ON THE RELATIONSHIP BETWEEN FRACTURE TOUGHNESS AND FRACTAL DIMENSION= IN AISI 4340 STEEL

CHAPTER 2

LITERATURE REVIEW

Fractal Geome= try

Concepts

The concep= t of fractal geometry is relatively young. Fractal geometry expands the concept of dimension and recognizes that there are an infinite number of dimensions in between the topological dimensions of 1, 2 & 3. I= n the particular case of irregular surfaces, for instance, it is possible to obta= in fractal dimension values between 2.0 and 3.0. This number, to a large extent, represents the degree of surface irregularity. A planar surface would be expected= to produce values near 2.0 while more tortuous surfaces would be expected to be closer to 3.0.⁷

Although r= ougher surfaces are thought to exhibit a greater fractal dimension, this statement= is not strictly true. The indust= ry standards that cover roughness usually define it as a number which represen= ts an average deviation from a mean line.&nbs= p; For example; the International Standards Organization (ISO 4287) def= ines the most commonly used roughness parameter R_a (roughness average= ) as⁸:

(2-1) R_a =3D 1/L _<=
![endif]>

This expre= ssion is illustrated graphically in Figure 2-1.

X direction

L

<= /span>

0

<= ![endif]>

Figure 2-1.= Representation of roughness averag= e (R_a) per ISO 4287.

= ;

Line 2

Line 1

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Figure 2-2.= Lines of similar R_a but different fractal dimension.

Consider t= he two lines in Figure 2-2. If the traditional engineering definition of roughness is applied to these lines t= hen R_a would be found to be very similar. Both profiles deviate about the sa= me amount from a mean line. If t= heir fractal dimensions were measured, however, these values would be expected to differ significantly. Line 2 = is considerably more tortuous (irregular) than line 1.

An accurate definition of fractal dimension, then, is not based strictly on roughness.<= span style=3D'mso-spacerun:yes'> Fractal objects possess at least t= hese two important qualities:

1) Scale invariant self-similarity: Fractal objects display identical = levels of irregularity no matter what the magnification of the object. In fact, given two images of the s= ame object at vastly different magnifications, the viewer would not be able to determine the actual scale of the object without a reference scale.

2) The geometric features of length, = area or volume change at a predictable linear rate on a log-log graph when the measurement device changes. F= or example, if an irregular line is measured with a step length of 1 cm (using dividers, for instance) the total length will be longer than if a step leng= th of 2 cm is used.

= ;

Measurement Through Richardson Plots<= o:p>

The scienc= e of measuring fractal curves made an important advance when Richardson attempted to correlate the l= ength of national boundaries with their respective military and economic conflicts. The boundary measurements were made from various maps, however, he quickly realized that these perimeter values varied widely depending on the scale of the map. He further learned that boundary dimensions also varied if the scale was kept constant and the measuring unit was altered.⁹

Figure 2-3= shows Richardson’s measurement process. Each of = the four identical images produces a different boundary length depending on the ruler length.

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Figure 2-3. Variation of boundary length with = ruler length. Smaller rulers produce larger perimeters [Ref. 4, pg 27, with kind permission of Springer Science & Business Media].

The variat= ion of perimeter with measurement scale can be demonstrated with a log-log graph.<= span style=3D'mso-spacerun:yes'> The typical result is a linear relationship as shown in Figure 2-4. Plots of this type are often used today in the fractal analysis of irregular lines or perimeters and are referred to as = Richardson plots.

Stride length too large for object size

<= /span><= /span>

Stride length near or less than boundary width.

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Figure 2-4. Richardson plot produces a straight line on a log-log graph. The slope gives the fractal dimension. Note that the devi= ations from linearity at the ends are due to measuring unit effects. [Ref 4, pg 29, with kind permission of Springer Science & Business Media].

The mathem= atical representation of a Richardson= graph (for a boundary line), and the one which is characteristic of fractal relationships, is given by:

(2-2) = L =3D Ks^(1-D)

The linearized version is:

(2-3) = Log(L) =3D Log (K) + (1-D)Log(s)

where in both (2-2) and (2-3):

L =3D Measured length. = K =3D A constant.

S =3D The measurement scale. D =3D The fractal dimension.

Other Measurement Techni= ques

Dimensional analysis

Mandelbrot= and co-workers¹⁰were the first to examine fracture surfaces using techniques derived from Richar= dson’s observations. For the case of fracture surfaces, the Richard= son relationship was modified to read as follows:

(2-4) A =3D Kp^2/D

The linearized version is:

(2-5) Log(A) =3D Log(K) + 2/D_=
s Log(p)

where in both (2-4) and (2-5):

A =3D Measured area. = K =3D A constant.

p =3D The perimeter length. D_{s =3D
The surface fractal dimension.}

2/D_s =3D The slope obtained from the Log(A) - = Log(p) plot.

In practic= e, this technique is applied by mounting and polishing a fracture surface in such a= way as to allow only the cross sections of "high spots" or "islands" of the fracture surface to show on a sectioning plane.<= span style=3D'mso-spacerun:yes'> Figure 2-5 illustrates an example = of a group of such islands.

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Low spot

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Figure 2-5. T= ypical cross section through a fracture surface revealing several islands and low spots.

Once a set= of islands is obtained, the surface may be digitized and a computer program ma= y be applied to obtain data measuring areas and perimeters for all the islands available. The data is then u= sed to obtain graphs of log area v log perimeter.

The Minkowsky dimension<= sup>11

The Minkow= sky dimension is determined for a line or boundary by sweeping out the feature = from beginning to end with circles of various radii. A graph of log area of the circle = v log radii of the circle produces a slope which gives the Minkowsky fractal dimension.

The Kolmogorov dimension= ¹²

The Kolmog= orov dimension is determined for a line or boundary by covering the feature with grids of different sizes. A g= raph of log grid size v log number of grids through which the feature passes produces a slope which gives the Kolmogorov fractal dimension.

Fourier analysis^13<=
o:p>

A log-log = graph is constructed of the magnitude v. frequency for the Fourier transform. This graph produces a straight line whose slope is related to the fractal dimension.

Selection of Measurement Technique and Inherent Differences.

The fractal dimension measurement techniques listed above are not the only methods available, however, they are the most widely used. Despite the fact that they all cla= im to measure a "fractal dimension" by definition, the only method that produces a true fractal dimension (Hausdorf dimension) is the Richardson plot⁴. The other methods have been develo= ped and employed in cases where the feature(s) are more easily measured using o= ne of the alternative techniques. In addition, in many cases the exact value of the fractal dimension is not nec= essary. The important aspect being studied= is the difference in fractal dimension between similar features or their rate = of increase. As long as the measurement technique is consistent among the features examined, and its strengths and limitations are understood, any of these techniques may be used. An excellent review of = the available methods and their inherent strengths and weaknesses is provided by Russ⁴.

Self-Similar v. Self-Aff= ine Surfaces

One of the problems encountered in the literature is the erroneous application of frac= tal dimension measurement techniques. It is common for many authors to use profile sections from fracture surfaces, apply the Richardson= technique to the resulting line and report a fractal dimension - property relationship. The error here = is that there is no evidence to suggest that the fractal dimension of a profil= e is equivalent to the fractal dimension of the surface.

The argume= nt initiated above is particularly troublesome in the case of fracture surfaces. These surfaces are thought to be self-affine rather than self-similar.^4,14 That is, a self affine surface exh= ibits different scaling behavior with orientation. The patterns formed on the X-Y (horizontal) plane are not thought to be the same as those formed on the X-Z (vertical) plane. Even though= some relationship may exist, they are not thought to be identical. In the case of fracture surfaces, = the difference in scaling behavior is attributed to differences in applied stresses, constraints, residual stresses and, perhaps, the characteristics = of the material itself.

If we are to apply the Richardson technique we must use a plane that is essentially parallel to the direction= of fracture. Such planes are kno= wn as zeroset planes and are considered to be suitable for a fractal dimension measurement which is representative of the fracture surface. Deviations of 5° or more from this zeroset plane may lead to large measurement errors.¹⁵=

Applicat= ion of Fractals to Material Surfaces

Fractal Mechanics

The previo= us section provided an introduction to the concept and mathematics supporting fractal dimension measurement. A number of excellent references are available in the literature which descri= be the methods, underlying theory and applications in greater detail. Considering that a thorough review= of the underlying mathematics is beyond the scope of the present study, the re= ader is referred to those papers for additional information^16-25. This document will elaborate in gr= eater detail on those papers and references which are most relevant to the questi= ons evaluated by this work.

Many of the references cited in this study employ one the two forms of the slit island analysis method (SIA by Richar= dson plots or dimensional analysis, a.k.a. the perimeter-area method). A thorough review of the mathemati= cal foundations of these methods and their application to fracture surfaces is provided by Meisel²⁶. Meisel also applies these methods to "artificial" fractal objects and determines their validity, strengths and weaknesses. The central finding in Meisel's wo= rk is that perimeter-yardstick calculations are essentially equivalent to perimeter-area calculations provided that the measuring units are chosen appropriately. Under carefully chosen conditions both will provide acceptable measures of the fractal dimension. Meisel offers guid= elines on the selection of length scales for obtaining satisfactory measurements.<= span style=3D'mso-spacerun:yes'> In particular he suggests that a f= airly large range of ruler lengths be employed for measurements such that scale dependent behavior may be detected in the fractal dimension graphs.

Application of Fractals to Ceramics<= /p>

A signific= ant portion of the work performed on fracture surfaces has been accomplished on ceramics^27,28. For= this class of materials, much of the research has been performed by Passoja, Mecholsky and co workers^29-33.&= nbsp; It is of particular interest that these studies did not simply limit themselves to identifying that a surface is fractal but continue on to sugg= est a structure-property relationships to explain this behavior. The fundamental relationship sugge= sted by these researchers is that fracture toughness may be related to the fract= ure surface topography as follows:

(2-6) KIC =3D A(D*)^1/2

where:

K_IC =3D Plane st= rain fracture toughness. A =3D a parameter characteristic of the material class.

D* =3D the fractional porti= on of the fractal dimension.

Mecholsky = et al.³⁰proposed that A is a function of E, Young's Modulus, and a_{0,
isa characteristic length such that A =3D E(a₀)^=
1/2. Mecholsky et al.³¹note=
that
other functions may fit the experimental data, however, equation 2-6 was
constructed on the basis of dimensional arguments. The resulting units are MPa root-m=
eter
as required for the plane strain fracture toughness, K_IC. The usual form of their equation is
then:}

(2-7) KIC =3D E(a₀D*)^1/2

Mecholsky = and co-workers have interpreted a₀ to be a length characteristic of = the material (a structure parameter). The actual meaning of this variable remains to be investigated further. The structure parame= ter does not seem to be related to grain size, distance between inclusions, distance between internal flaws or other typical microstructural features.<= span style=3D'mso-spacerun:yes'> The suggestion is made that a_=
0 could represent more esoteric "distance related" features such as free volume, glass-crystal clusters, stretched bonds or glass phase stretch= ed bonds³⁴in inorganic glasses and glass ceramics (see Figure 2-6)= .

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Figure 2-6. Relationship between D* and a₀ and interpretation of the structure parameter for various classes of materials. Note: units of a₀ in angstroms [Ref. 34].

Despite the uncertainty surrounding the interpretation of the structure parameter, the relationship (equation 2-7) appears to work well for a large variety of cer= amic materials. It is particularly applicable to establishing a relationship within classes of ceramic materia= ls (see Figure 2-7).

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Figure 2-7. Fracture toughness-fractal dimension relationships in graphical form= for various classes of ceramics. = [Refs. 14 and 34].

In additio= n to the fracture toughness-microstructure relationships discussed above, Mecholsky = and Freiman³² have shown that there is a relationship between the decimal portion of the fractal dimension (D*) and the flaw-to-mirror size r= atio at the origin of most ceramic fracture surfaces. The relationship has been given as= :

(2-8) C/r1 =3D D*

A graphical representation of their data gathering procedure and findings is shown in Figure 2-8.

Slope =3D -D

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Figure 2-8. Graphical representation of investigative technique and results establishing the relationship between C/r₁ and D* [Ref 32].=

= ;

This later= finding is particularly interesting considering that several relationships have been established between mirror, mist and hackle regions found at the origin of ceramic fracture surfaces³⁵.&nb= sp; Fractal dimension studies imply that the features that develop at the origin are somehow propagated throughout the fracture surface. It is assumed that all of these fe= atures are a reflection of the material's characteristics and can be described by fractal analysis methods.

Application of Fractals = to Polymers

The applic= ation of fractal fracture surface analysis to polymers and polymer composites is more limited than for metals and ceramics. In this particular class of materials fractal concepts have found greater use in the study of polymer structures at molecular³⁶ and physical levels³⁷. In this respect, fractal theory has been applied to polymer melts³⁸, polymerization reactions^{39, 40}, rheology⁴¹ and polymer matrix/second phase composite^{42, 43} interactions.

The focus = on polymer structures not withstanding, a number of significant studies have b= een performed which add to our knowledge of the fracture process. Some have emphasized bulk mechanic= al behavior^44-46while others have delved deep into atomic structure-fracture surface-property relationships as noted in the paragraphs that follow.

Joseph et = al.⁴⁷ have looked at the fracture surface-fracture toughness relationship using atomic force microscopy. Four relatively brittle polymers were used in this study. Measurements in the mirror, mist a= nd hackle regions (identical to those found in the fracture initiation regions= of ceramics) resulted in fractal dimension values which correlated directly wi= th fracture toughness.

Kozlov et = al⁴⁸. have studied the applicability of fractal fracture mechanics to polymers and polymer composites and have concluded that the concepts should be equally appropriate in this class of materials as it appropriate in metals and ceramics. In fact, they asser= t that the critical crack opening displacement method of measuring fracture toughn= ess may provide a suitable scale of fracture in polymeric materials.

Lyu and co= -workers⁴⁹ have applied fractal analysis methods to two engineering thermoplastics; the partially crystalline Polyetherketone-C (PEK-C) and the amorphous polyethersulfone (PES). They = report that these polymers behaved in a ductile manner. Presumably, this indicates that the materials fractured by crazing as is typical for thermoplastics. Their findings show a decrease in = D* with increasing plane-strain fracture toughness.

= ;

Application of Fractals = to Metals

Fractal an= alysis methods have been applied to metallic fracture surfaces from the inception = of these concepts. Mandelbrot, P= assoja and co-workers initially applied the slit-island method and Fourier analysi= s to the fracture surfaces of 300 maraging steel separated by impact^{10. They observed that in these ductile
fractures, the fractal dimension decreased with increasing toughness in a
predictable manner. This semi=
nal
work launched fractal surface analysis into the mainstream of materials
science. Since then, the conc=
epts
initiated by Mandelbrot have spread to varying degrees to all other materia=
ls
and their associated wide variety of fracture mechanisms.}

Researcher= s in almost all areas related to fracture have attempted to explain their partic= ular phenomenon of interest in terms of fractal concepts. Fractal analysis has been employed= to explain the limit of unstable crack velocities,⁵⁰ the mechanisms= of fatigue,^51-55 creep,⁵⁶ superplasticity,^{57,58.
brittle fracture,⁵⁹ and stress corrosion cracking.⁶⁰<=
span
style=3D'mso-spacerun:yes'> Success in each of these areas has=
been
limited. None of the studies =
have
been duplicated sufficiently in other materials or conditions for adequate
verification of each theory.}

Pande and = Richards⁶¹ examined the fractal characteristics of fractured surfaces in titanium using vertical SEM brightness profiles and the dimensional analysis method with slit-islands. Their initial conclusions indicated that the surface of this material could be described = using fractal geometry. A second pa= per released shortly thereafter by Pande et al.⁶² cautioned that previous findings had been too optimistic and that the fracture surfaces of titanium may not be fractal. = In this later paper Pande and co-workers attempted to relate the fractal dimen= sion to the dynamic tear energy (DTE) and found only a tenuous decreasing value = of D* with an increase in this measure of toughness. Gong and Lai⁶³ revisite= d this fracture toughness-fractal dimension relationship in titanium alloys by usi= ng J-R resistance curves and the = Richardson method on vertical profiles. = They reported an increase in fractal dimension with toughness.

A series of important fracture surface topography-fracture toughness investigations have been performed= by Bouchaud et al^{64, 65}on ductile aluminum alloy 7475. In these studies, and a review pap= er released later⁶⁶, the authors argue that there appears to be a universal "roughness exponent" (ζ, related to D* by D* =3D 3 ─ ζ ) which measures in the order of 0.80 ± 0.05. No change was observed with fractu= re toughness. Other investigators have found similar results in a variety of relatively brittle materials (plaster, Bakelite, porcelain, graphite, steel, and Al-Si) which revealed a roughness exponent in the order of 0.87 ± 0.07.⁶⁷

Application to Steels

Relationship between imp= act toughness and fractal dimension

Many resea= rchers have long suspected a toughness-fracture topography relationship in metals.= ^5,6,68 As noted earlier, however, the fir= st attempt to identify the relationship between toughness and "surface roughness" through fractal geometry was made by Mandelbrot et al.^=
10 The study used impact toughness specimens of type 300-grade maraging steel and the perimeter-area version of the slit-island method. This = study was duplicated by Ray and Mandal⁶⁹ (using a high stregth low all= oy steel) and Hilders and Pilo⁷⁰ (using a ferrite-pearlite steel) by employing charpy V-notch impact specimens and the perimeter-area slit-island method. The findings of all t= hree studies are shown in Figure 2-9.

Brittle<= /p>

Ductile<= /span>

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Figure 2-9. F= ractal dimensional increment v Charpy impact energy for 300-grade maraging steel (∆), HSLA steel (●) and ferrite-pearlite (○) steel [Refs = 10, 69 and 70].

= ;

Hilders an= d Pilo state in their work that their results are consistent with the work of Ray = and Mandal yet opposite of that reported by Mandelbrot et al. This discrepancy is explained on t= he basis of different micromechanisms of fracture. The specimens with increasing frac= tal dimension as a function of increasing impact energy were observed to fractu= re primarily through cleavage. T= he specimens with decreasing fractal dimension as a function of increasing imp= act energy were observed to fracture primarily through microvoid coalescence.

It is inte= resting to note that the results noted in Figure 2-9 are also consistent with the behavior of brittle materials reported by Mecholsky ^14,34 (see Figure 2-7; increasing fractal dimension with increasing plane-strain fract= ure toughness), consistent with studies on brittle polymer by Joseph et al.47 (increasing D* with increasing K_IC) and ductile polymers by Lyu = et al.⁴⁹ (decreasing D* with increasing K_IC) and general= ly consistent with the observations of Pande et al.⁶² in ductile titanium alloys (they reported a roughly decreasing fractal dimension with increasing dynamic tear energy-a measure of toughness).

Similar observations have been reported in research where the microstructure was not discussed but may be inferred by the test conditions. Hisiung and Chou⁷¹ repo= rt an increase in fractal dimension with increase in impact toughness in high strength low alloy (HSLA) specimens that should be fracturing through cleav= age (test temperatures ranging from -145 to -20°C). Hui et al.⁷² report a decrease in fractal dimension with increasing impact toughness in an experi= mental steel that should be fracturing through microvoid coalescence (test temperatures at -20= °C).

Not all st= udies using impact toughness are consistent, however. According to Wiencek and co-worker= s,⁷³ no relationship exists between the fractal dimension and impact toughness.<= span style=3D'mso-spacerun:yes'> In this case it should be mentione= d that the authors used vertical profiles rather than slit-islands. The writers note that they applied= a segment counting method similar to the box counting method (Kolmogorov frac= tal dimension). Despite impact te= sting spherodized steel specimens at temperatures of -196°C, -45°C and -20= °C, all D* values were found to be between 0.09 and 0.10.

Relationship between fra= cture toughness and fractal dimension

Mu and Lun= g⁷⁴ have studied the change in fractal dimension with fracture toughness in two medium carbon steels. In one = set of specimens the heat treatment was altered to vary the fracture toughness whi= le in the second set the temperature was regulated. In both cases the study revealed a decrease in fractal dimension with increasing fracture toughness as illustr= ated in Figures 2-10 and 2-11.

Figure 2-10 a= nd 2-11.= Plots of fracture toughness against fractal dimension under two conditions.&nb= sp; In Fig 2-10, the specimens have been heat treated differently. In 2-11 the temperature has been varied. [Ref 74].<= /span>

The basic relationship deri= ved from this study can be stated as:

(2-9) Ln(K_IC) =3D Constant+[(= 1-D_F)Ln(ε_i)]/2

Where the "constant&qu= ot; term incorporates the quantities: E, γ and ν.

The terms are defined as fo= llows:

K_IC=3D&nb= sp; Plane strain fracture toughness. = E' =3D Young's modulus/(1-ν²).

γ =3D Effective surface energy (γ_{true surface energy}+ = 47;_{Plastic
strain energy)}.

D_F =3D Surface fractal dimension. = ν =3D Poisson's ratio.

ε_{i =3D}a "step" of crack propagation.

Mu and Lung prompted additional studies and many attempts to explain their data. Many of the studies employed impact toughness specimens (see previous section) which provide a measure of tough= ness but no directly comparable data. There is no confirmed correlation between impact toughness and fract= ure toughness.⁷⁵

In follow-= on work Lung and Zhang explained that the negative correlation between D* and KIC could be explained in terms of the intricacies of the perimeter-area slit-island measurement method as well as the micromechanism of fracture⁷⁶.<= span style=3D'mso-spacerun:yes'> A quantitative "fractal" description of the mechanisms of transgranular and intergranular fracture w= as found to be consistent with their qualitative observations⁷⁷.

Most recen= tly the fracture toughness-fractal dimension has been examined by Su and Lei. They attempt to explain all fractu= re toughness-fracture surface relationships with a unified model based on prof= ile roughness⁷⁸. The f= inal form of their equation is:

(2-10) J_IC =3D (σ_{ys =}L)(A₀ + B₀ ln(ΔD/ƒ)

Where:

J_IC =3D Critical J-integral value at fracture initiation.

σ_ys =3D Yie= ld strength. L =3D a microstructural length. A0 and B₀ =3D Material constants

ΔD =3D D-1 for profile= s, D-2 for surfaces.

ƒ =3D volume fraction = of dimple nuclei.

The interp= retation in this case being that fracture toughness depends on several microstructur= e - surface "roughness" features and that a single parameter, namely = D, is insufficient to determine fracture toughness from fracture surface topography data.

Interpretation of fractal dimension relationships in metals

The interp= retation of toughness-fractal dimension graphs derived from fracture surfaces have b= een the focus of extensive research. This area of study appears to have attracted more attention than fra= cture toughness-fracture surface roughness correlations. This may be due to the fact that researchers are still attempting to understand the meaning of fractal curves and their subtle distinguishing characteristics.

One of the significant observations reported by many researchers is the reverse sigmoi= dal shape (RSC) of many Richardson= or perimeter-area type slit-island measurements. A classic work in this area was published by Underwood and Banerji⁷⁹. Their paper focused on the asserti= on that fractal plots were naturally reverse S shaped and that this shape cont= ains useful information about the fracture process. Underwood and Banerji approached t= his problem by converting the RSC shapes into linear plots which appeared to of= fer better resolution to differentiate different fracture mechanisms. The technique was applied to speci= mens of AISI 4340 steel heat treated and tempered at different temperatures. These researchers found that the f= ractal dimension was lowest within the tempering range which usually produces embrittlement in this type of steel. Their key findings are shown in Figures 2-12, 2-13 and 2-14. No fractographic information discu= ssing the fracture mechanism(s) observed was presented for any of the specimens in this study.

Other rese= archers have also devoted extensive effort into the examination of this often seen = RSC behavior. Some have explained= this fractal graph shape in terms of the interaction of multiple fractals (also known as "multirange fractals" which expose different fractal beh= avior at different scales)⁸⁰. Investigations by Shi et al.⁸¹ suggest that the shape of fractal plots are influenced not only by D*, but also by the configuration = of the "initiator length". As a result both factors should be incorporated into any models atte= mpting to relate fracture toughness and fractal dimension.

= ;

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Figure 2-12. F= ractal plots of AISI 4340 steel samples tempered at different temperatures [Ref. 7= 9].

= ;

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Figures 2-13 and 2-14. Linearized fractal plots and plot = of fractal dimension v. tempering temperature. According to Underwood and Banerji= the lowest values of D are found within the temper embrittlement region [Ref. 7= 9].

= ;

RSC behavi= or continue to be a much disputed characteristic of fractal graphs. Many researchers maintain that the= inflections points have some inherent meaning and must be considered⁸² while others claim that the points are an artifact of the measurement method. Russ⁴ and others have s= hown that it is possible to artificially induce inflection points at both ends of the curve shown in Figure 2-4. At the small ruler length side of the graph, the curve may bend as a result of= the measuring unit being as small as or smaller than the profile width. At the large ruler length side of = the plot the curve may deviate from a straight line because the ruler is approaching or surpassed the size of the measured object and the "step count" is no longer accurate. This last explanation is the most reasonable. However, there are conflicting dat= a..

It is the = view of Dauskardt et al.⁸³ that the former is the case. That is, that fractal curve inflec= tion points reveal something intrinsic about microstructures, particularly those= at the low ruler length side of the plots.&nb= sp; This team considered many types of metallic fracture surfaces in a v= ariety of steels. Through an analysi= s of artificially generated profiles and real fracture surfaces they conclude th= at fractal plots can indeed identify the influences of microstructure on fract= ure surface profiles. Two of their graphs which may be applicable to this study are illustrated in Figure 2-15= and 2-16.

<= ![endif]> = <= ![endif]>

Figures 2-15 and 2-16. Log-log R_L v. η graphs illustrate the pos= sible interpretation of fractal plot curves and their relationship to microstructures. Note: R_=
L=3DR₀η^(1-D) [Ref 83].

Dauskardt = et al. postulate that the slope fractal of graphs constructed with short steps may= be attributed to cleavage steps in brittle materials and slip steps in ductile materials. At larger step len= gths the plots are influenced by grain size in brittle materials and particle spacing in ductile materials. Williford has also commented on the perceived sigmoidal shape of the step size-total size fractal plot.^84-86 In a manner similar to Dauskart and co-workers, Williford attributes this shape to the influence of a variety of fracture mechanisms (see Figu= re 2-17).

Figure 2-17. Plot of step len= gth v energy of fracture. Interpretations of various sections along the curve also given.<= span style=3D'mso-spacerun:yes'> [Refs. 84-86].

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Based on h= is observations in metals and other materials, Williford also provided a unifi= ed qualitative model for fracture⁸⁵. His interpretation proposed = that materials may be divided into two basic categories; brittle and ductile. In each of these classes the mecha= nisms of fracture is said to depend on the characteristics of the microfracturing mechanism prior to full scale unstable crack propagation. If the fracture mechanism can be v= iewed as the nucleation of small cracks, their joining and propagation, the resul= ting fracture surface would be expected to be rather flat and smooth at low toughness. As the energy of fracture increases it is reasonable to expect that the excess energy be dissipated by creating more surface. This process would result in more tortuous fracture surfaces and resulting greater measured fractal dimensions.

Williford = provides a similar argument in the case of ductile fracture surfaces. In this case, as the fracture toug= hness increases, the surface is "stretched out". That is, the dimples that are crea= ted tend to elongate and widen resulting in an overall smoother surface. The proposal is illustrated in gra= phical form in Figure 2-18.

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Figure 2-18. S= urface dimension (D_s) vs. surface energy (γ). [Ref 85]<= /span>

Difficulty correlating d= ata.

Williford = made one of the first attempts to collate the known information about fracture toughness-fractal surface relationships and establish a unified model. Despite the noble attempt, doing s= o with the available information may not have been appropriate. Each of the data sets referenced in Figure 2-18 were collected and analyzed in different ways in their respecti= ve sources. The data for ceramic= s was obtained by Mackin et al.⁸⁷ using the perimeter-area relationshi= p on slit-islands and true K_IC values, the data for AISI 4340 steel w= as collected by Underwood and Banerji⁸⁸ using fracture profiles and= a linearization scheme for sigmoidal curves, the data for 300-grade maraging steel was obtained by Mandelbrot et al.¹⁰by using impact toughn= ess specimens and the perimeter-area relationship, and, finally, the data for titanium alloys was gathered by Pande et al.⁶² on dynamic tear energy specimens (a measure of toughness per ASTM E 604) using both vertical sections and the perimeter area relationship.

The diffic= ulties encountered in the literature by Williford persist in the present. Studies continue to emerge which c= ontain data that is difficult to correlate to the work of other authors. It is one of the aims of this work= to follow the methods prescribed by Hill et al.¹⁴, so that the data= may be analyzed by researchers in quantitative fractography. As it stands today, the goal of establishing a definite microstructure-bulk material properties-fracture surface topography relationship or a model which adequately describes the seemingly similar observations made in many materials remains elusive.

CHAPTER 3

Materials and methods<= /p>

Material=

A steel me= eting the compositional requirements of The American Institute of Steel<= /st1:PlaceName> and Iron (AISI) 4340 was used in this study. The alloy was selected on the basi= s of its ready availability, well documented mechanical behavior, relatively low cost, and technological significance. The alloy is in widespread industrial use and has, for decades, established the foundation for the development of many commercial high stre= ngth steels.

The raw ma= terial was delivered as two rolled plates measuring 1.52 cm x 35.7 cm x 55.9 cm. Each plate had been cut from the s= ame parent slab. A visual examina= tion revealed no abnormal pitting, cracking or unusual levels of corrosion.

The chemic= al composition of one plate was measured through Glow Discharge Spectroscopy (= Leco GDS-750A). The results were consistent with the requirements of AISI 4340 and are given in Table 3-1. <= /p>

Table 3-= 1 Chemical Composition of Raw Materi= al

Elemen= t	AISI 4= 340 (WT%)	Sample= (WT%)
Carbon	0.38 - 0.= 43	0.382 - 0= .412
Manganese	0.60 - 0.= 85	0.78 - 0.= 80
Silicon	0.15 - 0.= 35	0.278 - 0= .288
Phosphorus	£ 0.025	0.0082
Sulfur	£ 0.025	0.011 - 0= .012
Chromium	0.70 - 0.= 90	0.868 - 0= .879
Nickel	1.65 - 2.= 00	1.687 - 1= .702
Molybdenum	0.20 - 0.= 30	0.244 - 0= .249
Copper	£ 0.35	0.143 - 0= .145
Iron	94.75 - 9= 6.27	95.4 - 95= .5

Fracture Toughness Specimens

Initial Manufacturing and Hardening

The two pl= ates were used to manufacture 24 rectangular Compact Tension (CT) type fracture toughness samples according to the requirements of ASTM E 399⁸⁹.= All samples were configured to mee= t a standard L-T orientation. This designation has a fracture plane whose normal is in the longitudinal direct= ion (L) of the plate and an expected direction of crack propagation coincident = with the width (also known as the long-transverse or "T") direction.

In an effo= rt to minimize cracking and machining difficulties, the samples only had the pin holes drilled and its hole corners radiused prior to hardening (Austenitizi= ng + Oil Quenching only). Following hardening, the samples were electric discharge machined (EDM) to create che= vron type (V-shaped) starter notches.

The initia= l heat treatment (with the exception of tempering) was performed in accordance with AMS 2759/1⁹⁰, Heat Treatment of Carbon and Low alloy Steel Pa= rts, one of the many industrial standards which provide guidelines for heat trea= ting of AISI 4340. Consistent with= this document, the samples were Austenitized at 816°C and held for a minimum of 25 minutes. Again = per AMS 2759/1, the specimens were quenched in oil (at 16°C - 71°C) and cooled to room temperature. As a result of this heat treating process, the samples produced a hardness in = the range of 52 - 56 HRC.

Precracking

ASTM E 399 requires that fracture toughness samples have sharp cracks of a prescribed length. In this study, cracks= of the type required were produced by stress corrosion cracking. A solution of 5% HCl/water was app= lied by eyedropper to the starter notch of a sample. Five drops were sufficient to pene= trate and wet the notch. The area w= as sealed by a layer of teflon tape and the specimen was loaded to 2.2 kN. Visual monitoring revealed that th= is method could produce starter cracks of the desired length within 30 minutes. All samples were pre-cracked in the same manner.

Immediately following this operation, specimens were cleaned and neutralized by immersi= on in a ultrasonic cleaning chamber filled with water/soap solution for 30 minutes. Specimens were then = dried for another 30 minutes at 50°C and coated with a light oil.

Tempering=

The precra= cked set of 24 CT samples was divided into eight groups of three. The first group was set aside to f= orm an untempered subset. The remain= der were each tempered at 480°C as indicated in Table 3-2 to obtain hardness values in the range of 52 to 40 HRC.

Table 3-= 2

Tempering S= chedule for CT Samples

Group

#

Hardness = Before

(HRC ± 2)

Tempering= Treatment

Time & Temperature

Hardness = After

(HRC ± 2.0)

1

54

None

54

2

54

480°= C for 05 minutes

52

3

54

480°= C for 10 minutes

50

4

54

480°= C for 15 minutes

47

5

54

480°= C for 30 minutes

46

6

54

480°= C for 50 minutes

44

7

54

480°= C for 75 minutes

42

8

54

480°= C for 95 minutes

40

Notes:   a) Samples were given 10 minutes to c= ome up to temperature then soaked for the specified time.

b) Samples were immediately water quenched after tempering.<= /span>

c) Individual hardness values are given in Table 4-1.=

Samples of= 40 HRC were specifically desired due to the technological significance of AISI 434= 0 in this form. Many aviation and industrial components are used in this hardness condition (corresponding to= a UTS of 1,241 to 1,379 MPa).

Changes That Occur on Qu= enching and Tempering AISI 4340 Steel

A brief re= view of the microstructural changes that occur on quenching and tempering AISI 4340= is warranted at this point. The features that develop as a result of these processes significantly influence the overall mechanical properties as well as the fracture process^91.

The object= ive of most heat treatments involving medium carbon-low alloy steels is to produce= a microstructure consisting largely of tempered martensite. Martensite is the particular name assigned to the body centered tetragonal (bct) structure which results from= the diffusionless transformation of face-centered cubic (fcc) gamma iron (a.k.a. austenite) on quenching from any temperature over 727°C⁹².

The usual austenitizing temperature for AISI 4340 is approximately 816°C. At this temperature, and in t= he austenitic phase, carbon may be located along the edges and cube center of = the fcc crystal structure. Upon quenching, the structure desires to transform itself to the equilibrium bcc crystal structure, however, the process is inhibited from doing so by the presence of carbon atoms. The resulting intermediate bct unit cell structure has a total of four carbon a= toms positioned between the iron atoms in the "c" or vertical direction and an additional carbon atom at the lower and upper square faces.

This carbon-strained unit cell produces a steel which is very hard and brittle.<= span style=3D'mso-spacerun:yes'> At this stage, the nearly 100% martensitic steel is virtually useless unless it is softened and toughened = by tempering. The tempering temperature and time controls the decomposition of martensite as well as the formation of various other second phases.

In AISI 43= 40 steel, martensite forms in laths on quenching. The final decomposed microstructure (generally achieved by long term exposure to elevated temperatures) is a mixture of iron and iron carbide in the form of layers (pearlitic) or carbi= de spheres in a iron matrix (spherodized)⁹². In between these two extremes ther= e may exist a multitude of intermediate and stable carbides. These also tend to change as a res= ult of tempering. The driving forces= for these changes include a reduction in strain energy when moving from bct to = bcc structures, supersaturation of carbon in martensite, a reduction in interfa= cial energy in going from lath martensite to ferrite/cementite structures, and lowering of interfacial energy when coarsening ferrite/cementite structures= .

In practic= e, quenched steels having approximately 0.4% carbon also contain a percentage = of retained austenite. A level of austenite of about 2 to 4% by volume is not uncommon. The decomposition sequence of the initial quenched structure (martensite/retained austenite) as it proceeds towards the equilibrium structures (iron/iron carbide) involves several sta= ges. They are presently considered to b= e as follows⁹³:

Table= 3-2a

Step

Change

Controling Mechanism=

1<= /p>

Retained austenite transforms to martensite..

Diffusionless transformation.

2<= /p>

Carbon atoms redistr= ibute to (a) lattice defects and (b) carbon clusters.

Volume diffusion of = carbon atoms.

3<= /p>

Precipitation of = 49; and η transition carbides.

Diffusion of iron at= oms along dislocations.

4<= /p>

Precipitation of Hag= gs carbide.

-<= /p>

5<= /p>

Decomposition of rem= aining retained austenite to ferrite/cementite.

Volume diffusion of = carbon atoms in austenite

6<= /p>

Conversion of the segregated carbon and the transition carbides to cementite

Volume and pipe diff= usion of iron atoms.

Note: Summarized from ref 93, pg 669.

The temper= ing steps reported in Table 3-2a are not discrete sequential steps. Considerable overlap sometimes exi= sts and it is difficult to say specifically when one step ends and another begi= ns⁹³.

The evolut= ion of the tempering process and the particular steps completed at any given time depends on the alloy and the temperatures employed. The length of time required to com= plete a step or the location of an alloy along the sequence of steps is difficult= to establish without the use of sophisticated equipment and time consuming procedures. Typically, a thor= ough evaluation would require the combined use of metallography, scanning electr= on microscopy, transmission electron microscopy and energy dispersive X-ray spectroscopy.

A specific correlation between the tempers employed in this study and their correspond= ing detailed microscrotructural make-up was beyond the scope of this report. This work was necessarily limited = to characterizing some of the basic features of the alloy and microstructure employed in this research. As= a result, the reader is merely advised that a host of microstructural changes which could not be completely quantified are likely to have taken place dur= ing tempering and that these changes undoubtedly influenced the fracture process. The specifics of how= this might have occurred is considered throughout the Results and Discussion<= /i> section of this work.

Second phases in AISI 43= 40

Second pha= ses are among the most important features that develop on tempering steel. As noted in Table 3-2a, the epsilon (ε) and μ(eta) transition carbides begin to form early in the tempering process. These carb= ides are difficult to resolve under a TEM and are often discussed together as the (ε/η) carbides. The crystal structure of epsilon is hexagonal while that of eta is orthorhombic= .

Epsilon an= d eta carbides nucleate on martensitic subgrain boundaries and/or martensitic lath boundaries, grow by the diffusion of carbon and eventually transition to cementite. Prior to becoming cementite and at the early stages of cementite growth these particles are sub-micron sized. The martens= itic subgrains are between 1.0 to 0.1 microns, or less, in diameter and the grain boundary thickness is in the order of 20nm⁹³.

A study by= Speich⁹⁴ shows that for a carbon steel of ~0.4% carbon, tempered from a hardness of = 62 HRC (760 DPH) to a hardness of 40 HRC (400 DPH), the microstructure has tra= nsitioned through steps 1-5 of Table 3-2a. At ~40 HRC the steel is roughly at the beginning of the ferrite recovery and cementite spheroidization process. Most of the cementite nucleated and grown up to now is in the form of rods and located at martensitic lath boundaries.

AISI 4340 = steel contains as its primary alloying elements nickel, chrome and molybdenum. The central function of these ingredients is to improve hardenability, however, chrome and molybdenum are also strong carbide formers and may also participate in the formation of se= cond phases. Nickel provides some solution strengthening but does not undergo any reactions. When tempered at below ~540°C, chrome and molybdenum may enter the cementite phase by forming (Fe, M)_=
3C type carbides⁹³. <= /p>
In additio= n to the desirable martensitic and iron carbide phases, AISI 4340 steel may contain a wide variety of inclusions. T= he presence, quantity and size distribution of inclusions depend greatly on the steel manufacturing method. In modern day practice, foundrys go to great lengths to minimize inclusion content, however, no steel can be made to be completely clean in an economically feasible manner.

The most i= mportant types of inclusions are those consisting of sulfides, oxides nitrides and silicates⁹⁵. These= basic compositions are MnS, Al₂O₃, MnAl₂O_{4<=
/sub>,
FeAl₂O₄, MgAl₂O₄, TiN, SiO_=
2⁹⁶. Oxides are introduced into the mol=
ten
steel by adsorption from the atmosphere, sulfides are present in the initial
iron ore and Al and Ti are typically introduced as part of the deoxidation
process.}

Perhaps th= e most abundant inclusion type is manganese sulfide. Sulfur is naturally present in the initially mined iron mix and is difficult to remove. The usual manner of controlling the location of this potentially harmful element in steel is by adding manganese and forming the far more benign MnS. This inclusion may solidify as a initially spheroidal structure, however, hot working causes it to assume a geometry and orientation consist= ent with the deformation process and direction.

Microstructural Evaluati= on Through Metallography:

Two sample= s of the heat treated steel, one at 54 HRC (untempered) and another at 42 HRC (tempe= red, Group 7 of Table 3-2), were prepared in a polymeric mount to reveal their microstructural features. The= plane made ready for examination was equivalent to the plane of crack propagation (L-T, according to ASTM E 399). The samples were initially polished through a progression of 180, 240, 360, 400, 600, and 800 SiC grits in a standard metallographic polishing wheel. The wheel surface was changed to a= cloth and used to continue polishing through a progression of 1.0 micron and 0.3 micron aluminum oxide slurries. The last polishing step employed a cloth pad and a 0.05 micron diamond abrasive suspension.

The polish= ing procedure resulted in samples having an essentially flat, scratch free, mir= ror surface finish. The polished surfaces were then etched for 10-15 seconds with a traditional 2% nitric acid/water (a.k.a. 2% Nital) etch. The etchant revealed the microstructures provided in Figures 3-1 thr= ough 3-4.

The microstructural features observed at both extremes of the hardness range we= re similar. The samples displaye= d the martensitic laths typical of AISI 4340 steel heat treated to 40 HRC and abo= ve. No abnormal structures, inclusions= or chemical anomalies (segregation) were observed.

<= /span><= ![endif]><= ![endif]>

Figures 3-1 and 3-2. Microstructure of a 54 HRC sample = at approximately 400X and 1000X, respectively.

Both samples exhibited features typical of untempered martensite.

<= ![endif]><= ![endif]>

Figures 3-3 and 3-4. Microstructures of 42 HRC sample at approximately 400X and 1000X, respectively.

Features are typical of tempered martesite.=

Heat Treatment Uniformit= y

In an effo= rt to insure that all samples exhibited uniform heat treatment through their thickness, a microhardness traverse was performed in accordance with ASTM E= 384,⁹⁷ Knoop 500 g scale. The 54 HRC mounted and polished cross section employed for microstructural evaluation = was used in the traverse. The res= ults of this evaluation are shown in Figure 3-5.

<= ![endif]>

The mean microhardness value was measured to be 598 HK_500g. Figure 3-5 shows a fluctuating set= of points about this mean value. The fluctuations are considered normal for microhardness tests. The precision of the measurement is typically in the order of ±15 HK_500g. The initial heat treatment is cons= idered to have produced a relatively uniform microstructure and a homogeneous set = of mechanical properties through the sample thickness.

The microh= ardness test results were found to be consistent with the known heat treatment resp= onse of AISI 4340 steel. Jominy end quench tests show steady hardness values (less than 1.0 point variability on the HRC scale) for the first 19.0 mm from the quenched end⁹¹. Considering that the distance betw= een the CT sample sides and the center is no more than 6.35 mm, even hardening through the thickness should be expected.

Grain Size

Grain size measurement in hardened steels generally refers to the prior austenite grai= n size. This measurement is notoriously difficult to make in these alloys due to the problem of differentiating the etched grain boundaries from the martensitic laths.&nbs= p; Note, for example, the inability to distinguish grain boundaries from the bulk microstructure in Figures 3-2 through 3-4.

ASTM E112<= sup>98 is the industry standard for grain size measurement. This document suggests using a pol= ished and etched metallographic cross section to obtain grain boundaries suitable= for measurement. Krauss^{92
provides a number of methods for revealing grain boundaries, however, the
techniques are cumbersome and time consuming. An alternative method is provided =
by
Raymond⁹⁹.}

Raymond's = method can be performed on specimens where fracture surfaces have been formed by intergranular fracture. In th= is case, a set of images may be obtained through a scanning electron microscope (SEM) from the surface. A ser= ies of grain diameter measurements is collected and a chart, published in Raymond's work, can be employed to obtain an equivalent ASTM grain size measurement.<= /p>
Recall tha= t stress corrosion cracking (SCC) was used to create pre cracks in the fracture toughness samples. In AISI 43= 40 SCC occurs through the prior austenite grain boundaries. The resulting fracture surface is = intergranular in nature. Figures 3-6 throug= h 3-9 were taken from the pre cracked areas of fracture toughness samples heat treated to 54 HRC and left untempered.&nbs= p; A set of sixty diameter measurements were taken. The mean diameter was measured to = be 7.7 microns. This value results i= n an average ASTM grain size of 11.

<= /span>

MnS stringers

<= /span>

Fig. 3-7

Fig. 3-6

<= ![endif]><= ![endif]>

<= /span><= /span>

MnS stringers

Fig. 3-8

Fig. 3-9

<= ![endif]><= ![endif]>

Figures 3-6 to 3-9. Pre crack fracture surfaces obtain= ed through SCC of fracture toughness samples.= Granular features represent the sizes and shapes of the prior austen= ite grains. The microstructure contained a distribution of small MnS stringers typical of AISI 4340 (Fig 3= -9) and a few large stringers (Fig 3-7).

Microchemistry

The surfac= es illustrated in Figures 3-1 through 3-4 and 3-6 through 3-9 were inspected u= nder an SEM through energy-dispersive X-ray spectroscopy (EDS) in accordance with the methods recommended in ASTM E1508¹⁰⁰. The topographies were examined for evidence of abnormal composition, inclusions, unusual segregation or other similar chemical anomalies. T= he lenticular fissures illustrated in Figures 3-6 and 3-7 were given particular attention. Many similar crevi= ces were observed throughout a large number of the pre-cracked fracture surface= s.

Figure 3-1= 0 shows a typical EDS spectrum taken from the intergranular region of a fracture toughness specimen. The spect= rum is consistent with the composition reported in Table 3-1. The peaks associated with iron, ni= ckel, chrome and molybdenum are marked. Elements present in low concentrations reveal peaks that are barely discernible from the background radiation.= The smaller peaks are expanded vertically in Figure 3-11.

Figure 3-10. EDS Spectrum of Overall Intergranu= lar Region

<= ![endif]>

=

Figure 3-11. EDS Spectrum of Overall Intergranul= ar Region

(Spectrum same as Fig.= 3-10 expanded vertically)

<= ![endif]>

Figure 3-1= 2 shows an EDS spectrum typical of the data collected from fracture surface fissures such as those illustrated in Figures 3-7 and 3-8. Note the elevated levels of mangan= ese and sulfur compared to the overall spectra of Figures 3-10 and 3-11. Two superimposed and normalized spectrums showing the overall composition and that found within one of these surface anomalies is provided in Figure 3-13.

Figure 3-12. EDS Spectrum from Fracture Surface Fissures

<= ![endif]>

Figure 3-13. Superimposed and Normalized EDS Spectrums.

Overall Fracture Surfa= ce (red) v Fissure (yellow).

<= /span>

Mn

S

<= /span>

Higher Mn and S in fissure

<= ![endif]>

Tensile Testing

<= /span>Tensile tests were conducted in accordance with ASTM E8. Considering the vast amount of ten= sile test data available for AISI 4340, only three sets of two samples each were considered necessary to establish and verify average properties. Samples were manufactured and heat treated to hardness values of 40, 46 and 50 HRC as indicated in Table 2-2.<= span style=3D'mso-spacerun:yes'> Additional data was obtained from a number of sources in the literature^101,102 (see Table 3-14 for a summary).

<= ![endif]>

Fracture Toughness Testing

Fracture t= oughness tests were conducted in accordance with ASTM E 399. All samples were loaded to failure= in an Instron Series IX Automated Materials Testing System equipped with a 89 kN = load cell. A crosshead speed of 0.= 127 cm/min was selected. Displace= ment was measured by mounting an extensometer at the mouth of the CT samples. The tests collected load displacem= ent information as required by the standard.&n= bsp; Figures 3-15 and 3-16 show representative CT fracture surfaces after testing.

W

<= /span><= /span>

a

<= /span><= /span><= ![endif]><= ![endif]>

<= ![endif]><= ![endif]>

Figure 3-15. CT specimen #21. Surface texture is        Figure 3-16. CT specimen #1. Surface texture is

typical of high fracture toughness steels.        &= nbsp;           &nbs= p; typical of low fracture toughness steels.

The load-displacement graphs were employed as indicated in ASTM E 399 to determ= ine the maximum load to fracture, P_Q. The fracture surfaces were used to measure the size of the crack (pre-cracked dimension "a") prior to fast fracture. This data can = be inserted into equation 3-1 to calculate a conditional value for K_{IC (known
as K_Q). If this va=
lue
was found to be consistent with the sample thickness and mechanical propert=
ies
given by equation 3-3, the test was considered to be a valid plain-strain
fracture toughness test and K_Q=3D K_IC. The measurements and results discu=
ssed
above are provided in Table 3-1.}

(3-1) K_Q =3D (P_Q/B= W^1/2) ƒ(a/W)

(3-2) ƒ(a/W) =3D (2 + a/W)(0.866= + 4.64a/W - 13.32a²/W² + 14.72a³/W^{3 -
5.6a⁴/W⁴)}

(1 - a/W)^3/2
(3-3) B ³ 2.5 (K_{Q/s_YS)²}

Where:

K_{Q<=
/sub>        &=
nbsp; =3D Conditional stress intensity facto=
r.}

P_{Q<=
span
style=3D'mso-tab-count:1'>        &=
nbsp;}=3D Load to fracture

B        &= nbsp;    =3D Specimen thickness

W        &= nbsp; =3D Specimen width

ƒ (= a/W)   =3D Shape factor function.

a        &= nbsp;    =3D Crack length

s_{YS        &=
nbsp;}=3D Yield stress

Fractal Dimension Measurements

Specimen Preparation

One side o= f each of the fracture surfaces from the fracture toughness specimens was used to manufacture a sample suitable for fractal dimension measurement. The slit-island method was selecte= d.

Preparatio= n steps began by cutting the region of each CT sample which contained the fracture surface into a rectangular section measuring approximately 3.54 cm x 1.27 c= m x 1.5 cm. Each section was imme= rsed into a heated (88= °C) electroless nickel plating bath. A plating time of 15 minutes produced a uniform deposit measuring 6 microns thick.

Each of th= e plated fracture surfaces was carefully sanded in a standard metallographic polishi= ng wheel through a progression of 400, 600, 800 SiC grits. The wheel surface was changed to a= cloth and used to continue polishing through a progression of 1.0 micron, 0.3 mic= ron aluminum oxide slurries. The = last polishing step employed a cloth pad and a 0.05 micron diamond abrasive suspension.

The polish= ing steps required extreme care and gentle applications of pressure to avoid destroying the fracture surface. This method was successful in producing numerous slit-islands in all samples. The islands were very small and difficult to see by the unaided eye, however, they could be readi= ly identified under an SEM (BEI atomic number contrast mode).

An image representative of a plated, cut and polished specimen is shown in Figure 3-17. This sample is ready for insertion to the SEM.

<= /span><= ![endif]>

Figure 3-17. CT specimen cut, plated and polished to reveal slit islands near its center (circled).

Slit-island Image captur= e

The grindi= ng and polishing procedure described caused the removal of nickel plate from high spots on the fracture surfaces. This resulted in the production of numerous flat islands of steel su= rrounded by nickel plate. These island= s were readily identified under a scanning electron microscope (SEM) in backscatter electron imaging (BEI) mode.

A CamScan = MaXim 2040 SL Scanning Electron Microscope was applied to capture the necessary images. The instrument was eq= uipped with a LaB₆ electron source.&nb= sp; The SEM was set up to operate at 35-40 Kv and a working distance of approximately 15 mm. The BEI detector employed was of a four quadrant, annular, solid state design. The detector was positioned above = the sample and surrounded the instrument pole piece.

Three slit= islands were photographed from each sample. A magnification of 500-800X was found to produce sufficient resoluti= on and a clear view of each area of interest.= The islands were selected on the following basis:

1. The island must fill between 70 to= 90 %. of the image window

2. The island must have an undamaged contour (free from pits, voids, large scratches).

3. The island must not exhibit metallurgical irregularities on its perimeter (free of inclusions or MnS stringers).

4. The image must have good contrast between the base metal and the surrounding Ni plate

5. Slit islands were to be located wi= thin flat the plane-strain region of the sample (away from the angular shear lip= s).

&n= bsp;

Images mee= ting the above requirements were initially photographed at a resolution of 640 x 512= in a Tiff format. The pictures w= ere then converted to a JPEG format for ease of transfer to the image processing software. All islands are sho= wn in Appendix C.

Image Processing

Image proc= essing was performed in a standard desktop computer operating a software program k= nown as Image Pro (V. 3.0.01.00). = Image Pro included an accessory program known as Materials Pro (V. 3.1). The latter software is capable of performing a large variety of materials related image analysis, optimization and measurements. The combination of Image Pro/Mater= ials Pro allowed for the introduction of an image, semi-automatic selection of t= he island (through an area of interest) and measurement of the fractal dimensi= on.

In essence, Materials Pro performs the same operations an operator would perform in a manually executed Richardson plot. That is, once the user defines the perimeter to be measured, the software uses a large number of "virtual yardsticks" to obtain a plot of the log distance v log stride length relationship¹⁰³.&= nbsp; The software then calculates and inserts a best fit line into the data. The decimal part of the= slope of this plot is defined as D*; the fractal dimensional increment.

Preliminary Testing of t= he Fractal Dimension Measurement Method

As noted a= bove, the fractal dimension is reported as D* where D* is the mantissa of the fra= ctal dimension D=3D2.D*. This D* v= alue assumes that a linear measurement (such as that performed on island perimeters), i.e., 1-D, results in the same fractal dimension as a 2-D measurement. A series of stan= dard test images were collected. T= heir fractal dimension were determined in an effort to verify that the proposed measuring method would provide accurate measurements. These values obtained through the = test standard were compared against known reported or calculated data. The images covered the anticipated= range of results from D* =3D 0.000 to D* =3D 0.500 as indicated in Table 2-3.

The "textbook" and "Koch" test images noted in Table 3-3 we= re obtained by digitally photographing pictures in a textbook⁴(ill= ustrated in Appendix B). A similar met= hod would be required in many instances to obtain suitable measurements from the actual sample surfaces. Manua= l contrast enhancement was then applied to the photographs to create images of suffici= ent contrast for boundary detection in the image analysis system. A second digital image of the firs= t was actually used for measurement.

The contra= st enhancement consisted of manually darkening excessively light areas and lightening excessively dark areas. The objective in each of these cases was to provide an object which could be readily detected and processed by the software while being careful= to preserve its original contour.

Table 3-= 3

Test Images= and Resulting D* Values

Image Description

Known<= o:p>

Calcul= ated D*

Materi= als Pro

% Diff= erence

Black cir= cle

0.000

&nbs= p;

0.00

0

Textbook<= /p>

0.17

&nbs= p;

0.18

+6

Textbook<= /p>

0.23

&nbs= p;

0.22

-4

Sample Im= age 1a

&nbs= p;

0.19

0.21

+11

Sample Im= age 1b

&nbs= p;

0.18

0.19

+6

Sample Im= age 1c

&nbs= p;

0.20

0.20

0

Koch Tria= dic

0.262

&nbs= p;

0.25

-5

Koch Quad= ratic

0.500

&nbs= p;

0.48

-4

= ;

The fractal dimension of sample images 1a - 1c were measured manually. Eight separate perimeter measureme= nts were taken from each island using different lengths of measuring units, i.e. "yardsticks". A gra= ph of the log(measuring unit) v. log(perimeter) produced a linear relationship. Automatic processing of the data p= oints by a curve fitting program known as Curve Expert¹⁰⁰ produced the reported fractal dimension. T= he correlation coefficients for the manual curve fits were calculated to be in= the range of 0.95 - 0.98. Based o= n the test data, the Materials Pro image processing software was able to measure = the fractal dimension to within ±6% of its true value. The data were more consistent than= the manual method which could produce a discrepancy of up to 11%.

In obtaini= ng and processing digitized images every effort was made to control imaging parame= ters (brightness, contrast and gamma) in such a way as to insure that the true e= dges of the islands were detected by the software. The effects of each variable and h= ow the perimeter was subsequently detected could be observed in real time on a computer monitor. If the effe= cts were not carefully monitored on-screen, adjustments within the imaging parameters could produce errors from known values in the fractal dimension = of up to ± 10%. Despite t= he manual and imaging parameter enhancements, the resulting data was found to = be comparable in accuracy to the manually calculated and processed data.

Sample Measurements=

A minimum = of three BEI images were obtained from each of the twenty four samples. The images are provided in Appendix C. Each image was used "= as is" if the image analysis program could accurately distinguish its boundaries from the surrounding nickel plate. Otherwise, the island was printed = on paper for manual contrast enhancement, re-photographed and digitally analyzed. Fractal dimension r= esults are provided in Table 3-4.

= ;

= ;

= ;

= ;

= ;

= ;

= ;

= ;

= ;

= ;

= ;

= ;

Table 3-= 4

Fractal Dim= ension Results

&nbs= p;

D* for In= dividual Images

Sample #<= /p>

Image a

Image b

Image c

Image d

Image e

Image f

Mean

D*

1

0.20918

0.18508

0.20299

&nbs= p;

&nbs= p;

&nbs= p;

0.20

2

0.21019

0.21092

0.18999

&nbs= p;

&nbs= p;

&nbs= p;

0.20

3

0.17086

0.16807

0.22111

&nbs= p;

&nbs= p;

&nbs= p;

0.19

4

0.31824

0.28206

0.24483

&nbs= p;

&nbs= p;

&nbs= p;

0.28

5

0.29884

0.27498

0.21906

&nbs= p;

&nbs= p;

&nbs= p;

0.26

6

0.26536

0.29401

0.32258

&nbs= p;

&nbs= p;

&nbs= p;

0.29

7

0.24242

0.22037

0.30174

0.23678

&nbs= p;

&nbs= p;

0.25

8

0.27602

0.22414

0.22633

0.24019

0.21861

&nbs= p;

0.24

9

0.33127

0.23291

0.21328

0.24006

0.28716

0.23661

0.26

10

0.21310

0.19569

0.17148

&nbs= p;

&nbs= p;

&nbs= p;

0.19

11

0.27696

0.20276

0.17871

&nbs= p;

&nbs= p;

&nbs= p;

0.22

12

0.21841

0.21836

0.19293

&nbs= p;

&nbs= p;

&nbs= p;

0.21

13

0.20662

0.16948

0.15626

&nbs= p;

&nbs= p;

&nbs= p;

0.18

14

0.22060

0.21511

0.22641

&nbs= p;

&nbs= p;

&nbs= p;

0.22

15

0.21770

0.15615

0.18659

&nbs= p;

&nbs= p;

&nbs= p;

0.19

16

0.18952

0.12210

0.15969

&nbs= p;

&nbs= p;

&nbs= p;

0.16

17

0.12514

0.10469

0.13395

&nbs= p;

&nbs= p;

&nbs= p;

0.12

18

0.11616

0.14686

0.13636

&nbs= p;

&nbs= p;

&nbs= p;

0.13

19

0.13191

0.14458

0.07253

&nbs= p;

&nbs= p;

&nbs= p;

0.12

20

0.12714

0.13013

0.10179

&nbs= p;

&nbs= p;

&nbs= p;

0.12

21

0.12966

0.08658

0.07294

0.11991

0.04018

0.09440

0.09

22

0.15203

0.18989

0.18587

&nbs= p;

&nbs= p;

&nbs= p;

0.18

23

0.09862

0.09889

0.18028

&nbs= p;

&nbs= p;

&nbs= p;

0.13

24

0.13333

0.14525

0.19721

&nbs= p;

&nbs= p;

&nbs= p;

0.16

Note: Std deviation in D* provided in Ta= ble 4-1.

Sample Number	Hardness Bef./Aftr.<= /p> (HRC)	Crack Length (m)	Fracture Load (Newtons)	Shape Factor	K_IC (MPa Öm)	D^*	STD Dev in D*
1	56.3/56.3=	0.02667	6978.9	10.45	25.6	0.20	0.010
2	56.4/56.4=	0.02540	8153.2	9.66	27.6	0.20	0.010
3	55.7/55.7=	0.02692	7116.8	10.63	26.5	0.19	0.024

4	55.4/52.4=	0.02540	7561.6	9.66	25.6	0.28	0.030
5	55.5/52.0=	0.02464	8495.7	9.23	27.5	0.26	0.033
6	55.7/51.7=	0.02464	9340.8	9.23	30.2	0.29	0.023

7	53.9/49.7=	0.02438	16680.0	9.09	53.2	0.25	0.031
8	55.8/49.6=	0.02540	14233.6	9.66	48.2	0.24	0.021
9	56.1/49.9=	0.02286	21350.4	8.34	62.4	0.26	0.040

10	52.5/46.8=	0.02540	16457.6	9.66	55.7	0.19	0.017
11	51.5/48.0=	0.02616	16457.6	10.12	58.4	0.22	0.042
12	52.8/47.5=	0.02464	17792.0	9.23	57.6	0.21	0.012

13	52.5/45.7=	0.02286	28467.2	8.34	83.3	0.18	0.021
14	52.3/46.5=	0.02426	21795.2	8.98	68.6	0.22	0.005
15	51.5/46.8=	0.02426	21128.0	8.98	66.5	0.19	0.025

Group #	Hardness = Before (HRC ± 2)	Tempering= Treatment Time & Temperature	Hardness = After (HRC ± 2.0)
1	54	None	54
2	54	480°= C for 05 minutes	52
3	54	480°= C for 10 minutes	50
4	54	480°= C for 15 minutes	47
5	54	480°= C for 30 minutes	46
6	54	480°= C for 50 minutes	44
7	54	480°= C for 75 minutes	42
8	54	480°= C for 95 minutes	40

Step	Change	Controling Mechanism=
1<= /p>	Retained austenite transforms to martensite..	Diffusionless transformation.
2<= /p>	Carbon atoms redistr= ibute to (a) lattice defects and (b) carbon clusters.	Volume diffusion of = carbon atoms.
3<= /p>	Precipitation of = 49; and η transition carbides.	Diffusion of iron at= oms along dislocations.
4<= /p>	Precipitation of Hag= gs carbide.	-<= /p>
5<= /p>	Decomposition of rem= aining retained austenite to ferrite/cementite.	Volume diffusion of = carbon atoms in austenite
6<= /p>	Conversion of the segregated carbon and the transition carbides to cementite	Volume and pipe diff= usion of iron atoms.

Image Description	Known<= o:p>	Calcul= ated D*	Materi= als Pro	% Diff= erence
Black cir= cle	0.000	&nbs= p;	0.00	0
Textbook<= /p>	0.17	&nbs= p;	0.18	+6
Textbook<= /p>	0.23	&nbs= p;	0.22	-4
Sample Im= age 1a	&nbs= p;	0.19	0.21	+11
Sample Im= age 1b	&nbs= p;	0.18	0.19	+6
Sample Im= age 1c	&nbs= p;	0.20	0.20	0
Koch Tria= dic	0.262	&nbs= p;	0.25	-5
Koch Quad= ratic	0.500	&nbs= p;	0.48	-4

&nbs= p;	D* for In= dividual Images
Sample #<= /p>	Image a	Image b	Image c	Image d	Image e	Image f	Mean D*
1	0.20918	0.18508	0.20299	&nbs= p;	&nbs= p;	&nbs= p;	0.20
2	0.21019	0.21092	0.18999	&nbs= p;	&nbs= p;	&nbs= p;	0.20
3	0.17086	0.16807	0.22111	&nbs= p;	&nbs= p;	&nbs= p;	0.19

4	0.31824	0.28206	0.24483	&nbs= p;	&nbs= p;	&nbs= p;	0.28
5	0.29884	0.27498	0.21906	&nbs= p;	&nbs= p;	&nbs= p;	0.26
6	0.26536	0.29401	0.32258	&nbs= p;	&nbs= p;	&nbs= p;	0.29

7	0.24242	0.22037	0.30174	0.23678	&nbs= p;	&nbs= p;	0.25
8	0.27602	0.22414	0.22633	0.24019	0.21861	&nbs= p;	0.24
9	0.33127	0.23291	0.21328	0.24006	0.28716	0.23661	0.26

10	0.21310	0.19569	0.17148	&nbs= p;	&nbs= p;	&nbs= p;	0.19
11	0.27696	0.20276	0.17871	&nbs= p;	&nbs= p;	&nbs= p;	0.22
12	0.21841	0.21836	0.19293	&nbs= p;	&nbs= p;	&nbs= p;	0.21

13	0.20662	0.16948	0.15626	&nbs= p;	&nbs= p;	&nbs= p;	0.18
14	0.22060	0.21511	0.22641	&nbs= p;	&nbs= p;	&nbs= p;	0.22
15	0.21770	0.15615	0.18659	&nbs= p;	&nbs= p;	&nbs= p;	0.19

16	0.18952	0.12210	0.15969	&nbs= p;	&nbs= p;	&nbs= p;	0.16
17	0.12514	0.10469	0.13395	&nbs= p;	&nbs= p;	&nbs= p;	0.12
18	0.11616	0.14686	0.13636	&nbs= p;	&nbs= p;	&nbs= p;	0.13

19	0.13191	0.14458	0.07253	&nbs= p;	&nbs= p;	&nbs= p;	0.12
20	0.12714	0.13013	0.10179	&nbs= p;	&nbs= p;	&nbs= p;	0.12
21	0.12966	0.08658	0.07294	0.11991	0.04018	0.09440	0.09

22	0.15203	0.18989	0.18587	&nbs= p;	&nbs= p;	&nbs= p;	0.18
23	0.09862	0.09889	0.18028	&nbs= p;	&nbs= p;	&nbs= p;	0.13
24	0.13333	0.14525	0.19721	&nbs= p;	&nbs= p;	&nbs= p;	0.16