Materials and StructuresjMatériaux et Constructions, Volo 29, June 1996, pp 259-266

Size effects on concrete fracture energy :dimensional transition from order to disorder

Alberto Carpinteri and Bernardino ChiaiaDepartment c!fStructura/ Engineering, Politecnico di Torino, 10129 Torino, Ita/y

ABSTRACT

The nominal fracture energy of concrete structures isconstant for relatively Iarge structures, whereas itincreases with size for relatively small structures. If theenergy dissipation space is modeled as a monofractaldomain, with a non-integer dimension comprised be-tween 2 and 3, a unique siope in the bilogarithmic frac-ture energy versus size diagram is found, as was stated ina previous paper [1]. On the other hand, when the scalerange extends over more than one order of magnitude, acontinuous transition from siope + 1/2 to zero siope mayappear, according to the hypothesis of multifractality ofthe fracture surface [1]. This means that, at small scales, aBrownian microscopic disorder is prevalent whereas, atlarge scales, the effect of disorder vanishes, yielding amacroscopical homogeneous behavior. The dimensionaltransition from disorder to order may be synthesized by aMultifractal Scaling Law (MFSL) valid for toughness, inperfect correspondence with the MFSL valid forstrength, which has been described in a previous paper[2]. The MFSL for fracture energy is applied, as a best-fitting method, to relevant experimental results in theliterature, allowing for the extrapolation of fractureenergy values valid for real-sized structures.

RÉSUMÉ

L'énergie de fracture nominale des structures en béton estconstante pour des strnctures relativement grandes, alors qu'elle aug-mente avec les dimensions paur les strnctures relativement petites. Sil'espace de dissipation de lenergie est modelé comme domaine mono-

fractal avec une dimension non-intégrale comprise entre deux et .trois, on observe une seule pente dans le diagramme bilogarithmiqueénergie dejacture-dimensions, comme il a d4jà été observé dans unarticle précédent [1). D'autre part, si l'intervalle d'échelle s'étend àplus d'un ordre de grandeur, une transition continue de la pente+ 1/2 à la pente zéro peut apparattre, en accord avec l'hypothèse demultifractalité de la surface de fracture [1]. Cela montre qu'undésordre microscopique Brownien prédomine dans les petites échelles,alors que, dans les grandes, l'iffet de désordre disparait et donne lieuà un comportement macroscopique homogène. La transition dimen-sionnelle du désordre à l'ordre peut étre synthétisée par la Loid'Échelle Multifirlctale (MFSL) valable paur la ténacité, en parfaitaaotd avec la Loi d'Échelle Multifi'actale valable paur la résistance,qui a été décrite dans un article précédent [2]. La Loi d'ÉchelleMultifractale pour l'énergie de fracture est appliquée, commeméthode d'interpolation (I best-fitting», aux résultats d'expéri-mentation plus impartants contenus dans la littérature, car elle per-met d'extrapaler les valeurs de l'énergie de jacture valables paur lesstructures de dimensions réelles.

1. RllEM OEFINITION OF FRACTUREENERGY

The constitutive model that appears more suitable fordescribing the mechanical behavior of "quasi-brittle"materials like concrete, cerarnics and rocks is the CohesiveCrack Model [3], which is based on two distinct relation-ships (Fig. 1). The first one is the elastic-plastic stress-strain law, holding up to the ultimate tensile stress 0'1/ , andthe second is a stress-crack opening displacement law, also

called the cohesive law, which describes the softeningbehavior provided by the damaged process zone.

The area under the cohesive law O'(w) represents theenergy dissipated on the unitary crack surface and, bydefinition, is called the fracture energy f/F of the material :

f/F= WFIAUg (1)where WF is tlie total work necessary for the completefracture of the specimen and AZig is the area of the initialresisting ligament.

The cohesive law is generally assumed as a material

0025-5432/96 © RILEM 259

Materials and Structures/Matériaux et Constructions, VoI. 29, June 1996

Wc••.•.•••••--"--+ wcrack opening

Fig. 1- Cohesive CrackMode1 [3].

(JU .- ••••••••••••••• i

LlEll lEu

-str-am-:-''--"-- .• Ea

* )1(

characteristic, since it intimately depends on the materialmicrostructure and on the dissipation mechanismsinvolved in the fracture process (bridging, creep, aggre-gate interlocking, etc.) ; therefore, the fracture energy f/Fis usually considered as a material constant.

Starting fì:omits defrnition, the fracture energy f/F doesnot represent a Iocal toughness parameter, like the criticalstress-intensity fàctor KIC (tensional fracture toughness) ; itrather represents a rnean-field parameter, involving thewhoIe complexity of microscopical phenomena ahead ofthe crack tip, which take part in the total work-of-fracture(energy fracture toughness). The great advantage of such aglobal parameter is provided by the absence of linearityrequirements in the fractureprocess:no information on thesingular stress field at the crack tip is needed, and LinearElastic Fracture Mechanics can be neglected. On the otherhand, the physical meaning of f/F proves to be ambiguoussince it has been defined, according to equation (1), aspurelya surfaceenergy ([F][L]-l). On the contrary, it refersto a much more complicated process of dissipation, takingpIace in a higher dimensionalspace which includes all thepreviously mentioned micromechanisms of damage [1].

2. SIZE EFFECTS ON RllEM FRACTUREENERGY

The experimental determination of concrete fractureenergy f/F is nowadays governed by a RILEM Recom-mendation [4] and consists of a standard displacement-controlled three-point bending test. A huge number oftests preceded and followed the publication of the afore-mentioned Recommendation: unfortunately, the increaseof the measured value of f/F has alwaysbeen detected withincreasing specimen size,which is essentiallythe same trendobserved in the case of other toughness parameters (Krc or

J-integran, even in different materials. In the first extensiveround-robin, almost 700 beams were tested in 14 differentlaboratories, but the size range was rather small [5] ; afteranalyzing the results, Hillerborg concluded that f/F couldbe considered as a material constant, since its variationwith size was less than one-third of the correspondingvariation of strength. Subsequent investigations on widersize ranges [6] showed that the f/F variation with size is not

at all negligible, and that this scalingeffect has to be consi-dered together with the more familiar sizeeffect on tensilestrength [2].

A well-known interpretation of this scaling behavioris owed to Wittmann et al. [7, 8]. They state that theincrease with size of the fracture process zone (FPZ)width tlp is responsible for the variation of the nominalfracture energy f/F, which causes an increase of the criti-cal crack opening displacement W2 = Wc in a bilinearcohesive law (Fig. 2). Since the energy dissipation takesplace in the fracture process zone, whose width expandsduring crack propagation (R-curve behavior) at least upto a limit value (W2)/im (fiilly-developed process zone), itis reasonable to suppose that in larger specimens, wherethe FPZ can develop entire1y, a higher value of Wc isreached, thereby yielding a higher measured fractureenergy. The authors propose a local fracture energygF(X), proportional to the FPZ width and thus to Wc,whose integration along the fracture path provides asize-dependentfracture energy, in accordance with theexperimental.values-Beyond a well-defined structural

~~----~---~ w

r'(w2)lim -------------;;..------

acrack length

Fig. 2 - Bilinear cohesive law [7].

260

Carpinteri, Chiaia

size, the FPZ attains its highest width and remainsconstant; therefore, an asymptotic value of f/F is measur-ed in the limit of the Iargest sizes.

Mai and Cotterell [9] proposed a bilinear size-effectIaw for fracture energy, based on several investigations ofdifferent materials characterized by well-defined processzones:

f/F = f/e + Cb (b < b*) (2a)f/F = f/l' (b > b*) (2b)

where b* is a threshoId size, f/e is the "essential work offracture", C and f/F are two constants to be determinedfrom the tests, with the latter being the asymptotic valueof fracture energy valid for the largest sizes. An applica-tion of equations (2) is shown in Fig. 3.

ogooF o o o

:b*L- ~~ ~~ binitialligament length

Fig. 3 - Bilinear size effect law for fracture energy [9].

An extensive investigation on the fracture energy sizeeffect has been reported by Elices et al. [10-12]. Threecauses, not adequately taken into account by the RILEMtest, are considered to be responsibie for the variation oftoughness with size: the energy dissipation from hyste-resis in the testing equipment and in the lateral supports,the bulk dissipation in the most stressed regions of thesample and the dissipated energy at the end of the load-ing process, which is negIected due to the cutting of theP-o tail. It can be argued that this analysis, althoughpointing out some remarkable details of the RILEM test,does not focus on the intimate nature of the localized

~dg1

L- ~ lnb

Fig. 4 - Monofractai Scaling Law for nominai fracture energy.

energy dissipation, which seems mainly to controi thephenomenon.

Other models, alternative to the RILEM defmition,have been proposed by Bazant and Kazemi [13], basedon the Size Effect Law for strength [14] and on theR-curve concept, by Karihaloo and Nallathambi [15],based on the "Effective Crack Model", and by Jenq andShah [16], based on the "Two-Parameter FractureModel". Although they all show less sensitivity to thesize effect with respect to the RILEM fracture energydefinition, they represent less significant models from aphysical point of view.

3. MUlTIFRACTAl SCALING lAW FORFRACTURE ENERGY

As stated in a previous paper [1], the compIexity ofthe energy dissipation during a fracture process can beconsistently synthesized by considering this dissipationoccurring in a domain with a fractional topologicaldimension comprised between 2 and 3. By appIying areal-space renormalization procedure, the scaling behav-ior of the nominai fracture energy f/F can be described,in the bilogarithmic diagram, as follows :

lnf/F = Inf// + d~lnb (3)where bis a characteristic structural size, d~is the fraction-al topological increment, due to disorder, of the energydissipation space and f// is the renormalized fractureenergy, whose anomalous physical dimensions([F][L]-(l +d~)) imply that the energy dissipation is, insome way, intermediate between a purely surface dissi-pation (which is the RILEM hypothesis) and a bulk dis-sipation (which is the classicalapproach ofLimit Analysisand Damage Mechanics theories).

In any case, the monofractal scaling behavior describ-ed by equation (3) and showed in Fig. 4 does not ade-quately reproduce the experimental results, since an infi-nite nominal fracture energy wouId be predicted for thelargest sizes by equation (3), though an asymptoticconstant value of f/F has always been detected by thetests. Moreover, the topoIogy of the fracture surfacesappears experimentally multifractal [1] in the sense that,as in any natural fractal set, the presence of an internalmicrostructural scale leh and of an external macrostruc-turaI size b provides the progressiveIy-decreasinginfluence of disorder when increasing the scale of obser-vation. From the mechanical point of view, this geome-trical trend implies that the effect of microstructuraldisorder on the mechanical properties of the materialbecomes progressively less important for the larger speci-mens, whereas it represents the fundamental parameterat the smaller scales [17, 18].

A transition from a disordered (fractal) regime to anordered (homogeneous) one can therefore be emphasizedin the scalingbehavior of any mechanical quantity. In thecase of fracture energy, the former regime is ideallybound-ed by a "Brownian" microscopic disorder, correspondingto the highest possible disorder of the fracture domain(Iocalfractaldimension = 2.5), whilst the latter corresponds

261

Materials and Structures/Matériaux et Constructions, Volo29, June 1996

to the vanishing of fractality or, equivalently, to the macro-scopic homogenization ofthe microstructure (dfl-7 O) [1].A strong scale effect is provided by the influence of disor-der below the transition scale leh, whereas beyond lch, thesize effect rapidly vanishes, and the classical Euclideantheories (RILEM approach) become applicable since aconstant value of the mechanical quantity is attained.

On the basis of these hypotheses, a Multifractal ScalingLaw (MFSL) can be deduced for the nominal fractureenergy f/p, in perfect correspondence with the MFSL pro-posed for the nominal tensile strength [2]. The analyticalexpression for this Multifractal Scaling Law, represented inFig. 5, is the following :

[ 1

-112

f/)b) = f/; 1 + l~h

where f/; is the nominal asymptotic fracture energy validwithin the limit of infinite structural size (b -7 00). Thenon-dimensional term in square brackets represents thedecrease, due to the disorder, of the nominal fractureenergy with respect to the constant asymptotic value. Notethat the asymptotic requirements are satisfied by the formerexpression: if one tak:es the derivative of equation (4) andtak:es its limit for b -7 0+, the maximum slope of the sizeeffect law, equal to +1/2 (Brownian disorder), is obtained.

g;~----------------------

L-----------------.bFig. 5 - Multifractal Scaling Law for nominal fracture energy.

In the bilogarithmic diagram lnf/p vs. lnb, showed inFig. 6, the analytical expression becomes :

In (F= In g;+++ )~hb)) (5)

The asymptotes in the bilogarithmic diagram presenta peculiar physical meaning. The horizontal asymptote,corresponding to the larger structural sizes, representsthe so-called homogeneous regime of the scaling and isdescribed by the following expression :

lnf/p = lnf/p (6)whereas the oblique asymptote, corresponding to thefractal regime of the scaling, is described by :

(4)

1 ( f/; lln f/F = 2ln b + In vz;;,. (7)

A very strong difference can be emphasized betweenthe two asymptotes: besides the evident transition in thescaling behavior, they imply different physical mechanisrnsin the fracture process of the material. Whilst the horizon-tal asymptote, corresponding to the larger structures, isgoverned by f/; which is a pure1y surface energy, theoblique asymptote, corresponding to the smaller struc-tures, is controlled by the ratio of surface energy to thesquare root of a length, or, by a stress-intensity factor (KI)with physical dimensions [F][L)-3/2.

On the basis of the MFSL, the RILEM fracture energy,defined as a mean-field quantity, appears to be a physicallymeaningful parameter on1y in the homogeneous regime,whereas Linear Elastic Fracture Mechanics, characterizedby a local approach (KI),governs the collapse of unnotchedstructures on1y when the characteristic size a of micro-structural defects becomes comparable with the macrosco-pie size b of the specimen or, equivalently, when theinfluence of disorder becomes essential. Note that thisimplies a dimensional transition of toughness, which canbe reconducted to the non-integer dimensions of therenormalized fracture energy f/p* [1].

From these considerations, it can be argued that thefundamental Irwin's relation:

K/c= J f//cE (8)relating the critical value of the stress-intensity factor Klc,the fracture energy (or critical value of the energy releaserate) f/IC and the elastic modulus E of the material, is validon1y for the larger structures when a constant value of frac-ture energy can be defined. Therefore, on the basis of theMFSL, LEFM always governs the local collapse of a mate-rial but can be homogenized into an energy parameter (f/p)on1y at the larger scales. This is in perfect agreement withthe MFSL proposed for tensile strength [2], where theGriffith collapse, valid for the smaller sizes, causes anoblique asymptotic behavior towards infinite values ofstrength, whereas a constant (minimum) value of strengthcan be determined for the larger structures.

The intersections of the MFSL and of its asymptoteswith the logarithmic axes are reported in Fig. 6; in parti-cular, the intersection Q between the two asymptotesrepresents a very important point in the diagram. It is cha-racterized, in fact, by a value of the abscissa equal to thecharacteristic length leh of the material, which represents athreshold scale between the two different scaling regimes,analogously to the case of the MFSL for tensile strength[2]. This internallength, which is a parameter appearing inany sound modeling approach ofheterogeneous materials,is typical of any natural fractal domain, whereas mathema-tical fractals (monofractals) have no characteristic scale andtherefore show a linear and unbounded scaling.

It is reasonable to suppose that the value Ofleh is relat-ed to the size of heterogeneities in the material micro-structure, these being the aggregates in the case of concrete,

262

homogeneous (euclidean) regime

Carpinteri, Chiaia

IngF. Q

lng": ---------------_------F :

(g~)In FJt;"

Genera1ly speaking, one has to determinefor each material the proper range of scaleswhere the fractal regime is predominant,and consequently the minimum structuralsize beyond which the local toughnessfluctuations are macroscopically averaged,and a constant value of fracture energy canbe assumed.

~X-~~---r------~------------------~Inb1

dg=O.sVfractal (Brownian) regime

4. APPLICATION OF THE MFSlTO EXPERIMENTAl DATA

Fig. 6 - Multifractal Scaling Law: bilogarithmic diagram.

concrete

In the present section, the MultifractalScaling Law is applied in order to interpretthe results from experimental tests on dif-ferent concrete geometries. Two aims arepursued by this statistical analysis: from aphysical point of view, the goodness of fitof the size-scale behavior for the nominalfracture energy is provided, whereas froman engineering point of view, the methodallows for the extrapolation, from labora-tory-sized specimens, of a reliable value ofthe fracture energy valid for real-sizedconcrete structures.

A non-linear best-fitting method hasbeen implemented, based on theLevenberg-Marquardt algorithm [20]; inthis way, equation (4) is fitted to experi-mental data, giving as results for the parti-cular concrete mixture and test geometryconsidered the values çl" and leh. Notethat this regression procedure turns out to

be more statistically reliable the wider the size rangeconsidered in the tests: one order of magnitude of thestructural size range should be considered in any casebut, unfortunately, this requirement is rarely fulfilled inthe literature.

The first geometry to be considered is that ofWittmann et al. [7] and consists of a seriesof compact testsover a size range 1:4 (bmin = 150 mm, bmax = 600 mm,where b is the initialligament length). The concrete mix-ture provides an average compression strength fc equal to42.9 MPa, with a maximum aggregate size dmax = 16mm.This test geometry presents the advantage,with respect tothe standard RILEM test, of eliminating the specimen'sself-weight; nevertheless, it results in being less practicalto carry out. Only a two-dirnensional similitude is gua-ranteed, with the thickness t of the specimens beingconstanly equal to 120 mm. Six specimens have been test-ed for each representative size. The nominal values of thefracture energy are obtained by the ratio of the total workoffracture (areaunder the load-displacement curve) to theinitial area of the ligament (b x t). Note that the authorscut the end of the softening tail, intending that the hinge-mechanism due to bridging and interlocking betweenaggregateshas not to be taken into account in the tough-ness evaluation.

In/ch (concr)

______~~~----+-~~------~------------~lnb

Fig. 7 - Application of the MFSL to two materials with different microstructure.

the grains in the case of metals, the crystals or the poresin the caseof rocks, the fibers in a fiber-reinforced com-posite and even the polymeric chains in a plastic mate-rial. In the specific case of concrete, a linear relationshipwith the maximum aggregate size has been proposed [2] :

~=~max ~The decisive role of the aggregate on the toughness

characteristicsof concrete is widely recognized nowadays:the measured values of çp in the case of mortar are muchlower than in the case of concrete [3] ; also, day bricksshow a higher toughness if a higher percentage of sand(which represents a large aggregate when compared withday) is used in the mixture [19].

It can be stated that in the case of a finer-grainedmateriallike a ceramic composite, the MFSL is shifted tothe left with respect to the case of concrete, with thevalue of leh being much lower for ceramics than forconcrete (Fig. 7). Therefore, two specimens of differentmaterials, with the same structural dimension bl, besidesobviously showing two different values far the nominalfracture energy, have to be set into two different scalingregimes. With reference to Fig. 7, the concrete specimenbehaves according to the fractal regime, whereas theceramic one lies on the asymptotic branch of the MFSL,thus showing a homogeneous macroscopic behavior.

263

Materials and Structures/Matériaux et Constructions, Volo 29, June 1996

200 140QF QF120 o

150 100 o

80100

mb 60

50 40 ~p20 l lIbb b

o oo 100 200 300 400 500 600 700 o 50 100 150 200 250 300(a) (a)

2.4 140logQF QF2.2 120

2.0 1001.8 801.6 P P 60

~b logè

1.4 401.2 20 l1.0 O

-1 O 2 3 4 5 O 50 100(b)

Fig. 8 - Application of the MFSL to the data by Wittmann et al.[7]: linear diagram (a), and bilogarithmic diagram (b).

i 100 IQp

I 80!160

I 40

20

..-.--u-----0~ I

!I

bO~~~~~~~~~~~~~~~O 50 100 150 200 250 300 350 400

Fig. 9 - Application of the MFSL to the data by Elices et al. [12].

The application of the MFSL to the f/F values isshowed in Fig. 8a, whilst the bilogarithmic diagram isrepresented in Fig. 8b. The asymptotic fracture energyresults in f/F = 196.2 N/m, and the characteristic inter-nallength 1eh = 209.5 mm. Therefore, the asymptotic

o

150 200 250 300(b)

Fig. 10 - Application of the MFSL to the data by Perdikaris andRomeo [21]: Series I (a), and Series II (b).

toughness is about 40% larger than the smallest speci-rnen's value (121.5 N/m). Note that by applying theindirect method by Bazant and Kazemi [13], based onthe extrapolation from the tensile strength Size EffectLaw [14], one would obtain an asymptotic fractureenergy equal to 117 N/m, even smaller than the smallestspecimen's value! On the basis of equation (9), the non-dimensional pararneter ex results in ex = leh / dmax = 13.1.The correlation coefficient, which is a due to the good-ness offit, turns out to be R = 0.937.

Experimental results obtained by rigorously followingthe RlLEM Recommendation [4] are those by Elices et al.[12]. Three-point bending tests under crack openingcontrol have been carried out by the authors on bearnsmade of concrete with dl1lax = 10mm andfc = 33.1 MPa. fuin the previous case, only a two-dimensional similitude isprovided, with the thickness t being equal to 100 mm forall the bearns. The bearn height, assumed as the referencesize, ranges from 50 mm to 300 mm (range 1:6). Thenominal fracture energy is obtained from the total work offracture (considering also the weight of the beam and ofthe testing equipment [4]) divided by the initialligarnent((b - aO) x t, where aO = b/3 is the initial notch depth).

264

Carpinteri, Chiaia

The application of the MFSL is shown in Fig. 9; thebest-fitting values are tlp"" = 110.6N/m and 1eh = 133.1mmrespectively. The asymptotic fracture energy is thereforealmost 90% larger than the smallest specimen's value(57 N/m). The non-dimensional parameter a = 1eh/dmax isequal to 13.3, whilst the correlation coefficient turns outto beR = 0.982.

Another series of three-point bending tests, carriedout according to the RILEM standards, is that ofPerdikaris and Romeo [21]. Two mixtures of concretewere examined with the same granulometry (dmax = 6.4mm), butwith different compressive strengths. In the caseofSeries I, a result offc(I)= 35 MPa was obtained, whereasin the case of Series II, a higher strength was obtained(fc(II)= 75.5 MPa). The size range under examination wasequal to 1:4 for both Series and, as usual, onlya two-dimensional similitude was present, with the thickness ofthe bearnsbeing constant and equal to 127 mm.

The application of the MFSL to the data from Series I(low-strength concrete, LSC) is described in Fig. 10a: thebest-fitting values are tlpoo= 129.5 N/m and leh = 104.1mmrespectively,and the correlation coefficient results in R =

0.892. From the value of the internallength 1eh, the non-dimensional parameter a = 16.3 is obtained. The applica-tion of the MFSL to the data from Series II (high-strengthconcrete, HSC) is shown in Fig. 10b: in this case, the best-fitting procedure provides tlpoo = 160 N/m, 1eh = 320.9 mm,a = 50.1 and R = 0.921. In the latter case, the asymptoticfracture energy turns out to be about 130% higher thanthe smallest specimen's value (68 N/m), and the non-dimensional parameter a results in an anomalously largevalue with respect to the other cases (where a is generallycomprised between 10 and 20).

An interesting comparison can be drawn between thestrength and toughness properties of the two Series.Whilethe strength value ofSeries II is in fàct 114% larger than inSeries I, a much smaller difference (24%) between theasymptotic fracture energies of the two Series has beendetected. This confirrns the (relative)increaseofbrittlenesscorresponding to the increase of compactness and homo-geneity in the microstructure which, in turn, tends toincrease the strength value. The transition to a more brittlebehavior, in the case of the HSC, can be synthesized bymeans of Carpinteri's Brittleness Number 5 [22]: suppos-ing, in fact, that the tensile strength values are equal toone-tenth of the compressive strengths and setting b = 100mm for both Series, one would obtain S(Series I) = 0.00037and S(Series II) = 0.00021, thus concluding that the larger thestrength, the larger the brittleness.

An innovative test procedure has been recentlydeveloped by Carpinteri and Ferro in order to determinethe tensileproperties of concrete [23].Bone-shaped speci-mens are tested in direct tension, under displacement-controlled loading. The specimens are bounded with stiffplates at their ends, and the reacting section is subjected totractions always being centered with respect to the pro-gressively-damagedligament. This is provided by the useof three independent jacks, the first one acting axiallyandthe others on the two principal planes, in order to com-pensate for the load eccentricities due to the non uniform

damage through the reacting ligament. Therefore, nobending moment is present during the loading processoAlarger fracture energy, with respect to the three-pointbending tests, is obtained : this is due to the (infinite) stiff-ness of the boundary conditions, which may give rise tothe formation of two macrocracks instead of one as in theRILEM test, and to their subsequent bridging, thus result-ing in a long-tail softening.

The details of the experimental procedure areexhaustively described in [23] : the maximum aggregatesize is dmax = 16 mm, and the average compressivestrength is equal to 36.9 MPa. A two-dimensional simi-litude is present, with the thickness t of the "bones"being constantly equal to 100 mm. A 1:8 size interval hasbeen tested, ranging from b = 50 mm to 400 mm, with bbeing the width at the neck of the specimens.

The application of the MFSL is shown in Fig. 11 ; theasymptotic fracture energy results in tlpoo = 226.4 N/m,and the characteristic internallength 1eh = 328.8 mm. Onthe basis of equation (9), the non-dimensional parametera results equal to a = 1eh / dmax = 20.5. Thecorrelationcoefficient turns out to be R = 0.864.

2oor--------------------------,175 gF

15012510075

5025

bOL---~~--~~--~--~~--~o 50 100 150 200 250

Fig. 11 - Application of the MFSL to the data by Carpinteri andFerro [23].

5. CONClUSIONS : DUAUTY OF THEMUlTIFRACTAl SCAUNG lAWS FORSTRENGTH ANO TOUGHNESS

It is interesting to note the perfect duality existing be-tween the Multifractal Scaling Laws obtained for strength[2] and the toughness of disordered materials. The pro-gressive rarefaction, due to damage, of the resisting Iiga-ment, as well as the disordered evolution of the fracturesurfàce, can be modeled by means offractal domains, witha lacunar fractal (with topological dimension smaller than2) in the case of the ligament and an invasive fractal (withtopological dimension larger than 2) in the case of thefracture surface, respectively. By applying aRenormalization Group procedure, these hypothesesyield the decrement of nominal tensile strength and the

265

Materia/s and StructuresjMatériaux et Constructions, VoI. 29, June 1996

increment of nominal fracture energy respectively, withincreasing structural size.

The multifractal nature of the material microstructure,which has been experimentally detected by the authors [1]and has been indirectly confirmed by the observed trendsof the nominal mechanical quantities, provides a transitionfrom disorder to order in the scaling behavior of thesequantities, thereby yielding, for strength and toughness res-pectively, the foliowingMultifractal ScalingLaws:

o,(b) ~t,[l /;f' (lOa)

[ l-Vçp(b) = ç; 1 + l~h 2 (10b)

where the exponent ± 1/2 comes from the (thermodyna-mie) hypothesis of a Brownian microscopic disorder ofthe microstructure [17, 18]. It can be observed in equa-tions (10) that the unique difference between theMultifractal Scaling Laws is in the sign of the exponents,connected with the different kind of topology involvedin the phenomenon, that defines the asymptotic behav-ior of the considered mechanical quantity as the structu-ral size tends to zero.

An extensive experimental and statistical investiga-tion is stili necessary in order to verify the relationshipbetween the lengths lch in the two Scaling Laws and inorder to confirm the validity of equation (9), whichallows connecting such internallengths to a characteris-tic granulometric size. Size ranges of more than oneorder of magnitude should preferably be considered, andvarious mixtures of concrete should be compared.

ACKNOWlEDGEMENTS

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

REfERENCES[1] Carpinteri, A. and Chiaia, B., 'Multifractal nature of concrete

fracture surfaces and size effects on norninal fracture energy',Materials and Structures, 28 (182) (1995) 435-443.

[2] Carpinteri, A., Chiaia, B. and Ferro, G., 'Size effects on nominaltensile strength of concrete structures: multifractality of materialligaments and dimensional transition from order to disorder',Materials and Structures, 28 (180) (1995) 311-317.

[3] Hillerborg, A., Modeer, M. and Petersson, P.E., 'Analysis ofcrack forrnation and crack growth in concrete by means of frac-ture mechanics and finite elements', Cement and Concrete Research,6 (1976) 773-782.

[4] RILEM Technical Committee 50, 'Determination ofthe fractureenergy of mortar and concrete by means of three-point bend testson notched beams', Draft Recommendation, Materials andStructures, 18 (1985) 287-290.

[5] Hillerborg, A., 'Results of three comparative test series for deter-mining the fracture energy çF of concrete', Materials and Structures,18 (1985) 407-413.

[6] Swartz, S.E. and Refai, T.M.E., 'Influence of size effects on open-ing mode fracture parameters for precracked concrete beams inbending', Proceedings of the SEM(RILEM Int. Conf. Fracture ofConcrete and Rock (Shah and Swartz eds., Houston, 1987), pp.403-417.

[7] Wittmann, F.H., Mihashi, H. and Nomura, N., 'Size effect onfracture energy of concrete', Engineering Ftacture Mechanics, 35(1990) 107-115.

[8] Hu, X.Z. and Wittmann, F.H., 'Fracture energy and fracture pro-cess zone', Materials and Structures, 25 (1992) 319-326.

[9] Mai, Y.W. and Cotterell, B., 'Effect of specimen geometry on theessential work of plane stress ductile fracture', Engineering FractureMechanus, 21 (1985) 123-128.

[10] Guinea, G.V., Planas, J. and Elices, M., 'Measurement of thefracture energy using three-point bend tests : Part 1 -Influence ofexperimental procedures', Materials and Structures, 25 (1992) 212-218.

[11] Planas, J., Elices, M. and Guinea, G.V., 'Measurement of thefracture energy .ising three-point bend tests : Part 2 -Influence ofbulk energy dissipation', Materials and Structures, 25 (1992) 305-312.

[12] Elices, M., Guinea, G.V. and Planas, J., 'Measurement of thefracture energy using three-point bend tests : Part 3 -Influence ofcutting the P-o tail', Materials and Structures, 25 (1992) 327-334.

[13] Bazant, Z.P. and Kazemi, M.T., 'Size dependence of concretefracture energy determined by the RILEM work-of-fracturemethod', Iat.jour. Fracture, 51 (1991) 121-138.

[14] Baéant, Z.P., 'Size effect in blunt fracture: concrete, rock,metal',journal ofEngineering Mechanics, ASCE, 110 (1984) 518-535.

[15] Karihaloo, B.L. and Nallathambi, P., 'Size-effect predictionfrom effective crack mode! for plain concrete', Materials andStructures, 23 (1990) 178-185.

[16] Jenq, Y. and Shah, S.P., 'Two-parameter fracture model forconcrete',jour. Eng. Mech., ASCE, 111 (1985) 1227-1241.

[17] Carpinteri, A., 'Scaling laws and renormalization groups forstrength and toughness of disordered materials', International

journal ofSolids and Structures, 31 (1994) 291-302.[18] Carpinteri, A., 'Fractal nature of material microstructure and size

effects on apparent mechanical properties', Mechanics of Matenals,18 (1994) 89-101.

[19] Bosco, C. and Iori, L, 'Sulla meccanica della frattura dei laterizi:indagine sperimentale di elementi inflessi ed analisi di parametricaratteristici', Studi e Ricerche, 13 (1992) 269-300.

[20] Marquardt, D.W., 'An algorithm for least-squares estimation ofnonlinear parameters',jour. Soc. Industr. Appi. Math., 11 (1963)431-441.

[21] Perdikaris, P.c. and Romeo, A., 'Effect of size and compressivestrength on the fracture energy of plain concrete', Proceedings ofthe First International Conference on Fracture Mechanics ofConcrete Structures, FRAMCOS1, (Baéant ed., Breckenridge,1992) pp. 550-555.

[22] Carpinteri, A., 'Mechanical Damage and Crack Growth inConcrete: Plastic Collapse to Brittle Fracture', (Martinus NijhoffPublishers, Dordrecht, 1986).

[23] Carpinteri, A. and Ferro, G., 'Size effects on tensile fracture pro-perties: a unified explanation based on disorder and fractality ofconcrete microstructure', Materials and Structures, 27 (1994) 563-571.

266