and 2, to maintain always the tensile loadperfectly centred so as to a void the presence of bendingmoments which can alter the results.

The post-peak softening response can be obtained onlyif the loading process is deformation-controlled. Ruschand Hilsdorf [2J and then Hughes and Chapman [3J aswell as Evans and Marathe [4J showed that the stressversus strain curve obtained from direct tension ofconcrete is similar to that obtained from compression

0025-5432/94 cc.' RILEM

even in the post-peak regime. The localization of strainin direct tensile tests on concrete was discussed byHeilmann, Hilsdorf and Finsterwalder [5]. Strain gaugeswere glued at different positions on a 600 mm longconcrete specimen. The load was eccentrically applied inan attempt to produce stable fracture.

More recently, Petersson [6J carried out tensile testsin order to study the development of the fracture zonein concrete. The tests were performed in a very stiff tensiletesting machine constituted by three aluminium columnswith cylindrical heating elements. The aluminium columnsexpanded when they were heated and transmitted thetensile load to the specimen.

Gopalaratnam and Shah [7J proposed a test techniq uewith a uni versai joint to minimize in-piane and out-of-piane bending effects and a particular wedge-typefrictional grip system to assure that fracture occurredaway from the grips. With this instrumentation theyperformed tests on constant size specimens and observedthat the notches do not influence the average responseof the specimens and that the tangent modulus ofelasticity in tension is comparable with that in com-pression.

Reinhardt and co-workers [8J performed cyclic testson notched specimens and proposed softening constitutivelaws on the basis of their experimental results.

Phillips and Zhang [9J designed a stiff frame whichproved highly effective in order to obtain stable andcomplete stress-deformation curves on both unnotchedand notched plain concrete. They also established asubstantial homogeneity in the behaviour of the notchedspecimens compared with that of the unnotched ones.

564 Carpinteri and Ferro

Van Mier [IOJ studied the problem of the inftuence ofboundary rotation stiffness and indicated some conditionsfor test stability. Moreover, he described a vacuumimpregnation experiment using a low viscosity ftuorescingftuid which allows monitoring of the complete crackgrowth processo He could observe self-similar crackpatterns, and Saouma et al. [IIJ emphasized the fractalnature of the concrete fracture surfaces.

A completely new testing apparatus made up of threemutually orthogonal actuators is proposed in the presentpaper. In this way, it is possible to perform a true directtensile experiment on concrete, whereas usually tensionand bending are both present. The principal purpose isthat of avoiding any spurious effect that might providea fallacious explanation of the basi c size effects onapparent tensile strength and fictitious fracture energy.Once these secondary effects have been excluded, onlythe fractality [12, 13J of the heterogeneous microstructureremains to explain such fundamental trends.

2. OESCRIPTION OF SPECIMENS ANOTEST EQUIPMENT

The fìrst problem we were faced with was that of defìningthe shape and the size of the test specimens. The shapeis linked to the choice of the type of gripping mechanismand to the desirability of creating a preferential fracturezone. The size of the test specimen is, instead, limitedboth by the size of the aggregate used (lower limit) andby the potentialities of the available equipment (upperlimi t). In an initial cycle of tests the approach was to userelatively srnall aggregates to make it possible to performexperiments on test specimens of varying sizes (Fig. I).The test specimens were flared in the centre as shown inFig. I. The thickness of the test specimens, lO cm, was ,maintained constant for the entire set, while the transversedimension was 0.5, 1,2, and 4 tirnes that of the thickness.

The test specimen was glued to steel supporting platesfor attachment to the load-bearing system. An epoxyresin based bicomponent adhesive was chosen. Thisensures a high adherence both to concrete and to steel.The plates were secured to the lower and upper crossmembers with a series of bolts. The shape of the test

2S=20cmb=l? cm I

b=5~ '<b=40cm

---+---.+-b b

Fig. I Geometries of the specimens tested.b

b

specimen, the two ends ofwhich had a cross-section threetimes larger than the cross-section undergoing tensiletesting, made it possible to reduce the stress in the gluedarea.

The testing method has been developed taking intoaccount the following four factors: (i) the generaIimpossibility of making geometrically perfect test speci-rnens, since even if considerable care is taken in theirpreparation, they inevitably present certain defects inshape; (ii) the extreme difficulty of eliminating centringerrors even if recourse is made to introducing hinges inthe chain applying tensile force (it is by no means certainthat the test specimen is coaxial with the hinges); (iii) evensupposing that centring of the stress is performed asexactingly as possible, the development of cracks duringthe loading phase leads to a new cross-sectional asym-metry and hence produces a bending moment; and (iv)the use of closed-circuit electronically servo-controlledmachines becomes indispensable in order to maintain aconstant strain rate, and in arder to detect the descending(softening) branch of the load-extension curve, measuringthe extension by strain gauges (extensometers) directlyapplied to the test sample.

The procedure devised was to use servo-controlledmachines to keep the ftexural forces under control alongtwo mutually orthogonal planes. For this purpose it wasnecessary to ha ve some sort of electrical signal, capableof functioning as a control signal in a counter-reactionsystem, which was proportional to the bending momentdeveloped on the specimen cross-section.

Initially the new counter-reaction signals were obtainedby detecting the strains from electrical strain gauges seton two opposite sides of the specimen and connectedonto the adjacent sides of the measuring bridge. In thisway two electrical signals were obtained proportional tothe difference in the strains and hence proportional tothe bending moment acting along a piane passingthrough the axis of the specimen, each in the directionperpendicular to the faces of the specimen to which thestrain gauges were attached. The test proved particularlyuseful for pinpointing the problems involved in thesimultaneous operation of the three servomechanismsemployed. It did not, however, prove satisfactory asregards the response curves, which stili emerged asdivergent (Fig. 2).

It thus appeared logical to use, for control purposes,the sa me analogue signals supplied by the four exten-someters employed for measuring the strains, carryingout appropria te algebraical operations on them. If EI, E2,

E3, E4 are the strains of the single measuring points setout as shown in Fig. 3, the strains due to bending momentturn out to be: along the x axis,

(Ez + E4) - (EI + E3)E, =. 2

b

b

b

and along the y axis,

(EI + Ez) - (E3 + E4)E,,= ---. 2

Materials and Structures

20,---------------------------------~

565

16

Specimen: G503Cross-section:cm 5xlOStrain-gauge length: mm 50

-El E2E3 E4--- Average

4

OL-L-~~~~~~~~L_~ __ ~~~~-5 O 5 10 15 20 25 30 35 40 45 50 55 60 65 70

Extension 8, 11m

Fig. 2 Load versus extension diagrams coming from the fourextensometers. Single jack solution.

Fig. 3 gives the electrical scheme illustrating the use ofanalogue amplifiers for performing operations of additionand subtraction of the four extensometers signals andhow they are connected up to the three servocontrolledjacks.

Fig. 4 depicts the test scheme and shows the overallset-up of the equipment with one of the largest specimensunder load. The central jack was suspended from thecontrast structure via a ball joint and was connectedagain with a ball joint to the moveable cross-member.The jack applies the tensile force to the centre of thecross-rnember and, as already mentioned, the control isperformed by summation of the signals from the fourextensometers mounted on the specimen. The other twojacks are disposed one at the end of the main cross-member and the other at the end of an auxiliarycross-member set perpendicular to the main one. Both

x

SIS2S3S4

Specimen

CIC2C3x

c

RI R2

I M.T'S.2 !R3

I M.;S.3!M.T.S.1

Fig. 3 Counter-reaction electrical system.

jacks generate a couple acting along a principal planeof the specimen (Fig. 5).

The test is thus carried out by the centraljack imposingan extension on the specimen at a constant rate, whilethe other two jacks maintain the values Ex and Ey equalto zero. Fig. 6 presents the load-extension curvesobtained with the three jack methcd, and it can be seenhow they almost coincide up to the point where the loadgoes to zero in the softening phase.

3. EXPERIMENT AL RESULTS

The tests were carried out on four specimen sizes. Thedimensions of the minimum cross-section of the smallestspecimen were 5 cm x lO cm, lO cm x lO cm for thesecond size, 20 cm x lO cm for the third and 40 cm xlO cm for the largest size (Fig. 1). Four specimens ofeach size were cast. The concrete mix had a water/cernentratio of 0.5. The maximum grave l size was dmax =16 mm while the mean compression strength was f: =36.9 MPa.

For the smallest size, only one complete diagram isavailable, in that the other three were used to set thesensitivity of the three servocontrolled machines. For thesubsequent sizes, two complete diagrams are available.For the specimens of length 80 cm the hinges wereintroduced to bind the lower cross-head since, in the firsttwo specimens, a failure near the grip plates had occurred.For this size only a complete curve is available. On theother hand, for the largest specimens only the values ofthe tensile strength are available, because the length ofthe extensometers (40 cm) was greater than the criticalvalue and therefore snap-back behaviour occurred. Theresults of the tests are reported in Table 1.

LEGEND :EI E2 E3 E4SI S2 S3 S4AlA2

A3

Fr-actur-e mechanics extensometersExtensomeler output signalsAcluator used lo apply te.naion lo adAclualor used to reduce the bendìng momentin the Y - Y direct.ìcnActualor used lo reduce the bending momentin the X-X directionSignal conditioner amplifier

ft~;-a:~t~~;o~l~:ie~ul;~tt~~n~~~itionof theOperational amplifiers lo make other algebraicaloperations on extensometer output signalsFeed-back sìg nal for act.uator- AlFeed-back signal for actuator A2Feed-vbeck signal for actuator A3Al A2 A3 actuator output signals

C01

02 03RI=SI+S2+S3+S4R2=lS2+S4)-CSl+S3)R3= S1+S2)-(S3+S4)Cl 2 C3

566 Carpinteri and Ferro

if<P

i

lI

<PI

<P

~II

·speclmèn·• ·1· ••

<P!

Fig. 4 Testing scheme with the three jack solution.

Fig. 5 Three jack solution.

20,--------------------------------------.Specimen:CA5R2Cross-section:cm5 x lOStrain-gaugelength:mm5016

12

8

4

--- Average-El E2E3E4

O~~~~~~~~~~~~~~-JO 20 40 60 80 100

Extension0, umFig. 6 Load versus extension diagrams coming from the fourextensometers. Three jack solution.

Table 1 Experimental average values for tensile strength andfractu re energy

Specimen Cross-sectional Tensile Fracturewidth area strength energy(cm) (cm ') Uu (MPa) GF (N/m m)

5 50 4.25 0.083IO 100 3.78 0.10220 200 3.64 0.14240 400 3.11

Materials and Structures 567

80~----------------------------, -1.8~----------------,

60

20

40 80 120 160 200Extension 0, um

Fig. 7 Experimental load versus extension curves obtainedfrom three different specimen sizes.

The experimentalload versus extension curves for thethree specimen sizes are shown in Fig. 7. It is importantto note how the slope of the elastic portion is the samefor ali sizes. This is due to the fact that the lengths of theextensometers were proportional to the size of thespecimens (5 cm for the smallest ones, lO cm and 20 cmfor the other two). Moreover, the shape of these curvesis in any case the same and this fact means that this kindof test procedure is independent of the machine stiffness.In fact, unlike the results reported by Van Mier [lO] forthe specimens tested with fìxed grip plates, no bumpswere found. This is due to the load, which is alwayscentred when the control takes piace both in-piane andout-of-plane. In the tests discussed by Van Mier, on theother hand, the check was only in the specimen piane,and no check was performed for the out-of-planeeccentricity.

The values of the apparent tensile strength are plottedin Fig. 8 against the specimen size in a bilogarithmicpiane. The size effect is represented by the slope of thelinear regression of the points of the diagram. It is evidenthow the tensile strength decreases with increasingspecimen size. The fracture energy instead shows theopposi te trend: i.e., it increases with specimen size. Thevalues obtained experimentally are reported in Fig. 9 onthe bilogarithmic piane In CF versus In b. The variation

1.5~-------------------,- y = 1.997- 0.14 x , (j

1.4

1.3

1.2

• ln b1.1L......~........L~~--L~~.L.....~........L~~.....L~.o.-..o.....J3.5 4.0 4.5 5.0 5.5 6.0 6.5

Fig. 8 Size effect on apparent tensile strength.

- y =- 4.03+ 0.38x

-2.0

-2.2

-2.4

In b-2.6L~---.L~~..L..~----.l~~....L....~.........JL..-~....J3.5 4.0 6.54.5 5.0 5.5 6.0

Fig. 9 Size effect on fictitious fracture energy.

of CF as a function of size is represented by the slope of thelinear regression diagram.

4. FRACTAL NATURE OF MATERIALLIGAMENT AND SIZE EFFECTS ONNOMINAL TENSILE STRENGTH

It is well known that the nominai tensile strength ofmanymaterials undergoes very clear size effects. The usualtrend is that of a strength decrease with size, and this ismore evident for disordered (i.e. macroscopically hetero-geneous andjor damaged) materials. Griffith [14] ex-plained the strength size effect in the case of glassfilaments, assuming the existence of inherent microcracksof a size proportional to the filament cross-sectiondiameter. Some years later Weibull [15] gave a purelystatistical explanation ofthe same phenomenon accordngto the weakest-link-in-a-chain concept. Only recentlyhave the two views been harmonized, enriching theempirical approach of Weibull with the phenomeno-logical assumption of Griffith [16-18]. A statistical sizedistribution of self-similarity may be defined [18, 19] forwhich the most dangerous defect proves to be of a sizeproportional to the structural size. This corresponds tomaterials presenting a considerable dispersion in statis-tical microcrack size distribution (disordered materials).In this case, the power of the LEFM stress singularity,!-, turns out to be the slope of the strength versus sizedecrease in a bilogarithmic diagram. When the statisticaldispersion is relatively low (ordered materials) the slopeis less than !-, and tends to zero for regular distributions(perfectly ordered materials).

Although the above contains the fractal concept ofself-similarity, this is circumscribed only to the defect ofmaximum size, whereas the disordered nature of thematerial microstructure is completely disregarded. Thereal nature of the material will be described herein by amore complex fractal model, where the property ofself-similarity is extended to the whole defect population.This model represents a closer picture of reality and isconsistent with the fractal explanation of the fractureenergy size effect, which will be proposed in the nextsection. On the other hand, as will be shown, slope valueshigher than !- would represent, with both models, a

568 Carpinteri and Ferro

degree of disorder that is so high as to be absent usuallyin real materials.

Let us assume that the reacting section or ligament ofa disordered material at peak stress could be represented .as a fractal space of dimension IX = 2 - d", with 1 < IX ::; 2and, therefore, O ::; d" < 1. The dimensional decrementd" may be due to the presence of cracks and voids andthen, generally, to a cross-sectional weakening.

Let us consider two bodies, geometrically similar andmade up of the same disordered material. If the ratio ofgeometrical similitude is equal to b and the renormalizedtensile strength (T~ is assumed to be a material con-stant and to have the physical dimensions [forceJ x[lengthr(2-da), we have:

* _ Fl _ F2(T ------

u 12- da b2 - da

F, and F2 being the ultimate tensile forces acting on thetwo bodies, respectively.

On the other hand, the apparent nominaI tensilestrengths are, respectively,

where the latter, according to (1), becomes

(T~2) = (T~l)b-da

We can write the relationship between nominaI strengthsrelated to different sizes in Iogarithmic form:

In (Tu = In (T~l) - d" In b.

Equation 4 represents a straight line with siope -d" inthe In (Tu versus In b pIane (Fig. 8).

Confirmation of (4) has been provided by severaiprevious investigations, carried out on different metalli cor cementitious materials and with different specimengeometries: tension, bending, Brazilian, etc. These resultswere summarized by Carpinteri [18, 19]. The same trendshave been found also in the present experimental programaimed at evaluating the true tensile properties of concrete.As may be seen from Fig. 8, the slope of the strengthdecrease proves to be equal to 0.14, thus revealing amaterial ligament of dimension 1.86, i.e., a fractal setwhich is very close to a 2-dimensional surface. Ifthe forceis assumed as sustained by a ligament of dimension 1.86,a constant (universal) strength parameter may bemeasured (Fig. lO). The fractal nature of the materialligament emerges very clearly at the size scales of thespecimens. On the other hand, the property of self-similarity is very likely to vanish or change at higher orlower scales, owing to the limited character ofthe particlesize curve.

5. FRACTAL NATURE OF FRACTURESURF ACE AND SIZE EFFECTS ONFICTITIOUS FRACTURE ENERGY

The fracture surfaces of metals [20J and concrete [11Jpresent a fractal nature with a roughness producing a

dimensional increment with respect to the number 2.Even in this case we can detect an evident mechanicalconsequence, considering the size effects on fractureenergy CF' Since Hillerborg's proposal for a concretefracture test was published as a RILEM Recommendation[21J, several researchers have measured a fracture energyCF which increases with the specimen sizes and, morespecifically, with the size of the uncracked ligament. Sucha trend has been found systematically, and in each casethe authors of the papers describing these experimentshave tried to provide various empirical or phenomeno-logical explanations, without, however, endeavouring tointerpret their findings in a larger conceptual framework.On the other hand, if we wish to understand the experi-mental observations, it is necessary to abandon theclassical thermodynamic concept of surface energy of anideai solid, and to assume the energy dissipation to beoccurring in a fractal space of dimension IX = 2 + dG'with 2 ::;;IX < 3 and, therefore, O ::; dG < 1. This representsan attenuation of fracture localization due to materialheterogeneity and multiple cracking.

Let us consider two bodies, geometrically similar andmade up of the same disordered material, If the ratio ofgeometrical similitude is equal to b and the renormalizedjracture energy C~ is assumed to be a material con-stant and to have the physical dimensions [forceJ x[lengthr(2 +dc), we obtain:

C* _ E, _ E2

F - 12+dc - b2+dc'

L5

LO •...•...~ ...•......~~ .•••.•..~~--'-~~-'--'"~~ •...•...~ ...•...•3.5 4.0 4.5 5.0 5.5

Fig. 10 Renormalized value of tensile strength.

(1)

(2)

(3)

(4)

In b

6.0 6.5

(5)

E l and E 2 being the energies dissipated in the two bodies,respectively. On the other hand, the apparent fractureenergies are, respectively,

where the latter, according to (5), becomes

C~2) = C~l)bdc.

(6)

(7)

We can write the relationship between fracture energiesrelated to different sizes in logarithmic form:

In CF = In clf) + dG In b. (8)

Materials and Structures 569

-3.0.------~8(J- y =- 4.02+ 0.0072x

-3.5

In QF*

-4.0 • • •

-4.5 Il Qe* '" 0.0179N mm·1.38

In b-5.0L.......-~......L~~.L.....~.......L~~...L....~......J.~~-..J

3.5 4.0 4.5 5.0 5.5 6.0 6.5

Fig. 11 Renormalized value of fracture energy.

Equation 8 represents a straight line with slope dG in theIn CF versus In b plane (Fig. 9).

The same trends have been found in the present experi-mental investigation. Tensile testing performed at thePolitecnico di Torino provides a plo t slope equal to 0.38(Fig. 9), which allows a constant (universal) energyparameter to be obtained in the case where the dissipationis considered as occurring in a damaged space ofdimension 2.38 (Fig. 11).

6. CONCLUSIONS

The abundant literature on tensile strength and fractureenergy size effects as well as the newly introduced fractaltheories would appear to indicate the need for a dramaticchange in the conceptual framework and even in ourwhole way of thinking, if we want to consider andmeasure material constants in materials engineering andin fracture mechanics. This means that we have to giveup ideal reference areas when we consider the tensilestrength and fracture energy of disordered materials witha fractal microstructure. The so-called homogeneousmaterials (on the macroscopic scale or macrolevel)present, on the other hand, a very small deviation fromthe ideal case. Only on the microscopic scale or microlevelmight they present a higher degree of fractality [22]. Forconcrete and rocks, for which the micro- and mesolevelscoincide with the structural level, the tensile strength isgiven by a force acting on a surface having a fractaldimension lower than 2, and also the fracture energy isrepresented by a dissipation over a surface with a fractaldimension higher than 2.

In the case of tensile strength the dimensional decrementrepresents self-similar weakening of the reacting cross-section or ligament, due to pores, voids, defects, cracks,aggregate s, inclusions, etc. Likewise, in the case offracture energy, the dimensional increment representsself-similar tortuosity of the fracture surface, due toaggregates and inclusions, as well as self-similar micro-crack overlapping and distribution even in the directionorthogonal to that of the developing macrocrack [lO].

As regards the dimensional decrement da and thedimensional increment dG' experimentally they appearalways contained in the interval [O,~]. The dimensional

decrement da tends to the LEFM limit ~ only forextremely brittle and disordered materials (defect sizedistribution of self-similarity) and is connected to theWeibull parameter m in the case of planar similitude[18, 19]:

2d =-a m

(9)

Even the dimensional increment dG tends to the samelimit ~ for extremely brittle and disordered materials. Theexplanation for the latter bound could ari se for dimen-sional analysis reasons. A generalization of the brittlenessnumber defined by the first author [23-28] could in factbe the following:

(lO)

If we postulate that the reversal of the physical roles oftoughness and strength is absurd, the exponent of thecharacteristic linear size b must be positive:

da + dG < 1. (11)

The sum of the dimensional decrement (for a materialligament) and the dimensional increment (for a fracturesurface) must therefore be lower than unity. On the otherhand, when, for very disordered materials, we have da ~ ~,the upper bound of 11 becomes dG ;S ~.

The above fractal interpretations could be regarded bysome as purely mathematical abstractions, if not indeeddistortions of reality. The truth is that both classicalgeometrical domains and fractal geometrical loci areidealizations of reality. The question that should beanswered is the following: which model is closer to a realfracture trajectory in a concrete specimen: a straight lineor the von Koch curve? Of course the latter, even thoughthe fractal nature of the fracture trajectory is randomand valid only in a limited scale range. This means that,for size scales tending to infinity or, in other words, forvery large specimens, tensile strength (Ju and fractureenergy CF may appear constant by varying the specimensize [29, 30], whereas, for size scales where randomself-similarity holds, the so-called 'universal properties'of the system ((J~, Cn are constant, although they arerepresented by physical quantities with unusual dimen-sions. The last result represents the target of the so-called'renormalizatiori' procedure, i.e., the determination ofphysical quantities that are invariant under a change oflength scale [31, 32].

ACKNOWLEDGEMENTS

The present research was carried out with the financialsupport of the Ministry of University and ScientificResearch (MURST) and the National Research Council(CNR).

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570 Carpinteri and Ferro

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19. Idem 'Mechanical Damage and Crack Growth in Concrete:Plastic Collapse to Brittle Fracture ' (Martinus Nijhoff,Dordrecht, 1986).

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23. Carpinteri, A., 'Experimental determination of fracturetoughness parameters Kre and Jre for aggregativematerials ', Proceedings of the Fifth InternationalConference on Fracture (Pergamon Press, Oxford, 1981)pp. 1491-1498.

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Materials and Structures 571

RESUMEEffets d'échelle sur les propriétés de la rupture en traction:explication unitaire basée sur le désordre et le caractèrefractal de la microstructure

L'expérience a été conduite en utilisant trois vérinsindépendants placés orthogonalement. Celà a permisd'appliquer une charge de traction pure de façon que lescontraintes secondaires de fléxion, si elles sont contràlées,donnent des erreurs dont l'ordre de grandeur est com-parable au valeurs tolérées pour les appareils utilisésnormalement. Ce gendre de méthode expérimentale permetde suivre la partie descendante (écrouissage négatif) de lacourbe effort-déformation jusqu'au point OÙ la sectiontransversale de l'échantillon sous traction se casse. Le butprincipal est d'éviter tous les faux effets qui peuvent nous

conduire à des explications erronées à propos des effetsd'échelle récurrents sur l' al/ure de la résistance à la tractionapparente et de l'énergie de rupture fictive. Une fois queles effets secondaires ont été exclus, seuls le désordre et lecaractère fractal de la microstructure du béton permettentde justifier cette tendance fondamentale. Dans le cas de larésistance à la traction, la diminution dimensionnellereprésente un affaiblissement des liaisons du matériau, diìaux pores, aux vides, aux défauts, aux fissures, auxgranulats, aux inclusions, etc. D'une manière analogue, dansle cas de l'énergie de rupture, l'accroissement dimensionnelreprésente la sinuosité de la surface de rupture, ainsi quela superposition et la distribution des microfissures dans ladirection perpendiculaire à celle des microfissures enformation.