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Surface roughness of anisotropic fracture surfaces
Surface Roughnessof Anisotropic Fracture Surfaces A. Gokhale and W. J. Drury School of Materials Engineering, Georgia Institute of Technology, Atlanta, GA 30332 A new assumption-free method has recently been developed for estimating the surface roughness parameter (Rs) of fracture surfaces regardless of the relative extent of isotropy or anisotropy exhibited. An overview of the new method, as well as certain statistical aspects associated with sampling procedure, are presented. Computer-simulated sampling of idealized anisotropic surfaces is extensively used to investigate these issues.
planes oriented mutually at 120° about the global surface normal. The theoretical development of this method and certain associated mathematical considerations have been presented previously [1]. The goal of the current work is to provide an overview of the correct application of the new technique and to address some specihc questions related to specimen sampling:
INTRODUCTION
The increasing use of anisotropic materials such as composites in failure-critical applications emphasizes a need for quantitative failure analysis techniques capable of dealing with the directional (anisotropic) fracture surfaces that these materials produce. The well-known surface roughness parameter (Rs) is commonly used as a measure of fracture surface tortuosity and is generally applicable to all surfaces. Rs is usually determined through relationships between surface roughness and the mean roughness of fracture surface profiles revealed by metallographic sectioning. The measurement is relatively easy to perform on isotropic surfaces, where a single section yields a profile that is statistically representative of profiles at all sectioning orientations. However, the measurement of Rs for anisotropic surfaces presents difficulties because profiles at different section orientations have different appearances and roughnesses. A large number of sections at various orientations must be sampled for an accurate estimate of Rs in this case. Recently, a new method for measuring Rs accurately, regardless of surface anisotropy, has been developed [1, 2]. This method makes use of only three sectioning
• What is the maximum statistical bias? (Bias is the accepted nomenclature for that error inherent in a given technique itself. Bias cannot be eliminated or reduced by increased sampling, as opposed to sampling error, which can be reduced by increased sampling.) • How sensitive is the technique to misorientation of the sectioning planes (i.e., sectioning planes that are not oriented at precisely 120° to one another)? The surface roughness of a fractured fiber composite was measured to provide an example of the use of this technique.
BACKG ROU N D
The surface roughness parameter, Rs, is equal to the true average area of a fracture
Originally presented at the 1990 IMS Convention, Cincinnati, Ohio, USA. 279 © ElsevierSciencePublishingCo., Inc., 1993 655 Avenueof the Americas,New York,NY10010
MATERIALSCHARACTERIZATION30:279-286(1993) 1044-5803/93/$6.00
280
A. Gokhale and W. J. Drury Normal
to average
Z Fracture profile of length ~.o
Appaxem
'
projected
area 'A' 9
(a)
I.
(b)
FIG. 1 Definitionsof (a) fracture surfaceroughness parameter, Rs, and (b) profile roughness parameter, RL.
surface divided by its apparent projected area [Fig. l(a)]. This may be expressed as: S
Rs = X '
(1)
where S is the true surface area and A is the projected area on a plane parallel to the average topographic plane of the fracture surface. Surface roughness is dimensionless and can range from one (for a flat surface) to infinity. A range of 1.5-3.0 is common for actual fracture surfaces. Because the true surface area cannot be directly measured efficiently, Rs has previously been estimated most often through parametric relations between Rs and the tortuosity of surface profiles revealed by metallographic sectioning of the fractured specimen. Numerous such parametric equations have been proposed, each based on a slightly different set of assumptions [3-6]. In most cases, the general form of the relations can be expressed as: Rs -- M . RL + B,
equations may not be accurate for a specific surface. 2. In general, applicability is limited to isotropic surfaces. Therefore, the equations may not apply in the case of the anisotropic surfaces generated by the fracture of directionally reinforced or strengthened materials such as composites. 3. The form of Eq. (2) leads to the incorrect conclusion that surface roughness and profile roughness are linearly related with only the specific values of the relation constants (M and B) in question. This can lead to incorrect attempts to relate various mechanical properties to profile roughness on the grounds that changes in RL are directly proportional to changes in Rs. The input data for the general method are the profile roughness parameter, RL, and a structure factor, W, which is related to the angular orientation distribution of line segments on the fracture profile: Rs = RL. ~.
RL" ~/is the mean value of the product RL" ~/ in a set of vertical sectioning planes about a global surface normal. [It should be emphasized that Eq. (3) involves the mean of the product (RL. ~) rather than the product of the means (RL. ~).] A reliable estimate of this mean can be obtained from three such planes arranged symmetrically at 120° to one another as shown in Fig. 2.
1. They involve assumptions that are difficult to verify experimentally. Thus, the
Z
@pl + 240*
(2)
where M and B are constants related to a specific parametric expression, and RL is the profile roughness parameter. RL, which will be discussed later, can be easily measured from the profiles of the fracture surfaces. The primary drawbacks to the parametric equations of the type given by Eq. (2) are as follows:
(3)
SeCtioning ~bpl @pl + 120 °
L
,,_ _
s0o,o
Sectioning plane II
FIG. 2. Sectioning planes arranged symmetrically around the surfacenormal mutually at an angle of 120°.
Anisotropic Fracture Surfaces
281
, Z %'
[qbPt~÷~" ~ ~ ~ q b P t 120 °
FIG. 3. Sectioning planes that are arranged mutually at angle of 120 ° but do not have a c o m m o n intersection.
nifled image of the profile using an electronic cursor on a digitizing tablet. The instrument records a series of coordinate points of locations at regularly spaced intervals along the profile path, as shown in Fig. 4. Thus, the profile is represented by a series of line segments of a given length (ruler length, 11) and varying angular orientations. This set of information, essentially a digital map of the profile, is used to determine the values of the RL and k~ parameters.
PROFILE ROUGHNESS PARAMETER,RL The steps involved in estimating Rs through this technique are as follows: 1. Electroless or electrolytic metal plating of the fracture surface provides a protective layer against surface distortion during subsequent metallographic procedures. Plating also helps retain edge sharpness during polishing. 2. Metallographic sectioning is performed along three planes mutually at an angle of 120°. Note that, although the three sections are oriented symmetrically about the global surface normal, they need not be coincident, as shown in Fig. 3. The sections can be taken individually from any convenient location on the surface, provided that each resulting profile is representative of all profiles at that orientation. Next, the sections are polished using standard metallographic procedures for light microscopy. 3. Fracture profiles are digitized using either semiautomatic or automatic digital image analysis equipment. Regardless of the actual equipment used, the general procedure involves manually tracing the mag-
fracture
profile
The profile roughness is the ratio of the total profile length to its apparent projected length [Fig. l(b)]: RL -
)~o
L'
(4)
Total profile length, ~0, is equal to the sum of the lengths of all the line segments recorded for the digitized profile. Projected length is the total horizontal length spanned by the irregular profile. Both measures can be easily extracted from the digital data.
STRUCTURE FACTOR, ~p In addition to the total profile length, the coordinate data can be used to determine the angle (a) between each segment and the vertical axis (values of et range from 0 to 180°). The frequency distribution of these angles is the profile segment angular distribution function, f(a), such that:
Iof(a)d(ot) = 1.
(5)
The tt values are easily grouped into a histogram of K classes in the range 0-180 °, each class having a width or interval A. If one defines the height of the ith histogram bar as hi, it can be seen that hiA is the fraction of profile segments having orientation angles tt in the range (i - 1)A to iA: h/A
iA
I(i-1)a f(ot)d(a).
(6)
digitized coordinate points FIG. 4. Digitization of fracture profile.
Presented in histogram form, the structure factor is defined as:
A. Gokhale and W. J. Drury
282 K
= a • ~, aihi,
(7)
/=1
where a i = sin [ ( i - 1 ) A ]
semi-circular wave generating curve
+ [2-(i-1)A]
(8) Equation (8) represents a set of numerical factors associated with corresponding histogram terms hi. The actual values of the ai terms will depend on the number of classes in a given histogram.
COMPUTER CALCULATIONS
Because of its rotational symmetry, the RL, ~, and RL.~ values obtained from an isotropic surface are statistically identical regardless of the orientation of the vertical sectioning plane. Thus, a single representative sectioning plane provides sufficient information for an accurate estimation of Rs for an isotropic surface. In the case of an anisotropic surface, RL, ~, and RL. ~ are dependent on the orientation of the vertical plane; different values will be produced from sections at different orientations. Many plane orientations are required to characterize this type of surface. Because of the considerable metallographic effort involved, such sampling is not practical for real materials, and a method requiring as few sections as possible is desired. The extent of a surface's anisotropy dictates the extent of orientation dependence of RL" ~. The most extreme anisotropy and, therefore, the strongest orientation dependence of RL. ~ occurs with a surface generated by translating an arbitrary planar curve in the direction perpendicular to the plane of the curve (Fig. 5). Such "ruled" surfaces have the following properties: 1. RL and ~u both equal 1.0 on the vertical section perpendicular to the planar curve. 2. RL and ~ have maximum values on the vertical section parallel to the planar curve
FIG. 5. Example of a semicircular r u l e d surface.
(i.e., in which the profile is the generating planar curve itself). 3. At intermediate orientations, RL and take on intermediate values. The maximum bias associated with the use of three symmetrical vertical sectioning planes will arise on this type of surface. Therefore, to determine the maximum bias possible when estimating Rs using the technique described previously, sampling of a ruled surface generated by a semicircular wave (Fig. 5) was simulated with a computer. For the purposes of the simulation, it was assumed that the surface was inhnite in extent to eliminate edge effects. The true Rs for this surface is n/2. The simulation proceeded as follows: 1. The first vertical sectioning plane orientation, q)pl, was chosen in the range 0360 ° about the global surface normal. 2. The orientations of the second and third planes were fixed by the first: 120 °, q0p3 = (l)p1 + 240 ° . ~0p2 = ~0pI q-
3. Values of RL, ~, and the corresponding RL.~ for each plane were calculated. 4. RL. ~ for the three symmetric planes was calculated. This procedure was repeated for each possible orientation of the initial plane to generate a frequency distribution of RL. ~ values, which is presented in Fig. 6. The key features of this distribution are as follows: 1. The expected value of RL. ~ is equal to the known Rs; ~/2. 2. All values of RL.~ fall in the narrow range 1.545-1.595, that is, 1.57 + 0.025. Ruled surfaces generated by a sine wave and
Anisotropic Fracture Surfaces
o
283
THEORETICAL EXPECTED (R ~ ) = m2
~o 4
AVERAGE (R ~t'~jS POPULATION = 1570
sine wave generating curve i
o
!'O
12
1'4
16
1'8
2L.O
CALCULATED SAMPLE AVERAGE
FIG. 6. Frequency of RL" ~ values for systematic sampling by three vertical sections mutually at 120°.
by a rectangular wave (Fig. 7) were also examined. Surfaces with a roughness greater than 5.0 were not examined. The m a x i m u m bias encountered was for a rectangular wave ruled surface with Rs = 5.0. In this case, it was observed that 95% of RL. ~ values were within +6% of the correct value. This particular surface represents an extreme; real fracture surfaces are not expected to have roughnesses greater t h a n 5.0 and should not exhibit a degree of anisotropy worse than a rectangular ruled surface. Thus, in the three extremely anisotropic surfaces examined, three vertical sections mutually at 120 ° provided RL. tV values with a bias of less t h a n +6% at 95% confidence. This sampling technique is concluded to be sufficient for any arbitrary fracture surface that m a y be viewed as a superimposition of different ruled and isotropic surfaces. In practice, it may be difficult to ensure that the sampling planes are oriented precisely at 120 ° to one another. Thus, it is of interest to k n o w w h a t effect such misorientation has on the distribution of RL. values. This was examined for the rectangular ruled surface by varying slightly the orientation of the second and third sectioning planes about their nominal positions. For the sampling plane orientations (q)pl, (~pl + 115 o, q)pl + 245o), it was observed that the expected value of RL. u/was still equal to the correct Rs with 95% of the values lying between 4.56 and 5.34, representing a potential bias of 9%.
rectangular wave generating curve (2m = 0.5)
FIG. 7. Examples of ruled surfaces: (a) ruled surface having sine wave generating curve, and (b) ruled surface having rectangular generating curve.
In general, it was observed that the bias was more sensitive to extreme misorientation of a single plane than to equivalent misorientation shared between two planes. A 6 ° misorientation of a single plane yields greater potential bias than two planes misoriented at +3 °. Thus, it is important to ensure during sectioning that no single plane is highly misoriented. It is clear from these results that care should be taken to produce sections that are oriented as accurately as possible.
PRACTICAL EXAMPLE
A composite material consisting of continuous unidirectional alumina fibers in an A1-Li matrix was fractured in uniaxial tension applied transverse to the fiber axis. The anisotropic fracture surface is presented in the scanning electron microscope fractograph in Fig. 8. The surface roughness of this specimen was measured using the procedure outlined previously. Figure 9(a-c) presents the RL values and the fla) histograms determined for each of the three vertical sections sampled. The structure factors were calculated from the fiR) histograms and
A. Gokhale and W. J. Drury
284
Table 1 Calculated Values of ai Coefficients
Fic. 8. Scanning electron micrograph of fracture surface. the ai coefficients in Table 1. The results are included in Fig. 9. From these data, the estimated surface roughness of the composite is: Rs = 1/3[(2.16)(1.27) + (1.61)(1.20) + (1.54)(1.19)] = 2.17.
(9)
DISCUSSION
Most stereological and fractographic measurements require data taken by sampling from metallographic sectioning planes. The orientation of these planes is often an important consideration in determining the accuracy or generality of the results. In a case where so-called random sectioning is required, the resulting randomly selected plane orientations could prove awkward in the actual performance of the sectioning procedure. Such is not the case with the method discussed here. Aside from the simple requirement that the sections are vertical, the orientation of the trio of planes can be speciflcaUy selected to be as convenient as possible, provided that the angular relationship between the planes is satished. In the case of the example given, the initial section was chosen to be the plane perpendicular to the fiber axis. The subsequent cuts were then made at orientations +120 ° from this first plane. Had the initial plane been chosen to be parallel to the fiber axis, the subsequent planes would have been oriented relative to this plane, and the expected value of the Rs
i
ai
i
1 2 3 4 5 6 7 8 9
1.565 1.5232 1.4508 1.3599 1.2625 1.1694 1.0906 1.0336 1.0037
18 17 16 15 14 13 12 11 10
estimate would be unchanged. Producing sections at 120 ° does not present a considerable obstacle, a sectioning jig with slots at the correct orientations is easily fabricated. In certain situations, it may be desired to estimate Rs with an accuracy greater than that provided by the three section technique discussed here. The expression presented in Eq. (3) is general for all orders of symmetrical vertical sectioning. For example, bias can be considerably reduced by using five symmetrical sections (mutually at 72° to one another). RL'~ is still utilized to estimate Rs in this case, with the average now being taken over five planes instead of three. Naturally, the sampling effort increases dramatically as the number of sectioning planes increases. It is well known that RL and f(~) for a given profile are sensitive to the ruler length at which the profile was digitized. RL tends to increase as ruler length decreases because of the interaction of the line segments with smaller and smaller features along the profile path. The Rs estimate provided by RL. ~ is therefore valid only for a specified ruler length. However, recent work has shown that as the ruler length becomes very small, the profile roughness reaches an upper limit and does not continue increasing [7]. Thus, a limiting "true" value of Rs can be achieved at small ruler lengths. Both RL and ~ are measures of profile characteristics that are independent of one another. Therefore, whether or not a surface is isotropic or anisotropic, Rs is the product of two independent variables and cannot be
Anisotropic Fracture Surfaces = 0
285
degrees
= 120 d e g r e e s
R L = 2.16
R L = 1.61
= 1.27
'-£ = 1.20
O"
c~
o
20#m
O
f(~)
....
c5
~5
f(a)
o
c~ c5
o~"
o.
0
30
60
a
90
120
150
180
(degrees) (a)
30
60
90
120
150
180
a (degrees) (b)
degrees
= 240
R L = 1.54 '£ = 1.19 <3
g a
f((z)
a
a el oo 0
0
30
60
a
90
120
150
1BO
(degrees) (c)
linearly related to either in the m a n n e r implied by Eq. (3). This result makes it clear that attempts to relate mechanical properties to surface m o r p h o l o g y as characterized by profile r o u g h n e s s are ill f o u n d e d .
CONCLUSIONS
Certain practical considerations of the surface r o u g h n e s s estimation of anisotropic surfaces using a new assumption-free m e t h o d have b e e n discussed. It has b e e n d e m o n strated that the m a x i m u m sampling bias associated with this technique is less t h a n 6% with a 95% c o n h d e n c e . In addition, the sensitivity of the technique to misorientation of the three symmetrical sectioning planes
FIG. 9. (a) RL value and f(a) histogram for ~ = 0O. (b) RL value and f(a) histogram for ~0 = 120°. (c) RL value and f(a) histogram for ~0 = 240°.
was investigated. As a result, it is e m p h a sized that care should be taken w h e n sectioning to p r o d u c e planes as accurately oriented as possible. Planes having misorientation of +5 ° increase potential bias to 9% with a 95% confidence.
The authors gratefully acknowledge the financial support provided by the National Science Foundation through grant number DMR-9013098 for the development of stereology and quantitative fractography.
References 1. A. Gokhaleand E. E. Underwood, A general method for the estimation of fracture surface roughness; Part
286
1: Theoretical aspects, Met. Trans. 21A:1193-1199 (1990). 2. A. Gokhale and W. J. Drury, A general method for the estimation of fracture surface roughness; Part II: Practical considerations, Met. Trans. 21A:1201-1207 (1990). 3. K. Wright and B. Karlssen, Topography of non-planar surfaces, Proc. 3rd European Symposium for Stereology, M. Kalisnik, ed., Ljubljana, Yugoslavia, pp. 247250 (1981). 4. M. Coster and J. L. Chermant, Recent developments in quantitative fractography, Int. Met. Rev. 28:228-250 (1983).
A. Gokhale and W. J. Drury 5. K. Wright and B. Karlssen, Topographic quantification of non-planar localized surfaces, ]. Microscopy 130:37-51 (1983). 6. E. E. Underwood and K. Banerji, Statistical Analysis of Facet Characteristics in a Computer Generated Fracture Surface, Proc. 6th Int. Cong. for Stereology, M. Kalisnik, ed., Gainesville, FL, Acta Stereologica 2(Suppl. 1):75-80 (1983). 7. E. E. Underwood and K. Banerji, in ASM Metals Handbook, 9th ed., vol. 12, Metals Park, OH, pp. 211-215 (1987).
Received July 1992; accepted January 1993.
planes oriented mutually at 120° about the global surface normal. The theoretical development of this method and certain associated mathematical considerations have been presented previously [1]. The goal of the current work is to provide an overview of the correct application of the new technique and to address some specihc questions related to specimen sampling:
INTRODUCTION
The increasing use of anisotropic materials such as composites in failure-critical applications emphasizes a need for quantitative failure analysis techniques capable of dealing with the directional (anisotropic) fracture surfaces that these materials produce. The well-known surface roughness parameter (Rs) is commonly used as a measure of fracture surface tortuosity and is generally applicable to all surfaces. Rs is usually determined through relationships between surface roughness and the mean roughness of fracture surface profiles revealed by metallographic sectioning. The measurement is relatively easy to perform on isotropic surfaces, where a single section yields a profile that is statistically representative of profiles at all sectioning orientations. However, the measurement of Rs for anisotropic surfaces presents difficulties because profiles at different section orientations have different appearances and roughnesses. A large number of sections at various orientations must be sampled for an accurate estimate of Rs in this case. Recently, a new method for measuring Rs accurately, regardless of surface anisotropy, has been developed [1, 2]. This method makes use of only three sectioning
• What is the maximum statistical bias? (Bias is the accepted nomenclature for that error inherent in a given technique itself. Bias cannot be eliminated or reduced by increased sampling, as opposed to sampling error, which can be reduced by increased sampling.) • How sensitive is the technique to misorientation of the sectioning planes (i.e., sectioning planes that are not oriented at precisely 120° to one another)? The surface roughness of a fractured fiber composite was measured to provide an example of the use of this technique.
BACKG ROU N D
The surface roughness parameter, Rs, is equal to the true average area of a fracture
Originally presented at the 1990 IMS Convention, Cincinnati, Ohio, USA. 279 © ElsevierSciencePublishingCo., Inc., 1993 655 Avenueof the Americas,New York,NY10010
MATERIALSCHARACTERIZATION30:279-286(1993) 1044-5803/93/$6.00
280
A. Gokhale and W. J. Drury Normal
to average
Z Fracture profile of length ~.o
Appaxem
'
projected
area 'A' 9
(a)
I.
(b)
FIG. 1 Definitionsof (a) fracture surfaceroughness parameter, Rs, and (b) profile roughness parameter, RL.
surface divided by its apparent projected area [Fig. l(a)]. This may be expressed as: S
Rs = X '
(1)
where S is the true surface area and A is the projected area on a plane parallel to the average topographic plane of the fracture surface. Surface roughness is dimensionless and can range from one (for a flat surface) to infinity. A range of 1.5-3.0 is common for actual fracture surfaces. Because the true surface area cannot be directly measured efficiently, Rs has previously been estimated most often through parametric relations between Rs and the tortuosity of surface profiles revealed by metallographic sectioning of the fractured specimen. Numerous such parametric equations have been proposed, each based on a slightly different set of assumptions [3-6]. In most cases, the general form of the relations can be expressed as: Rs -- M . RL + B,
equations may not be accurate for a specific surface. 2. In general, applicability is limited to isotropic surfaces. Therefore, the equations may not apply in the case of the anisotropic surfaces generated by the fracture of directionally reinforced or strengthened materials such as composites. 3. The form of Eq. (2) leads to the incorrect conclusion that surface roughness and profile roughness are linearly related with only the specific values of the relation constants (M and B) in question. This can lead to incorrect attempts to relate various mechanical properties to profile roughness on the grounds that changes in RL are directly proportional to changes in Rs. The input data for the general method are the profile roughness parameter, RL, and a structure factor, W, which is related to the angular orientation distribution of line segments on the fracture profile: Rs = RL. ~.
RL" ~/is the mean value of the product RL" ~/ in a set of vertical sectioning planes about a global surface normal. [It should be emphasized that Eq. (3) involves the mean of the product (RL. ~) rather than the product of the means (RL. ~).] A reliable estimate of this mean can be obtained from three such planes arranged symmetrically at 120° to one another as shown in Fig. 2.
1. They involve assumptions that are difficult to verify experimentally. Thus, the
Z
@pl + 240*
(2)
where M and B are constants related to a specific parametric expression, and RL is the profile roughness parameter. RL, which will be discussed later, can be easily measured from the profiles of the fracture surfaces. The primary drawbacks to the parametric equations of the type given by Eq. (2) are as follows:
(3)
SeCtioning ~bpl @pl + 120 °
L
,,_ _
s0o,o
Sectioning plane II
FIG. 2. Sectioning planes arranged symmetrically around the surfacenormal mutually at an angle of 120°.
Anisotropic Fracture Surfaces
281
, Z %'
[qbPt~÷~" ~ ~ ~ q b P t 120 °
FIG. 3. Sectioning planes that are arranged mutually at angle of 120 ° but do not have a c o m m o n intersection.
nifled image of the profile using an electronic cursor on a digitizing tablet. The instrument records a series of coordinate points of locations at regularly spaced intervals along the profile path, as shown in Fig. 4. Thus, the profile is represented by a series of line segments of a given length (ruler length, 11) and varying angular orientations. This set of information, essentially a digital map of the profile, is used to determine the values of the RL and k~ parameters.
PROFILE ROUGHNESS PARAMETER,RL The steps involved in estimating Rs through this technique are as follows: 1. Electroless or electrolytic metal plating of the fracture surface provides a protective layer against surface distortion during subsequent metallographic procedures. Plating also helps retain edge sharpness during polishing. 2. Metallographic sectioning is performed along three planes mutually at an angle of 120°. Note that, although the three sections are oriented symmetrically about the global surface normal, they need not be coincident, as shown in Fig. 3. The sections can be taken individually from any convenient location on the surface, provided that each resulting profile is representative of all profiles at that orientation. Next, the sections are polished using standard metallographic procedures for light microscopy. 3. Fracture profiles are digitized using either semiautomatic or automatic digital image analysis equipment. Regardless of the actual equipment used, the general procedure involves manually tracing the mag-
fracture
profile
The profile roughness is the ratio of the total profile length to its apparent projected length [Fig. l(b)]: RL -
)~o
L'
(4)
Total profile length, ~0, is equal to the sum of the lengths of all the line segments recorded for the digitized profile. Projected length is the total horizontal length spanned by the irregular profile. Both measures can be easily extracted from the digital data.
STRUCTURE FACTOR, ~p In addition to the total profile length, the coordinate data can be used to determine the angle (a) between each segment and the vertical axis (values of et range from 0 to 180°). The frequency distribution of these angles is the profile segment angular distribution function, f(a), such that:
Iof(a)d(ot) = 1.
(5)
The tt values are easily grouped into a histogram of K classes in the range 0-180 °, each class having a width or interval A. If one defines the height of the ith histogram bar as hi, it can be seen that hiA is the fraction of profile segments having orientation angles tt in the range (i - 1)A to iA: h/A
iA
I(i-1)a f(ot)d(a).
(6)
digitized coordinate points FIG. 4. Digitization of fracture profile.
Presented in histogram form, the structure factor is defined as:
A. Gokhale and W. J. Drury
282 K
= a • ~, aihi,
(7)
/=1
where a i = sin [ ( i - 1 ) A ]
semi-circular wave generating curve
+ [2-(i-1)A]
(8) Equation (8) represents a set of numerical factors associated with corresponding histogram terms hi. The actual values of the ai terms will depend on the number of classes in a given histogram.
COMPUTER CALCULATIONS
Because of its rotational symmetry, the RL, ~, and RL.~ values obtained from an isotropic surface are statistically identical regardless of the orientation of the vertical sectioning plane. Thus, a single representative sectioning plane provides sufficient information for an accurate estimation of Rs for an isotropic surface. In the case of an anisotropic surface, RL, ~, and RL. ~ are dependent on the orientation of the vertical plane; different values will be produced from sections at different orientations. Many plane orientations are required to characterize this type of surface. Because of the considerable metallographic effort involved, such sampling is not practical for real materials, and a method requiring as few sections as possible is desired. The extent of a surface's anisotropy dictates the extent of orientation dependence of RL" ~. The most extreme anisotropy and, therefore, the strongest orientation dependence of RL. ~ occurs with a surface generated by translating an arbitrary planar curve in the direction perpendicular to the plane of the curve (Fig. 5). Such "ruled" surfaces have the following properties: 1. RL and ~u both equal 1.0 on the vertical section perpendicular to the planar curve. 2. RL and ~ have maximum values on the vertical section parallel to the planar curve
FIG. 5. Example of a semicircular r u l e d surface.
(i.e., in which the profile is the generating planar curve itself). 3. At intermediate orientations, RL and take on intermediate values. The maximum bias associated with the use of three symmetrical vertical sectioning planes will arise on this type of surface. Therefore, to determine the maximum bias possible when estimating Rs using the technique described previously, sampling of a ruled surface generated by a semicircular wave (Fig. 5) was simulated with a computer. For the purposes of the simulation, it was assumed that the surface was inhnite in extent to eliminate edge effects. The true Rs for this surface is n/2. The simulation proceeded as follows: 1. The first vertical sectioning plane orientation, q)pl, was chosen in the range 0360 ° about the global surface normal. 2. The orientations of the second and third planes were fixed by the first: 120 °, q0p3 = (l)p1 + 240 ° . ~0p2 = ~0pI q-
3. Values of RL, ~, and the corresponding RL.~ for each plane were calculated. 4. RL. ~ for the three symmetric planes was calculated. This procedure was repeated for each possible orientation of the initial plane to generate a frequency distribution of RL. ~ values, which is presented in Fig. 6. The key features of this distribution are as follows: 1. The expected value of RL. ~ is equal to the known Rs; ~/2. 2. All values of RL.~ fall in the narrow range 1.545-1.595, that is, 1.57 + 0.025. Ruled surfaces generated by a sine wave and
Anisotropic Fracture Surfaces
o
283
THEORETICAL EXPECTED (R ~ ) = m2
~o 4
AVERAGE (R ~t'~jS POPULATION = 1570
sine wave generating curve i
o
!'O
12
1'4
16
1'8
2L.O
CALCULATED SAMPLE AVERAGE
FIG. 6. Frequency of RL" ~ values for systematic sampling by three vertical sections mutually at 120°.
by a rectangular wave (Fig. 7) were also examined. Surfaces with a roughness greater than 5.0 were not examined. The m a x i m u m bias encountered was for a rectangular wave ruled surface with Rs = 5.0. In this case, it was observed that 95% of RL. ~ values were within +6% of the correct value. This particular surface represents an extreme; real fracture surfaces are not expected to have roughnesses greater t h a n 5.0 and should not exhibit a degree of anisotropy worse than a rectangular ruled surface. Thus, in the three extremely anisotropic surfaces examined, three vertical sections mutually at 120 ° provided RL. tV values with a bias of less t h a n +6% at 95% confidence. This sampling technique is concluded to be sufficient for any arbitrary fracture surface that m a y be viewed as a superimposition of different ruled and isotropic surfaces. In practice, it may be difficult to ensure that the sampling planes are oriented precisely at 120 ° to one another. Thus, it is of interest to k n o w w h a t effect such misorientation has on the distribution of RL. values. This was examined for the rectangular ruled surface by varying slightly the orientation of the second and third sectioning planes about their nominal positions. For the sampling plane orientations (q)pl, (~pl + 115 o, q)pl + 245o), it was observed that the expected value of RL. u/was still equal to the correct Rs with 95% of the values lying between 4.56 and 5.34, representing a potential bias of 9%.
rectangular wave generating curve (2m = 0.5)
FIG. 7. Examples of ruled surfaces: (a) ruled surface having sine wave generating curve, and (b) ruled surface having rectangular generating curve.
In general, it was observed that the bias was more sensitive to extreme misorientation of a single plane than to equivalent misorientation shared between two planes. A 6 ° misorientation of a single plane yields greater potential bias than two planes misoriented at +3 °. Thus, it is important to ensure during sectioning that no single plane is highly misoriented. It is clear from these results that care should be taken to produce sections that are oriented as accurately as possible.
PRACTICAL EXAMPLE
A composite material consisting of continuous unidirectional alumina fibers in an A1-Li matrix was fractured in uniaxial tension applied transverse to the fiber axis. The anisotropic fracture surface is presented in the scanning electron microscope fractograph in Fig. 8. The surface roughness of this specimen was measured using the procedure outlined previously. Figure 9(a-c) presents the RL values and the fla) histograms determined for each of the three vertical sections sampled. The structure factors were calculated from the fiR) histograms and
A. Gokhale and W. J. Drury
284
Table 1 Calculated Values of ai Coefficients
Fic. 8. Scanning electron micrograph of fracture surface. the ai coefficients in Table 1. The results are included in Fig. 9. From these data, the estimated surface roughness of the composite is: Rs = 1/3[(2.16)(1.27) + (1.61)(1.20) + (1.54)(1.19)] = 2.17.
(9)
DISCUSSION
Most stereological and fractographic measurements require data taken by sampling from metallographic sectioning planes. The orientation of these planes is often an important consideration in determining the accuracy or generality of the results. In a case where so-called random sectioning is required, the resulting randomly selected plane orientations could prove awkward in the actual performance of the sectioning procedure. Such is not the case with the method discussed here. Aside from the simple requirement that the sections are vertical, the orientation of the trio of planes can be speciflcaUy selected to be as convenient as possible, provided that the angular relationship between the planes is satished. In the case of the example given, the initial section was chosen to be the plane perpendicular to the fiber axis. The subsequent cuts were then made at orientations +120 ° from this first plane. Had the initial plane been chosen to be parallel to the fiber axis, the subsequent planes would have been oriented relative to this plane, and the expected value of the Rs
i
ai
i
1 2 3 4 5 6 7 8 9
1.565 1.5232 1.4508 1.3599 1.2625 1.1694 1.0906 1.0336 1.0037
18 17 16 15 14 13 12 11 10
estimate would be unchanged. Producing sections at 120 ° does not present a considerable obstacle, a sectioning jig with slots at the correct orientations is easily fabricated. In certain situations, it may be desired to estimate Rs with an accuracy greater than that provided by the three section technique discussed here. The expression presented in Eq. (3) is general for all orders of symmetrical vertical sectioning. For example, bias can be considerably reduced by using five symmetrical sections (mutually at 72° to one another). RL'~ is still utilized to estimate Rs in this case, with the average now being taken over five planes instead of three. Naturally, the sampling effort increases dramatically as the number of sectioning planes increases. It is well known that RL and f(~) for a given profile are sensitive to the ruler length at which the profile was digitized. RL tends to increase as ruler length decreases because of the interaction of the line segments with smaller and smaller features along the profile path. The Rs estimate provided by RL. ~ is therefore valid only for a specified ruler length. However, recent work has shown that as the ruler length becomes very small, the profile roughness reaches an upper limit and does not continue increasing [7]. Thus, a limiting "true" value of Rs can be achieved at small ruler lengths. Both RL and ~ are measures of profile characteristics that are independent of one another. Therefore, whether or not a surface is isotropic or anisotropic, Rs is the product of two independent variables and cannot be
Anisotropic Fracture Surfaces = 0
285
degrees
= 120 d e g r e e s
R L = 2.16
R L = 1.61
= 1.27
'-£ = 1.20
O"
c~
o
20#m
O
f(~)
....
c5
~5
f(a)
o
c~ c5
o~"
o.
0
30
60
a
90
120
150
180
(degrees) (a)
30
60
90
120
150
180
a (degrees) (b)
degrees
= 240
R L = 1.54 '£ = 1.19 <3
g a
f((z)
a
a el oo 0
0
30
60
a
90
120
150
1BO
(degrees) (c)
linearly related to either in the m a n n e r implied by Eq. (3). This result makes it clear that attempts to relate mechanical properties to surface m o r p h o l o g y as characterized by profile r o u g h n e s s are ill f o u n d e d .
CONCLUSIONS
Certain practical considerations of the surface r o u g h n e s s estimation of anisotropic surfaces using a new assumption-free m e t h o d have b e e n discussed. It has b e e n d e m o n strated that the m a x i m u m sampling bias associated with this technique is less t h a n 6% with a 95% c o n h d e n c e . In addition, the sensitivity of the technique to misorientation of the three symmetrical sectioning planes
FIG. 9. (a) RL value and f(a) histogram for ~ = 0O. (b) RL value and f(a) histogram for ~0 = 120°. (c) RL value and f(a) histogram for ~0 = 240°.
was investigated. As a result, it is e m p h a sized that care should be taken w h e n sectioning to p r o d u c e planes as accurately oriented as possible. Planes having misorientation of +5 ° increase potential bias to 9% with a 95% confidence.
The authors gratefully acknowledge the financial support provided by the National Science Foundation through grant number DMR-9013098 for the development of stereology and quantitative fractography.
References 1. A. Gokhaleand E. E. Underwood, A general method for the estimation of fracture surface roughness; Part
286
1: Theoretical aspects, Met. Trans. 21A:1193-1199 (1990). 2. A. Gokhale and W. J. Drury, A general method for the estimation of fracture surface roughness; Part II: Practical considerations, Met. Trans. 21A:1201-1207 (1990). 3. K. Wright and B. Karlssen, Topography of non-planar surfaces, Proc. 3rd European Symposium for Stereology, M. Kalisnik, ed., Ljubljana, Yugoslavia, pp. 247250 (1981). 4. M. Coster and J. L. Chermant, Recent developments in quantitative fractography, Int. Met. Rev. 28:228-250 (1983).
A. Gokhale and W. J. Drury 5. K. Wright and B. Karlssen, Topographic quantification of non-planar localized surfaces, ]. Microscopy 130:37-51 (1983). 6. E. E. Underwood and K. Banerji, Statistical Analysis of Facet Characteristics in a Computer Generated Fracture Surface, Proc. 6th Int. Cong. for Stereology, M. Kalisnik, ed., Gainesville, FL, Acta Stereologica 2(Suppl. 1):75-80 (1983). 7. E. E. Underwood and K. Banerji, in ASM Metals Handbook, 9th ed., vol. 12, Metals Park, OH, pp. 211-215 (1987).
Received July 1992; accepted January 1993.