A Study of the Fundamental Relationships between

Deformation-Induced Surface Roughness and Strain

Localization in AA5754

M.R. STOUDT, J.B. HUBBARD, M.A. IADICOLA, and S.W. BANOVIC

Three-dimensional, matrix-based statistical analysis methods were developed and integrated

with high-resolution topographical imaging, to assess how microstructural changes influence the

evolution of plastic deformation and strain localization in a commercial AA5754-O aluminum

sheet in three in-plane strain modes. Analysis of the raw surface data revealed that the general

composition of the surface roughness was highly sensitive to strain mode and strain level. The

microstructural conditions that promote strain localization were assessed by extending a profilebased surface roughness parameter (Rt) to matrix form. Both analyses revealed that different

strain modes produce characteristic dissimilarities in the deformation at the grain level. The

localization data can be well characterized with a two-parameter Weibull distribution, suggesting that strain localization is a stochastic process that can be modeled reliably with Weibull

statistics. This study clearly demonstrates that an accurate and straightforward probabilistic

expression that captures the microstructural subtleties produced by plastic deformation can be

developed from rigorous analyses of raw topographic data. Because variations in surface

morphology profoundly influence the reliability of the numerical models used to predict strain

localization, incorporating expressions of this type could greatly enhance the accuracy of these

models.

DOI: 10.1007/s11661-009-9881-6

 The Minerals, Metals & Materials Society and ASM International 2009

I. INTRODUCTION

THE introduction of new alloys into the automobile

body is an expensive and time-consuming process. For

this reason, numerical predictions of the formability of

metal sheet have become integral components in the

automotive design process, generating a strong demand

for accurate and reliable predictions of mechanical

behavior under a wide range of deformation conditions.

The inability to reliably model the evolution of the

surface inhomogeneities produced during sheet metal

forming creates a significant obstacle that impedes the

widespread incorporation of these alloys. One of the

most common approaches to assessing the formability,

and thus the suitability, of an alloy for a particular

application is the forming limit diagram. It is well

known that macroscopic deformation in a typical metal

stamping occurs through a combination of strain modes,

or paths (e.g., biaxial, uniaxial, and plane strain).

Considering that a combination of strain modes will

likely promote failure at an overall strain that is lower

than would be expected if the deformation occurred in a

single strain mode, it is essential that the relative

influence of each component of the macroscopic strain

combination be properly identified and accurately

assessed. Such a characterization is a complicated

process requiring numerous high-resolution measurements of the deformation under each strain mode.[1]

Consequently, the limiting strains in a typical stamping

are most often estimated via numerical simulations

specifically designed to predict the onset of macroscopic,

or gross, strain localization. These complex simulations

are usually based on phenomenological constitutive

relations that assume a homogeneous response to an

imposed macroscopic strain at the microstructural level

until the onset of localization.[2]

Accordingly, significant

deviations from the homogeneous response are then

considered to indicate the onset of a critical localization

event (i.e., a direct precursor to failure such as the

formation of cracks or splits, necks, etc.).[3]

Even though models based on this approach tend to

correctly indicate the general trends, they often fail to

consistently predict the precise strains at which localization occurs.[4] The addition of revised plasticity and

kinematic hardening models and the results from studies

of the influence of various material parameters (such as

grain size and grain orientation effects, surface roughening effects, and other damage mechanisms) on strain

localization have enhanced the reliability of the numerical models.[5–12] Despite the significant improvements

that have been made to these models, inconsistencies

still exist between the mechanical behavior that is

predicted and the behavior that is observed experimentally. One possible source for these discrepancies is that

the models used to predict the mechanical behavior are

M.R. STOUDT, M.A. IADICOLA and S.W. BANOVIC, Materials

Research Engineers, and J.B. HUBBARD, Research Associate,

are with the NIST Center for Metal Forming, Materials Science and

Engineering Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899. Contact e-mail: stoudt@nist.gov

Manuscript submitted October 22, 2008.

U.S. GOVERNMENT WORK

NOT PROTECTED BY U.S. COPYRIGHT

METALLURGICAL AND MATERIALS TRANSACTIONS A
fundamentally deterministic and they simply cannot

account for all of the variability that is possible in every

microstructural component involved in the plastic

deformation process of a polycrystalline material. In

many cases, the complexity of the microstructural effects

is often greatly simplified, to reduce the required

computational time.

Clearly, plastic deformation in a polycrystalline alloy

is extremely complex and the evolved surface is the

result of many factors.[13]

Most mechanical deformation

models incorporate microstructural and surface roughness effects to some extent, as a means of estimating the

local deformation conditions that promote strain localization. Unfortunately, the fidelity of the roughness data

used in these models is questionable, given that these

data are routinely based on the erroneous assumption of

a linear relationship between the surface roughness and

the plastic strain.[11,14,15] The literature also indicates

that the strain mode has little or no influence on the

measurable surface roughness, if the grain size is

constant.[16]

The results of a recent evaluation of deformationinduced surface roughness with plastic strain plainly

demonstrated that the linear relationship between the

surface root-mean-squared (RMS) roughness (Sq) and

the plastic strain reported in the literature is a statistical

artifact resulting from inadequate sampling.[17] That is,

such a simple expression relating strain and deformation-induced surface roughness does not accurately

reflect the true deformation behavior of the free surface

of a polycrystalline material. The findings of this

study[17] also reveal that the linear relationship between

surface roughness and strain is a consequence of the

methods used to acquire and analyze the roughness

data. The assessments of the roughening behavior in the

literature are largely derived from sets of linear profiles;

they express the surface roughness as the simple average

of a single scalar roughness parameter that describes the

dispersion of heights within the profiles about the mean

(e.g., Ra, the arithmetic average roughness, or Rq, the

RMS average roughness). The character of the entire

surface at a particular strain level (i.e., XRq) is then

represented by this ‘‘average’’ roughness parameter.

When it is calculated from the mean of individual

profiles, the ‘‘roughness’’ becomes highly sensitive to the

sampling conditions and to the influence of the natural

ordering (i.e., sequences or correlations) present in the

profile data.[18] In addition, this approach generates an

ensemble of Rq distributions in which each sample in the

ensemble has its own mean and variance. The total

number of possible ensembles and the magnitude of the

statistical uncertainty associated with each ensemble

then become a function of the sampling parameters.

(That is, XRq 61⁄4 Sq, under these sampling conditions.)

If one considers a deformed surface as a composite

consisting of the topographical characteristics produced

by each mechanism that was active during the deformation process, it then becomes reasonable to assert that

integrating high-resolution topographical measurements

and analysis methods that minimize the statistical uncertainty will improve our understanding of how changes

in microstructure influence the evolution of plastic

deformation and, ultimately, critical strain localization.

This then raises the following question: Can carefully

designed surface topography measurements be used to

reliably predict critical strain localization? Such a prediction is predicated on accurate assessments, or maps, of

the morphological conditions that promote strain localization. These maps must be derived from multiple sets of

surface roughness data scrupulously acquired from a

range of deformation conditions that capture the scope of

the microstructural variations and minimize the statistical uncertainty in the roughness data.

This article presents a study of strain localization

behavior in three in-plane strain modes for a commercial

AA5754-O aluminum sheet, as determined from detailed

topographic analyses. The motivation for this research is

the concept of mapping microstructural changes that

occur as a function of plastic deformation, such as has

been shown by Frost and Ashby,[19]

Greenwood,[20]and

Mohamed and Langdon.[21] The measurement and

sampling protocols used in this study are based on

high-resolution topographical imaging techniques and

rigorous matrix-based, three-dimensional statistical

analysis methods. This article is part of a larger effort

designed to understand and predict the factors that

determine the multiaxial flow surface for automotive

alloys. This more general study includes detailed characterizations of the relative changes that occur in the

crystallographic texture with strain level in the same

three in-plane strain modes,[22] the parametric analyses

of the relative influences of strain with the strain

mode,[1]

and the evolution of multiaxial flow surfaces.[23]

The relationships among the individual elements of this

general study will be presented in future studies.

II. EXPERIMENTAL

A. Material

The aluminum alloy selected for this study, AA5754-

O, was developed primarily for automotive applications.

Like most alloys in the 5xxx series, AA5754 is substitutionally strengthened and demonstrates good overall

formability. It also exhibits discontinuous yielding of the

Porte ́vin-Le Chaˆtlelier (PLC) type (i.e., PLC banding)

that has been characterized in the literature.[23–26] As

reported by The Aluminum Association (Arlington,

VA), AA5754 typically contains 2.8 pct mass fraction

Mg for solid-solution strengthening and approximately

0.5 pct mass fraction Mn for grain refinement and

stability.[27] Metallographic examination revealed that

the grain structure was relatively equiaxed in the rolling

plane and slightly elongated along the rolling direction

(RD) of the sheet. This is indicative of the recrystallized

microstructure normally associated with the O-temper.

The mean grain size for this alloy was approximately

40 ± 20 lm.[22]

B. Generation of Surface Roughness

A set of 30 9 30 cm (12 9 12 in.) blanks was sheared

from 1-mm-thick sheet stock for testing. When the

METALLURGICAL AND MATERIALS TRANSACTIONS A
tensile axis of rolled sheet specimens lies in the transverse orientation to the RD, the evolved surface

roughness tends to be slightly more severe than when

the tensile axis is parallel to the RD.[17] Therefore, the

scope of this study focused on samples deformed in the

transverse direction.

Industry customarily uses aluminum sheet stock in the

as-received condition, O-temper, with a specified maximum acceptable initial surface roughness that varies with

the particular application. While it is important to

represent actual forming conditions, the mill scale

present on most commercial aluminum alloys completely

obscures the fine surface features that evolve during the

initial stages of the deformation process, thereby making

high-resolution measurements of these characteristics

extremely difficult. As such, the specimens in this

evaluation were polished to reveal the surface character

at low strains and to produce more consistent surface

roughness measurements. Preparation consisted of

mechanically polishing one side to a 6-lm diamond

finish using a metallographic polishing technique that

was based on standard metallurgical procedure.[28]

After polishing, the blanks were deformed in three

in-plane strain modes defined in terms of the in-plane

principal true strain ratio q 1⁄4 e2

e1

 . The first strain ratio

was q = 1 (hereafter referred to as equibiaxial); the

second strain ratio was q = 0.5 (hereafter referred to

as uniaxial); and the third strain ratio was q = 0

(hereafter referred to as plane strain). The uniaxial

samples were machined to an ASTM E08-91 standardsheet-type tensile specimen geometry. The specimens

were then strained according to standard test methods[29] at a constant displacement rate that produced a

strain rate of 6 9 104 s

1 to nominal true strain values

of 5, 10, and 15 pct. Both the equibiaxial and plane

strain deformations were imposed using an augmentation[30]

of the Marciniak flat-bottom ram test.[3]Samples

were strained in equibiaxial tension using a typical strain

rate of 5 9 104 s

1 to nominally 5, 10, 15, and 20 pct

true first-principal strain (e1). A third set of samples was

strained to similar true strain levels in the plane strain

condition. Additionally, one sample was taken to failure

in each strain mode. Note that, for purposes of this

study, ‘‘failure’’ refers to the maximum uniform strain

outside of the region in which gross localization

occurred. Therefore, the surface roughness data acquired

at this condition establish the maximum uniform surface

roughness obtainable for the AA5754-O alloy in each

strain mode. Coupons were cut from the center of each

deformed specimen for topographic analysis.

C. Surface Roughness Measurements

The surface topography was quantified for each strain

level in each strain mode with scanning laser confocal

microscopy (SLCM). Each measurement in this evaluation consisted of five well-spaced SLCM images taken

from the specimen surface. The imaging conditions were

the same for each evaluation; they consisted of an

objective lens (10 times) and a nominal total z-scan

depth of approximately 40 lm. These parameters

generated sets of 640 9 512-pixel intensity images with

12-bit resolution. The spacing between sampling points

in the (x, y) plane was fixed by the objective lens at

1.56 lm; the spacing between the focal planes within

each image was approximately 100 nm. This produces

images with nominal physical dimensions (x, y, z) of

1000 9 800 9 40 lm. The SLCM stores each topographic image data as a raw tagged image file format

(TIFF) depth map that contains the full set of imaging

parameters and binary pixel values. A computer code

was developed that uses the format standards within a

TIFF image to convert the bit values into a simple

numerical matrix of surface heights. The resulting

matrix was then trimmed to a square 512 9 512-pixel

(800 9 800-lm) 262,144-element array, to facilitate the

matrix-based mathematical operations.[17]

After conversion, the extreme values (taken to be

values in the height data greater than ±6r, where r is

the standard deviation for all the heights in that matrix)

were filtered from the data sets. This step was necessary

because some of the statistical parameters used to

interpret the surface data are highly sensitive to outlier

data points. Filtering consisted of setting the magnitude

of any extreme value equal to the mean for that

particular surface. Note that the number of affected

data points for a given surface was typically fewer than

20 points (i.e., <0.008 pct of the total number of data

points). The residual matrix was used as the source for

all subsequent assessments of the surface character. It

was essential for these images to be well separated, to

ensure that the surface data contained in each image was

statistically independent (i.e., no overlapping image

data) and that the data properly represented the full

range of surface characteristics.

III. RESULTS

A. Analysis of Topographic Data

The evolution of surface deformation is presented in

Figure 1 as a function of the strain mode and strain level.

The scanning laser confocal micrographs shown in this

figure reveal the complexity of the surface features that are

produced by the deformation process in a polycrystalline

alloy. When viewed along the columns, the representative

micrographs in this figure exhibit the evolution of surface

deformation (i.e., at 5 pct nominal true strain, 10 pct

nominal true strain, and at the ‘‘failure’’ strain) in the

equibiaxial, uniaxial, and plane strain modes. The estimated true strain levels for the failure condition are 30, 19,

and 18 pct for the equibiaxial, uniaxial, and plane strain

conditions, respectively. When viewed across the rows,

the micrographs reveal the influence of the strain mode on

the evolved surface at each strain level. The RD and

tensile axis for the uniaxial and plane strain conditions are

also indicated in each figure. Note that the 6-lm diamond

surface contains scratches and other small remnants from

the polishing procedure that may not be consistent with

the directions indicated.

Probability density functions (PDFs) were calculated

for each set of raw height data by extracting the

individual data points from the five matrices acquired at

METALLURGICAL AND MATERIALS TRANSACTIONS A
each strain condition. An example of this construction is

shown in Figure 2 for the data in the failure condition.

The failure condition was selected for this plot because

the variations among the individual data sets were

generally larger at the higher strain levels. The envelopes

in this figure are the upper and lower bounds of the PDF

range for the five sets of surface height data acquired in

each strain mode. As such, these envelopes reflect the

magnitude of the variability in the raw surface height

data for each strain mode. Because the areas of each

envelope are the same, the differences in the shapes of

the envelopes also graphically illustrate the influence of

the strain mode on the dispersion of heights about the

mean (Sq).

The results of an analysis of the surface height data

are shown in Table I. Note that this table contains both

the true strain and the effective strain values. While the

results of these analyses are expressed in terms of a true

Fig. 1—Scanning laser confocal micrographs showing the surface structure of AA5754-O at three strain levels and in three strain modes. Images

(a) through (c) are in equibiaxial strain; (d) through (f) are in uniaxial strain; and (g) through (i) are in plane strain. Images (a), (d), and (g)

show the surfaces after 5 pct nominal true strain; (b), (e), and (h) show the surfaces after 10 pct nominal true strain; and (c), (f), and (i) show

the surfaces at the maximum uniform strain.

METALLURGICAL AND MATERIALS TRANSACTIONS A
strain level as a function of the strain mode, to facilitate

a direct comparison of the influence of the strain mode

on localization, a comparison of the localization behavior in terms of a normalized (i.e., von Mises) uniaxial

strain[31]

can be achieved with the effective strain values.

Note that analyses based on these two strains will not

necessarily produce identical results. As shown in

References 17 and 32, the standard deviation r and

higher statistical moments can be determined directly

for any matrix. Because the topographical matrices were

leveled, the mean for each matrix is zero. The Sq values

are consistent with other observations,[13,33] in that they

exhibit a proportional response to an increase in the

plastic strain. While this proportionality is present in

each strain mode, the Sq magnitude appears to reach a

saturation value in the equibiaxial and plane strain

conditions.

The surface skew Ssk (or third moment, r3) is a

measure of the overall symmetry of the distribution of

the raw height data about the mean plane; because it is

normalized by the standard deviation, it is a dimensionless quantity. For a Gaussian distribution, Ssk = 0 and,

as shown in Table I, the skew values are all slightly

negative, indicating that a valley is somewhat more

probable than a peak. Unlike the Sq values, the skews

do not appear to reflect any systematic change with

strain level or strain mode. The surface kurtosis Sku (or

fourth moment, r4) is a measure of the overall shape of

the height distribution about the mean plane and, like

the skew, it is also normalized by the standard deviation

and is dimensionless. For a Gaussian distribution,

Sku = 3, and the Sku values shown in Table I are all

somewhat greater than 3. This indicates that the raw

height distributions have slightly sharper central peaks

and longer tails than would occur if the distribution

were Gaussian. As was the case with the skew values, no

systematic change with strain is apparent in the kurtosis

data.

B. Analysis of Localization Data

Analyses of all of the values contained within each set

of raw height data provide details about the evolved

surfaces with respect to the strain mode. They also

reveal generalities about the relative trends resulting

from the plastic deformation specific to a particular

strain mode. However, these data cannot quantify the

magnitude of a specific feature within the data. Quantification of the relationship between the surface roughness and strain localization requires an analysis that is

based on a measure that describes the relative changes in

the local surface heights. Data of this type can be

obtained via a roughness parameter that is sensitive to

the changes in the peaks and valleys on the surface. One

such parameter is the maximum height of a profile Rt,

Table I. Analysis of Raw Surface Height Data

Strain Path

True

Strain

Effective

Strain

Mean

(lm)

Uncertainty*

(lm)

Sq

(lm)

Uncertainty

(lm) Skew Uncertainty Kurtosis Uncertainty

As polished 0.00 0.00 8.26 9 109 1.38 9 108 0.921 0.027 0.234 0.021 3.468 0.155

Equibiaxial 0.05 0.10 1.79 9 108 2.00 9 108 1.308 0.055 0.338 0.075 4.004 0.194

0.10 0.21 1.95 9 108 3.84 9 108 2.136 0.102 0.465 0.055 4.149 0.163

0.15 0.28 1.23 9 108 7.98 9 108 2.606 0.078 0.305 0.043 4.203 0.149

0.20 0.41 5.11 9 108 3.90 9 108 3.611 0.039 0.291 0.058 4.164 0.131

0.30 0.62 1.19 9 107 1.04 9 107 3.886 0.180 0.060 0.121 3.720 0.165

Uniaxial

(RD ^)

0.05 0.05 1.21 9 109 2.04 9 108 0.927 0.039 0.375 0.022 3.783 0.114

0.10 0.12 3.74 9 109 5.89 9 108 1.619 0.101 0.563 0.101 4.289 0.266

0.15 0.17 8.62 9 109 6.27 9 108 2.320 0.117 0.673 0.040 4.794 0.152

0.19 0.20 3.51 9 108 6.85 9 108 3.137 0.647 0.385 0.105 4.560 0.341

Plane strain

(RD ^)

0.05 0.07 1.22 9 108 2.15 9 108 0.952 0.052 0.298 0.054 3.737 0.347

0.10 0.11 4.50 9 108 3.70 9 108 1.455 0.053 0.509 0.064 4.298 0.127

0.15 0.17 2.67 9 108 6.27 9 108 2.510 0.520 0.456 0.134 4.258 0.666

0.18 0.19 1.17 9 108 5.28 9 108 2.304 0.272 0.330 0.101 4.409 0.281

*Refers to the statistical uncertainty.



The skew and kurtosis for a Gaussian distribution are normalized by the standard deviation and are dimensionless.

0

0.03

0.06

0.09

0.12

0.15

-25 -20 -15 -10 -5 0 5 10 15 20 25

Probability Density

Uniaxial

Equibiaxial

Plane

Strain

Surface Height, μm

Fig. 2—Range of probability density distributions constructed from

maximum and minimum raw height data values contained in the five

topographies acquired at maximum uniform strain in each strain

mode.

METALLURGICAL AND MATERIALS TRANSACTIONS A
which is defined as the vertical distance between the

highest and lowest points of a profile within a particular

evaluation length.[14] That is:

Rt 1⁄4 Rp þ Rv 1⁄21

where Rp is the distance between the highest point of the

profile and the mean line and Rv is the distance between

the lowest point of the profile and the mean line within a

particular evaluation length. (Note that heights lower

than the mean plane are traditionally considered negative.) The Rt parameter is ideal for characterizing the

surface conditions that promote critical strain localization and for quantifying the magnitudes of those surface

conditions for several reasons. (1) This parameter is

highly sensitive to a local change in the surface height

and it is acquired through a straightforward calculation.

(2) The length of the profile over which Rt is determined

can range from a few data points within a profile to the

entire profile, so the Rt parameter can track the changes

in the surface height on a local or overall basis. (3) The

magnitudes of the local surface extremes will change

with strain to the point at which they reach a value that

correlates with the onset of critical strain localization.

(4) The Rt parameter can be easily determined for any

two points within a particular region during a numerical

simulation, thereby making changes in the local surface

conditions directly integrable into formability models.

The statistical uncertainty associated with profilebased analysis can be minimized by extending the Rt

parameter to a matrix form.[17] This is accomplished by

creating a Euclidean distance matrix.[35] The construction of this matrix required subdividing a 512-row 9

512-column source matrix into smaller matrices consisting of 64 element cells (8 pixels/row 9 8 pixels/column).

This created a differential matrix consisting of 4096

elements (i.e., 64 9 64 cells). The maximum difference

in the surface height for a given cell, Rt(i,j), was

determined from the 64 values contained within that

cell. For this reason, Rt(i,j) is the maximum displacement

normal to the mean plane in the source matrix at the

coordinates of that cell. Because the matrix format

preserves all of the spatial coordinates for an Rt cell by

construction, both the magnitude and the location of

each Rt(i,j) value are quantified by this method. Consequently, this technique links any feature in the source

image to the corresponding change in magnitude of the

local surface height and the level of resolution can easily

be adjusted by altering the cell dimensions. Given that

the mean grain size of this alloy is 40 lm, the 12.5 9

12.5-lm cell used in this assessment was an acceptable

level of resolution.

The results of the Rt matrix constructions are shown

in Figure 3. The data used to construct the maps in

Figure 3 are identical to those presented as images in

Figure 1. The main distinction between Figures 3 and 1

is the change in resolution resulting from the Rt cell

construction. The range of Rt magnitudes was mapped

into a standard 8-bit color scale in which blue is low and

red is high, so that the color of an individual cell directly

reflects the local magnitude of Rt. The maps shown in

Figure 3 exhibit the variations in Rt intensity produced

by the different strain conditions in exactly the same

manner as in Figure 1. That is, when viewed across the

rows, the maps reveal the local Rt intensity as a function

of the strain mode at three different strain levels and,

when viewed down the columns, the maps reveal the

changes in the local Rt intensity as a function of the

strain level for the three strain modes. While it differs

with the strain mode, the number of high-intensity Rt

cells increases progressively with the strain level, with

the highest number of these cells occurring in the failure

condition. Note that the scales for the Rt maps are

different for each strain path and the significance of this

difference will be addressed later in this article.

Upper- and lower-bound probability density envelopes were constructed for the Rt data in the same

manner as was used for the height data (Figure 2).

Again, the data for the failure condition are shown.

There are two distinct differences between the PDF

envelopes shown in Figures 4 and 2. First, the overall

widths of the PDF envelopes are substantially greater

for the Rt data than for the height data. This indicates a

higher variability within the individual Rt data. Second,

and more important, the overall shapes of the distributions are markedly different. The PDF envelopes shown

in Figure 4 all exhibit a comparatively shorter left-hand

tail and a substantially longer right-hand tail than those

shown in Figure 2, which denotes that any analysis of

the Rt data that is based on Gaussian statistics would

not produce meaningful results.

C. Weibull Analysis

The general form in Figure 4 is consistent with that of

an extreme value distribution (i.e., the limiting distribution for a maximum or minimum of a large collection of

random observations from the same distribution).[36]

Clearly, these data can be analyzed with several distributions,[37] but the Weibull distribution is one of the

more well known[38,39] and was selected for this reason.

Weibull statistics are used extensively in reliability

predictions to estimate failure, because such predictions

are concentrated on the incidence of statistically rare

events (i.e., the tail region). The Weibull distribution can

be used to describe a unimodal distribution that has one

disproportionally long tail, such as those shown in

Figure 4.

[40–42]

A Weibull distribution has two forms: a two-parameter and a three-parameter distribution. Most Weibull

analyses are based on a two-parameter distribution,

because the third parameter, a ‘‘guaranteed,’’ or threshold, value (i.e. a value after which no events occur) is

typically not known a priori. The Weibull distribution is

continuous and has a probability density function of the

form:[42]

fPDFð Þ1⁄4 x; a; b b

a

x

a

 ð Þ b1  e x ð Þa

b

1⁄22

In this equation, (x ‡ 0) and (fPDF(x; a, b) = 0 for

x<0). The two parameters, a, a scale parameter, and

b, a shape parameter, must be real and >0. Because of

the extensive use in reliability prediction, the Weibull

METALLURGICAL AND MATERIALS TRANSACTIONS A
distribution is more commonly expressed in terms of

the cumulative distribution function (CDF):

fCDFð Þ1⁄4 x; a; b 1  e x ð Þa

b

1⁄23

Note that, for a given strain level, the CDF is the

probability P of the occurrence of an Rt magnitude that

is less than or equal to any given Rt (i.e.,

fCDFðRtgivenÞ 1⁄4 PðRt  RtgivenÞ). In addition, the CDF

is bounded by the following conditions: fCDF(0) = 0,

and fCDF(¥) = 1.

The five sets of Rt values acquired in each strain

condition were fitted with a Weibull distribution, using

commercially available statistical software,[43] and the

mean a and b parameters for a given strain condition

were determined from each set of a and b values using

regression analyses. The results of the Weibull analysis

are shown in Table II. The a and b values are plotted as a

Fig. 3—Set of plots exhibiting the propensity for strain localization as a function of strain level and strain mode. Each plot is the Rt roughness

data calculated for corresponding image shown in Fig. 1.

METALLURGICAL AND MATERIALS TRANSACTIONS A
function of the strain level for the equibiaxial, uniaxial,

and plane strain modes in Figures 5 and 6, respectively.

Note that the uncertainties in the a and b parameters

were typically smaller than the plot symbols in both

figures. The considerably smaller range exhibited by the

b values in Figure 6 suggests that the shape parameter

may be insensitive to changes in the strain state.

A Weibull distribution that describes the overall

surface character can be estimated for each strain level

and strain mode by substituting the mean a and b

parameters calculated for those strain conditions into

Eq. [3]. Because each Rt matrix contained 4096 values, a

random data set with the same number of values was

extracted from the Weibull distribution function for

each strain condition, to represent the overall surface

behavior. The results illustrate the relationships that are

likely to occur between the strain level, strain mode, and

Rt magnitude. The characteristics of these estimated

Weibull surface distributions are shown as the cumulative probability of the Rt magnitude plots for the

equibiaxial, uniaxial, and plane strain conditions in

Figures 8, 9, and 10, respectively. All three figures

demonstrate systematic changes in the estimated Rt

distributions with increasing strain. In the case of the

equibiaxial and plane strain conditions, the overall

shape of the CDF exhibits no appreciable changes after

reaching approximately 20 pct in equibiaxial strain and

15 pct in plane strain mode. While saturation behavior

also represents the behavior at failure in the uniaxial

strain mode, none is apparent (Figure 10). At 0 pct and

5 pct strain, both the uniaxial and plane strain conditions do not appear to produce significant changes in the

Rt Magnitude, μm

Uniaxial

Equibiaxial

0

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 10 0 20 30 40 5

Plane

Strain

Probability Density

Fig. 4—Range of probability density distributions constructed from

maximum and minimum Rt data contained in the five topographies

acquired at maximum uniform strain in each strain mode.

Table II. Analysis of Weibull Fit to Strain Localization (Rt) Data

Strain Path True Strain Mean Variance Skew* Mode Coefficient of Variation

As polished 0 2.944 1.137 0.165 2.883 0.362

Equibiaxial 0.05 4.338 3.100 0.294 4.083 0.406

0.10 7.537 8.761 0.255 7.188 0.393

0.15 9.349 14.055 0.280 8.844 0.401

0.20 13.349 28.121 0.269 12.674 0.397

0.30 13.446 24.674 0.186 13.090 0.369

Uniaxial (RD ¥) 0.05 3.0264 1.3408 0.2253 2.9132 0.383

0.10 5.2469 4.3256 0.2662 4.9859 0.396

0.15 7.6222 11.4687 0.4063 6.8647 0.444

0.19 11.1392 22.1056 0.3416 10.3030 0.422

Plane strain (RD ¥) 0.05 2.7052 1.5291 0.4433 2.3957 0.457

0.10 4.7749 4.1568 0.3559 4.3918 0.427

0.15 8.0626 12.6252 0.3958 7.2942 0.441

0.18 7.9014 10.7269 0.3195 7.3697 0.415

*The skew for a Weibull distribution is normalized by the standard deviation and is dimensionless.

0

2

4

6

8

10

12

14

16

0 0.05 0.10 0.15 0.20 0.25 0.30

Alpha (scale) Parameter Value

Nominal True Strain

Biaxial

Uniaxial

Plane Strain

Fig. 5—Weibull scale (a) shown as a function of nominal true strain.

METALLURGICAL AND MATERIALS TRANSACTIONS A
shape of the Rt distribution, whereas significant changes

are evident in the equibiaxial condition for the same

strain levels.

The uncertainty present in the mean a and b values

(Table II, Figures 5 and 6) produced a range of Rt

probabilities for each strain condition shown in

Figures 7, 8, and 9. As noted previously, the largest

uncertainties were observed at the failure condition, for

each strain mode. The degree of variation in the

estimated Rt distributions at failure is exhibited in the

width of the uncertainty envelope associated with each

strain mode in Figure 10. Unlike those shown in

Figures 2 and 4, the envelopes in this figure are

estimates of the largest uncertainties possible for the

two Weibull parameters, as calculated from the amin,

amax, bmin, and bmax in each strain mode. Note that the

three envelopes in this figure do not exhibit significant

changes in overall shape; this is consistent with the small

0 % 5 %

10 %

15 %

20 %

30 %

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35

Cumulative Rt Probability Distribution

Rt

Fig. 7—Influence of equibiaxial strain on the mean cumulative probability distribution of Rt. Each curve is the mean of the five Weibull

scale and shape parameter values determined for that strain level, as

shown in Figs. 5 and 6.

0 %

5 %

10 %

15 %

19 %

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30

Cumulative Rt Probability Distribution

Rt

`

Fig. 8—Influence of uniaxial strain on the mean cumulative probability distribution of Rt. Each curve is the mean of the five Weibull

scale and shape parameter values determined for that strain level, as

shown in Figs. 5 and 6.

0 %

5 %

10 % 15 %

19 %

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25

Cumulative Rt Probability Distribution

Rt

Fig. 9—Influence of plane strain on the mean cumulative probability

distribution of Rt. Each curve is the mean of the five Weibull scale

and shape parameter values determined for that strain level, as

shown in Figs. 5 and 6.

Beta (shape) Parameter Value

0

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0

Nominal True Strain

0 0.20 .05 0.10 0.15 0.25 0.30

Biaxial

Uniaxial

Plane Strain

Fig. 6—Weibull shape parameters (b) shown as a function of nominal true strain.

METALLURGICAL AND MATERIALS TRANSACTIONS A
variations observed in the shape parameter b. For this

reason, variations in the scale parameter a have the

greatest influence on the probability distribution. Given

that the CDFs shown in Figure 10 are the probable Rt

values predicted by the estimated Weibull surface

distributions at failure, each envelope in this figure

reflects the probability that a critical strain localization

event will occur for this alloy in that strain mode. For

example, an Rt value with a magnitude of 15 lm

signifies that the probability for critical strain localization lies between approximately 60 and 68 pct in

equibiaxial strain, 72 and 90 pct in uniaxial strain, and

96 and 99 pct in plane strain. Similarly, a 90 pct

probability for critical strain localization corresponds

to an approximate Rt value range between 11 and

12.5 lm for equibiaxial strain, 15 and 19 lm for

uniaxial strain, and 19.5 and 21 lm in plane strain.

The probability distribution envelopes in Figure 10 are,

therefore, the basis for the color maps used in Figure 3.

That is, the maximum of the color range (i.e., the dark

red) in Figure 3 corresponds to the magnitude of Rt on

the CDF (rounded to the nearest whole number) at

which the lower bound of the probability envelope

reached a value of 95 pct for each strain mode, that is,

the minimum Rt value at which the probability of the

occurrence of a critical strain localization event is

approximately 95 pct. Similarly, the color of each cell

in Figure 3 directly corresponds to the local probability

for critical strain localization.

IV. DISCUSSION OF RESULTS

The results of this study demonstrate that rigorous,

matrix-based statistical analysis methods and appropriate

surface roughness parameters can be combined to

create an effective technique for characterizing the

surface inhomogeneities generated by plastic deformation. All of the topographic data acquired for this

evaluation had a measurement resolution sufficient to

assess the morphological conditions that promote strain

localization developed by the three strain modes shown

in Figure 1. These high-resolution data were used to

map those morphological changes in two ways: one

based on the raw height data and one based on the

magnitude of the local extremes within the height data,

with the primary intent being to determine whether

carefully designed surface topography measurements

can be used to reliably predict critical strain localization.

The matrix methods used throughout are designed to

handle large data sets efficiently. This enabled each

surface condition to be based on five statistically

independent topographic images (each consisting of

262,144 samples), as opposed to most traditional surface

analyses, which are based on the average of a few linear

profiles. This approach effectively maximized the statistical reliability of each surface characterization, resulting in extremely accurate analyses of the surface data

and, because of this extraordinarily high number of

samples, any differences between the individual data sets

were directly attributed to variations in the surface

character resulting from the local strain conditions, and

not to measurement error.

The raw topographical maps reveal the relative

influences of strain level and strain mode on the overall

composition of the surface morphology. Thorough

examination of the metallurgical history of the

AA5754-O alloy revealed that, while this alloy exhibits

PLC banding and some of the surface structure is likely

due to inhomogeneities resulting from this PLC behavior, the morphologies shown in Figure 1 were primarily

generated by dissimilarities in the deformation at the

grain level. Because the deformed surfaces are highly

sensitive to strain level and strain mode, the numerical

results from the topographical analysis (Table I) reflect

the variability in the local strain conditions produced by

the strain level and strain mode. At low levels of plastic

strain, the amount of deformation that occurs within

each grain depends on the individual orientation,[44,45]

the local Taylor factor,[46] and the constraints imposed

by neighboring grains located at or just below the

surface.[47] That is, in those grains in which the

orientation favors slip, the deformation occurs by

primary slip in the interior regions. Conversely, in

grains in which the slip conditions are not as favorable,

the deformation tends to localize in the grain boundary

regions because of the additional shear displacements

required to produce grain rotation and maintain

grain-to-grain contiguity.[48] Recalling that the surface

topography contains information about each active

deformation mechanism, the composition of the surface

roughness must change accordingly with strain at the

grain level such that neighboring surface grains with the

same level of macroscopic strain can exhibit appreciably

different surface roughnesses. The higher dislocation

densities produced by localized work hardening makes

deformation by primary slip increasingly more difficult,

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35

Plane

Strain

Uniaxial

EquiBiaxial

Cumulative Probability Distribution

for Critical Strain Localization 

Rt

Fig. 10—Influence of strain mode on the cumulative probability distribution for critical strain localization. Each band is the maximum

uncertainty envelope determined from the five Weibull scale and

shape parameter values calculated at the maximum uniform strain in

each strain mode.

METALLURGICAL AND MATERIALS TRANSACTIONS A
with additional plastic strain resulting in the activation

of additional deformation mechanisms. Furthermore,

plastic deformation is a volume-conserving process, and

each strain mode in this evaluation imposes different

degrees of freedom and effective levels of strain for the

deformation of the sheet.[31] Therefore, the local conditions required to activate a particular deformation

mechanism must differ with the strain mode, thereby

directly influencing the composition of the measurable

surface roughness. For this reason, the micrographs

shown in Figure 1 also exhibit the manner in which the

physical constraints imposed by the strain modes

exacerbate the local differences in the plastic flow.

Considering that Rt is effectively the scalar projection

of the height gradient, the strain localization maps in

Figure 3 reflect the magnitude of the changes in the local

peak-to-valley height for the surfaces shown in Figure 1.

Furthermore, because it is based on a group and not on

individual pixels, an individual Rt cell is a coarsegrained measure of the differential topography on a

local level. In contrast to the maps based on the raw

surface data, the Rt maps provide a basis for quantifying

the magnitude of the microstructural dissimilarities that

promote critical strain localization (e.g., cracks, splits

and tears, and necking).

These maps also tend to visually accentuate more

macroscopic surface features in the data, especially at

the failure condition in the uniaxial strain mode

(Figure 3(f)), in which PLC banding may have had the

strongest effect on the surface character.[49]

A discussion

of the intricacies of PLC banding lies outside this

investigation; however, the aspect of this behavior that

is germane to this particular study is the evolution of the

macroscopically scaled planar bands that emerge in a

periodic fashion on the specimen surface.[50,51] Even

though recent research by Halim et al.

[52] suggests that

failure and PLC band formation may be independent, it

is generally believed that PLC banding enhances strain

localization by focusing deformation into the regions

surrounding the PLC bands.[53]

The different widths of the uncertainty envelopes

associated with the estimated Rt distributions in

Figure 10 suggest that PLC banding may also have

had an influence on the variability of the predicted Rt

behavior by increasing the variability in the Weibull

scale parameter a. In this figure, the envelopes associated with the equibiaxial and plane strain conditions are

considerably smaller with respect to the uniaxial condition, which also exhibited the greatest range in a. This is

consistent with the literature that describes the relationship between PLC banding and strain mode.[54]

Numerous cells in Figure 3 exceeded the Rt magnitude predicted by the Weibull analysis to produce

critical localization in Figure 10, but failure did not

occur. This behavior implies that the simple occurrence

of an individual cell with a critical Rt magnitude is

insufficient for initiating critical strain localization. That

is, failure may involve a threshold event or possibly a

nucleation and growth process involving several cells

that exceed the critical Rt magnitude. Further examination of Figure 3 revealed that, as the strain level

increases, groups of cells with similar relative probabilities

for strain localization (i.e., structure), do appear to form

within the Rt maps. Clearly, this structure is associated

with the color scale used for this analysis; the existence

of this structure, however, cannot be assumed an artifact

a priori. An assessment of the influence of the Rt

structure on strain localization requires detailed analyses of the spatial arrangements that may exist within the

Rt data. Research is in progress to determine whether

structure is, indeed, present in the Rt data and whether

or not it has any influence on strain localization.

V. CONCLUSIONS

Rigorous three-dimensional matrix-based statistical

analysis methods were developed and integrated with

high-resolution topographical imaging to assess critical

details related to the way in which changes in microstructure influenced the evolution of plastic deformation and strain localization in a commercial AA5754-O

aluminum alloy sheet in three in-plane strain modes.

The analysis of the surface data involved mapping the

topographic data in two distinctly different but complementary forms. The first examined the characteristics that were present in the raw surface data at

different strain levels in the three strain modes. The

results revealed that the general composition of the

surface roughness was sensitive to the strain level and

strain mode. However, the evaluation of the raw

surface data did not reveal any substantive information

about the propensity for strain localization in any

strain condition. The second extended a profile-based

surface roughness parameter (Rt) to a matrix form,

thus enabling a three-dimensional quantification of the

microstructural conditions that promote strain localization. This calculation filtered the raw surface data

and accentuated more macroscopic features in the

topographies. In addition to revealing that the conditions for strain localization were sensitive to the strain

mode, the results indicated that the Rt data could

be well characterized with a two-parameter Weibull

distribution.

The results from both analyses show that both the

raw and the filtered surface data exhibited dissimilarities

in the deformation at the grain level that were produced

by the individual strain modes. The influence of the

strain path was reflected in the widths of the envelopes

associated with the raw height distributions, the Rt

distributions, and the uncertainties associated with

Weibull predictions of the probabilities for the onset

of strain localization in the three strain modes. Furthermore, the widths of the uncertainty envelopes associated

with the Rt distributions and the Weibull predictions

reflect a dependence on the strain mode that suggests

that PLC banding might have had an influence on the Rt

data.

The results of this evaluation clearly demonstrate that

the topographical conditions that promote critical strain

localization can be assessed directly from rigorous

analyses of accurate surface roughness data and that

an accurate and straightforward probabilistic expression

capturing the subtleties produced by these microstructural

METALLURGICAL AND MATERIALS TRANSACTIONS A
conditions can be developed from such an assessment.

These results also suggest that the propensity for strain

localization is a stochastic process that can be reliably

predicted with Weibull statistics. Numerous Rt cells

that exceeded the predicted critical Rt magnitude were

‘‘observed’’ in the Rt maps at strains at which failure

did not occur. This implies that failure could involve a

nucleation and growth process requiring several Rt

cells that exceed the critical magnitude. As noted

throughout this article, local variations in surface

structure are known to have a profound influence on

the reliability of the primarily deterministic numerical

models used to predict the onset of localization.

Therefore, the incorporation of a simple expression

that reliably estimates the probability of critical strain

localization (such as the one developed in this study)

could greatly enhance the accuracy of these numerical

models.

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METALLURGICAL AND MATERIALS TRANSACTIONS A