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Page 1: Fractal Characteristics of Fracture Surfaces

Fractal Characteristics of Fracture Surfaces

George K. Baran* School of Dentistry, Temple University, Philadelphia, Pennsylvania 19 140

Claude Roques-Carmes and Dahloul Wehbi Laboratoire d e Microdnalyse des Surfaces, Ecole Nationale Supkrieure de Mkcanique et des Microtecliniclues,

25030 Besanqon, France

Michel Degrange Iaboratoire d’Etude des Biomateriaux, Facultk d e Chirurgie Dentaire Paris L: 92 120 Montrouge, France

Quantitative fractography is often used to study material failure mechanisms. During calculation of surface or profile roughness parameters, the magnification used in obtaining fractographic data is found to influence the value of the parameters. Fractal geometry has been developed into a tool capable of defining surface and profile topography without sensitivity to magnification, and several studies have related fractal dimension (OF) to other physical or mechanical properties. In this study, we obtained the frac- tal dimension of profiled fracture surfaces of one glass and three proprietary dental porcelains. The fracture toughness ( K J of these materials was also measured using the indenta- tion-strength method. Results show the surfaces to be frac- tal. No quantitative relationship between fractal dimension and toughness was found. Differences in K,, were demon- strated between some materials. It is postulated that the size range within which fractal dimension can be defined as con- stant is dependent on the toughening mechanism, and that the relationship between K,, and D, cannot be identical for all materials.

I. Introduction

ATASTROPHIC failure of nominally brittle material compo- C nents occurs when a crack propagates to a size sufficient to cause instability at the crack tip, and K , , the stress intensity fac- tor, exceeds Klc , the fracture toughness of the material. The direction of crack propagation is governed on macroscopic length scales by the distribution of stresses within the material, on the microscopic scale by material microstructure, and on the atomistic scale by the crystallography of the material.

Typically, fractographic examination of fracture surfaces provides information on the flaw responsible for crack nucle- ation, the direction of crack propagation, and in some instances, on speed of crack propagation. Quantitative fracto- graphic techniques seek to describe fracture surfaces in terms of area, length, size, spacing, distribution, and orientation of fea- tures.1.2 A basic characteristic of a surface or a profile of a sur- face is its roughness. Examples of fracture research areas relating fracture surface roughness to toughening phenomena are the modeling of crack deflection mechanisms3 and the phys- ical propertylcrack velocity correlations in p ~ l y m e r s . ~ Once

D. R. Clarke--contributing editor

Manuscript No . 196833. Received March 22, 1991; approved June 17, 1992. G.R.B. supported in part by a USPHS Senior Fellowship Grant No . DE 05474

from the National Institute of Dental Rescarch, National Institutes of Health, and by Temple University.

’Member, American Ceramic Society.

roughness data are digitized, they may be processed to yield a multitude of roughness parameters .’

Because microstructure plays an important role in dctermin- ing roughness, the technique used to measure this variable must have a resolution capable of detecting features of interest. I t is often necessary to use an empirical approach in determining the resolution needed to acquire data that can be correlated to other material properties. An additional complication first reported by Kostron‘ is that, in certain cases, the measured values for the size and area fractions of microstructural features vary with the magnification used to examine the material.

This “magnification effect,” or variation in measurement result as a function of the size of the measuring unit, provides the impetus for use of fractal concepts in describing roughness. Recent developments in the theory of geometric measurements of integral and fractional dimension have been primarily due to the work of Mandelbrot,’ who has defined a fractal as “a shape made of parts similar to the whole in some way.” It was found that certain irregular curves (e.g., coastlines) could be charac- terized by a noninteger fractal dimension, and that this value was scale-invariant. Fractal dimension “D,.” is defined through

( 1 ) D , = [(log L,, - log L)llog SI + D

where t is measured length, S is the scale of measurement, L,, a constant, and D the topological dimension (equal to 1.0 for lines and 2.0 for planes).

The application of fractal theory to fractography has recently become important. Mandelbrot er ul.’ studied fracture surfaces of steels and found D , to correlate inversely with the impact fracture energy (toughness). Similarly, Pande ( I t a[.’ studied the fracture behavior of Ti alloys and obtained an approximate inverse correlation between fractal dimension and dynamic tear energy. Banerji and Underwood“’ investigated the fracture resistance of 4340 steel as a function of tempering temperature and found the fracture surface to be fractal, with an ernbrittled steel having the lowest D,. Similar trends in the change of D , and three classical roughness parameters were observed after varying tempering temperature. However, in a study of the frac- ture of brittle ceramic materials, Mecholsky et ul.” found a direct correlation between (DF - D ) and K,, for single- and multiphase materials. ‘ I Finally, when fracture surface rough- ness of certain steels was plotted versus the scale of measure- ment, it was found that changes in the fractal plot could be correlated with relevant metallurgical and microstructural features.12

If fractal profiles and surfaces are self-similar or self-affine, it may be possible to apply macroscopic techniques to study microscopic or even atomistic fracture behavior of some mate- rials with the aim of increasing resistance to fracture. It was our intention to use a new technique for calculating the fractal

2687

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2688 .lournti/ of ihe Arnerirun Cerumir, Soc,ierj-Baran ef al. Vol. 75 , No. 10

dimension of profiles through fracture surfaces of single- and multiphase ceramic materials, and to examine the possible rela- tionship between fractal dimension and fracture toughness.

11. Theoretical Background

Several approaches may be used to obtain the fractal dinien- sion of fracture surfaces. One involves study of the profile of sections through a surface, where micrographs are examined and profile lengths measured in a straightforward application of Eq. ( I ) . This method was employed by Pande et ul.’ in their study of Ti-alloy fracture surfaces. Another technique, used by Mandelbrot et ul.’ and Mecholsky et a/.,” is the slit island method, which involves serial polishing of a specimen inibed- ded in some mounting medium. As polishing proceeds, “islands” corresponding to sections through the highest rough- ness features appear, and the perimeter and area of these islands are measured. The fractal dimension is calculated from the slope of the log perimeter vs log area plot. Criticism of the slit island m e t h ~ d ’ . ’ ~ . ’ ~ has focused on the fact that island contour roughness does not scale directly with surface roughness, pri- marily because the observed surface section is at an angle to the fracture surface. An increase i n surface roughness, i.e., in slope of the peaks, is expected to inversely affect the roughness of the island contour.“ Lung and Mu14 have shown that results obtained using slit-island methods are strongly dependent on the length of the “yardstick” (S) used to measure island perime- ter, perhaps because the perimeter of the island may itself be fractal. It should be noted, however, that Mecholsky et d.,” investigating the fracture of polycrystalline aluminas and glass ceramics, found agreement between values of D, generated by the slit island method and those found through Fourier analysis of fracture surface profiles.

In this study, an algorithm called the “variation method” was used to calculate fractal dimension.” While a full discussion of this approach is not appropriate here, we summarize as follows. If the graph of the function f is fractal, f is nowhere differen- tiable on the whole interval [0, I]. Considering a point “x” in [0, I] , and the centered interval [x + F, x - F], we evaluate all possible valuesf(x‘) of the function for x‘ in [x + 8, x - E]. The &-oscillation v, of the functionfin x is given by

v, = maximumf(x‘) - minimumf(x‘)

where

(1X’ - XI) 5 E (217:)

Now the &-variation offis

( 3 )

It has been proved’’ that

(4) v, == &(2 0 1

D, = Iim [2 - (log VJog F)]

and

(5) Samples of the function J’ may be provided by digitized data from a profilometer. An advantage of this method is that its accuracy has been verified on profiles with known fractal dimension, e.g., the Kieswetter curve, the Weierstrass-Man- delbrot function, and the trace of fractional Brownian motion, yielding errors of 0.4%4).7%.“

111. Experimental Procedure

One experimental glass and three dental porcelains were selected for study. These are described in Table I . Porcelain samples for testing were prepared by compacting a porcelain--

water slurry into molds made of a refractory investment mate- rial (Symphyse Ceramic Investment, Marseille, France), then sintering in the mold according to the manufacturer’s recom- mended firing schedule. Typically, this consisted of heating in a partial vacuum for about 6 min, with temperature increasing from 600” to 930°C during that time. The resulting porcelain buttons were ground under water through 800-grit Sic paper to eliminate traces of investment and achieve parallelism of the two flat faces on the approximately I .2-mm-thick, 13-mm- diameter disks.

Glass samples were prepared by hot-pressing the powder in a cylindrical mold and piston assembly made of the same invest- ment material. The mold was evacuated to 5.3 X MPa, and a pressure of 1 MPa was applied during heating to 800°C. The resulting glass cylinders (approximately 25 mm long and 13 mm in diameter) were sectioned using a diamond-charged cutoff wheel into disks approximately 1.5 mm thick, then ground in the same manner as the porcelain samples.

All disk specimens were polished on one face, using dia- mond paste to a 0.33-pm finish. Glass specimens were given a stress-relief anneal at 425°C for 15 min, a temperature some 35°C above the T, of the glass. The diamond-polished faces were coated with Au-Pd for contrast and to enhance the visibil- ity of cracks. Immediately prior to mechanical testing, a Vick- ers diamond indenter under a load of 2.94 N was used to generate a flaw at the center of the polished face. After mea- surement of the crack size, a drop of oil was placed onto the flaw, and the specimen inserted into a biaxial flexure jig for test- ing.” Loading was applied at a strain rate of 0.1 mmimin until failure. The fracture toughness was calculated according to Chantikul et ul.” as

K,, = ~ ( E / H ) ” ’ ( U ~ P ” ~ ) ~ ’ ~

where q is a constant = 0.59, E is Young’s modulus, H i s hard- ness, u, is the failure stress of the indented specimen, and P is indentation load. The ultrasonic method was used to determine Young’s modulus.

Subsequently, portions of the fractured specimens were mounted so as to expose a fracture surface for profilometry (Talysurf model 5-60, Taylor-Hobson, London, England). A 2-mm length of a fracture surface was surveyed, with measure- ments occurring every 2 pm. Horizontal and vertical resolution was 2 pm.

IV. Results and Discussion

Values of fracture toughness (KIJ for the brittle materials studied are presented in Table I. The values represent the means of 10 viable tests. We define “viable” here as values from speci- mens where fracture originated from the indentation flaw and not from an internal or edge flaw located elsewhere in the speci- men. Values for the K,, of porcelains are somewhat lower than those reported by Morena et a/.,’” who used the indentation method to measure toughness. Values of D, for the materials studied are also shown in Table I . These represent the mean of three determinations, obtained from three different fracture sur- faces arbitrarily chosen from a group of fractured test speci- mens. Figure I shows fracture surfaces of the experimental glass. Figure 2 shows the fracture surface of dental porcelains. Figures 3-6 show representative profiles of fracture surfaces used for calculating fractal dimension, and Fig. 7 shows a log- log plot whose slope is taken to be the fractal dimension of the sample Cer. Ging.

Fracture surfaces of brittle ceramic materials tested to failure in a controlled manner exhibit a number of features evolving from the site of the failure-inducing flaw. Typically, a smooth region (mirror) is found adjacent to the flaw, followed by rougher areas where changes in crack speed and direction occur. The mirror area, within which only the primary crack is presumed to propagate, has been related to the fracture stress (u,) of materials through the relationship

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October 1992 Frrictul Clitrrirctrristic~s of' Frtrc.t~ire Surfirces 2689

Table 1. Means and Standard Deviations" of Measured Values of K,: and Db'. Matcrial Cimipo\ition Miinulacturcr K, , (0 = 10) I ) , (I1 ~ 31

Glass P2V25 25 SiO,, 22 CaO, 22 B,O,, IS Na,O, Mobay Chcrnical. Baltimore, MD 0.63 (0.05) I .07 (0.0057)

Un . body Feldspar, quartz Unitek, Monrovia, CA 0.8 (0.03) I . 16 (0.0288) Un. Inc. Feldspar, quartz Unitek, Monrovia, CA 0.77 (0.028) I I .3 I (O.OX62) Cer. ging. Feldspar, quartz Ceramco, Ncw Brunswick, NJ 0.74 (0.044) I .33 (0.0152)

12 F, 3 MgO, I AI,O,

*SD in parenlhescs. 'Homogeneous suhscta a( the 5% conhdciicc levcl arc connected by d i d linc. 'Units for K , , arc MPa.m' '.

uf = Ar "' (7 ) where A is a mirror constant and r the distance between the cen- ter of the failure-initiating flaw and the mirror boundary. In this study, the glass failed at a mean stress of 50.2 MPa. After com- paring this strength with published data obtained from soda- lime glasses,'" we inferred that the failure stress suggests an outer mirror radius of approximately 2-3 mm for glass P2V25P We therefore expected that some of the radial fracture surface length of 6 mm in our glass specimens would consist of the mir- ror region, i.e., be smooth within the limitations of the profi- Iometric technique used here. Surface profiles of glass P2V25P, however, were noticeably rough and were found to have an average D, of 1.07. Closer examination of this glass showed it to be heterogeneous, with a crystalline precipitate as shown in Fig. I (B). These irregularities were presumably responsible for our failure to detect mist and hackle on fracture surfaces.

The variation method indicates that the fracture surfaces of the brittle materials tested here are fractal. The analysis of vari- ance was performed and found significant differences (at the 5% level) among values of KIL. Tukey's multiple comparison

Fig. 1. Fracture surface of P2V25P glass. View is of cross section along radius of disk-shaped specimen. Note the heterogeneity shown in Fig. 2(B). Space bar is in units of micrometers.

test2' was applied to identify similar data subsets, and these are joined with a solid bar in Table I .

Figure 8 shows a plot of K,, vs D,. In this representation, means are located at the intersection of two solid lines. The line parallel to the x-axis shows the full range and mean of measured D, values. The line parallel to the y-axis shows the full range of and mean K,, values. Insofar as the relationship between K,< and D,; is concerned, the data are inconclusive.

Fig. 2. Fracture surfaces of (A) Un. Body, (B) Un. Inc., and (C) Cer. Ging. dental porcelains. View is of cross section along radius of disk- shaped specimens. Space bar is in units of micrometers.

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2690 Jourriul o j ihe Anirric (it1 Cerumic Soc~ietj~--Rurun et al. Vol. 75, No. 10

600 1 2 0 0 1800

DISTANCE (microns)

Fig. 3. Representative profile of the fracture surface of glass P2V25P

600 1200 l800

DISTANCE (microns)

Fig. 4. Representative profile of the fracture surface of porcelain Un. Body.

I I I 1 I , I 600 1200 1800

DISTANCE (microns)

Fig. 5. Inc.

Representative profile of the fracture surface of porcelain Un

600 1200 1800

DISTANCE (microns)

Fig. 6. Ging.

Representative profile of the fracture surface of porcelain Cer.

I I I I I

ling.

~- - 2 8 33 120 513 1200

Los I /€

Fig. 7. Ging.

Log-log plot for a fracture surface profile of porcelain Cer.

Published reports concerning the relationship between fractal dimension and toughness do not agree. For example, Pande et ul.’ (using the slit island and fracture surface profile methods) and Mandelbrot et a/.# (using the slit island method) report a decrease in D, with an increase in toughness of metals, while Mecholsky et al.” (using the slit island and fracture surface profile methods) show an increase in D , with an increase in K, for ceramics. Davidson2’ (using the fracture surface profile method), studying aluminum-matrix-Sic-particle-reinforced composites, found that fracture toughness did not correlate with any measure of fracture surface roughness. Richards and Dempsey” (using the slit island method) also found no correla- tion between fractal dimension and tensile strength, elongation, or microstructural features of a titanium alloy. Fahmy et (using an image analysis technique based on the slit island method) investigated the fracture of brittle V,Au materials and found an increase in fractal dimension with fracture toughness.

Because differences in K,, were found among the brittle materials used in our study, and recalling that other studies dis- agree on the relationship between toughness and the fractal dimension of the fracture surface, we suggest that considering toughness alone is not appropriate when testing for correlation with fractal dimension. Although all crack propagation in mate- rials eventually reverts to a problem of interatomic bond break- ing, it is known that toughness parameters also include a surface area contribution. Combining the results of Griffith25 and Irwin” we obtain

K = (2Ey*,)”’ (9) where K is the stress intensity factor, E is Young’s modulus, and y*& the fracture surface energy, where it is recognized that this energy may be partially converted into plastic deformation (after Orowan), sound, and heat. If we neglect these compo- nents, then GiIman2’ first related fracture surface energy to atomic size and interatomic spacing:

1 ’ i

? o.,} 1 Cer. (ling. i L

P2V25P

1 .o 1.2 1.4

FRACTAL DIMENSION

Fig. 8. Plot of fracture toughness (K ,J vs fractal dimension ( D J . Bars intersect at means. Horizontal bars show lowest, mean, and high- est of three independent measurements of D,. Vertical bars show low- est, mean, and highest values of 10 independent measurements of K,‘.

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October 1992 Fructul Chuructuis f ics of Fracture Surfcices 269 1

y, = EU~/(dI l?r~) (10) where y\ is the surface energy, a is atomic radius, and d(, the crystal lattice spacing. Subsequently, Mackin et a/.” argued that a term called a,, (similar to d,, in Eq. ( I 0)), defined as the characteristic atomic length, can be differentiated for single- crystal, polycrystalline, and glass-ceramic materials, with the result that a,, is interpreted as the limiting length for the scaling process that yields a fractal fracture surface. The scaling pro- cess requires a scheme, and it is here that the toughening mech- anism becomes important. For example, if we regard a material that is toughened by crack deflection around a dispersed second phase, and a,, defines a limiting length for scaling at the atomic or “molecular” level, then another length for limiting scaling must exist whose magnitude will depend on the distance between second-phase particles. Of the two types of materials studied here (glass and porcelain), the porcelains may be tough- ened by crack deflection around dispersed crystals of leucite (K20.AI,0,.4Si0,); see, for example, the fractographs in Morena et al.“ Glass P2V25P also contains crystalline precipi- tates. Because the two different lengths (i.e., interprecipitate distance and dl,) can define two different scaling domains, they may yield two superimposed fractal profiles. In principle, this concept could hold true for both ductile and brittle material fracture. It is therefore likely that the attempt to discover a sim- ple K,, vs D, relationship for all materials will be confounded, even if it were proved that all techniques cited in obtaining the fractal dimension yielded similar results.

V. Conclusions

In summary, we have characterized the fracture surfaces of one glass and three proprietary dental porcelains as fractal. If we regard only the three porcelains studied here, as shown in Fig. 8, it appears qualitatively that fractal dimension decreases with an increase in fracture toughness. This result is opposite to reports concerning other brittle materials,”.24 where different techniques for obtaining fractal dimension were used. No rela- tionship between toughness and fractal dimension was found in our study, possibly due to the different toughening mechanisms operating in glasses and dental porcelain. The sensitivity of fractal analysis suggests the possibility of identifying several crack-path-determining features acting simultaneously, as the analysis is capable of identifying a size range over which a unique D, remains constant. Finally, this study points out the need for additional investigation of the techniques used to obtain fractal dimension, as well as roughness data for materials with well-understood crack-path-determining parameters.

Acknowledgments: for helpful discussions and comments.

The authors are indebtcd to C. Tricot and P. Meakin

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