The question of the universality of a dimension of fracture surfaces is discussed, and it is shown that such a general parameter may exist at least for a particular class of materials.

Ficker, T.

English

CTU Open Journal Systems (Czech Technical University, Prague / České vysoké učení technické v Praze)

1 IntroductionFracture surfaces are valuable sources of information onthe structural composition and physical properties of materi-als. For these reasons they are a subject of interest for manyresearch laboratories. Since the publication of the basic workby Mandelbrot and his co-workers [1] many authors havetried to correlate the fractal dimensions of fracture surfaceswith the mechanical properties of materials. This effort hasbeen impacted by the great complexity of these surfaces,especially in the case of composite porous materials. Ce-mentitiousmaterials have complex fracture surfaces that havebeen extensively studied [2–4].The values of the fractal dimensions of a range of materi-als show only a narrow scatter, ranging from ~2.0 to ~2.2.Repeatedly determined dimensions of different fracturedsamples of the same materials have often resulted in identicalvalues and this has led some authors [5] to the idea of auniversal co-dimension (Hurst exponent) H 08. that charac-terizes fracture surfaces as a whole. Though this idea mayinvoke certain doubts at first sight [6], it should be carefullyconsidered before it is rejected or accepted.The aim of this paper is to investigate the concept of a uni-versal co-dimension of fracture surfaces. As will be shown, thisconcept may be verified experimentally using fracture sur-faces of porousmaterials in connection with their compressivestrength. For this purpose, it is necessary first to derive corre-sponding relations for fractal porosity and fractal strength,and then to apply them to a particular material, in our casecement gel.2 Fractal porosityThe large class of porous materials possesses at least onecommon feature, namely, they are composed of grains (parti-cles, globules, etc.) ofmicroscopic size l. The grains are usuallyarranged fractally with number distribution N(l)N lLlD( )  , l<L. (1)The porosity P of the clusterPlLD 		13(2)modifies its form if the porous material consists of more thanone (n  1) fractal clusterPlLiiDin i 		130. (3)However, relation (3) does not take into account the caseof a composite material in which the fractal clusters of charac-teristic sizes Li can be stochastically scattered and mixed withother phases so that the size of the investigated samplemayconsiderably exceed the cluster sizes Li . In order to gen-eralize relation (3), let use suppose that there are mi fractalclusters with dimension Di in the sample. Their volume frac-tions i m Li3 3 enable us to calculate the porosity of thewhole sample, as followsPlLiiiDin i 		130 . (4)Eq. (4) includes all possibilities of fractal, non-fractal( )D 3 or mixed arrangements of the solid environmentsurrounding the pores. Provided there are n 1componentsdistributed over the whole sample (Li  , mi 1,  i 1),Eq. (4) then converts back to (3).3 Fractal compressive strengthOne of the most widely used relations for compressivestrength  of a porous material is that of Balshin [7], thoughother relations [8] have also been proposed for this purpose.Balshin considered an ideal case when pores are not filledwith an incompressible liquid and compressive strength  isdirectly dependent on the compactness ( )1	 P of the material  	0 1* ( )P k. However, as soon as the virtual incompres-sibility of the pore liquid is included together with some otherfactors, certain remaining strength s0 appears as a constantwhen the porosity reaches a critical value Pcr, i.e. ( )P scr  0.The generalized Balshin function then reads   	  	 	   	 0 0 0 01 10 1 1* ( ) ,,PPs P b sb Pkkcrcr 0 0 00   **,Pkcr, (5)where P is total porosity and  is compressive strength of thesample with a virtually incompressible fluid filling at least apart of its pore space.Combining (4) and (5), the compressive strength of po-rous matter appears as a function of the fractal structure  		0300iiiDin klLb si. (6)©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 27Acta Polytechnica Vol. 47  No. 6/2007Dimension of Fracture SurfacesT. FickerThe question of the universality of a dimension of fracture surfaces is discussed, and it is shown that such a general parameter may exist atleast for a particular class of materials.Keywords: fractal dimension, fracture surfaces, porous materials, compressive strength.4 Dimensions of a fracture surfaceGenerally, in the case of amixed structure containing bothfractal and non-fractal regions some of the dimensions Di areassociated with volume (mass) fractals (0<Di<3) and somewith non-fractal solid phases (Di 3). Similarly, when analyz-ing the fracture surfaces of such multicomponent materials,projections (Di*) of the volume components (Di) appear onthese surfaces. These projected surface “patterns” preservethe fractal or non-fractal characters of their volume parentobjects, but their dimensions Di* are smaller than those of theoriginal objects. Provided a fracture surface has its own di-mension S and its morphology is “typical” rather than “spe-cial”, the relation between Di* and Di can be expressed [9] asfollows D D S D Si i i* *max , ( , 	 	  0 3 3 , (7)where 3 	 S is the codimension of the fracture surface. Ifthe fracture surface were a perfect Euclidean plane (S  2),expression (7) would lead to the well-known equationD Di i* 	1. Using (7), the exponent 3 	 Di in Eq. (6) can bereplaced by S Di	* and the generalized Balshin strengthfunction now reads  		000iiiS DinklLb si*. (8)This function may contain many parameters, so that it isvery difficult to fit it to the experimental data, because theremay be more than one “reliable” set of parameters 0,  i in1,li, Li, S, Di*, b, k, s0. Fortunately, the structure of a porous ma-terial often contains only one type of grain or – at least –one grain type of fractal arrangement (i 1) dominates overthe solid remainder (i 0), which is usually of a non-fractalcharacter (D D S0 03  * )     	 	0 1110 00 11lLb sS Dk*( )exp ,ln,*S DAsALlk		10111  	( ) .b 0(9)To our knowledge, the fractal analyses of fracture surfacespublished so far have not distinguished between the dimen-sions Di* of the projected objects and the dimension S of thefracture surface itself. Here, for the first time clear differencesbetween these two kinds of dimensions are specified anddiscussed. Although the fracture surface dimension S as aseparate parameter and independent of the inner fractalstructure may seem to be rather vague, it nevertheless has aclear interpretation as the fracture surface dimension thatwould be directly measurable if the sample were fully compact(non-porous, i.e. b 0, s0 0 ), and, therefore, non-fractal( )*D D Si i  3 , which means a perfect Euclidean body.Nevertheless, the surface itself may be a fractal with non-inte-ger S, which should even be expected in most cases.The foregoing equations and discussion have shown thatfracture surfaces bear information about the multi-compo-nent volume structure that is projected on the surface as thespectrum of surface patterns Di in*0that preserve theirfractal or non-fractal origins. The fracture surface has itsown dimension independent of from the dimensions of theprojected surface patterns. These facts require a strict distinc-tion between dimensions Di* and S, and also among Di*themselves. This can be done either locally (e.g. using a mi-croscopic technique) or globally within restricted fractal scalesl Li i, if, of course, there are no overlaps between the scales.5 Experimental resultsTo illustrate the soundness of the concept of fractal com-pressive strength (9) and the concept of, their application tohydrated Portland cement is presented in the following para-graphs. Seventy-two samples of hydrated ordinary Portlandcement paste of various water-to-cement ratios r (0.4, 0.6, 0.8,1.0, 1.2, 1.4) were prepared. After 28 days of hydration thesamples were subjected to three-point bending and were frac-tured. The fracture surfaces were used for further fractal anal-ysis. The 3-D digital reconstruction of the fracture surfaces(Fig. 1) was performed using a confocal microscope andthen a series of horizontal sections (contours) were analyzedwith resolution 0.04m2/pixel by means of the standard box--counting method [10–12] to obtain a representative D* forthe particular self-affine surface. The box-counting analyseswere performed in the length interval 02 30. , m m . The28 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/Acta Polytechnica Vol. 47  No. 6/2007Fig. 1: Digital 3-D reconstruction of fracture surface (confocal mi-croscopy: 0.34 mm × 0.21 mm; total magnification 600×)second parts of the fractured samples were cut into smallcubes and subjected to destructive tests to determine theircompressive strength values.Samples prepared with different water-to-cement ratiosr possess different porosity. From cement technology it iswell-known that with increasing ratio r the porosity increasesexponentially. Naturally, this will change the dimensions ofthe projected patterns D*. Six groups of samples with differ-ent r means six different D* at which we are able to measurethe dependence ( )*D and check it according to Eq. (9). Theresult is shown in Fig. 2, along with all the fitting parameters.Since the assumed analytical form (9) of dependence ( )*Dhas been reproduced well, we may conclude that compressivestrength is one of those mechanical quantities whose value is“coded” in the surface arrangement of the fractured samplesof porous materials.Dimension S of the fracture surfaces of hydrated cementpaste proved to be near to 2.24, indicating a fractally cor-rugated surface onto which another fractal componentD* . ; . 2075 2170 is projected ( 1). The range of values ofD* is in full accord with other authors [2–6]. The contributionof the non-fractal solid phase (0 0 ) is negligible (0).As far as dimension D of the original volume (mass) fractal ofthe cement paste used is concerned, its value is rather highD  2835 2930. ; . but this was expected due to the low dimen-sion D* of the corresponding projection and the high value ofthe co-dimension, i.e. Hurst exponent H, of the fracture surfaceitself H S 	 3 076. . The value of D near to 2.90 agrees withthe dimension of the cement gel within the hydrated cementpaste having r 04. and exposed to ~55 % RH [13], [14].6 ConclusionIt has been shown that the fracture surface possesses itsown dimension S which is independent on the dimensions ofthe patterns projected on this surface. Dimension S seems todepend more on the fracture process itself than on structuralcomponents. This feature may partly support the claims ofseveral authors [5] that there is a universal Hurst exponentH 08. for a wide class of materials. Now it is clear that if sucha universal exponent really exists, it is not as a co-dimensionof the surface projected patterns3 	 D* but as a co-dimensionof the fracture surface itself H S 	3 . In our case H 076. ,which is almost identical with the previously anticipated [5]value 0.8.Dimension S can be determined not only as the difference3 	 	( )*D D but also as a “byproduct” when measuring a con-venient physical quantity, in our case compressive strength.Dimension S of fracture surfacesmay be interpreted as the di-mension that would be directly measurable if the sample werefully compact (non-porous and non-fractal), i.e. a perfectEuclidean body. Nevertheless, the fracture surface itself maybe a fractal with non-integer S, which should be expectedmost cases.7 AcknowledgmentThis work was supported by an Internal Grant assigned bythe Faculty of Civil Engineering, Brno University of Technol-ogy, in 2007.References[1] Mandelbrot, B. B., Passoja, D. E., Panllay, A. J.: FractalCharacter of Fracture Surfaces of Metals, Nature (Lon-don) Vol. 308 (1984), p. 721–722.[2] Yan, A., Wu, K.-R., Zhang, D., Yao, W.: Effect of FracturePath on the Fracture Energy of High-Strength Concrete,Cem. Concr. Res. Vol. 25 (2003), p. 153–157.[3] Wang, Y., Diamond, S.: A Fractal Study of the FractureSurfaces of Cement Pastes and Mortars Using a Stereo-©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 29Acta Polytechnica Vol. 47  No. 6/2007Fig. 2: Dependence of compressive strength on fractal dimension with cement pastescopic SEM Method, Cem. Concr. Res. Vol. 31 (2001),p. 1385–1392.[4] Prokopski, G., Langier, B.: Effect of Water/Cement Ra-tio and Silica Fume Addition on the Fracture Toughnessand Morphology of Fractured Surfaces of Gravel Con-cretes, Cem. Concr. Res. Vol. 30 (2000), p. 1427–1433.[5] Bouchaud, E., Lapasset, G., Plan s, J.: Fractal Dimen-sion of Fractured Surfaces: a universal value?, Europhys.Lett. Vol. 13 (1990), p. 73–79.[6] Carpinteri, A., Chiaiai, B., Invernizzi, S.: Comments on“Direct fractal measurement of fracture surfaces”, Int.Sol. Struct. Vol. 37 (2000), p. 4623–4625.[7] Balshin, M. Y.: Zavisimost mechaniceskich svojstvporoskovych metalov ot poristnosti i predelnyje svojstvaporistych metalokeramiceskich materialov, Dokl. Akad.Nauk SSSR (in Russian), Vol. 67 (1949), p. 831–834.[8] Narayanan, N., Ramamurthy, K.: Structure and Proper-ties of Aerated Concrete: a Review, Cem. Concr. Res.Vol. 22 (2000), p. 321–329.[9] Falconer, K. J., Professor of Mathematics, University ofSt. Andrews, Scotland – private communication.[10] Ficker, T.: Electrostatic Discharges and MultifractalAnalysis of their Lichtenberg Figures, J. Phys. D: Appl.Phys. Vol. 32 (1999), p. 219–226.[11] Ficker, T.: Fractal Morphology of Partial Microdis-charge Spots on Dielectric Barriers, IEEE Trans. Diel. El.Insul. Vol. 10 (2003), p. 700–707.[12] Ficker, T.: Electrostatic Microdischarges on the Sur-face of Electrets, J. Phys. D: Appl. Phys. Vol. 38 (2005),p. 483–489.[13] Winslow, D., Bukowski, J. M., Young, J. F.: The FractalArrangement of Hydrated Cement Paste, Cem. Concr.Res. Vol. 25 (1995), p. 147–156.[14] Boddoe, R. E., Lang, K.: Effect of Moisture on FractalDimension and Specific Surface of Hardened CementPaste by Small-Angle X-ray Scattering, Cem. Concr. Res.Vol. 24 (1994), p. 605–612.Prof. RNDr. Tomáš Ficker, DrSc.phone: +420 541 147 661e-mail: ficker.t@fce.vutbr.czDepartment of PhysicsUniversity of TechnologyFaculty of Civil EngineeringŽižkova 17662 37 Brno, Czech Republic30 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/Acta Polytechnica Vol. 47  No. 6/2007

Dimension of Fracture Surfaces

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Dimension of Fracture Surfaces

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