materials ^Department of Civil Engineering, University of ... · 2. Fractal modelization of heterogeneous microstructures The characteristic features of critical phenomena, such as

Multifractality of fracture patterns under tensile

and compressive stresses in cement-based

materials

A. Carpinteri," B.Chiaia,* K.M. Nemati^

"Department of Structural Engineering. Politecnico di Torino,

70724 Tbrmo, /Wy

^Department of Civil Engineering, University of California,

AbstractThe scaling properties of the fracture surfaces generated by uniaxial tensile testsand of the crack networks induced by compressive loadings are investigated. Inthe case of the tensile tests, a laser profilometer was adopted to scan the fracturesurfaces. In the case of the compression tests, after injection of a fusible alloyinside the specimen, scanning electron microscope was used to detect the stress-induced crack patterns. The highly localized energy dissipation in tensile fractureis shown to occur over a fractal domain with dimension larger than 2.0. Also, themicrocracks network induced by compression loadings possesses fractaldimension much larger than 2.0 in the bulk. Fractality permits to explain someaspects of the fracture behaviour, like the stable crack growth encountered in thetests and the size-scale effect undergone by the toughness parameters.

1 Introduction

1.1 Tensile failure in concrete structures

Heterogeneous materials like concrete, ceramics and rocks, show a similarmechanical behaviour (at the global level) under uniaxial tensile loadings, despitethe mechanisms of failure are different at the microstructural level. Strain-softening occurs after the peak load, involving progressive failure by damage-localization and breakdown of the microstructural components. Moreover, thesematerials are all characterized by considerable size-effects and by a markeddifficulty in the reproducibility of the tests. The continuum mechanics approachfails to predict these aspects, and also Linear Elastic Fracture Mechanics isinadequate because of the scale of heterogeneities.

Transactions on Engineering Sciences vol 13, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533

22 Localized Damage

Starting from the pioneering works of Griffith, energy-based failure criteriaproved to be more appropriate than stress criteria to describe tensile failure ofbrittle materials. Afterwards, LEFM has been extrapolated and non linear modelshave been introduced to explain peculiar phenomena occurring in the presence ofmeso-level heterogeneities. The constitutive model that best describes the tensilebehaviour of concrete is the Cohesive Crack Model by Hillerborg et al. [1], whichis based on two different relationships. The first one (Fig. la) is the elastic-plasticstress-strain law, holding up to the ultimate tensile stress o%, and the second one(Fig. Ib) is a stress-crack opening displacement law, also called the cohesive law,which describes the softening behaviour provided by the damaged process zone.

rfGp = J cr

strain(a)

crack opening

(b)

Figure 1. Cohesive Crack Model [1].

The area under the cohesive law a(w) represents the energy dissipated onthe unitary crack surface and, by definition, is called the fracture energy Qp of thematerial. The cohesive law is assumed as a material characteristic, since itintimately depends on the microstructure and on the dissipation mechanismsinvolved in the fracture process (bridging, creep, aggregate interlocking):therefore, the fracture energy Qp is usually assumed as a material constant.

According to its definition, the fracture energy Qp does not represent a localtoughness parameter, like the critical stress-intensity factor Kj£ (tensile fracturetoughness). It rather represents a mean-field quantity, involving the wholecomplexity of microscopical phenomena ahead of the crack tip, which contributeto the total work-of-fracture (energy fracture toughness). The great advantage ofsuch a global parameter is provided by the absence of linearity requirements inthe fracture process: no information on the singular stress field at the crack tip isneeded, and Linear Elastic Fracture Mechanics can be neglected. On the otherhand, the physical meaning of Qp reveals to be ambiguous, being it defined as apurely surface energy ([F][L]~*). On the contrary, it refers to a much morecomplex process of dissipation, taking place in a higher dimensional fractalspace [2] which includes all the previously mentioned micromechanisms ofdamage.


Localized Damage 23

1.2 Compressive failure in concrete structures

The composite nature of concrete represents the major difficulty in developing a

fracture criterion for concrete in compression. It is nowadays clear that a sound

explanation of the behaviours observed in the laboratory and in real-sizedstructures can be obtained only if the internal microcracking phenomenon is takeninto account [3]. Similar crack patterns in similar uniaxial compressive tests havebeen detected, at different scales, in different materials like glass, concrete, rocksand masonry. The similarity in the cracking behaviour of different brittlematerials emphasizes common fundamental fracture mechanisms.

(a)

(b)

low frictionboundaries

high frictionboundaries

Figure 2. Compression-induced fracture patterns in concrete specimens [4].

The dominant mechanism of brittle fracture in compression is basicallyMode I cracking, the same as in tension [3]. Tensile stresses break interatomicbonds and separate the surfaces. However, this does not totally exclude Mode IIor Mode III cracking, or their combination, but it is believed that these Modescome into play in the last stages of the failure process, whilst in the pre-peakregime they may occur only at a local scale as second-order effects. Macroscopicfailure modes frequently observed in compression tests, such as conical,pyramidal or wedge-shaped modes, are mainly due to the boundary confinementof the loading platens [4]. Under the usual testing conditions, the stress field in thespecimen is highly anisotropic and not uniaxial at all. The so-called shear failureis nowadays considered a secondary (macro) effect which occurs after the axialsplitting has begun. Shear band and shear displacement are not fundamentalmechanisms of failure, but only the consequence of boundary restraint (Fig. 2).


24 Localized Damage

In uniaxial compressive tests, macroscopic cracks essentially start parallelto the direction of axial loading (Fig. 2). Convincing theoretical explanations arelacking for the propagation of a parallel crack during a uniaxial test in ahomogeneous material. Stress-based criteria would request the presence of tensilestresses. Further, energy-based criteria fail if no source of energy release ispresent, as it is the case of a hairline crack parallel to the direction of load, whichapparently alters neither the strain energy nor the total potential energy. The onlyreliable explanation for this behaviour is provided by the intrinsic heterogeneityof the material, and especially by the presence of cracks, pores and voids evenprior to the application of the load. Griffith postulated that brittle materialscontain randomly oriented flaws which significantly alter stress-strain fields in thebulk. As well as pre-existing flaws act as stress-raisers in tensile tests, they can beconsidered tensile stress producers in compression. In fact, the presence of voidsin concrete strongly reduces its strength. That is, 5% of voids can lowercompressive strength by as much as 30% compared to the same mixture whenfully compacted.

Any constitutive or geometrical heterogeneity alters the stress-strain field,and thus may promote crack initiation. On the other hand, heterogeneity may alsostop or slow down crack propagation. This duality represents an awkward aspectto be modeled. It can be concluded that the interplay between the tensile stress-inducing capacity and the crack-arresting capacity of the micro heterogeneities isthe key for a better understanding of brittle failure in compression.

2. Fractal modelization of heterogeneous microstructures

The characteristic features of critical phenomena, such as hierarchy of interactingdefects, stochasticity and far-from equilibrium instabilities, are present in bothtensile and compressive failure modes. The irregularity of the crack trajectories isa well documented and discussed phenomenon. The two potential sources ofdisordered damage patterns are the intrinsic instability of the cracking processand the effect of the (random) pre-existing material inhomogeneities. Anextensive documentation exists on the stability of crack trajectories to smallperturbations and it is nowadays widely demonstrated that the quenched disorderof brittle fracture patterns cannot be ascribed to crack-tip instabilities. It can beconcluded that the main source of randomness and chaotic behaviour is theinherent disorder of the material micro structure.

Whenever entropy wins over energy, the resulting structure will bedominated more by randomness than by strict Euclidean order and we mightexpect to find fractal patterns with self-similar scaling symmetry [5]. Multifractalpatterns prevail in Nature when the interplay between two length scales arises,whereas mathematical fractals, lacking any internal length, can be characterizedby a unique value of the self-similarity dimension.

The application of Fractal Geometry to cementitious materials is relatively


Localized Damage 25

recent in comparison with the application to rocks and metals, but has proven tobe very appropriate, due to the multi-scale heterogeneity of these materials [6].This is reflected into the hierarchical process of failure. Cracks usually originatefrom pre-existing micro-defects in the matrix or at the matrix/aggregate interface(micro-level), then develop at the scale of the matrix, (meso-level), and finallyextend to the structural scale (macro-level), resulting in a typically disorderedpattern with multiple cracking randomly diffused at the interfaces, through thematrix and through the aggregates.

The main aspect to be highlighted, next to the phenomenological evidence,is the physical significance of fractality. The deep connection between Physicsand Topology is nowadays accepted in the framework of critical phenomena. Allthe physical systems undergoing catastrophic transformations, like phasetransitions, earthquakes and brittle fracture, show at the critical point fluctuationsthat are self-similar at all length scales, thus resulting in the (theoretical) absenceof any internal characteristic length (or, which is the same, in the infinitecorrelation length of the phenomenon).

N=2

A =1.262

[A = 2.00[(a) (b)

Figure 3. Invasive fractals: random von Koch curve (a) and Peano curve (b).

In the case of fracture patterns, invasive fractals (i.e. fractal sets withdimension A strictly larger than their euclidean projection) represent adequatemodels of their topology. The random von Koch curve (Fig. 3a) can be consideredthe archetype of the fracture trajectories obtained as intersections of tension-generated fracture surfaces with orthogonal planes. The Peano curve (Fig. 3b)represents the limiting invasive fractal in the plane (fractal dimension A=2),


26 Localized Damage

because establishes a continuous correspondence between the straight line and theplane. The space-filling ability of these sets provides positive scaling exponentsfor the mechanical quantities (e.g. the fracture energy) defined over them [6].

3. Experimental investigation on the fractality of tensile fracture

surfaces

3.1 Experimental set-up

Bone-shaped concrete specimens have been tested in uniaxial tension and theresulting fracture surfaces have been subsequently analysed [2]. A laserdisplacement sensor (3mW semiconductor, class Illb) has been used to scan thesurfaces along parallel profiles. The wavelength of the laser beam was 780nm,allowing for an accuracy of ±2|um. This yields a significant improvement withrespect to previous investigations [7] and represents, for our purposes, the optimalresolution in the case of cement-based materials. For an optimal measurementset-up, the sensitivity of the laser was adjusted to fit the gain-level properties ofthe concrete surface (color, light absorption and reflection, and so on).

Figure 4. Acquisition of fracture profiles: experimental set-up.

The laser sensor is mounted on an advanced motion controller system, at adistance of approximately 25 mm from the concrete surface (Fig. 4). Themechanical system shifts the sensor along two perpendicular directions (x and ycoordinates), usually parallel to the main sides of the cross section, withhorizontal steps equal to 0.01mm. After each step, the guide stops and the profileelevation (z coordinate) is detected by the sensor and stored in the computer alongwith the horizontal coordinates. Due to inertial forces, a ±1 uin error is present inthe determination of the horizontal coordinates.

In principle, given the coordinates of a limited area on the surface, theprocedure is fully authomatized, and a 3D topography of the fracture surface isobtained. On the other hand, a lOxlOcm^ area (which is the minimumrepresentative area for a normal concrete) consists of 100 millions digitized


Localized Damage 27

points, that is, 300 millions stored coordinates. This results in an enormouscomputational effort when the fractal dimension has to be measured on PCs. Thus

it was decided, in a first stage, to analyse single profiles (2D topography). Six

profiles have been digitized on each specimen, three along each perpendiculardirection, in order to average the effects of highly correlated directions ofpropagation.

3.2 Determination of the fractal dimension of fracture profiles

Deterministic and statistic methods can be exploited for the measurement of thefractal dimension of fracture patterns. In principle, they are all equivalent in thelimit of the smallest scales, but they provide significantly different values whenapplied to digitized profiles. Most of the methods (e.g. the Divider method andthe Spectral method) are tailored to the case of one variable function graphs.Moreover, the Spectral method is rigorously demonstrated only in the case ofself-affine scaling [7]. Thus, it was decided to adopt the 2D Box-Countingmethod which, on the contrary, applies to any (invasive or lacunar) fractal domainreferred to the 2D plane. Both the tensile fracture profiles and the compressivecrack networks can be analysed by this method.

Figure 5. Application of the Box-Counting method to a digitized concrete profile.

The Box-Counting method is based on the concept of the Minkowski-Bouligand dimension. The Minkowski-cover of a set is obtained by covering theconsidered domain by means of a collection of regular euclidean figures, withdimension equal to the space in which the fractal is embedded. The fractaldimension is obtained by computing the logarithmic density of the measure ofthese coverings, as their linear size decreases. In principle, the concept ofgeneralized cover allows one to adopt any kind of convex shapes to perform thecovering. On the other hand, from a computational point of view, square or


28 Localized Damage

rectangular grids are always used. In order to study the complex patterns resultingfrom the experimental tests, a general-purpose version of the method has beendeveloped [8]. As the linear size 8 of the covering boxes decreases, their numberN increases and their area E decreases (Fig. 5). The Box-dimension A is given by:

~ = 2- Urn -\

(1)

In practical applications, instead of dealing with a limit, linear regression iscarried out in the logTV vs. loge plot or in the logE vs. loge plot. The slopes 9 andi) are respectively obtained and the Box-dimension is computed as A=-(p or asA=2-$. Further details on the algorithm can be found in [2].

In the case of fracture profiles, direction-dependent rescaling of the grid hasto be performed, because of the multifractal character of the set. Self-affinescaling is provided, in fact, by the interplay between the (mainly horizontal) crackadvancement and the orthogonal direction. Thereby, anisotropic grids are used, inorder to enhance the statistical fluctuations in one direction. In this way, it ispossible to detect the local fractal dimension in the limit of the smallest grids, aswell as the multifractal transition to homogeneous behaviour (Fig. 6).

P

1 1

1.6

a 0.8

0.6

Brownian disorder, A = 1 .5

Euclidean order, A = 1 .0

(b)

1 2 3 4 5log (grid size)

0 1 2 3 4 5 6 7scale of observation, mm

Figure 6. Double-log plot from the Box- Counting method (a), and multifractaltransition from Euclidean order to Brownian disorder (b).

The multifractal character of the fracture profiles clearly emerges from theBox-Counting analysis. Not a unique slope can be traced in the double-logdiagram (Fig. 6a), but the fractal dimension A progressively decreases as the scaleof observation increases. The densifying ability of the domain vanishes at coarsescales (macro-level), where a smooth euclidean profile is measured. On thecontrary, as the grid size tends to zero (micro-level), the fractal dimensionincreases, ideally tending to 1.5 which is the dimension of a unidimensionalBrownian motion in the space vs. time domain (Fig. 6b). This trend, already


Localized Damage 29

captured by previous investigations [2, 7], is even more pronounced due to thehigher resolution in the data acquisition. Local dimensions A^ax ranging from1.28 to 1.41 were measured at the smallest scales, testifying the tendency to 1.5.

It is worth pointing out that discreteness effects come into play when thegrid size is close to the digitization step (i.e. 10|im). Therefore, data in the upperleft side of the double-log diagram have to be neglected (Fig. 6a). This representsan unavoidable drawback of any experimental measure of fractality on real sets,regardless of the data-acquisition method and of the measurement algorithm.Discreteness effects arise also in the case of SEM images of compression-inducedcrack networks, strongly affecting the image analysis procedure (Fig. lOf).

zj(mm)

x (njim)10 20 30

(a)

I x(mim)40 50 0 10 20 30 40 50

(b)

0 10 20 30 40 50(c)

A =11.77

I x (rrtm)

0 10 20 ,30 40 50(d)

Figure 7. Profile elevations (a, b) and plot of the increments (c, d) for two profiles.

Confirmation of the Brownian character of the fracture surfaces comes fromthe analysis of the elevations growth rate (derivative) of the profiles. In the idealcase of geometrical Brownian motion, the plot of the increments possessesHausdorff dimension equal to 2.0 and its power spectrum is absolutely flat [5]. Nocorrelation is present between two subsequent increments and self-similar scalingcomes into play (Gaussian white noise), meaning that the system of the


30 Localized Damage

increments lacks any characteristic scale.The trail of the Brownian motion can be intended as the integral of a white

noise and possesses fractal dimension equal to 1.50. Self-affine scaling isprovided in this case, and an internal length characterizes the graph. In fact, if werescale the horizontal axis by a factor 'V and the vertical axis by a factor 'fe ',we reproduce the original distribution of the motion. Short-range correlations arepresent in the graph, as well as long-range ones. Indeed, short-range correlationsprevail, meaning that the position of each point in the graph is dependent on theposition of all the other points, but mainly on the position of the surroundingones. Therefore, the power spectrum (S(f) ~/~ ) steeply decreases as the spatialfrequence increases.

In the case of concrete fracture profiles, analysis of the growth rates (Figs.7c,d) yields dimensions ranging from 1.71 to 1.94, which confirms the hypothesisof Brownian disorder underlying the fracture phenomenon. Recent applications ofthese concepts to turbulence [9] lead also to affirm that a connection must existbetween all the chaotic dissipative phenomena. Extrapolation of the results to 3Dsets is straightforward, since 2D profiles can be considered intersections of the 3Dsurface with vertical planes. Indeed, 3D correlations cannot be revealed by theprofile analysis. It can be concluded that the Box-dimensions measured on 2Dprofiles represent a lower bound of fractality in concrete fracture surfaces.

4. Experimental investigation on the fractality of compression-

induced damage patterns

4.1 Experimental set-up and image processing

The experimental technique (Fig. 8) designed to induce and preserve the cracks inconcrete under applied compressive load is described in detail elsewhere [10].Wood's metal is a non-wetting and fusible alloy with an effective surface tensionof about 400N/m [11]. It consists of 42.5% Bi, 37.7% Pb, 11.3% Sn, and 8.5%Cd. It has a melting point range from 71.1°C to 87.8°C, and is solid at roomtemperature. Wood's metal has a Young's modulus of 9.7GPa and a density of9.4g/cnA The advantage of such an alloy is that it can be intruded into voids andstress-induced microcracks while the specimen is held at the desired stress leveland then solidified to preserve the geometry of the microcracks induced.

The concrete sample used in this experiment was a 102mm in diameter by203mm long cylinder with an ultimate compressive strength of 51.7MPa. Theconcrete cylinder was partially confined by stainless steel wire of 0.3mmdiameter and pre-tension of 130KN that was wound one third of the way fromeach end at a pitch of winding of 4 pitches per centimeter to generate biaxialcompression, and was loaded to about 80-85% of the ultimate compressivestrength. After crack induction and subsequent solidification of the Wood's metal,the concrete cylinder was sectioned along its long axis. An axial slab


Localized Damage 31

(approximately 3mm thick) was sliced parallel to the direction of the load. A

specimen was extracted from this slab. The extracted specimen was polished with

120#, 220#, 320# and 600# silicon carbide using a rotating grinder. The specimenwas then lapped with a wheel grinder and polished with 600# silicon carbide.Further polishing was performed with 100-, 50- and 10-micron aluminum powderon a glass plate. The final stage involved was treating specimens with 5-, 3- and I/4-micron diamond paste using a special polishing equipment (according to ASTM1993). After each stage of polishing, the specimen was immersed in acetone andplaced in an ultrasonic machine in order to remove the residual silica film on itssurface, thus preparing it for the next stage of polishing. The specimen wasexamined using a JEOL JSM-35CF scanning electron microscopy (SEM) in

conjunction with a KONTRON SEM-IPS image analyzer.

II ^ 01500Psi

/-*~\concrete cylinder

steel wire

loading piston

nitrogenpressurebottle

Figure 8. Schematic diagram of test assembly.

The sample was examined in the SEM with backscattered electrons (BSE).The images were acquired by the image analyzer at magnifications of xlOO,xlOOO, x 10000, X20000, and X30000. The images were digitized into an array of512x512 pixels with 255 gray levels. SEM produces a multiphase image fromeach observation. In order to recognize and isolate Wood's metal, which is therepresentative of pores and stress-induced fractures in concrete, the imageanalyzer can make a histogram for all of the different phases in the image basedon their gray levels, with 0 representing the darkest phase and 255 representingthe brightest phase. From this histogram two threshold levels can be establishedto encompass the brightest phase in the image, namely the Wood's metal. Thethreshold for Wood's metal identification was set between 170 and 255. The nextstep is to eliminate objects from the background that don't fall between thesethreshold levels, i.e., cementitious phases and aggregates. Once this thresholdingis accomplished, what is left in the image is the crack network and pores shown


32 Localized Damage

by Wood's metal. At this point the aim is to eliminate objects on the basis of theirarea in pixel units. The lower and upper limits of the objects to be eliminated hasto be established to include small pores, small non-continuous cracks, etc. Thestep is to transform the thresholded image into a skeletonized binary image bymeans of a binary thinning process. For every thinning step, pixels that are notrelevant to the connectivity of an object are removed from the object margins, i.e.,converted into background pixels. The connectivity of objects is thus maintained.This process can be continued until all objects are reduced to a width of one pixelthat approximates the skeletons.

4.2 Determination of the fractal dimension of the crack networks

The application of the Box-Counting method to compression-generated damagepatterns is different from that on the tensile fracture profiles. Because the crackadvancement direction is highly inclined and curved, the orthotropy directionscontinuously change. Moreover, the high resolution of the SEM introduces a finercut-off in the scaling behaviour (about two orders of magnitude finer than in thecase of the tensile profiles). Thus, SEM allows for the detection of crackbranching and secondary cracks, whereas the post-mortem digitization of thetensile fracture surfaces yields a series of fracture profiles which can beinterpreted as one-variable function graphs. In the case of compression cracknetworks like those in Fig. 9a, it seems more convenient to use isotropic squarecoverings, because damage at that scale is self-similar rather than self-affine.

Comparison between tensile and compressive patterns has to be made withcare. In particular, it is not straightforward to measure the (local) fractaldimension of a localized crack on the thinned images. Instead, very interestingresults come from the fractal analysis of the global damage patterns inside thespecimen, which are directly related to the total dissipated energy. The globalfractal dimension of the crack networks obtained thresholding the Wood's metalfrom the surrounding bulk can be called the damage dimension Aj (Figs. 9 c,d). Itcan be considered as a synthetic and quantitative measure of the complexityassociated to the scale-invariant patterns of micro-fracture. A similar fractaldescription has been implemented for modeling the microcracks network inducedby tensile-splitting forces on notched concrete specimen [8].

An opposite trend with respect to the tensile fracture profiles is obtainedwhen looking at the compressive damage patterns with increasing SEMresolution (Fig. 10). Chaotic self-similar patterns, resembling 2D diffusion frontsin porous media, are revealed in the low-resolution images (0.304pixels/jLim ,which is comparable to the laser resolution). High values of the damagedimension are measured at this scale (Figs. 10 a,d), ranging from 1.65 to 1.79. Aplane-filling network would possess fractal dimension equal to 2.0, as in the caseof the Peano curve (Fig. 3b). Extrapolating to 3D domains, this would yield a


Localized Damage 33

space-filling domain of microcracks, thereby implying a purely volumetric energydissipation.

(c) (d)

Figure 9. SEM gray-level image (a), thresholded Wood's metal (b), andapplication of the Box-Counting method to the crack network (c, d).

As the resolution increases by one order of magnitude (3.04pixels/|um ),the fractal dimension decreases, dropping to values ranging between 1.20 and1.55. This is due to the transition from a fully developed web of microcracks inthe matrix to the local properties of isolated microcracks (Figs. 10 b,e).Nevertheless, damage patterns maintain self-similar character in most of thescaling regime. At very high resolutions (30.4pixels/jLim ) the width of the cracksis very large and the subsequent image-processing may lead to anomalous thinnedcrack patterns (Figs. 10 c,f). Also, very low fractal dimensions are provided. Itwas thus decided to skip the highest-resolution images from the fractal analysis.


34 Localized Damage

Figure 10. Compression-induced crack networks with different SEM resolutions.

If the complex character of the energy dissipation taking place at the crackboundaries is investigated, the local fractal dimension of the crack lips has to bemeasured (as in the case of tensile profiles), instead of the global damage


Localized Damage 35

dimension Aj of the 2D network. For this purpose, a different technique proved tobe more effective in the binary thinning of the thresholded images [8]. Instead ofaveraging white pixels (corresponding to the Wood's metal) through the width(skeletonizing process, Fig. lib), binary thinning can be performed by isolatingthe boundaries of the cracks (Fig. lie).

The Box-Counting analysis on the crack lips reveals self-affine scaling atthe local level, and values of the fractal dimension very close to those of thetensile fracture profiles.

=1.32

(a) (b) (c)

Figure 11. Binary thinning of a thresholded image (a): averaging (b) through thewidth (skeletonizing), and outlining the boundaries of the cracks (c).

5. ConclusionsThe experimental determination of the fractal properties of any real set has totackle with the inherent discreteness of the digitized data. Care has to be takenwhen evaluating the results, especially if multifractal domains are considered.Any data acquisition technique provides an extrinsic characteristic size which issuperposed to the intrinsic length of the domain. Thus, a lower cut-off in thescaling regime has to be clearly defined. The representative range of fractality hasto be individuated for any material.

Indeed, the invasive fractal character of the fracture patterns in concrete hasbeen fully confirmed by the experimental investigations. Stress-induced crackingpossesses self-similar scaling properties when the global damage network isconsidered. When a single tensile profile is measured, or when the compression-induced crack lips are considered, self-affine scaling comes into play, leading to amultifractal behaviour. In this case, Brownian disorder (characterized by a +1/2dimensional increment) represents a theoretical limit for complexity.

From a fracture mechanics point of view, fractal crack domains implysmoothing of the stress singularity at the crack tips and provide essential reasonsfor the stable crack growth encountered when testing disordered materials [12].From a structural point of view, the positive size-scale effect on the values of the


36 Localized Damage

nominal toughness (Kj£ or Qp) of real structures can be adequately explained bythe complexity of energy dissipation occurring, at the material level, in higher-dimensional spaces [6]. Moreover, multifractality can account for the progressivehomogeneization of the material properties as the structural size increases [2].

References

1. Hillerborg, A., Modeer, M. & Petersson, RE. Analysis of crack formation andcrack growth in concrete by means of fracture mechanics and finite elements.Cement and Concrete Research, 1976, 6, 773-782.

2. Carpinteri, A. & Chiaia, B. Multifractal nature of concrete fracture surfacesand size effects on nominal fracture energy, Materials and Structures, 1995,28, 435-443.

3. Vonk, R.A. A micromechanical investigation of softening of concrete loadedin compression, Heron, 1993, 38, 1-94.

4. Van Vliet, M.R.A. & Van Mier, J.G.M. Concrete under uniaxial compression,Report 25.5-95, TU Delft, 1995.

5. Mandelbrot, B.B. The Fractal Geometry of Nature, W.H. Freemann &Company, New York, 1982.

6. Carpinteri, A. Fractal nature of material microstructure and size effects onapparent mechanical properties, Mechanics of Materials, 1994, 18, 89-101.

7. Carpinteri, A. & Chiaia, B. Multifractal nature of fracture surfaces and sizeeffect on nominal fracture energy, Localized Damage 94 (Aliabadi,Carpinteri, Kalisky & Cartwright eds.), pp 537-544, ComputationalMechanics Publications, Southampton, 1994.

8. Vervuurt, A., Chiaia, B. & Van Mier, J.G.M. Damage evolution in differenttypes of concrete, Journal of Materials in Civil Engineering (ASCE), 1996, tobe published.

9. El Naschie, M.S. Kolmogorov turbulence, Apollonian fractals and theCantorian model of Quantum space-time, Chaos, Solitons and Fractals,1996, 7.

10. Nemati, K.M. Generation and interaction of compressive stress-inducedmicrocracks in concrete, Ph. D. Thesis, University of California at Berkeley,1994.

11. Yadev, G.D., Dullien, F.A.L., Chatzis, I. & Macdonald, I.F. Microscopicdistribution of wetting and non-wetting phases in sandstone duringimmiscible displacements, SPE Annual Technical Conference and Exhibition,Dallas, 1984.

12. Carpinteri, A. & Chiaia, B. Crack-resistance behavior as a consequence ofself-similar fracture topologies, International Journal of Fracture, 1996, tobe published.


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materials ^Department of Civil Engineering, University of ... · 2. Fractal modelization of heterogeneous microstructures The characteristic features of critical phenomena, such as