Analysis of deformation-induced surface morphologies in steel sheet

M.R. Stoudt *, J.B. Hubbard

Materials Science and Engineering Laboratory, National Institute of Standards and Technology, 100 Bureau Drive, Stop 8553,

Gaithersburg, MD 20899-8553, United States

Received 4 February 2005; received in revised form 12 May 2005; accepted 23 May 2005

Abstract

The surfaces of strained mild steel sheet specimens were examined with scanning laser confocal microscopy. Rigorous assessments were employed to establish the form of the roughness data. The small deviations from the ideal conditions observed in these

assessments indicated that the stochastic process responsible for surface roughening does not conform to stationary, Gaussian,

Markov statistics. As physical considerations make violations of the stationary and Markov properties unlikely, it was concluded

that even small discrepancies in the Gaussian fit have substantial influences on the quality of roughening behavior characterizations.

This also confirms that deformation-induced surface roughness occurs by a small number of statistically dominant mechanisms.

Analyses of the mean amplitude (Rq) and spatial distributions both exhibited sharp changes in behavior at approximately 4% strain.

Since the physical meanings of the two analyses are independent, it was concluded that this common behavior correlated with a

change in dominant roughening mechanism.

Published by Elsevier Ltd on behalf of Acta Materialia Inc.

Keywords: Plastic deformation; Analytical methods; Ferritic steel; Microscopy

1. Introduction

Inhomogeneous surface deformation (e.g., orangepeel, banding, or roping) is a significant obstacle impeding the widespread use of the many alloys intended to

increase automobile fuel economy. In addition to being

cosmetically unacceptable, these surface markings have

a profound detrimental affect on the formability: They

localize strain, promoting component failure from tearing or wrinkling, and they alter the friction between the

metal sheet and the die surfaces during metal forming

[1,2]. As the surface roughening behavior essentially

establishes the suitability of an individual alloy for a

particular application, numeric predictions of the evolution of surface roughness have become more central in

the automotive development process. At present, there

are significant discrepancies between the surface

morphology predicted by numeric methods and what is

actually measured on the surface of real materials. These

discrepancies suggest that the available surface roughness

data are not adequate for the modeling requirements and

that a better understanding of the measurement issues

associated with the roughening process is necessary to

increase the accuracy of the formability predictions.

Reports in the literature indicate that the roughening

behavior of a free surface follows a relationship where

the surface roughness is proportional to the amount of

plastic strain [3–7]. There is also a consensus that this

relationship holds regardless of the influence the

material crystal structure may have on the deformation

[8,9]. In contrast, plastic deformation in a polycrystalline alloy is an extremely complex process [10–12]. The

roughness that occurs at a free surface strongly depends

upon both the grain size [13–15] and orientation effects

in the sheet [16–18], as well as upon the rate at which

the deformation occurs [19,20]. These inconsistencies

raise many questions about the validity of this simple

1359-6454/$30.00 Published by Elsevier Ltd on behalf of Acta Materialia Inc.

doi:10.1016/j.actamat.2005.05.038

* Corresponding author. Tel.: +1 301 975 6025; fax: +1 301 975 4553.

E-mail address: stoudt@nist.gov (M.R. Stoudt).

Acta Materialia 53 (2005) 4293–4304

www.actamat-journals.com
empirical relationship, and any prediction of the overall

roughening behavior upon which it is based.

The literature clearly establishes that the accuracy of

any roughness evaluation is dependent on the technique

used for data acquisition and that all surface measurement methods introduce errors of some form into the

data [5,21]. Radhakrishnan [22] showed how an improper ratio between the size of the measurement probe and

the scale of the measured features would introduce

substantial errors by masking or filtering the subtleties

in the roughness profile. The characteristics of these

errors are well known and appropriate measures for

their correction have been established [5].

The methods used to interpret roughness data are an

error source of greater significance. Most assessments of

roughening behavior in the literature are derived from

linear profiles (i.e., individual measurements of the

surface taken in one dimension). In such an assessment,

the profile is postulated to be a characteristic of the

entire surface [23]. Most of the literature accounts

express the roughness of the entire surface through an

estimation of the mean of the amplitude distribution.

The most common method is the arithmetic mean, or

the Ra parameter [21]:

Ra 1⁄4 1

L

Z L

0

jzðxÞj dx. ð1Þ

In this equation, L is the length over which the profile is

evaluated and z(x) is the magnitude of the profile height

at any point along the evaluation length. The mean

amplitude distribution can also be expressed through

another parameter, Rq, which is based on the root mean

squared (rms). The rms roughness is expressed as:

Rq 1⁄4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

L

Z L

0

z2ðxÞ dx

s

. ð2Þ

Both of these parameters are used routinely in industrial

applications to quantify the changes in surface morphology, and they are often used interchangeably even

though they have different physical meanings [24]. While

Ra tends to be slightly more common, it is not directly

based on any statistical treatment. Conversely, Rq is

defined as the standard deviation of all the heights

contained in the profile. For this reason, it is regarded

as a more functional representation of the mean of the

amplitude distribution.

As shown by Eqs. (1) and (2), both parameters

collapse an entire roughness profile into a single value.

This compresses complex surface information into an

expression that can be rather coarse with respect to

the length scale of the surface details involved [14].

Extrapolation of this compressed value to a representation of an entire surface could introduce substantial

error, as the results are highly dependent upon the statistical methods used for the analysis. Because these are

average values, is it also possible for distinctly different

surface morphologies to produce the same roughness

value. Fig. 1 is a recent observation of such behavior

[25]. Clearly, any prediction of the roughening behavior

that is based solely on a mean roughness parameter is

likely to be misleading. While the overall objective is a

robust prediction of the mechanical behavior for any

alloy under a given set of forming conditions, this

particular effort evaluates the changes in roughening

behavior with plastic strain for a mild steel sheet. It also

examines the underlying principles used to interpret a

typical surface roughness measurement.

2. Experimental procedure

Most of the surface roughness data reported in the

literature are derived from measurements performed

Fig. 1. Two heat treatments of an aluminum alloy exhibiting the same mean roughness values with distinctly different surface morphologies and

frictional behaviors.

4294 M.R. Stoudt, J.B. Hubbard / Acta Materialia 53 (2005) 4293–4304
on samples with polished gauge sections. While polished

surfaces tend to produce more consistent surface roughness measurements, the surfaces of the steel sheets used

in industry are not pristine. In fact, automotive manufacturers customarily specify a maximum acceptable

initial surface roughness for the metal sheet that is

dependent upon the specific application (e.g., Ra 6 0.5

lm for an outer hood panel). Therefore, all of the data

in this analysis were acquired from steel sheet in the asreceived condition to be more representative of actual

forming conditions.

2.1. Material

Cold-rolled AISI 1010 steel sheet with a nominal

1.0 mm thickness was selected for this analysis because

it is a generic alloy that demonstrates good formability,

and the deformation behavior has been well established

in the literature [26,27]. The 1.0 mm sheet thickness is

used routinely in automotive applications. The grain size

and rolling direction of the AISI 1010 steel were verified

through a series of microstructural examinations. Specimens were mounted and polished according to standard

metallographic procedure [28] for these studies. The

final step of mechanical polishing was 0.05 lm colloidal

silica after which the samples were immersed in solutions of 4-volume fraction picral and 2-volume fraction

nital to reveal the microstructure.

2.2. Generation of surface roughness

Flat sheet, tensile specimens were punched from sheet

stock with the tensile axis perpendicular to the rolling

direction of the sheet. The critical dimensions were

measured for all samples. Fiducial lines were lightly

engraved at the extremes of the gauge section of each

specimen with a silicon carbide scribe to facilitate a

more accurate assessment of the plastic strain. The

spacing between grid lines was determined with a

linear-encoded, measuring stage microscope that had a

resolution of ±500 nm. The specimens were pulled in

uniaxial tension to pre-determined levels of plastic strain

with a computer-controlled universal tensile machine

that continuously monitored the applied stress and the

total strain. The crosshead displacement rate was

1.0 mm/s for all of these experiments. When the desired

strain level was attained, the specimen was removed

from the grips, and the fiducial lines were re-measured

with the microscope to assess the actual amount of plastic strain in the gauge area of the specimen.

2.3. Surface roughness measurements

The changes in the surface topographies were quantified by examining the specimen surfaces in both the as

received and in the strained conditions with scanning

laser confocal microscopy (SLCM). The optics of the

SLCM used for this study were designed to optimize

the imaging conditions for opaque materials by

operating exclusively in a reflective imaging mode. More

information on SLCM imaging and the surface roughness measurement methodology can be found in

[29,30]. All of the SLCM images in this analysis were

created with a 635 nm laser source, a 10· objective lens

and a typical z-scan depth of approximately 20 lm.

These parameters generated a 640 · 512 pixel intensity image of the surface with both outstanding optical

depth of field and spatial resolution. The corresponding

(x,y, z) dimensions of the images were 1000 lm ·

800 lm · 20 lm, respectively, and the spacing between

the individual focal planes within each image was

approximately 100 nm. Topographic maps were generated from the intensity images with the controlling software. A series of linear roughness profiles with a typical

length of 750 lm were collected from the topographic

maps. Each roughness profile was corrected for flatness

and long wavelength effects resulting from specimen tilt

or other mechanical influences. This ‘‘leveling’’ was

accomplished with a routine in the controlling software

that first calculated the best multiple regression equation

for the plane from which the topographic image was acquired and then subtracted the individual points in the

profile from the regression equation. The controlling

software also interpolated the profile data by a factor

of five. This produced a roughness profile that contained

a minimum of 2400 data points and an approximate

sampling interval of 300 nm. A roughness measurement

consisted of five, well separated and randomly placed

profiles in the parallel orientation followed by five profiles in perpendicular orientation with respect to the rolling direction of the sheet.

3. Results and discussion

3.1. Microstructure

The as received microstructure of the 1010 steel sheet

was determined to consist of relatively equiaxed ferrite

grains, approximately 10 lm in diameter (Table 1), and

pearlite colonies. The volume fraction of pearlite (determined using ASTM E562 [31]) was found to be 0.039 ±

0.001. Other than the pearlite, the microstructure was

Table 1

Ferrite grain characteristics determined using ASTM E112 (24)

Viewing plane Mean grain

diameter (lm)

ASTM

grain size

Perpendicular to normal direction 9.9 ± 1.3 10.4

Perpendicular to rolling direction 8.4 ± 0.6 10.9

Perpendicular to transverse direction 8.6 ± 0.7 10.8

M.R. Stoudt, J.B. Hubbard / Acta Materialia 53 (2005) 4293–4304 4295
completely consistent with other observations reported

in the literature [32].

3.2. Analyses of height distributions

The roughness measurement data were acquired from

the SLCM topography images. Fig. 2 shows a typical

SLCM surface topography of the 1010 steel sheet in

the as-received condition. The plot directly beneath is

the surface profile of the trace shown in the upper figure.

As noted previously, the mean of the amplitude distribution of a roughness profile can be expressed through

either the Ra or the Rq value. The Rq parameter was selected to represent the mean of the amplitude distribution for this evaluation.

In general, the differences in behavior between the

parallel and perpendicular orientations to the rolling

direction of the sheet were relatively small. However,

the magnitudes of the changes in the amplitudes for

the perpendicular orientation were consistently larger

than those observed for the parallel orientation. Fig. 3

shows the changes in surface roughness of the as-received surfaces as a function of measured strain in both

the parallel and the perpendicular orientations to the

rolling direction of the sheet. Unlike a polished surface

where the mechanical action removes the deformed layer

produced by cold rolling and minimizes the variability in

surface hardness, the initial surface condition for this

study was the as-received condition. Accordingly, the

initial surface roughness was non-zero and all of the

roughness data include the localized inhomogeneities

in surface character that are produced by the deformed

surface layer [33]. For this reason, the roughness data

were corrected for the initial roughness by subtracting

the mean value of the initial roughness from all the subsequent roughness measurements and those corrected

values are denoted by the ‘‘D’’ preceding the Rq parameter. The presence of the initial roughness had no apparent influence on the overall trends after correction;

however, the magnitudes of the roughness data do not

directly represent the absolute changes in the surface

roughness. Note the error bars in Fig. 3 correspond to

the statistical uncertainty and not to the measurement

error in the roughness data. The measurement uncertainty was also corrected and the procedure used for

the propagation of error in these calculations is given

in [34].

The slope of the DRq versus strain relationship

shown in Fig. 3 defines the roughening rate for the

1010 steel under the aforementioned uniaxial loading

conditions (i.e., the rate of change in the relative heights

of the features on the surface as a function of plastic

strain). The change in roughening rate exhibited at 4%

strain in Fig. 3 is not consistent with the monotonic

roughening behavior reported in the literature.

However, it is important to note the following about

the literature accounts: They were typically acquired

from polished surfaces and, therefore, do not generally

account for the influences that variations in surface

hardness and grain shape have on the roughening

behavior [33]. They were also largely obtained with

contact profilometry, which as noted earlier, may not

have insufficient measurement sensitivity to resolve the

small-scale surface features.

Fig. 2. A scanning laser confocal microscopy image showing the

surface topography of mild steel sheet in the as-received condition and

the corresponding height profile of the 750 lm trace.

Fig. 3. The influence of uniaxial plastic strain on the surface

roughening behavior of mild steel sheet as exhibited by the corrected

rms roughness parameter, DRq.

4296 M.R. Stoudt, J.B. Hubbard / Acta Materialia 53 (2005) 4293–4304
Thus, the atypical roughening rate behavior exhibited

in Fig. 3 was attributed to a combination of enhanced

measurement resolution and the influence of an initial

roughness. Considering the substantial impact that

measurement or interpretative errors may have on the

results, the deviation from the behavior reported in the

literature does raise an important question: How well

does a simple analysis that is based on multiple linear

profile measurements and the Rq parameter represent

the stochastic process, or processes, that generates a

surface topography? As noted earlier, the fundamental

postulate in the interpretation of any roughness

measurement is that the surface profile represents the

intrinsic character of the overall surface. For this reason,

the validity of this postulate becomes the central issue in

the answer to this question. If it is valid, it then becomes

meaningful to characterize the data in a roughness

profile with random process statistics.

Most statistical characterizations of surface roughness utilize time series analysis methods to some degree.

These characterizations tacitly assume that the distribution of the height profile is both Gaussian and stationary. While this may enable a more straightforward

statistical analysis, the literature does not provide any

foundation to support such an assumption. In fact, the

literature clearly shows that the appropriate statistical

tools are determined by the character of the surface

[35]. The form of the surface can usually be determined

with a simple classification scheme such as the one put

forth by Nayak [36]. That is, establish whether the surface is stationary (i.e., homogeneous), random, isotropic, etc. The surfaces of the 1010 steel were classified

through a rigorous, systematic statistical evaluation

thereby minimizing any uncertainty that could arise

from an inappropriate assumption or from an inappropriate analytical method.

3.3. Classification of surface form

3.3.1. Assessment of Gaussian behavior

The accounts of roughening behavior in the literature

place great emphasis on the significance of the changes

exhibited by the Rq parameter; however, the properties

of the assumed Gaussian form upon which it is based

are rarely assessed [37]. Given the large number of data

points contained in each profile in this evaluation, the

Gaussian behavior could be straightforwardly assessed

by fitting the data to the normal distribution function

and analyzing the measured probability density of the

profile heights [38]:

pðzÞ 1⁄4 1

ffiffiffiffiffi

2p p r

exp ð1=2Þ ðz  lÞ

r

2 !. ð3Þ

In this equation, p is the probability density of realizing

a particular height (z), z is the magnitude of the height at

a location along the profile length, and l and r are the

mean and standard deviation of all the height values

in the profile, respectively. Fig. 4 shows the probability

density function (PDF) distributions as a function of

plastic strain level in the uniaxial strain condition. The

PDF data were adjusted so that the maximum probability density was zero at the mean value (i.e., the peak of

the bell curves). The Gaussian analysis in the perpendicular orientation exhibits the greatest magnitudes in the

numerical results from the two orientations and they

are presented in Table 2. Each value shown is the mean

value and the associated statistical uncertainty (UNC)

of the five individual measurements performed at that

strain level.

The quality of the fit to a Gaussian distribution is

assessed through an analysis of the higher central

moments of the distribution about the mean. The skew

(third moment, r3) is a measure of the asymmetry of

the distribution. If the distribution had an ideal

Gaussian form, the skew would equal zero [38], and as

shown in Table 2, the magnitude of the skew is slightly

less than zero. This indicates that heights below the

mean are somewhat more probable than are heights

above the mean. The kurtosis (fourth moment, r4) is a

measure of the overall shape of the distribution. An

ideal Gaussian distribution has a kurtosis equal to three

[38,39]. As shown, the kurtosis is slightly greater than

three indicating that the curves have a slightly sharper

peak and longer tails than what would occur if the

distribution were ideal. Given that the surfaces were

not assumed ideally Gaussian, deviations from the ideal

condition are not surprising. However, the relatively

Fig. 4. Normalized probability density functions (PDF) showing the

change in the surface height profile distributions as a function of

uniaxial plastic strain in mild steel sheet.

M.R. Stoudt, J.B. Hubbard / Acta Materialia 53 (2005) 4293–4304 4297
small magnitudes of these deviations suggest that the

surfaces exhibit an overall Gaussian form [21].

The trends exhibited by the PDF curves in Fig. 4

demonstrate that the range of probable heights (i.e.,

the width of the distribution) increases substantially as

a function of plastic strain. This is consistent with the

behavior observed in the height profile data, although

the change in slope at 4% strain is not apparent in this

figure. Since the PDFs have been normalized, their

progressive flattening does suggest a direct relationship

between the magnitude of the deviation from the mean

and the plastic strain. While the quality of a Gaussian

fit is expected to change with the type and condition

of the material of interest, these results do reveal the

some general trends about a Gaussian analysis: At low

levels of roughness, the probability density at the peak

is quite high, resulting in short tail regions. As the

roughness level increases, the probability density at the

mean is considerably lower than that for the smooth

surface, thereby increasing the statistical significance of

the tails of the PDF.

3.3.2. Assessment of stationarity

The second key assumption regarding the general

character of a surface is that it is stationary. A surface

can be regarded as stationary if some statistical description of that surface is invariant with respect to a translation along the surface [36]. That is, a change in the

length of a measurement does not affect the information

contained within the measurement. If this condition

does not hold, the statistics associated with standard

time series analysis are not appropriate, and a different

class of statistical tools must be employed to describe

the variations in a profile [40]. As a result, establishing

the degree of stationarity is essential for any substantive

evaluation of surface roughness. Unlike the straightforward test for Gaussian behavior, a determination of

stationarity, in the strict sense, is quite complex and

cannot be verified with a simple statistical test: It

requires detailed analyses of single, two-point, and multiple-point correlation functions. It should be noted that

most commercial statistical software packages do not

permit direct analyses of non-stationary two-point

correlation functions [41].

The literature is not clear as to whether a surface can

be regarded as a stationary process, or whether a roughness profile actually contains intrinsic characteristics of

the surface, from which it was taken. For example,

Sayles and Thomas [42] empirically demonstrated that

real surface topographies cannot be treated as resulting

from stationary Gaussian processes. From this, they

concluded that real surfaces follow a form that corresponds to highly non-stationary Gaussian distributions.

This conclusion signifies that surface roughness measurements are strongly dependent upon the actual

length-scale of the measurement. It also invalidates most

surface roughness studies by asserting that the measurements upon which they were based did not reflect the

true character of the surfaces. In contrast, Berry and

Hannay [43] argued that because Sayles and Thomass

conclusions could not be supported by the experiments

they cited, Sayles and Thomass approach has no predictive value when applied to individual topographic

measurements. This contradiction exemplifies how

stationary statistics cannot be assumed a priori: this

property must be empirically verified.

The Weiner–Levy process [44] utilized by Sayles and

Thomas can be generalized to the simple relationship

between the variance and the sampling length shown

in Eq. (4). This expression evaluates the validity of the

stationarity assumption by comparing the surface

roughness statistics from multiple measurements of

different lengths.

r2 1⁄4 Lh

. ð4Þ

In this expression, r2 is the variance of the amplitudes in

the profile and L is the length of the measurement. The

exponent, h, indicates the degree of deviation from

stationarity. That is, when h = 0, the variance is independent of sampling length and the statistics can be

considered stationary to a first order approximation.

When h = 1, the profile is described by Gaussian nonstationary statistics of the Weiner–Levy type.

An additional set of roughness profiles were obtained

from the steel surfaces for this determination. Unlike the

constant measurement length used in the previous

profiles, this set of measurements incorporated a range

of sampling lengths: 750, 375, 187, 93 and 46 lm. Fig.

Table 2

Gaussian analysis of perpendicular uniaxial strain data

Uniaxial strain Mean, l Unc. SD, r Unc. Max. PDF Unc. Skew, r3 Unc. Kurtosis, r4 Unc.

0.00 4.53 0.14 1.48 0.05 0.27 0.01 0.66 0.07 3.51 0.21

0.01 12.23 0.17 1.63 0.06 0.25 0.01 0.22 0.13 3.55 0.13

0.02 11.26 0.18 2.16 0.06 0.18 0.01 0.20 0.13 3.50 0.27

0.04 13.39 0.56 2.90 0.06 0.14 0.01 0.17 0.16 2.83 0.16

0.08 15.06 1.21 3.12 0.16 0.13 0.01 0.13 0.07 2.82 0.22

0.12 14.48 0.67 3.19 0.09 0.13 0.01 0.17 0.11 3.22 0.14

0.16 18.97 1.42 3.77 0.14 0.11 0.01 0.08 0.15 3.48 0.14

0.24 11.90 2.96 3.65 0.26 0.12 0.02 0.39 0.16 3.15 0.08

4298 M.R. Stoudt, J.B. Hubbard / Acta Materialia 53 (2005) 4293–4304
5 is a log plot exhibiting how the variance of the amplitude distribution changed with the profile length in the

0% strain condition and in the 16% strain condition.

These particular values are shown because one reflects

a surface condition with a small amplitude distribution

(i.e., the initial roughness) and the other reflects a

surface condition with a large amplitude distribution

(consisting of the initial roughness plus an additional

component generated by the plastic strain), respectively.

The exponent, h, was determined by fitting the variance

data to a power law and those results are shown in

Table 3. While the correlation coefficients in the 0%

strain condition are slightly lower than those obtained

in the 16% strain condition, the quality of the fit for

both data sets is sufficient to estimate the degree of

stationarity for the surfaces. The larger slope in the

16% strain condition could be an indication that the

degree of stationarity decreases slightly with strain or

that additional long wavelength surface features were

introduced during straining.

If a roughness profile truly reflects the character of a

surface, then the data contained within a single roughness profile can be used to verify the relationship

between the variance and sampling length expressed in

Eq. (4). That is, similar results should be obtained by

dividing the length of any one profile into several

equally sized segments, as the segments themselves will

be representative of the overall surface. Hence, the

750-lm profiles were divided into equal segments with

lengths of 375, 187, 93, 62, and 46 lm. The changes in

the variance of the amplitude distribution are shown

as a function of segment length for the 0% strain and

16% strain conditions in Fig. 6. As before, the exponent,

h, was determined by fitting the variance data to a power

law and those results are shown in Table 4. Note that

the variance of the entire profile (i.e., 750 lm) is shown

as a single data point. The correlation coefficients

indicate that overall quality of the fit was relatively good

and that these results were remarkably similar to those

acquired from the individual measurements.

Even though the coefficients from the two analyses

are comparable, the statistical approaches used to

Fig. 5. A log plot exhibiting the change in profile amplitude variance

as a function of measurement length at 0% strain and at 16% strain in

mild steel sheet. The slopes were used in conjunction with a generalized

Weiner–Levy relationship to gauge stationarity.

Table 3

Measurement-based stationarity analysis results

Plastic

strain level

Profile

orientation

Slope Correlation

coefficient

0.00 Parallel to RD 0.154 0.863

0.00 Perpendicular to RD 0.139 0.831

0.16 Parallel to RD 0.213 0.928

0.16 Perpendicular to RD 1.961 0.939

Table 4

Segment-based stationarity analysis results

Plastic

strain level

Profile orientation Slope Correlation

coefficient

0.00 Parallel to RD 0.088 0.920

0.00 Perpendicular to RD 0.119 0.901

0.16 Parallel to RD 0.110 0.991

0.16 Perpendicular to RD 0.273 0.983

Fig. 6. A log plot exhibiting the change in segment amplitude variance

as a function of measurement length at 0% strain and at 16% strain in

mild steel sheet. The slopes were used to verify the stationarity.

M.R. Stoudt, J.B. Hubbard / Acta Materialia 53 (2005) 4293–4304 4299
obtain them are distinctly different. Thus, the good

agreement between the two data sets suggests that the

generalized Weiner–Levy process approach is robust.

It also indicates that the surfaces exhibit a high degree

of translational invariance with respect to their statistical properties and that these properties are indeed intrinsic to the surfaces as a whole. There is a twofold

significance of these results. The first is that the topography data can be generally considered as representative

of the overall surface character. The second is that the

location of the profile origin has no influence on any

statistical interpretation of the measurement.

3.4. Spatial correlation analyses of height variations

A roughness assessment based exclusively on the

mean of the amplitude distribution is not equipped to

provide any information about the spatial distribution

(i.e., arrangement) of the features on the surface. That

is, any rearrangement in the spatial order of the profile

heights will have no influence whatsoever on the magnitude of any of the roughness parameters that are derived

from the PDF [45]. Thus, a different set of analytical

tools must be employed to evaluate the spatial distribution. By establishing that the surfaces of the 1010 steel

have both a Gaussian and, to a first order approximation, a stationary form, it can then be assumed that

any distribution of heights within in a surface profile will

be unaffected by a change in the position of the profile

origin. This assumption provides some flexibility regarding how the arrangement of heights within a profile can

be represented.

A two-point correlation function is one of the more

convenient approaches to characterize spatial arrangement and it has the following general form [37]:

qðsÞ 1⁄4 limL!1

1

L

Z L

0

ðzðxÞ zðx þ sÞÞ dx. ð5Þ

As shown schematically in Fig. 7, L is the sample length,

z(x) is the height at any point along the profile, and

z(x + s) is the height at the corresponding lag (i.e.,

offset) length, s. Normalizing Eq. (5) by the variance

of the roughness profile (r2) yields the autocorrelation

function (ACF) [21]. Note that the value returned by

this function for a lag of any given length is actually

the mean of all the values evaluated over the entire

roughness profile for that lag length.

Even though each profile may be considered as representative of the overall surface, the data in an individual

profile are highly specific to the local region of the

topography from where it was acquired. Thus, the high

degree of variability between individual measurements

makes an averaging approach similar to that used for

the Rq analysis (i.e., a simple average of the ACF values

obtained from a series of individual profiles) inappropriate to represent the correlation behavior of an entire

surface. Additionally, evaluating the arrangement of

heights within single roughness profiles reveals little

about the correlations that may exist over the entire

surface. Since a two-point correlation function is a

transform of the profile data, only small variations are

produced in the overall shape of the ACF for any given

strain condition. For this reason, it becomes possible to

average the individual values calculated for each specific

lag across a series of roughness profiles taken from the

same surface. In addition, averaging the large amount

of data available for each ACF calculation minimizes

the relative significance of any singular occurrence that

may be present in any individual profile. This creates a

two-point correlation function ‘‘ensemble’’ that describes the distribution of heights for that surface condition. Analysis of the ACF ensembles reveals any general

trends that may occur in the surface topography resulting from the plastic strain. An example of this method is

shown in Fig. 8 that illustrates the behavior of a

typically observed ACF ensemble. The dark line in this

figure is the mean ACF value determined from the five

individual profiles taken at each lag in the 24% strain

condition. The grey lines on either side of the solid line

are the standard uncertainties (i.e., 1r) calculated for the

five values at each lag [34]. Curves similar to Fig. 8 were

obtained for each strain level with roughness profiles in

both the parallel and perpendicular orientations to the

rolling direction. Small variations were observed in the

ACF behaviors between the two orientations, but for

brevity, only the results from the parallel orientation will

be presented.

An autocorrelation function contains a considerable

amount of information about the spatial relationships

in a roughness profile. That is, when the ACF = 1, the

features in the profile are defined as being perfectly

correlated with themselves at that lag length. Similarly,

when the ACF = 0, the features are defined as having

no correlation, or completely random. The transition

between the correlated and random state defines the

correlation length. It is the most common characteristic

derived from the ACF and it is usually assumed proportional to the decay of the ACF. One approach for

estimating the correlation length is by fitting the ACF

to a simple exponential and calculating the lag where

the function reaches 0.1 [37]. This approach assumes

Fig. 7. A schematic diagram defining the protocol used to calculate a

2-point height correlation function.

4300 M.R. Stoudt, J.B. Hubbard / Acta Materialia 53 (2005) 4293–4304
that the ACF becomes statistically unreliable where the

quality of the fit to an exponential becomes significantly

degraded. Since additional analyses of long-range correlations are not practical with the data from an individual

ACF, the transition from the correlated to the uncorrelated state is typically assigned to the lag where the ACF

assumes a value of 0.1 (i.e., q(s) = 0.1). In contrast, the

ACF ensemble represents the behavior of the entire

surface in one direction, and not just the behavior within

a single profile. Therefore, statistically significant information is likely to be present in the region of the ACF

that lies between 0.1 and 0. A more general method

for estimating the correlation length is required to assess

the correlation of features occurring at the longer lag

lengths.

Assuming this process has Markov properties, the

correlation length is finite by definition. Therefore,

several statistical approaches can be adopted to

determine the correlation length. One of these is by

integrating the ACF (Eq. (5)) from the origin to some

threshold value, s* [46]. This is shown in the following

equation:

CL 1⁄4

Z s

0

qðsÞ ds. ð6Þ

Since the features in the ACF are considered correlated

until the ACF = 0, both the upper and lower boundaries

of the uncertainty envelope (shown in Fig. 8) were

integrated to the point where they first assumed a value

of zero. The results of that evaluation are presented in

Fig. 9A. For comparison purposes, the integration

procedure was repeated using a cutoff value of 0.1 and

those results are shown as Fig. 9B. While the

uncertainty envelope of the ACF is rather small, the

uncertainties associated with the correlation lengths

are substantially larger. This is primarily due to the discrepancies between the upper and lower lag-lengths at

which the ACFs were terminated. As expected, this

effect is not as significant when the function was cutoff

at 0.1. However, the general shapes of the correlation

length uncertainty envelopes are similar in both cases.

Fig. 8. The ACF ensemble in the parallel orientation to the rolling

direction for mild steel sheet with 24% uniaxial plastic strain. The dark

line is the mean ACF value determined from the five individual values

calculated at each lag. The grey lines on either side define the

uncertainty envelope associated with the five lag values.

Fig. 9. The correlation lengths estimated from the ACF ensembles for

mild steel sheet in the parallel orientation to the rolling direction.

(A) Integrated to where the upper and lower bounds of the ACF

uncertainty envelope (shown in Fig. 8) assume a value of zero. (B) The

same ACF ensemble integrated to a cutoff value of 0.1.

M.R. Stoudt, J.B. Hubbard / Acta Materialia 53 (2005) 4293–4304 4301
The results of the correlation examinations reveal two

key points. The first point concerns the magnitude of the

change that occurs in shape of the uncertainty envelope

at approximately 4% strain for both cutoff values in

Fig. 9. This is particularly interesting when compared

to the Rq behavior exhibited in Fig. 3. As the slope of

Rq versus strain shown in this figure reflects the roughening rate as a function of plastic strain, the change in

width of the uncertainty envelope in Fig. 9 reflects the

variability in the spatial distribution of the surface

features as a function of plastic strain. Since the physical

meanings of these two data sets are completely independent, the change in behavior that occurs at the same

strain level in both figures cannot be a statistical artifact;

rather, it suggests the existence of a set of common

underlying physical mechanisms.

As noted earlier, plastic deformation in a polycrystalline alloy is an extremely complex process and the

measurable surface roughness depends on several

factors. For this reason, the resulting surface topography is an amalgam of each active mechanism during

the deformation process. Additionally, each component

in this deformation has a corresponding effective characteristic length scale that can be used to distinguish it.

These range from the hundred-nanometer scale associated with slip steps to the several hundred-micron scale

associated with slip localization.

At low levels of plastic strain, the amount of deformation that occurs within each grain depends on the

individual orientation, the local Taylor factor [47], and

the constraints imposed by neighboring grains at or

below the surface [14]. That is, the deformation in a

grain with a favorable orientation for slip will occur

by primary slip in the interior regions of that grain.

However, in a grain where the slip conditions are not

as favorable, the deformation will tend to localize in

the grain boundary regions due to the additional shear

displacements required to produce grain rotation and

to maintain grain-to-grain contiguity. This anisotropy

produces an overall roughness character that is a mixture of both primary slip and near grain boundary

deformation. Accordingly, neighboring surface grains

possessing the same level of macroscopic strain can exhibit appreciably different amounts of measurable surface

roughness. As the plastic strain levels and dislocation

densities increase, localized work hardening makes

deformation by primary slip more difficult, thereby

activating additional slip systems, such as those that

promote secondary slip, pencil glide and slip localization. Therefore, the common behavior exhibited at 4%

strain most likely reflects a transition threshold between

principal roughening mechanisms.

The second point concerns the shape of the ensembleaveraged ACF profile. It is well established in the statistical literature that if a stochastic process has stationary,

Gaussian, Markov properties it is unique [48] and it

must possess an ACF that decays as a simple exponential function of lag length. The fact that this profile does

not exhibit such behavior specifically implies that one of

these three fundamental conditions has been violated.

This has significant consequences for this study as well

as for all analyses of surface roughness. A violation of

the stationarity condition directly disputes the validity

of the fundamental postulate that serves as the basis

for all surface roughness analyses: namely, that a surface

profile is representative of the intrinsic character of the

overall surface. If this does not hold, then a roughness

profile cannot reflect any inherent property of the

surface from which it was taken, thereby implying that

the commonly used methods to interpret surface

roughness are not meaningful. The results from the

analysis of the variance with the generalized Weiner–

Levy approach demonstrated that the roughness data

in this study was stationary to a first order. Therefore,

it can be concluded that the stationarity condition was

not violated. If the stochastic process had Markov properties, then the roughness profiles must have a finite

correlation length. Even though the ACF ensemble

profiles exhibited long wavelength features, there is no

justification for the assumption that the correlation

length associated with those features diverges. Thus,

the Markov condition also was not violated. Therefore,

it must be concluded that the deviation from a pure

exponential decay exhibited by the ensemble-averaged

ACF profile is most likely related to the small deviations

from Gaussian behavior present in the roughness data.

The significance of this conclusion is twofold: First,

non-Gaussian behavior implies a violation of the central limit theorem of statistics [49]. That is, surface

roughness profiles cannot result from linear superpositions of large numbers of statistically independent

roughening events. In this case, a violation of the central limit theorem of statistics means that the deformation of a free surface must occur by a small number of

statistically dominant mechanisms. This is consistent

with the literature in that measurable deformation-induced surface roughness, particularly at small strain

levels, is largely produced by crystallographic slip

and near grain boundary deformation. Consequently,

the sharp deviation in behavior exhibited at 4% strain

in Fig. 3 and in Fig. 9 probably reflects a change in

deformation mechanism. If so, the quality of a fit to

an ideal Gaussian profile should improve above the

4% strain level. A simple test of the data shown in

Table 2 revealed that the mean skew for the larger

strain values (i.e., 4–24%) was approximately 0.18 as

compared to approximately 0.37 for the mean skew

for the smaller strains (0–2%). Similarly, the mean

kurtosis was approximately 3.10 for the larger strains

and approximately 3.52 for the smaller strain values.

These results indicate that the divergence at 4% strain

is not a statistical artifact and that it almost certainly

4302 M.R. Stoudt, J.B. Hubbard / Acta Materialia 53 (2005) 4293–4304
corresponds to an increase in the number of active

deformation mechanisms. Second, the small deviations

from Gaussian behavior provide direct evidence that

Rq cannot completely characterize the roughening

behavior of a polycrystalline material. Mean amplitude

distribution parameters are all based on the presumption of ideal Gaussian behavior and, as these results

demonstrate, plastically deformed surface profiles do

not exhibit an ideal Gaussian form. While this presumption could hold for surfaces with more homogeneous roughening characters, such as machined

surfaces, plastic deformation of a free surface appears

to be too complex to be properly characterized by

such a simple approach.

The results of this study demonstrate that small

deviations from the ideal surface character, such as

those produced by crystallographic slip, may have a

pronounced influence on the accuracy of the analytical

methods used to interpret the roughness data. Because

of the density of data points used in this study, and the

high measurement resolution provided by the SLCM

technique used for the data acquisition, both the statistics associated with these small variations and the

methods used to evaluate them were determined to

be robust. However, additional studies are warranted

to determine (a) the influence of additional roughness

profiles on the width of the uncertainty envelope associated with the ACF ensemble and (b) how the individual roughening mechanisms influence surface profile

statistics.

4. Conclusions

The inability to accurately predict the surface deformation morphology with numeric methods raises a serious question as to how well the analytical tools

commonly used to evaluate roughness data actually represent the real surface. To address this question, roughness data were acquired from the surfaces of uniaxially

strained mild steel sheet specimens with a high-resolution scanning laser confocal microscopy (SLCM) technique. A conventional plot of the rms roughness (Rq)

versus plastic strain exhibited a sharp change in slope

at approximately 4% strain. This change in roughening

rate was determined to be inconsistent with the literature

consensus of monotonic roughening behavior and it was

attributed to a combination of the enhanced measurement resolution of the SLCM and an influence of the

initial surface roughness.

A two-point correlation function (ACF) ensemble approach was developed to evaluate the spatial distribution of the surface features induced by the plastic

strain. Integration of the uncertainty associated with

the ACF ensemble to the points where the upper and

lower bounds first assumed a value of zero enabled an

estimation of the correlation length envelope for the

ensemble. Like the plot of the rms roughness, the correlation length envelope plot also exhibited a sharp change

at 4% strain. Considering that the physical meanings of

the two data sets are completely independent and that

plastic deformation at a free surface of a polycrystalline

material is highly complex, the change in behavior at

this strain level most likely represented a transition in

principal roughening mechanisms.

While most roughness analyses tacitly assume that

the measured surface profile conforms to stationary,

Gaussian statistics, this study employed a set of rigorous

analyses to establish the actual form. Small deviations

from the ideal Gaussian behavior were observed in the

skew and kurtosis data; however, the results indicated

that a Gaussian height distribution provides a reasonably accurate characterization of the surface roughness

for this material. A generalized Weiner–Levy relationship was utilized to empirically assess the degree of stationarity by evaluating the statistical variance of the

roughness profile as a function of the measurement

length. Even though small deviations were observed,

the roughness data exhibited a high degree of translational invariance symmetry with respect to the statistical

properties. From this it was determined that the surfaces

were largely stationary and that the characteristics in the

roughness data could be considered as representative of

the surfaces as a whole.

The shape of the ACF ensemble did not exhibit a

simple exponential decay, which indicated that the

stochastic process responsible for surface roughening

did not conform to simple stationary, Gaussian,

Markov statistics. Thus, the behavior exhibited by

the ACF profile implies that one or more of these three

conditions was violated. A lack of stationary means

that a roughness profile is not representative of the

surface from which it was taken, whereas a violation

of the Markov condition implies the existence of extremely long-range (i.e., exceeding several hundred

microns in length) spatial correlations in these profiles.

Inasmuch as a violation of either of these conditions

was deemed unlikely, it was concluded that the pronounced deviation from the simple exponential form

exhibited by the ACF profile was associated with the

small discrepancies in the Gaussian fit. This conclusion

indicated that deformation-induced surface roughness

must occur by one, or just a few, statistically dominant

mechanisms. This agrees with the literature which

claims that deformation-induced surface roughness largely consists of near grain boundary deformation and

crystallographic slip. Additional studies are warranted

to determine (a) the influence of additional roughness

profiles on the width of the uncertainty envelope associated with the ACF ensemble and (b) how specific

roughening mechanisms influence surface profile

statistics.

M.R. Stoudt, J.B. Hubbard / Acta Materialia 53 (2005) 4293–4304 4303
References

[1] Dieter GE. Mechanical metallurgy. 3rd ed. New York

(NY): McGraw-Hill; 1986.

[2] Becker R. Acta Mater 1998;46(4):1385–401.

[3] Wilson WRD, Lee W. J Manuf Sci Eng 2001;123:279–83.

[4] Dai YZ, Chiang FP. J Eng Mater Technol 1992;114(Oct):432–8.

[5] Valkonen AE. Plastic deformation and roughness of free metal

surfaces. Ohio State University; 1987.

[6] Abe T, Nagaki S, Akase T. Bull JSME 1985;28(236):251–8.

[7] Kienzle O, Mieztner K. Stahl Eisen 1961;81(14):950–2.

[8] Osakada K, Oyane M. Bull JSME 1971;14(68):171–7.

[9] Guangnan C, Huan S, Shiguang H, Baudelet B. Mater Sci Eng A

1990;128:33–8.

[10] Bishop JFW, Hill R. Philos Mag 1951;42(327):414–27.

[11] Kocks UF. Metall Trans 1970;1:1121–43.

[12] Hansen N. Metall Mater Trans A 1985;16:2167–90.

[13] Armstrong RW. In: Baker TN, editor. Yield, flow and

fracture of polycrystals. London: Applied Science Publishers;

1983. p. 1–31.

[14] Stoudt MR, Ricker RE. Metall Mater Trans A 2002;33:

2883–9.

[15] Beaudoin AJ, Acharya A, et al. Acta Mater 2000;48:3409–23.

[16] Wu PD, Lloyd DJ. Acta Mater 2004;52:1785–98.

[17] Hurley PJ, Humphreys FJ. Acta Mater 2003;51:1087–102.

[18] Rabbe D, Sachtleber M, et al. Acta Mater 2003;51:1539–60.

[19] Thomson PF, Shafer BV. Int J Machine Tool Design Res

1982;22(4):261–4.

[20] Inagaki A, Komatsubara T, Inagaki H. Z Metallkd

1999;90(6):427–33.

[21] Thomas TR. Rough surfaces. 2nd ed. London: Imperial College

Press; 1999.

[22] Radhakrishnan V. Wear 1970;16:325–35.

[23] Murthy TSR, Reddy GC. Wear 1982;83:203–14.

[24] King TG, Spedding TA. Wear 1982;83:91–108.

[25] Stoudt MR, Hubbard JB, Banovic SW. Test Meas World

2004;24(5):A7–A10.

[26] Properties and selection of metals, vol. 1, 8th ed. Metals Park

(OH): ASM International; 1961.

[27] Smith WF. Structure and properties of engineering alloys. New

York (NY): McGraw-Hill; 1981.

[28] VanderVoort GF. Metallography principles and practice. Materials Park (OH): ASM International; 1999.

[29] Udupa G, Singaperumal M, Sirohi RS, Kothiyal MP. Meas Sci

Technol 2000;11(3):305–14.

[30] Lange DA, Jennings HM, Shah SP. J Mater Sci 1993;28(14):

3879–84.

[31] Annual Book of ASTM Standards, Section 3, Metals test methods

and analytical procedures, vol. 3.01. Metals-mechanical testing;

elevated and low-temperature tests; metallography. Philadelphia

(PA): ASTM; 2002. p. 547–53.

[32] Metallography and microstructures, vol. 9, 9th ed. Metals Park

(OH): ASM International; 1987.

[33] Stoudt MR, Banovic SW, Quarrick T. In: Baker MA, editor. 45th

MWSP conference proceedings, vol. XLI. Warrendale (PA): Iron

& Steel Society; 2003. p. 507–17.

[34] Taylor BN, Kuyatt CE. Guidelines for evaluating and expressing

the uncertainty of nist measurement results. NIST Technical Note

1297, 1994 Edition, NIST; 1994.

[35] Longuet-Higgins MS. Philos Trans R Soc Lond A

1957;250:157–74.

[36] Nayak PR. J Lubrication Technol (Trans ASME) 1971;93:

398–407.

[37] Whitehouse DJ, Archard JF. Proc R Soc Lond A

1970;316:97–121.

[38] Walpole RE, Myers RH. Probability and statistics for engineers

and scientists. New York (NY): Macmillian; 1972.

[39] Mendenhall W, Sincich T. Statistics for engineering and the

sciences. 3rd ed. San Francisco (CA): Dellen; 1992.

[40] Montgomery DC, Johnson LA, Gardiner JS. Forecasting and

time series analysis. 2nd ed. New York (NY): McGraw-Hill;

1990.

[41] Hanselman D, Littlefield B. Mastering Matlab 6. Upper Saddle

River (NJ): Prentice-Hall; 2001.

[42] Sayles RS, Thomas TR. Nature 1978;271(2):431–4.

[43] Berry MV, Hannay JH. Nature 1978;273:573.

[44] Hsu H. Probability, random variables, & random processes. New

York (NY): McGraw-Hill; 1996.

[45] Stoudt MR, Hubbard JB, Banovic SW. SAE Trans: J Mater

Manuf 2005;5:50–8.

[46] Berne BJ, Pecora R. Dynamic light scattering. New York

(NY): Wiley; 1976.

[47] Taylor GI. J Inst Met 1938;62:307–24.

[48] Doob JL. Ann Math 1942;43:351.

[49] Feller W. An introduction to probability theory and its applications, vol. 2. 2nd ed., New York (NY): Wiley; 1971.

4304 M.R. Stoudt, J.B. Hubbard / Acta Materialia 53 (2005) 4293–4304