Fractal characterization of fractured surfaces

OOOl-6160/87 $3.00+O.OO Pergamon Journals Ltd

Acra metal/. Vol. 35, No. 7, pp. 1633-1637,1987 Printed in Great Britain

FRACTAL CHARACTERIZATION FRACTURED SURFACES C. S. PANDE,’

L. E. RICHARDS,’

N. LOUAT,

OF

B. D. DEMPSEY’

and A. J. SCHWOEBLE’ ‘Naval Research Laboratory, Washington, DC 20375-5000, and *Crystal Growth and Materials Testing, Lanham, Maryland, U.S.A. (Received 10 October 1986)

Abstract-It has recently been claimed [B. B. Mandelbrot, D. E. Passoja and A. J. Pullay, Nature 308, 721 (1984)] that fractured surfaces are fractal in nature, i.e. “self similar” over a wide range of scale and that the fractal dimensions of the surfaces correlate well with the toughness of the material. We have investigated these concepts be measuring the fractal dimensions of fractured surfaces of two titanium alloys and attempting to correlate them with dynamic tear energy. We find that, although fractured surfaces can be classified as approximately fractal, the previous conclusions were too optimistic. An approximate correlation between fractal dimension and dynamic tear energy is obtained. R&sum&--II a Bt&r&cemment avan& [B. B. Mandelbrot, D. E. Passoja et A. J. Pullay, Nature 308, 721 (1984)] que les surfaces de rupture sont de nature fractale, c’est-&dire “autosimilaires” sur une vaste Bchelle et que les dimensions fractales de la surface correspondent bien au durcissement du mat&au. Nous avons ttudit ces concepts en mesurant les dimensions fractales des surfaces de rupture de deux alliages de titane et en essayant de les corrkler avec l’energie dynamique d’arrachement. Nous trouvons que, bien que l’on puisse donner aux surfaces de rupture un caracttre fractal approcht, les conclusions pr&dentes itaient trop optimistes. Nous obtenons une corrtlation approchee entre la dimension fractale et l’energie dynamique d’arrachement. Zusammenfassung-Vor kurzem wurde behauptet [B. B. Mandelbrot, D. E. Passoja und A. J. Pullay, Nature 308, 721 (1984)], daO Bruchfllchen fraktales Verhalten aufweisen, d.h. da13 sie iiber einen weiten MaDstabsbereich ‘selbst-iihnlich’ sind und dal3 die fraktalen Dimensionen der OberflHche gut mit der ZIhigkeit des Materiales korrelieren. Wir haben diese Konzepte iiberpriift, indem wir die fraktalen Dimensionen der BruchflIchen zweier Titanlegierungen gemessen haben und die Ergebnisse versuchsweise mit der Energie des dynamischen ZerreiDens korreliert haben. Wir finden, daB die friiheren Schliisse zu optimistisch waren, wenngleich Bruchfl&hen als niherungsweise fraktal klassifiziert werden kiinnen. Es ergibt sich eine nzherungsweise Korrelation zwischen fraktaler Dimension und der Energie des dynamischen ZerreiDens. 1. INTRODUCTION

In recent years, quantitative analyses of fractured surfaces have become an integral part of the study of deformation and rupture of materials. Such surface analyses often provide information which is complementary to that obtained by other metallurgical methods. An excellent review of the recent development in quantitative fractography is given by Coster and Chermant [l]. One of the newer methods described briefly in the above mentioned review is the use of fractals to characterize fractured surfaces. Such characterization provides a number D, called fractal dimension which is a measure of the roughness of the surface. Mandelbrot et al. [2] have recently described experimental techniques to obtain the fractal dimension of fractured surfaces of steel and have found that D, correlates with the impact energy of the fractured specimen. Fractal characterization thus provides in principle a measure of surface roughness with which material parameters can be correlated. Although the word “fractal” is of recent origin (coined by Mandelbrot [3]) the notion of fractal geometry is not new. The geometric measure theory of sets of integral and fractional dimension is

sometime referred to as Hausdorff or Hausdorff-Besicovitch dimension. Indeed much of the mathematical work in this area often containing highly ingenious arguments is due to Besicovitch [4]. Mandelbrot, however pioneered their use to model a wide variety of scientific phenomena ranging from tubulence in fluids, to geographical coastlines and surfaces [3]. Often such use has been applied or suggested empirically, without rational any justification. In case of fractured surfaces, for example, it is tacitly assumed that the surface is (statistically) invariant over a wide range of scale transformation, i.e. the surface features are self similar independent of the scale of magnification used to observe them. If this is indeed so, then for such surfaces the degree of roughness or irregularity can be given by a number D, called the fractal dimension such that 2 D, where D is the topological dimension.) The question we ask in our present investigation is; whether fractured specimens indeed show the self similar behaviour described above. Our results indicate that although fractured surfaces can

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approximately be classified as fractals, previous conclusions were far too optimistic. This is not surprising, since firstly no adequate justification for the self-similarly of fractured surfaces was provided; and secondly it was never clearly stated why the experimental measurements described by Mandelbrot et al. [2] which do not measure surface areas directly, provide a true value of D, for the fractured surface. The aim of the present paper is to examine critically fractal measurement techniques, in titanium specimens by several different methods and establish if D, correlates with material properties. 2. MEASUREMENTS OF FRACTAL DIMENSION

Consistent with the notion of self-similarity, define fractal dimension D, as L = ,r,~-u’-D)

we (1)

where L is the measured length, area etc., E is the scale of the measurement (for example E would be the openings of the dividers used to measure length for a wiggly curve), L, is a constant and D is the topological dimension. D, 2 D. For a straight line we have DF = D = 1, and for a plane surface D, = D = 2. For rough surfaces, etc. however, note that L + 00 as E + 0, since D, > D. Equation (1) also be written as

i.e. the slope of log L vs log E curve gives D, . In case of surfaces, it is difficult to measure actual surface area of a rough surface as a function of scale and two indirect methods have been used. 2.1. Vertical section method This method involves taking a vertical section through the fractured surface, and obtaining the fractal dimension of the profile by implementing equation (1). The two dimensional problem (surface) is thus reduced to one dimension (line). For this type of measurements the fractured specimens were mounted in conducting (carbon) bakelite. The mount was then cut vertically, to produce two complementary fracture surface cross sections. These were mechanically polished using Sic papers, given a final polish with 1.0 pm Alz03 power and subsequently observed in the scanning electron microscope, or in an optical microscope. The micrographs at a magnification of around 500 x were used to obtain fracture surface cross section profile (utilizing a Hewlett-Packard 9845B Computer with a digitizing pad) and analyzed by a computer program which “measures” the length of the digitized cross section profile using different scale lengths. Fractal dimension is then obtained from the slope of the log L vs log E curves, using a least square fitting procedure. The method is quite simple and

several of these measurements for D, can be made in rapid succession by further grinding. Statistical reliability of the procedure was enhanced by the use of random sections, at several angles. Underwood and Banerji [5] found that a simple serial section was equally as effective as sections taken over all possible orientations, greatly reducing the number of sections required. Figure 1 shows typical fractal curves obtained by the methods discussed above in an alloy of titanium (Ti-6Al-2V) with varying amounts of Zr. Figure 2 shows a similar curve for another titanium alloy, titanium 6211 (Ti-6Al-2Nb-lTa-0.8Mo) tensile test specimen. The surfaces in (c) and (d) are complementary and have almost identical fractal dimension. Surfaces in (a) and (b) were obtained by sectioning surface (c). In each case, a line, roughly straight, is obtained. On closer inspection it is found that there is in fact a systematic departure from linearity, and the points can be seen to lie on an arc of a circle in (c) and (d). One can also fit the data to parabolic forms. Such curves are shown in Fig. 3. The curvature is thus continuous and hence cannot be dismissed as “noise” [2]. We will consider this point in detail in Section 4. In each case slope was obtained from least square fitting method. Since the slope varies continuously, there is some uncertainty in the value of D, obtained by this procedure. 2.2. Slit island method In this method, originally due to Mandelbrot et al. [2] the fractured specimen was mounted in graphitefilled bakelite and polished in stages, parallel to the fractured surface. Islands of metal appeared in the bakelite surface and grew in size as the polishing progressed. The perimeter and area of these islands were measured using an electronic planimeter, or by digitizing the island perimeter on a computer. These measurements were made on photographs of islands taken by means of a scanning electron microscope or

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optical microscope at magnifications ranging from 75 to 1000 x . Log perimeter was plotted vs log area of each island measured (see Fig. 4). The slope of a straight line fitted to these data points by a linear least square program was used to calculate fractal dimension of the fracture surface. We followed the method used by Mandelbrot et al. [2] for calculating D, from the slope.

3. CORRELATIONS WITH MATERIAL PROPERTIES Having determined D, by several methods and after considering all possible sources of error, we now ascertain whether any correlation exist between D,, so measured, and material properties. In the titanium specimen with Zr, the five fractured specimens have various amounts of Zr ranging from 0 to 6% by weight. It would thus be interesting to see if D, correlates with Zr content. Further, the dynamic tear energy (DTE) (which is a measure of toughness) of these specimens were known from a previous investigation [6]. DTE in that investigation was found to decrease continuously with increasing Zr content. For details from DTE measurements see Ref. [7]. In Fig. 5, we plot D, determined by the cross section and the slit island methods as a function of DTE. It is seen that both methods gives a rough correlation between the D, and DTE. Further the two correlation appears to be similar, although the magnitude of D, for the

same specimen obtained tially different. 4. DISCUSSION

by two methods

are substan-

AND CONCLUSIONS

We consider first, the slit island method introduced by Mandelbrot et al. [2] and used in our study of fractals. Mandelbrot et al. state that the fractal dimension D, measured by this technique agreed very well with the value obtained by fracture profile

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which is encouraging. In our present study, however, we find a large scatter in the data. Repeated checks revealed that the scatter was not due to any error of measurement, but was inherent in the method itself. Apart from this, there are at least two other reasons for questioning the relevance of the slit island method as a measure of the fractal dimension of the fractured surface. (a) We recognise that an essential assumption of this approach is that island shape (say ratio of length to breadth) is invariant with size. A systematic failure of this constraint will lead to spurious values for D,; as reference to Fig. 6 shows in the present case. This figure gives average area as analysis,

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a function of form factor (defined as ratio of length D, over breadth D, of the islands). Each point of the plot is an average of over 15 islands. The islands become progressively rounder with decreasing area. (b) A second and more fundamental reason is that roughness exhibited by an island contour cannot be expected to scale directly with the roughness of this surface. This difficulty arises because the observed surface is a section at grazing incidence to the fracture surface. Thus, the magnitude of an observed indentation, say, on an island contour depends on the amplitude of the relevant surface displacement but inversely on its slope. Now, increasing surface rough-

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ness implies increasing amplitude and slope in surface hillocks. The observed amplitude of deviation of island slope from a smooth curve is thus oppositely effected by these two characteristics with the consequence, as stated above, that roughness of surface and contour cannot be expected to correlate. We conclude then that the slit island method is fundamentally flawed as a measurer of the fractal dimension of a fracture surface. Turning to the cross sectional method, we note first that since the section is essentially normal to the fracture surface the essential difficulty of the slit island method does not pertain. The connection between surface roughness and that of the profile developed by the sectioning process has not been determined in detail, however, it is clear that they are at least symbatic. Further, the amplitudes (and slopes) in the two representations would seem to be linearly related. We suppose then that there is also a linear relation between roughness in the two cases. Reference to Fig. 3 shows that the relation between log (length) and log (scale) is not linear as would be expected if the section profile examined were self similar with regard to scale. Self similarity can be expected to fail here in two aspects. Firstly at very small scale, that is to say where the operative scale becomes comparable with the size of the basic flats or tiles with which the surface is composed. Secondly, at very large scale. A fracture surface found under the simple stresses normally employed in fracture testing is approximately planar. This characteristic limits the longest scale which can be successfully employed. The degree of this limitation is not immediately clear but it would seem that the largest scale should certainly not exceed the height of the largest hillocks. Measurements using a range of scales which exceed both of these limits should give a curve which is sigmoidal. That is to say one for which (d lx&/d InE) + 0 at both extremes in log (scale) [8]. An examination of the present data (Figs 2 and 3) shows that the range of scales employed has not exceeded the upper limit. It would seem however that this is not so at lower limit and it is tempting to adduce the observed deviation from linearity at small scale to this cause. It should be noted that a systematic departure from linearity has been noted previously by Coster and Chermant [l], and by Underwood and Banerji [8]. Manderlbrot et al. [2] has dismissed these departures from linearity as “noise” and of no consequence. We disagree with this interpretation since our own results show these systematic deviations in a majority of fractured specimens. As stated above there is a underlying cause, which needs careful consideration and which challenges the naive notion of self-similarity in fractured surfaces. The fact is, that a single parameter D, is insufficient to characterize the curves shown in Figs 2 and 3. Underwood and Banerji [8] have used a new mathematical function to describe the curves shown in Figs 2 and 3. A careful check of their data however reveals

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that the equation is inadequate at small scales. Nonetheless the procedure employed by them is potentially useful and is at present under investigation by us. Finally we comment on the correlation between D, and dynamic tear energy detained by us (Fig. 5). It is interesting to note that the correlation observed by two are essentially similar, although the magnitude of the fractal dimensions determined by the two techniques are different. Further, the correlation is a negative one (i.e. higher DTE for smaller DF). We could explain the difference in values of D, obtained due to difficulties in the two techniques mentioned before. The negative correlation is however difficult to explain. If a high fractal number is supposed to be characteristic of a rough as opposed to a smooth surface we would conclude that the rough surfaces are to be associated with brittle materials. This is contrary to experience. It should be noted that Mandelbrodt et al. [2] also noted a negative correlation between fractal dimension and impact energy. So in that sense, our results are essentially in agreement with theirs. In summary, we have measured fractal dimensions of a series of titanium alloys with varying amount of Zr and have shown that a rough correlation exists between fractal dimension and toughness of the material. The intriguing question so far unanswered is the origin and the validity of the negative correlation. This study also points to the fact that the methods of measuring fractal dimension of surfaces are rather empirical and need further experimental and theoretical investigation. In particular the connection between D, obtained by the slit island method and fracture surface roughness is rather tenuous, as shown in this study. It is clear that a more precise method of measuring D, is needed. Acknowledgements-This program was funded bv the Office of Naval Research (ONR Contract. No. N0001486WR242781. We are erateful to Dr Bruce McDonald of ONR fo; his interest-and encouragement and to Mr Ralph Judy Jr for very kindly supplying all the fractured specimens used in this study.

REFERENCES 1. M. Coster and J. T. Charmant, Int. metall. Rev. 28, 228 (1983). 2. B. B. Mandelbrot, D. E. Passoja and A. J. Pullay. Nature 308, 721 (1984). 3. B. B. Mandelbrot, Fractal Geometry of Nature. Freeman, New York (1982). 4. See for example and K. J. Falconer, The Geometry of Fractal Sets, Cambridge Univ. Press (1985). 5. E. E. Underwood and K. Banerji, Proc. 6th Int. Cong. for Stereolodogy (edited by M. Kalisnik), pp. 75-80. Acta Stereologica, Ljuljana, Yugoslavia (1983). 6. R. W. Judy Jr, unpublished work. 7. ASTM Standard E 604-83, Standard Test Method for Dynamic

Tear

Testing

of Metallic

Materials,

1985.

Annual Book of ASTM Standards Vol. 03.01, ASTM. 8. E. E. Underwood and K. Banerji, Muter. Sci. Engng In Press.