On the Relationship Between Deformation-Induced Surface

Roughness and Plastic Strain in AA5052—Is it Really Linear?

M.R. STOUDT, J.B. HUBBARD, and S.D. LEIGH

Three different heat treatments of aluminum alloy AA5052 were subjected to various levels of

uniaxial plastic strain. The resulting surfaces were then evaluated using both scanning laser

confocal microscopy (SLCM) and stylus profilometry. Three regression approaches were used

to assess the quality of a linear and a curvilinear fit for the roughness data as a function of true

plastic strain. While there were differences among the regression results, the analyses revealed

that a linear model was more statistically appropriate for the finest grain size. As the grain size

increased, the surface morphology became more complex and a quadratic model became more

suitable. Since the relative area fractions of grain boundary–localized roughness and slip-induced roughness are grain size dependent, the higher order fit between the roughness and plastic

strain reflects substantial changes in the ratio of these areas. The differences between the SLCM

and profilometry results were attributed to the natural filtering that occurs during contact

profilometry. This filtering skewed the roughness data toward the largest surface displacements,

thereby reducing the measurement fidelity to the point where the only possible outcome was the

linear relationship.

DOI: 10.1007/s11661-011-0694-z

The Minerals, Metals & Materials Society and ASM International (outside the USA) 2011

I. INTRODUCTION

DISCREPANCIES between measured surface

roughness values and those predicted by numerical

modeling call into question the reliability of the material

property data used by models that predict the mechanical behavior. While this article focuses on the reliability

of the mechanical property data in sheet metal formability models, such inconsistencies can also affect the

accuracy of computational models for material design.[1,2] More specifically, the fact that discrepancies

exist implies either that the current approaches for

acquiring and analyzing deformation-induced surface

roughness are not appropriate for the task or that the

measured data lack sufficient measurement fidelity.

Undoubtedly, the accuracy of any surface roughness

evaluation strongly depends on the technique used to

acquire the topographic data, and while all surface

measurement methods introduce error to some extent,

careful examination of the characteristics of these errors

have established appropriate measures for correction.[3,4] When performed properly, most surface characterization techniques provide sufficient measurement

resolution, so the more likely source of discrepancy is

the manner in which the surface data are assessed. Many

of the assessments of roughening behavior reported in

the literature are derived from sets of linear profiles and

use time series–based statistical sampling protocols to

evaluate the roughness data.[5] These assessments typically express the surface roughness as the simple average

of a scalar roughness parameter that describes the

dispersion about the profile mean (e.g., the arithmetic

mean roughness, Ra, or the root-mean-squared roughness, Rq). This averaged mean roughness parameter is

then typically extrapolated to represent the characteristics of the entire surface at a given particular strain level

(i.e., a surface-Ra or a surface-Rq value). The key factor

is that the surfaces are assumed to have a statistically

stationary form (i.e., the statistical properties are invariant with respect to translation in position on the

surface), because stationarity is a requirement if the

extrapolation is to have any meaning. There are two

fundamental problems with this approach: (1) the

commonly used scalar parameters do not sufficiently

account for the complexity of the deformation in a

polycrystalline material[6,7] and (2) extrapolation will

compound any measurement uncertainty or statistical

errors present in the roughness data. For these reasons,

representations of the surface character derived from

surface roughness data of this type could easily over

simplify the true behavior and produce misleading

results. The high likelihood for error in these surface

roughness analyses raises questions about the validity of

the linear relationship presumed between deformationinduced roughness of a free polycrystalline surface and

plastic strain that prevails in the surface roughness

literature.[8–13] The literature also indicates that when

the complexities in the deformation process are properly

considered, such as in the molecular dynamic simulation

results reported by Perron et al.,

[14] the relationship

between the computed surface roughness and strain is

distinctly nonlinear.

U.S. GOVERNMENT WORK

NOT PROTECTED BY U.S. COPYRIGHT

M.R. STOUDT, Research Engineer, and J.B. HUBBARD, Guest

Researcher, are with the Material Measurement Laboratory, NIST

Center for Automotive Lightweighting, National Institute of Standards and Technology, Gaithersburg, MD 20899-8553. Contact

e-mail: stoudt@nist.gov S.D. LEIGH, Statistician, is with the

Statistical Engineering Division, Information Technology Laboratory,

National Institute of Standards and Technology.

Manuscript submitted March 11, 2010.

Article published online April 29, 2011

2668—VOLUME 42A, SEPTEMBER 2011 METALLURGICAL AND MATERIALS TRANSACTIONS A
A recent study evaluated the nature of the relationship between surface roughness and plastic strain in

AA6111-T4 aluminum using surface roughness data

acquired with scanning laser confocal microscopy

(SLCM) and two different analysis protocols.[15] The

first protocol was consistent with the method reported in

the literature in that a surface-Rq value was determined

from sets of widely spaced linear roughness profiles.

Specifically, each profile data set consisted of eight

regularly spaced columns that were extracted from

numerical matrices of topographical data acquired at

the different strain conditions. The second protocol

determined the surface-Rq value (Sq) with a single

calculation that included all the data points within the

same numerical matrix. When the surface data derived

from the two different protocols were plotted against

plastic strain, the large uncertainties associated with the

first protocol (i.e., the profile-based approach) produced

a linear correlation coefficient of 0.95, which would seem

to indicate an excellent overall fit. However, the data

from the second protocol exhibited a distinctly nonlinear form that clearly did not support the literature

consensus of a universal linear relationship, because the

associated uncertainties in the data were minimized by

the matrix-based calculation. Further analysis of these

results revealed that when a surface-Rq value is derived

from a set of individual profiles, it is highly influenced

by the sampling conditions and by the presence of the

natural correlations that exist within the profile data. In

contrast, when Sq is obtained from a single matrixbased calculation, any influence arising from correlations within the roughness data is eliminated. In fact,

after the correlations in the profile data were eliminated,

the same data produced a nonlinear behavior that was

nearly identical to that exhibited by the matrix-based Sq

data. There were two main conclusions drawn from this

study. The approach used to interpret the roughness

data as independent profiles does not accurately represent the true roughening behavior of the free surface of a

polycrystalline material even though the correlation

statistics may suggest an excellent fit, and the linear

relationship between Rq and plastic strain reported in

the literature is a statistical artifact resulting from

inadequate sampling.

In 2002, Stoudt and Ricker[7] reported that heightbased mean roughness parameters that reduce a roughness profile to a single value, such as Ra, may adequately

quantify the macroscopic changes that occur on a

surface with plastic strain in a commercial, polycrystalline aluminum alloy. In light of the findings in Reference

15 and the citations of Reference 7 supporting claims of

a universal linear relationship between surface roughness and plastic strain,[16,17] it is appropriate to reevaluate the conclusions reported by Stoudt and Ricker.

As such, this article compares roughness data of

uniaxially strained AA5052 that were acquired with

contact profilometry to data from the same set of

surfaces that were acquired by SLCM. The approach

used to evaluate these data is similar to the one used in

Reference 15, with the key distinction being that the

profile data is actual profilometer data. That is, the

profile data were not extracted from a matrix. For this

reason, many of the issues associated with stylus

profilometry such as filtering and mechanical damage

are relevant here, whereas they were inconsequential in

Reference 15.

II. EXPERIMENTAL

A. Material

As described in Reference 7, aluminum alloy AA5052

has a nominal Mg mass fraction of 2.5 pct and an

approximate Cr mass fraction of 0.25 pct to control grain

size and to enhance strength.[18] Three heat treatments

were developed to create different grain sizes: (1) 96 hours

at 813 K (540 C) (hereafter referred to as the HT

condition); (2) 2 hours at 623 K (350 C) (hereafter

referred to as the O condition); and (3) the as-received,

or H32 condition. The designation H3x refers to a

stabilized condition produced through a low-temperature

thermal treatment designed to retain a small amount of

the residual strain to stabilize the mechanical properties

and the ductility.[18]

The grain size was measured in each

heat-treated condition in both the rolling direction (RD)

and the long transverse direction (TD) using a linearintercept technique similar to that prescribed in ASTM

Standard E-112.[19] The grain size data, as reported by

Stoudt and Ricker,[7]

are shown in Table I.

B. Generation of Surface Roughness

Flat sheet, tensile specimens were machined from

1.0-mm-thick sheet stock with the tensile axis oriented

parallel to the RD. All of the specimens in this analysis

were obtained from the same sheet to minimize variations in the crystallographic texture and the heat

treatment.

Sample preparation consisted of polishing the entire

specimen surface to a 0.25-lm diamond finish using

standard metallographic practice.[20] A grid of fiducial

lines spaced nominally at 5-mm intervals was lightly

engraved on the gage section of each specimen with a

silicon carbide scribe to facilitate assessment of the local

plastic strain. The initial spacing between the lines on

the grid was determined with a linear-encoded, measuring stage microscope with a resolution of ±0.5 lm. The

specimens were pulled in uniaxial tension to predetermined levels of plastic strain with a computer-controlled

universal tensile machine. The crosshead displacement

rate was 1.5 mm/s for all experiments. After the desired

strain level was attained, the grid was remeasured with

the microscope and the true plastic strain was calculated

Table I. Analysis of the Grain Size

Heat

Treatment

Grain

Size

(RD)

(lm)

Standard

Error

(lm)

Grain

Size

(TD)

(lm)

Standard

Error

(lm)

Aspect

Ratio

(LT/RD)

H32 53.48 1.42 51.51 0.67 1.092

O 134.52 4.65 124.64 8.53 1.079

HT 159.31 7.66 154.63 6.37 1.030

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 42A, SEPTEMBER 2011—2669
from the grid displacement. The true strain values

shown in Table II differ slightly from those originally

reported in Reference 7, because the strain data were

evaluated more comprehensively for this study. The

recomputed strain values and the associated error are

shown in Table II.

C. Contact Profilometry

Some of the samples used to generate the data shown

in Reference 7 were re-strained to higher strain levels

and could not be re-evaluated. As such, the set of strain

levels is smaller for this analysis than what was

originally reported.

The first method used to acquire surface data was

contact profilometry.[7]

For these measurements, roughness profiles were acquired in the region of the specimen

where the plastic strain level was closest to the target

value. Each surface measurement consisted of five 1.75-

mm profiles made with a 5-lm-diameter stylus in both

the RD and TD orientations to generate ten profiles in

total. A space of approximately 1.0 mm was left

between profiles along the grid section to ensure that

there was no overlap. Each profile consisted of 8064

data points, which created a sampling interval of

approximately 0.22 lm/point. Each profile was corrected for offset effects resulting from specimen tilt or

other mechanical influences by computing the optimal

regression equation of the profile and then subtracting

the regression from each point in the image matrix.

D. Scanning Laser Confocal Microscopy

The second method acquired surface data by SLCM.

Each surface measurement consisted of five, well-spaced

SLCM topography images taken from the center region

of the specimen. All of the SLCM images were created

with a 635-nm red laser source. For each image, the

spacing between sampling points in the (x, y) plane was

fixed by the objective lens at 1.562 lm/point, and the

spacing between the individual focal planes in the z

direction was nominally 100 nm. These sampling

parameters generated sets of (640 9 512) pixel images

with physical dimensions (x,y,z) of (1000 9 800 9

50) lm. As with the profile data, it was essential that

these images be well separated to ensure that the surface

data contained in each image were uncorrelated (i.e., no

overlapping of any image data) and that the data

properly represented the full range of surface characteristics. The specifics of the procedure to convert the

SLCM images to simple matrices of topographic heights

are described in Reference 15. After conversion, the

numerical matrices were trimmed to square 262,144

element arrays (i.e., 512 pixels 9 512 pixels) to facilitate

the matrix-based mathematical operations. Each matrix

was leveled using a similar procedure to that used to

minimize offset effects in the profile data. In this case,

the optimal multiple regression equation of the Euclidean plane was computed for each matrix and was then

subtracted from each point in the image matrix.

Because some of the statistical parameters used to

interpret the surface data are highly sensitive to outliers,

the extreme values were filtered from all of the profilometer and SLCM data sets. For this evaluation, an

extreme value was defined as any value that was greater

than ±6r, where r is the standard deviation for all the

heights in that particular data set. Filtering consisted of

setting the magnitude of any data points that met this

criterion equal to the mean for that particular data set.

Note that the number of affected data points for a given

data set was typically less than 20 points. The residuals

were used as the source for all subsequent assessments of

the surface character.

III. ANALYSIS METHODS AND RESULTS

A. Evaluation of Surface Roughness Data

The increasingly complex relationship between plastic

strain and the evolved surface morphology is exhibited

in Figure 1. The high-resolution intensity SLCM images

in this figure illustrate how much the surface character

changes at low, medium, and high strain levels in the O

heat treatment.

Prior to evaluation, we merged the individual roughness measurements using a superposition technique. The

superposed data were rank-sorted and binned into

histograms that represented the entire surface height

distributions for each heat treatment and strain condition. This technique was repeated to produce two

complete sets of data from the same surfaces: one

acquired by SLCM and one by contact profilometry.

Probability density functions (PDFs) were calculated

from the histograms of the height data. As shown

previously, raw topographic data have a character that

can generally be described with Gaussian statistics.[6,15,21,22] This is convenient since Rq is defined as

one standard deviation of the normal distribution

Table II. Analysis of the Error in Strain

Heat

Treatment

True

Strain

Standard

Uncertainty (r)

Relative

Standard Uncertainty

of Mean (Pct)

H32 0 0 0

H32 0.0362 0.0048 0.0469

H32 0.0486 0.0020 0.0164

H32 0.0889 0.0030 0.0146

H32 0.0946 0.0065 0.0264

O0 0 0

O 0.0424 0.0022 0.0165

O 0.0778 0.0058 0.0283

O 0.0919 0.0008 0.0033

O 0.1226 0.0046 0.0139

O 0.1658 0.0068 0.0169

O 0.1796 0.0078 0.0162

O 0.1994 0.0068 0.0140

HT 0 0 0

HT 0.0286 0.0006 0.0105

HT 0.0449 0.0005 0.0045

HT 0.0905 0.0045 0.0210

HT 0.1082 0.0012 0.0046

HT 0.1326 0.0131 0.0053

HT 0.1415 0.0014 0.0375

2670—VOLUME 42A, SEPTEMBER 2011 METALLURGICAL AND MATERIALS TRANSACTIONS A
(i.e., Rq ” r).[23,24] Even though it is more common in

the surface roughness literature, Ra has no definitional

association to the moments of the height distribution.

Since the calculations here are based on the merged

distribution, r becomes an estimate of the Rq value for

the entire surface, or Sq. The expression used to

determine Sq is[23]

Sq 1⁄4 1

N

 X

N

j1⁄41

Zj  Z  2

" #1=2

1⁄21

In this equation, Z is a set of height values, Z is the

mean of Z, and N is the total number of values in the

calculation. We added the subscripts ‘‘slcm’’ and ‘‘prof’’

to the Sq data to distinguish the surface data acquired

by confocal microscopy from those acquired by contact

profilometry.

Figure 2 shows the relationship between Sqslcm and

true plastic strain for the three heat treatments. The

profilometry data (Sqprof) are presented in exactly the

same fashion in Figure 3. As noted previously, the

uncertainties associated with the Sq data in both figures

were minimized by the analysis protocol, and as a result,

the error bars for the Sq data are substantially smaller

than the plot symbols. Dashed lines connect the raw Sq

data to illustrate the changes in the surface roughness as

a function of plastic strain for each heat treatment, and

with the possible exception of HT in Figure 3, none of

the heat treatments exhibits any obvious linear trend.

This is the case for both the SLCM and the profilometry

data.

Tensile Direction

4.5 %

TD

RD

(a) 9.6 %

TD

RD

(b)

17.1 %

TD

RD

(c)

100 μm

21.2 %

TD

RD

(d)

Fig. 1—A series of scanning laser confocal micrographs showing the progressive complexity of the relationship between plastic strain and the

surface morphology of AA5052 in the O heat treatment. (a) The surface after the application of 0.045 true strain. (b) The surface after the application of 0.096 true strain. (c) The surface after the application of 0.171 true strain. (d) The surface after the application of 0.212 true strain.

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 42A, SEPTEMBER 2011—2671
One difference between the SLCM and profilometer

data is the initial roughness at zero strain. As shown in

Figures 2 and 3, the initial roughness is approximately

0.7 lm for the SLCM data, whereas the profilometer

value is considerably smaller at approximately 0.03 lm.

Further examination shows that the offset between the

SLCM and profilometer roughness data becomes negligible after the strain exceeds approximately 0.05. The

slight shadowing visible in the topography of the aspolished, zero strain condition, shown in Figure 4(a),

indicates that the difference in magnitudes is most likely

due to a specular distortion resulting from the highly

reflective surface in this condition. Since the variation in

dispersion of the incident light generates the contrast, or

signal, at each pixel point and the image is then created

from that intensity map, imaging of flat, highly reflective

surfaces with optical techniques is particularly difficult

because these surfaces often reflect the incident laser

with little or no dispersion. This tends to introduce

anomalies in the surface measurement. The relatively

small offset produced by this curvature artificially

broadened the range of the height distributions from

approximately ±0.5 lm (profile) to approximately

±4 lm (SLCM) and, consequently, increased the magnitude of Sq. This is reflected in the different shapes of

the PDFs for the as-polished condition shown in

Figure 4(b). Eliminating the curvature from linear

profile data is relatively straightforward because the

data are two-dimensional and the distortion only

propagates in one measured dimension. However, eliminating the curvature from the SLCM data is more

problematic, because the data are three-dimensional and

distortions generally do not propagate symmetrically.

The fact that correction measures are dependent on the

length scale of the surface data involved and on the type

of instrument used to acquire it[25] exacerbates the

problem. That is, if the parameters used to correct for a

distortion are inappropriate, the method will introduce

additional distortions in the height data. Analyses of the

different protocols reported in the literature to correct

for form errors are outside the scope of this evaluation.

We adopted the X-Y phase-correct Gaussian (PCG)

high-pass filter protocol to correct the data, because it is

the protocol recommended by the ASME Surface

Roughness Standard B46[23] for form errors of this

type. This protocol has been shown to effectively remove

long wavelength two-dimensional distortions from

three-dimensional data with minimum influence on the

magnitude of the short wavelength (roughness) components.[26] After a series of parametric evaluations, we

achieved the most effective filtering with a filter mask of

107.8 lm and a long wavelength cutoff of 34.4 lm. The

change in the shape of the PDFs in Figure 4(c) shown

before and after the application of the PCG filter

demonstrates that the application of this high-pass filter

considerably reduced the influence of the distortion. As

shown in Figure 4(d), the influence of the distortion is

strongest when the magnitude of the roughness is small.

That is, the zero strain, as-polished surface condition

has a poor signal-to-noise ratio. Furthermore, the

convergence of the two lines in this figure indicates that

as the magnitude of the roughness increases (signal), the

relative influence of the distortion (noise) progressively

diminishes to the point where the differences between

the behaviors of the raw and filtered data are negligible.

Based on the change exhibited in Figure 4(c), and the

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 0.05 0.1 0.15 0.2

SLCM

H32- Condition

O- Condition

HT- Condition

Surface Roughness, SqSLCM, μm

True Plastic Strain

Fig. 2—The relationship between Sq and true plastic strain for the

three heat treatments of AA5052. The surface data in this figure

were acquired with scanning laser confocal microscopy. Note that

the dashed lines connect the raw Sq data to illustrate the relative

magnitude of the changes in the surface roughness as a function of

plastic strain for each heat treatment. Because of the finer sampling

conditions and higher density of data points, the behaviors shown in

this figure are considered more representative of the actual relationships between Sq and plastic strain for the AA5052.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 0.05 0.1 0.15 0.2

Profilometry

H32- Condition

O- Condition

HT- Condition

Surface Roughness, SqProf, μm

True Plastic Strain

Fig. 3—The relationship between Sq and true plastic strain for the

three heat treatments of AA5052. The surface data in this figure

were acquired with contact profilometry. As in Fig. 2, the dashed

lines connect the raw Sq data to illustrate the relative magnitude of

the changes in the surface roughness as a function of plastic strain

for each treatment.

2672—VOLUME 42A, SEPTEMBER 2011 METALLURGICAL AND MATERIALS TRANSACTIONS A
fact that the distortion is not observable in any of the

images shown in Figure 1, the difference between the

profilometer and the SLCM height distributions was

attributed to differences in the composition of the

measured surface data and not to the influence of any

form errors.

One objective of this study is to determine whether the

relationship between plastic strain and surface roughness changes when the surface data are acquired by the

SLCM. Considering that any filtering could influence

the character of those relationships, and that the only

SLCM data in this study that are appreciably affected by

the distortion are those in the zero strain condition, we

elected to compare both data sets in the same condition

(i.e., flattened with a best regression fit). This decision

was based on an additional analytical objective: to

assess the trends in the behavior without introducing

any additional bias.

B. Evaluation of the Statistical Model

The data available here for a statistical comparison of

the goodness-of-fit between a straight-line model and a

curvilinear model are derived from a limited number of

strain states. In light of this, the curvilinear alternative

to the linear model will be a simple quadratic expression.

The formulas used for the statistical comparison to

assess linearity are

 Surface Height, μm

-2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0

(a) (b) Probability Density

 Surface Height, μm

0

2

4

6

8

10

12

14

-4 -3 -2 -1 0 1 2 3 4

Profilometer

SLCM

Raw Roughness Data

(c)

0

0.5

1.0

1.5

2.0

2.5

After Hi-pass Filter

Raw Data

-4 -3 -2 -1 0 1 2 3 4

SLCM Topography Data

Probability Density

 Surface Height, μm

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

σmeasured (μm)

σreal (μm)

With Image Distortion

Without Image Distortion

(d)

50 μm

Fig. 4—The influence of specimen curvature on the height distribution. (a) A typical topography for the AA5052 in the as-polished/zero strain

condition. (b) Probability distribution functions for the surface heights acquired by contact profilametry and by scanning laser confocal microscopy. (c) Probability distribution functions for the confocal data before and after a phase-correct Gaussian high-pass filter. (d) The behavior of

the surface data with and without a small distortion.

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 42A, SEPTEMBER 2011—2673
Sq 1⁄4 A0 þ A1e 1⁄22

Sq 1⁄4 A0 þ A1e þ A2e

2 1⁄23

The objective of this analysis is to compare the quality

of a simple linear fit (Eq. [2]) to a simple curvilinear fit

(Eq. [3]) of Sq as a function of the true plastic strain, e.

Several approaches are suitable for this purpose; however, considering the limited range of the data, and the

fact that both the line and the quadratic are linear in the

coefficients, a standard analysis of variance (ANOVA)

model comparison is a straightforward method to

indicate the more appropriate fit.[27]

An F-test can be used to compare the performance of

any pair of nested multilinear (or polynomial) models on

a given data set. A quadratic model is an extension of a

linear model in that it consists of a linear model with an

additional term. As shown in Eq. [4], the F-test refers to

the ratio of the residual sum of squares (RSS) of the

partial linear model, minus the RSS of the full quadratic

model, divided by the full model RSS normalized by

appropriate degrees of freedom (n – 3) to the corresponding F distribution:[28]

Fn3 1⁄4 RSSlinear  RSSquadratic  

RSSquadratic

n3

  1⁄24

The t-values associated with the coefficients of Eqs. [2]

and [3] are shown for the SLCM data in Table III and

for the profile data in Table IV. In general, a value ‡2.0

for any individual t-value indicates a statistically significant nonzero coefficient within a 95 pct confidence level

(as defined by the Student’s-t distribution[29]

). As shown

in Table III, all of the t-values are ‡2.0 for the linear

model, which suggests a credible linear trend with strain.

However, the A2 t-values for the O and HT conditions

are also ‡2.0. This suggests that the data from these heat

treatments could be quadratic as the linear terms are not

distinguishable from zero (t<2). In contrast, the tvalues from the profile data (Table IV) do not indicate

that either model is consistently applicable for any heat

treatment. As such, the results from the simplest

regression analysis are inconclusive.

Table V summarizes the ANOVA model comparisons. The data were assessed using the analysis of

variance (ANOVA) model comparison, which is embedded in many standard statistical analysis packages for

multilinear and polynomial fitting. This analysis used

the built-in ‘‘model ANOVA’’ command in the software

package S-plus* to perform the linear-quadratic comparisons on each of the Sq-strain curves under the two

different (operating) regimes discussed here.[30] The

small P-values in the table (<0.05) support rejection of

the linear model (the null hypothesis) and acceptance of

the quadratic model. In the case of the Sqslcm data, the

results shown in Table V establish that the linear fit is

the statistically more appropriate fit for the H32

condition, whereas the quadratic fit is statistically more

appropriate for the O and for the HT conditions. In fact,

the P-value of 0.004 for the HT data strongly supports a

preference for the quadratic model for this grain size.

However, the results for the Sqprof data do not indicate

any significant improvement with a quadratic fit—with

the O condition being the possible exception, where the

P-value of 0.047 is 0.05.

There is a fundamental, but often overlooked assumption invoked with use of ordinary least squares (OLS)

regression. That is, by using OLS to fit the expression

Y = f(X), one implicitly assumes that there is no error

in the X variable (strain) and all the error is in the Y

variable (Sq). Since most scientific measurements incur

error in both the X and Y values, standard OLS

methodology is generally considered applicable only if

Table IV. Analysis of the Profilometer-Based Sq Data

Linear Model

t-Values

Quadratic Model

t-Values

Heat

Treatment

A0

(Intercept) A1

A0

(Intercept) A1 A2

H32 0.875 1.398 0.020 1.856 –1.464

O –1.040 9.486 0.605 1.065 2.617

HT 0.311 38.710 1.042 9.356 1.260

Table V. Analysis of Variance (ANOVA) Results Assuming

No Error in Strain

Heat

Treatment

Data

Source

ANOVA

P-Value

Significant

Improvement

with Quadratic Fit?

H32 confocal 0.517 no

O confocal 0.026 yes

HT confocal 0.004 yes

H32 profiles 0.281 no

O profiles 0.047 marginal

HT profiles 0.801 no

Table III. Analysis of the SLCM-Based Sq Data

Linear Model

t-Values

Quadratic Model

t-Values

Heat

Treatment

A0

(Intercept) A1

A0

(Intercept) A1 A2

H32 13.450 5.279 10.780 0.589 0.780

O 2.980 10.950 5.602 1.611 3.141

HT 4.048 10.860 12.320 1.611 5.821

*Certain commercial equipment, instruments, or materials are

identified in this article to foster understanding. Such identification

does not imply recommendation or endorsement by the National

Institute of Standards and Technology, nor does it imply that the

materials or equipment identified are necessarily the best available for

the purpose.

2674—VOLUME 42A, SEPTEMBER 2011 METALLURGICAL AND MATERIALS TRANSACTIONS A
the relative error in Y is at least one order of magnitude

greater than the relative error in X.

As shown in Table II, the relative standard uncertainty of the means of the strain values ranges between

0.4 and 4.7 pct relative error (RE). In addition, some of

the error values lie in the 1 to 3 pct range, and none of

the data exhibit a clear, consistent pattern of increase or

decrease as a function of strain. Recalling that Sq is one

standard deviation of the superposed height distribution, one can use a propagation-of-error approximation

to estimate the RE in Sq (i.e., RE 1⁄4 Sq= ffiffiffiffi

2n p ð Þ s

  100,

where n is the sample size upon which the standard

deviation is based). This expression reveals that the RE

is 0.25 pct for the Sqprof data and 0.06 pct for the Sqslcm

data.** Both Sq relative error values are considerably

smaller than the relative error in the strain data. As

such, reversing the X-Y orders of magnitude is required

for OLS.

There are two standard approaches to regression

analysis in situations where the error magnitudes are

reversed (i.e., RE(X)>RE(Y)). The first method is to

simply reverse the roles of X and Y variables. That is,

model the strain as a linear, or curvilinear, function of

Sq. Even though the order of magnitude rule may not

consistently hold for the Sqprof data, it does hold for the

Sqslcm data, and in all cases, RE(strain) > RE(Sq).

Consequently, the error structure of the reversed X and

Y data is in better accord with the OLS assumption, and

one can apply the standard model ANOVA comparison.

The ANOVA on the Sqprof HT data produces a clear

preference for the linear model with either variable

ordering (P = 0.80 for X-Y compared to P = 0.72 for

Y-X), whereas the Sqslcm HT data exhibit a borderline to

strong preference for the quadratic model (P = 0.004

for X-Y compared to P = 0.054 for Y-X). In the case of

the H32 data, both variable orderings support a linear

model for the Sqprof data (P = 0.28 for X-Y compared

to P = 0.13 for Y-X). Similar results were obtained for

the Sqslcm data (P = 0.52 for X-Y compared to

P = 0.89 for Y-X). The most significant variable

ordering difference was observed in the O condition.

The Sqprof data exhibit a borderline quadratic preference

(P = 0.047) for X-Y, but a strongly quadratic preference (P = 0.001) for Y-X. In contrast, the Sqslcm data

exhibit a strong quadratic preference for X-Y, but a

strong linear preference for Y-X. Even though the trends

were reversed for the O condition, the similarities

observed in the trends for the HT and the H32 heat

treatments indicate that reversing the order of the strain

and Sq variables produced comparatively consistent

results.

A second standard approach to regression when the

error magnitudes are reversed is to apply Errors in

Variables methodology.[31] Since an exact, closed-form

theory for a specific nonstandard case (e.g., when the

estimated standard errors fluctuate in a haphazard

manner, such as that exhibited by the strain data) can

be difficult to implement, one approach is to use

orthogonal distance regression (ODR). ODR employs

an optimization algorithm that estimates the polynomial

coefficients by minimizing the squared sums of the X and

Y errors together, whereas OLS only minimizes the

errors in the Y variable.

Many software packages include ODR analysis. The

results presented here are derived from the ODRPACK

adaptation available in DATAPLOT.[32] Analyses of

the t-statistics of the ODR coefficients along with the

magnitudes of the residual standard error revealed the

following. In the HT condition, a no-intercept linear

model was more appropriate for the Sqprof data,

whereas a quadratic model was more appropriate for

the Sqslcm data. In the H32 condition, a quadratic was

the more appropriate model for the Sqprof data, which

contradicts both sets of OLS results. The ODR results

were consistent with the OLS results for the Sqslcm data

in that a linear model was more appropriate. As was the

case with the reversed OLS results, the ODR results for

the O condition were less obvious. A quadratic model

was slightly more appropriate for the Sqprof data,

whereas neither model was clearly dominant in the

Sqslcm data. Table VI is a summary of the most

appropriate model as determined by each of the three

regression analyses.

Regardless of the regression method used, the most

obvious difference in the form of the regression model for

Table VI. Summary of the Regression Analyses Results

Heat

Treatment

Data

Source

Ordinary Least-Squares

Regression (OLS)

Reversed Ordinary Least-Squares

Regression (r-OLS)

Orthogonal Direction

Regression (ODR)

H32 confocal linear linear linear

O confocal quadratic linear ambiguous

HT confocal quadratic ambiguous quadratic

H32 profiles linear linear quadratic

O profiles ambiguous quadratic quadratic

HT profiles linear linear linear

**The relative error in the Sq data was substantially reduced by the

superposition of the five individual data sets into a single distribution.

If the data were not superposed, the mean relative error would be

@0.56 pct for the Sqprof data and @0.14 pct for the Sqslcm data.



Dataplot is a free, downloadable statistical software package

maintained by the National Institute of Standards and Technology.

Additional information about this package can be obtained at the

Dataplot web page.

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 42A, SEPTEMBER 2011—2675
the Sq data was observed in the HT heat treatment. This

difference is illustrated for the two acquisition techniques

in Figure 5. In this figure, the best linear and quadratic

models are plotted as Sq vs true plastic strain using the

Sqprof and Sqslcm data sets. The preference for a

particular model is graphically obvious for each data set.

IV. DISCUSSION OF RESULTS

The differences in the results of the regression analyses

raise two fundamental questions: Which data set provides the better representation of the true relationship

between Sq and plastic strain, and why does the

regression model change from linear to quadratic as a

function of the grain size?

The answer to the first question lies in the manner in

which the surface data are sampled. The stylus profilometry technique has many associated issues that could

influence the quality of the surface data. The most

germane to this evaluation is mechanical filtering. Even

though great care was exercised during the acquisition

of the profile data, natural filtering of the surface data in

a way that alters the magnitudes of profile characteristics

(i.e., the peaks and valley values) is unavoidable in any

contact technique. Radhakrishnan[33] showed how the

size ratio between the profile tip and the surface features

could adversely influence the accuracy of the profilometer measurement. As shown schematically in Figure 6,

x

x

x

x

x

x

0 0.05 0.1 0.15 0.2

0

Quadratic Model

0.5

1.0

1.5

2.0

2.5

3.0

True Plastic Strain

(b)

x x

x

x

x

x

x

0 0.05 0.1 0.15 0.2

0

0.5

1.0

1.5

2.0

2.5

3.0

True Plastic Strain

(d)

x x

x

x

x

x

x

0 0.05 0.1 0.15 0.2

0

0.5

1.0

1.5

2.0

2.5

3.0

True Plastic Strain

(c)

Surface Roughness, SqSLCM, μm

x

x

x

x

x

x

0 0.05 0.1 0.15 0.2

0

0.5

1.0

1.5

2.0

2.5

3.0

Linear Model

True Plastic Strain

(a)

Surface Roughness, SqProf, μm

Linear Model Quadratic Model

Profile Data

SLCM Data

Profile Data

SLCM Data

Fig. 5—The best linear and quadratic models shown in individual plots of Sq vs true plastic strain using both the contact profilometry and the

scanning laser confocal microscope data sets. The difference in the preference for a particular model is graphically obvious for each data set.

Stylus Tip

Traced Profile

Original Surface

Measurement 

Error

Fig. 6—A schematic showing how the size ratio between the profile

tip and the surface features could adversely influence the accuracy of

a contact profilometer measurement.[33] Such filtering would tend to

artificially skew the roughness data toward the component(s) that

produce the largest surface displacements (i.e., grain boundary–

localized roughness).

2676—VOLUME 42A, SEPTEMBER 2011 METALLURGICAL AND MATERIALS TRANSACTIONS A
mechanical filtering tends to distort the true character of

the topography data by artificially accentuating the

magnitudes of the peaks, reducing the magnitudes of the

valleys, and suppressing the contributions to the magnitude of Sq by the more subtle features in the profile.

The resulting bias in the surface measurement could

easily degrade the fidelity to the point where the only

possible outcome is the simplest relationship between Sq

and plastic strain.

The SLCM data are not mechanically filtered, because

they are derived from a noncontact optical technique.

Furthermore, the technique developed to extract and

evaluate the surface data from the topographical images

captures a higher degree of the three-dimensional

character of the surface features. This results in a more

accurate representation of the original surface than that

provided by the profilometer data. The basis for this

assertion is the combination of a smaller probe (635 nm)

and the aforementioned spacing between the measured

surface heights in each of the five SLCM images. Recall

that the spacing between points was 1.56 lm in both the

x- and y-directions and encompassed an image area of

6.4 9 105 lm2

. While the 0.22 lm/point spacing in the

profilometer data is better than the SLCM in the

x-direction, the offset between the 10 profilometer traces

(the y-direction) was approximately 1 mm. That is, a

four orders of magnitude improvement in the sensitivity

of the sampling in the y-direction achieved through

SLCM enables a substantially more robust statistical

analysis of the surface data. Areal scans are certainly

possible with profilometry; however, this approach was

not practical in this situation for two reasons. First, the

traces cannot be closer than the radius of the profile tip,

which in this case is 5 lm and is at least 3 times greater

than the distance in the SLCM. Second, an additional

offset must be included to compensate for the surface

that is destroyed by the ceramic tip during the acquisition of the data. The combination of the higher density

of samples in the SLCM data and the lack of mechanical

filtering provides substantially greater surface detail

than the profile data (1.3 9 106 samples/strain condition vs 80 9 103 samples/strain condition). As a result,

the same level of surface detail is simply not obtainable

with profilometry under these conditions. Therefore,

despite the presence of a small offset at zero strain, the

trends exhibited by the SLCM data are considered the

better representation of the true relationships between

Sq and plastic strain for this evaluation.

The composition of the data in the sample is the key

to the answer for the second question, why does the

regression model change with grain size. Plastic deformation in a polycrystal occurs by a highly complex

process, and the measurable surface roughness is generally composed of two types: within-grain roughness

and grain boundary–localized roughness. The amount

of deformation that occurs within each grain depends on

several factors: the orientation and the Taylor factor[34]

of the individual grains, and the constraints imposed by

neighboring grains at or below the surface. That is, for a

grain in a favorable orientation for slip, the deformation

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 localize in the

grain boundary regions due to the additional shear

displacements required to produce grain rotation and to

maintain grain-to-grain contiguity.[35] The resulting

grain-to-grain anisotropy produces an overall roughness

character that is a mixture of both primary slip and near

grain boundary deformation. Moreover, neighboring

surface grains possessing the same level of macroscopic

strain can exhibit appreciably different amounts of

measurable surface roughness.[6] Since the anisotropy

is strongly influenced by the number of grains in a

particular sample, the grain size becomes a critical factor

in the deformation process. In fact, the component that

has the greatest influence on the magnitude of the

surface roughness in a polycrystal is the roughening

resulting from the grain-to-grain anisotropy. This is also

referred to as grain boundary–localized roughness.

An example of how grain size effects can influence the

relationship between plastic strain and the character of

the deformed surface is shown in Figure 7. The three

topographies in the figure were all acquired at the same

magnification after the application of approximately 0.1

true uniaxial strain. As shown in Figure 7(a), the fine

grain size in the H32 condition produces a high number

of grain boundaries per unit area. Since the surface data

in this condition consist almost entirely of grain

boundary–localized surface roughness, the Sqslcm values

primarily reflect the influence of that component. This is

exhibited in the relationship between Sqslcm and plastic

strain, which was linear for all the regression methods.

As the grain size increases (Figures 7(b) and (c)), the

area fraction of grain boundaries decreases and the ratio

of grain boundary–localized roughness to within-grain

roughness changes accordingly. The corresponding relationship between Sq and plastic strain also becomes

more curvilinear in nature. The surface morphology of

the largest grain size (HT, Figure 7(c)) is distinctly

different from that exhibited in the H32 heat treatment

(Figure 7(a)) at the same strain level. The relative

fraction of within-grain roughness in the figure is

substantially higher than the grain boundary–localized

roughness, and this produces a more complex relationship between Sqslcm and plastic strain for the HT heat

treatment. Note that these results are based on images

acquired with a 10 times objective lens. If the SLCM

data were acquired at a different magnification, the

relative ratios of grain-boundary–localized to withingrain roughness, as well as the relationship between

Sqslcm and plastic strain exhibited in the H32 data, are

likely to be considerably different.

The composition of the surface data is not the sole

basis for the change in regression model with grain size.

It was the analysis technique that we developed,[15]

which superposes the individual topographic samples

into a large single distribution and subsequently minimizes the associated statistical uncertainty in Sq that

revealed the subtle influences of the composition on the

character of the surface roughness. This influence is

obscured in the profilometry data by the aforementioned

lack of sampling sensitivity in the x-y plane. As a result,

the regression analyses do not indicate any discernable

trend between Sqprof, plastic strain, and grain size.

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 42A, SEPTEMBER 2011—2677
Because the SLCM data capture more of the influence of

the composition on the roughness, these measurements

appear to produce the more reliable trends.

V. CONCLUSIONS

Three different heat treatments of AA5052 were

subjected to various levels of uniaxial plastic strain,

and the resulting surfaces were evaluated using two

different data acquisition techniques: SLCM and stylus

profilometry. The behaviors of the surface data from

both sources were analyzed using a technique that

minimizes the statistical uncertainty in the Sq data and

compared as a function of true plastic strain. The trends

in the Sq data were then evaluated with three different

statistical methods to determine the more appropriate

regression model for the relationship between the Sq

data and true plastic strain. The first method was a

standard OLS and model comparison ANOVA. The

magnitudes of the error in the strain were found to be

substantially larger than that in the Sq data. The roles of

the X and Y variables were then reversed and a second

OLS/ANOVA was performed on those data. The third

method, ODR, which simultaneously accounts for the

error in both the strain and Sq data, was also employed.

Even though the sizes of the available data sets did cause

some differences among the results from the different

regression methods, these analyses demonstrated that a

linear fit was statistically more appropriate for the finest

grain size (the H32 heat treatment) and that, as the grain

size increased, a quadratic fit became the better statistical model. The ANOVA performed on the profilometry data did not reveal any discernable trends.

The differences observed between the SLCM and the

profilometry behaviors were attributed to two sources:

the manner in which the surface data were sampled and

the composition of the data in the surface measurements. The mechanical filtering inherent to the contact

profilometry technique introduced a bias that degraded

the fidelity of the surface data and altered the relationship between Sq and plastic strain. In contrast, the

SLCM data are not mechanically filtered and the

technique developed to extract and evaluate the surface

data from sets of topographical images captured more

of the three-dimensional character of the surface features. The higher density of surface samples in the

SLCM measurements produced a more robust representation of the original surface than the profilometer

measurements. Therefore, even though a small offset

was observed in the SLCM data at zero strain, the

trends exhibited by these data are considered the more

reliable representation of the true relationships between

Sq and plastic strain for this evaluation.

The composition of the topographic data in these

analyses changed because the relative area fractions of

grain boundary–localized roughness and within-grain

roughness are both dependent on the grain size. The fine

grain size produced a more linear relationship between

Sq and plastic strain, because the topographies consist

almost entirely of grain boundary–localized roughness.

H32 Condition O Condition HT Condition

Tensile Axis

100 μm

100 μm

100 μm

Surface Height, μm

(a)

(b)

(c)

Fig. 7—An example of how the grain size influences the composition

of surface roughness data. The three topographies shown in this figure were all acquired at the same magnification after approximately

0.1 true uniaxial strain. The ratio of grain boundary roughness and

slip-induced roughness changes with increasing grain size in these

figures. Similarly, the composition of the surface roughness in the

HT condition (C) is distinctly different for that in the H32 condition

(A).

2678—VOLUME 42A, SEPTEMBER 2011 METALLURGICAL AND MATERIALS TRANSACTIONS A
That is, the measurable roughness is primarily influenced by a single component that generally scales

linearly with strain. An increase in grain size reduced

the relative area fraction of grain boundaries, thereby

altering the ratio between grain boundary roughness

and within-grain roughness. This was illustrated in a

side-by-side comparison of the topographies produced

by 10 pct true uniaxial strain. The complex surface

morphology in the largest grain size required a more

complex relationship between Sq and plastic strain to

describe the behavior.

The answer to the question of whether the relationship between surface roughness and plastic strain is truly

linear seems to be ‘‘it depends.’’ Considering all of the

factors that could influence the composition or the

interpretation of surface roughness data in any given

measurement, a priori assumptions regarding the form

of the relationship that exists between surface roughness

and plastic strain are simply not appropriate. Such

assumptions become particularly precarious when the

character of the surface roughness data is a key factor in

a numerical model designed to predict the behavior

during sheet metal forming.

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