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Materials Science and Engineering A

journal homepage: www.elsevier.com/locate/msea

The fundamental relationships between grain orientation, deformation-induced

surface roughness and strain localization in an aluminum alloy

M.R. Stoudt a,∗, L.E. Levinea, A. Creuziger a,b, J.B. Hubbarda

a NIST Center for Metal Forming, Materials Science and Engineering Laboratory, National Institute of Standards and Technology, 100 Bureau Drive, Gaithersburg, MD 20899,

United States

b Kent State University, Kent OH 44242, United States

article info

Article history:

Received 15 February 2011

Received in revised form 17 June 2011

Accepted 7 September 2011

Available online 24 September 2011

Keywords:

EBSD

Mechanical characterization

Aluminum alloys

Sheet forming

Plasticity

abstract

Polycrystalline AA6022 tensile specimens were cut from sheet stock, mechanically polished, and unia

ally strained in situ under a scanning laser confocal microscope (SLCM) using a sub-sized universal testi

apparatus. Prior to deformation, electron backscatter diffraction (EBSD) was performed on the gauge s

tions of one specimen in the rolling direction of the sheet and one in the transverse direction. Maps 

the largest displacements in the surface morphology were constructed from the SLCM data and overl

onto maps derived from the crystallographic orientation data to examine the strength of the influen

that grain orientation effects have on critical strain localization. The roles of Taylor factors, grain boun

ary misorientation, largest Schmid factors, grain sizes, coincident site lattice orientations, and local gr

breakup were considered. The largest surface displacements were observed to be concentrated at tri

junctions where there is a large difference between the Taylor factors of the individual grains. The hi

degree of correlation between the density and location of these large surface displacements and the lo

plasticity conditions indicate that a critical localization event is most likely to initiate in grain bounda

regions where unfavorable slip interactions produce the largest plastic strains.

Published by Elsevier B

1. Introduction

In recent years, numerical predictions of formability have

become major components of the design process in the automotive industry. This transition to digital design has increased the

demand for accurate and reliable property data that describe the

mechanical behavior of metal sheet under a wide range of deformation conditions. This need is particularly acute for new high

strength/low weight alloys designed to reduce gross vehicle weight

and increase overall vehicle fuel economy. However, the inability

to reliably model the evolution of the surface heterogeneities produced during sheet metal forming remains a significant obstacle

impeding the widespread incorporation of these poorly understood

materials.

Formability simulations are extremely complex and are typically based on phenomenological constitutive relations that

assume the response to an imposed macroscopic strain is homogeneous at the microstructural level up to the onset of localization

[1]. Accordingly, a significant deviation from the homogeneous

response can signify the onset of a critical localization event (i.e., a

direct precursor to failure modes such as the formation of a crack

∗ Corresponding author. Tel.: +1 3019756025; fax: +1 3019754553.

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

or split, a neck, etc.) [2]. The incorporation of revised plasticity a

kinematic hardening models, as well as the results from numero

studies examining the influence of metallurgical parameters su

as grain size, grain orientation, and surface roughness, has grea

enhanced the reliability of the data used by the numerical mod

that predict limiting strains [3–9]. Despite these improvemen

large inconsistencies still exist between numerically predict

mechanical behavior and what is observed experimentally. In lig

of the fact that plastic deformation in a polycrystalline alloy 

extremely complex, and that the evolved surface results fro

many factors, fundamental studies that either relate metallurgic

factors for a particular alloy to a performance limiting param

eter (e.g., strain localization) or improve the reliability of t

material data are essential to improve reliability of the predict

formability.

This paper is part of a series of assessments of the stra

localization behavior in commercial aluminum sheet [10–12]. T

approach for this research is inspired by the work of Frost and Ash

[13], and others [14], in that changes in the microstructure, or 

this case the surface morphology, can be “mapped” with respe

to the level of plastic deformation. The results from the earl

studies demonstrated that integrating high-resolution topograph

cal imaging and rigorous matrix-based statistical analysis metho

could capture the subtleties of the microstructural conditions th

promote a failure event in a relatively simple probabilistic expre

0921-5093/$ – see front matter. Published by Elsevier B.V.

doi:10.1016/j.msea.2011.09.050
sion. A key component in this approach is extending a linear surface

roughness parameter (the peak-to-valley surface roughness) to a

matrix form. This particular roughness parameter, Rt, was selected

for two reasons: (1) the magnitudes of the local surface extremes

must change with increasing strain to the point where they reach a

value that directly reflects the conditions required to initiate critical

strain localization, (2) the Rt parameter is based on the difference in

the magnitudes of two individual values (as opposed to the mean

of multiple values) so it is highly sensitive to minute changes in

the local surface heights. The previous results also revealed that

failure did not occur when an individual Rt value reached the critical condition. Rather, failure required multiple critical Rt values

in close proximity to one another. This implies that an additional

condition, or set of conditions, is required to initiate failure – the

most likely of which is derived from variations in the local grain

orientation.

Even though numerous evaluations of the relationship between

crystallographic texture and the mechanical properties have been

reported in the literature, few have directly evaluated the influence of crystallographic texture on the character of the deformed

surface. Raabe et al. [15] evaluated this relationship by examining

the influence of the degree of misorientation among sets of neighboring points on the surface as a function of plastic deformation.

Their results demonstrated that the heterogeneity of the deformed

surface correlated with the changes in crystallographic orientation

produced by the plastic strain.

The present study examines the strength of the influence that

grain orientation effects have on critical strain localization. The

approach adopted for this research integrates Rt-based localization maps with grain orientation analysis and it is expected that

combining these two data sets will reveal new details about how

the location of the largest Rt values correlates with the variations

in crystallographic orientation. The overall objective is to develop a

reliable and accurate tool that can be used to probe the fundamental

relationships between the local microstructure and deformationinduced surface roughness.

2. Experimental

2.1. Material

The aluminum alloy selected for this study, AA6022-T43, was

developed primarily for automotive applications. AA6022 is precipitation strengthened with magnesium and silicon, and the T43 heat

treatment was designed to enhance the overall formability [16].

According to the literature, AA6022 typically contains (asmass fraction) 0.61 Mg, 0.9 Si, 0.13 Fe, 0.07 Cu, 0.07 Mn, 0.02 Ti, and 0.01 Zn

[17]. Metallographic examination revealed that the mean grain size

for this alloy was nominally 30 m and that the grain structure was

relatively equiaxed in the rolling plane and slightly elongated along

the rolling direction of the sheet (RD). This is consistent with the

microstructure normally associated with a solution heat-treated,

naturally aged (T) condition [18,19].

Sub-sized, flat tensile specimens were cut from a single 1 mm

thick sheet with a water jet cutting tool. The nominal dimensions

of the specimens used for these experiments are shown in Fig. 1.

Specimens were cut with the tensile axis both parallel to the rolling

direction of the sheet (hereafter referred to as RD) and perpendicular to the rolling direction of the sheet (hereafter referred to as

TD).

Industry customarily uses sheet stock in the as-received condition with a specified maximum acceptable initial surface roughness

that varies with the particular application. While it is always

important to emulate material performance under actual forming

conditions, the mill scale that forms on the surface of most commercial aluminum alloys during processing completely obscures

Fig. 1. The geometry of the tensile specimen used with the sub-sized screw driven

universal testing apparatus. The approximate locations of the EBSD and SLCM scans

are indicated.

the fine surface features that evolve during the initial stages of

the deformation process, making high-resolution measurements

of these characteristics extremely difficult. So, the specimens in

this evaluation were polished to reveal the surface character at low

strains, to produce more consistent surface roughness measurements, and to facilitate orientation measurements using electron

backscatter diffraction (EBSD). Surface preparation was based on

standard metallurgical practice [20] and consisted of mechanically

polishing one side to a 0.25 m diamond finish. The specimens were

then placed in a vibratory polisher with a colloidal silica suspension

for 2 h. The result was a brilliant surface with no mechanical damage layer that could influence the grain orientation measurements.

Upon completion of the polishing, the significant dimensions of

each specimen were determined with a linear encoded measuring microscope with a (X, Y, Z) resolution of ±0.5 m. In addition,

two fiducial lines were lightly scribed outside the uniform gauge

length (see Fig. 1) of the specimen to facilitate accurate strain measurements. The distance between these marks was also determined

with the measuring microscope.

Electron backscatter diffraction measurements were performed

on the gauge sections of the tensile samples prior to deformation. Orientation data were acquired as a beam scan in the center

region of the specimen (see Fig. 1) with a JEOL1 6400 scanning

electron microscope (SEM) using a LaB6 filament, and HKL EBSD

software. EBSD maps were recorded using a 20 keV excitation voltage, a 28 mm working distance, a 400× magnification, and a 70◦

specimen tilt. Each map was acquired with the tensile axis oriented

parallel to the vertical axis of the scan a with 3 m step size. Sampling the relatively large gauge area of these specimens required

stitching the series of 25 (5 maps × 5 maps) EBSD maps together.

Stitching the maps together required manual alignment of each

map. As such, an offset of 1–3 pixels (pixels in this case are equivalent to the EBSD step size of 3 m) is likely between each map. After

stitching, a ‘clean up’ procedure was applied to the images to fill in

unindexed points and delete points, which is due to mis-indexing

of the EBSD pattern did not match the average grain orientation.

The clean up procedure used was an initial removal of all spikes

(i.e., single pixels of an orientation different than the surrounding

8 pixels) and then an iterative fill-in by 8, 7, 6, 5, and 4 nearest

neighbors in sequence. The hit rates for the as-received condition

were high for the two samples investigated here (93.0% for TD and

86.8% for RD) so the clean up procedure should not bias the grain

1 Certain commercial equipment, instruments, software, or materials are identified in this paper 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.
orientation measurements. In addition, the TD sample was reanalyzed after deformation, which resulted in a lower hit rate (66%)

due to the plastic deformation.

2.2. Mechanical deformation

Uniaxial straining of the specimens was performed with a subsized screw driven universal testing apparatus that was designed

for in situ mechanical experiments in an SEM. Because of its size,

this rig easily fit on the stage of a scanning laser confocalmicroscope

(SLCM). After carefully mounting a polished specimen in the grips,

the tensile apparatus was positioned under the 10× objective of

the SLCM. Each grip in this testing system is mounted on a separate

crosshead and the gearing of the load train is designed to actuate

both crossheads symmetrically about the center of the load frame.

This arrangement creates a region of the specimen that is stationary

with respect to the displacement of the grips. That is, there is a

small region (shown in gray in Fig. 1) in the geometric center of

the specimen that does not move during straining. Therefore, using

this tensile stage enables acquisition of high-resolution images of

the specimen surface as a function of the deformation level. The

key step in this process is positioning this stationary region in the

exact center of the field of view.

Each specimen in this analysis was strained with a crosshead

displacement of 0.1 mm/s. A set of calibration experiments was

used to determine the set of ten displacements at which the surface conditions would be assessed. When the output of the linear

variable displacement transformer (LVDT) reached one of these

predetermined displacement values, the stage was held static for a

period of 420 s to allow the load to reach a steady state value. The

classic shapes of the load drops followed immediately by increased

strength during subsequent straining are attributed to dislocation

relaxation processes under constant load and simultaneous migration of solute atoms to the dislocation cores, as described recently

for a 7000 series alloy [21]. Once the load stabilized, the topography

could be assessed with SLCM.

2.3. Surface roughness measurements

All of the SLCM images were acquired in the aforementioned

stationary region of the specimen and they were created with a

635 nm red laser source. The spacing between sampling points in

the (x, y) plane was fixed by the objective lens at 1.562 m/point,

which generated 640 pixel × 512 pixel image maps with nominal

physical dimensions (x, y, z) of 1000 m × 800 m × 50 m. The

spacing between the individual focal planes in the z dimension was

nominally 400 nm. As described previously [22], the SLCM stores

each topographic image as a raw depth map in tagged image file

format (TIFF) that contains the complete set of imaging parameters and binary pixel values. The raw binary depth maps were

converted into simple matrices describing the x, y and z dimensions of the physical surface, which were then trimmed to square

512-row × 512-column arrays to facilitate the matrix-based mathematical operations. Using the aforementioned x–y pixel spacing,

the values in each matrix correspond to an 800 m × 800 m area

of the surface.

Each image was corrected for flatness by computing the optimal equation of the Euclidean plane for each matrix with multiple

regression analysis and then subtracting that plane from each point

in the image matrix. Next, the extreme values (defined as the values

in the height data greater than ±6, where is the standard deviation of all the height values in that matrix) were screened from

the data sets. This step was required because some of the statistical

parameters used to interpret the surface data are highly sensitive

to outlier data points. Any individual height value that exceeded

the ±6threshold was reset to the mean value for that particular

Fig. 2. Stress versus displacement curves for the RD and TD specimens. The dro

in stress indicate the levels at which the surface roughness was evaluated. Note t

the displacement offset exhibited by the RD specimen was attributed to backlash 

the load train at the start of the test.

surface. Note that the number of affected data points for a giv

surface was typically less than 20 points (or <0.008% of the to

number of data points within the matrix). The resulting residu

matrices were used as the source for all subsequent assessments 

the surface character.

3. Analysis methods and results

3.1. Deformation-induced topography analysis

The mechanical behavior of the AA6022-T43 in the RD and T

orientations is shown in Fig. 2. The stress drops observable in ea

of the two curves directly correspond to the displacements at whi

the surfaces were assessed. A small offset is observable in the R

orientation displacement data, which resulted from backlash in t

load train at the start of the test. Considering that the strain valu

were based on the measured displacement of the fiducial mar

and not on the raw displacement data shown in the figure, th

offset is of little consequence.

The evolution of the deformation-induced surface is shown f

the TD specimen in Fig. 3 and for the RD specimen in Fig. 4. Both fi

ures exhibit the character of the stationary region at 10%, 15%, 2

and 26% engineering strain. Note that all of the images have an ass

ciated error of approximately ±0.5% strain. In addition, the highe

strain value shown, 26% strain, exhibits the surface conditions 

the onset of critical strain localization (i.e., necking). This is the k

surface condition because it represents the topographical structu

just before the initiation of the failure event. Since the images pr

sented in both figures show the surface morphologies at essentia

the same strain levels, these figures also illustrate the strength 

the influence that tensile axis orientation has on the overall chara

ter of the surface. While some slip-based deformation is observab

at the lower strains, the surfaces shown in Figs. 3 and 4 are mos

composed of grain boundary-localized deformation. This is sligh

accentuated in the TD orientation (Fig. 3). As the strain increas

the composition of the deformation becomes a more even distrib

tion of grain boundary-localized and slip-based deformation. Th
Fig. 3. A scanning laser confocal micrograph showing the surface morphology evolution as a function of uniaxial strain for the AA6022 alloy in the T43 heat treatment: (a)

at 10% strain, (b) at 15% strain, (c) at 22% strain and, (d) at 26% strain (just prior to fracture). The tensile direction is perpendicular to the rolling direction (TD orientation) in

this figure.

behavior is consistent with the observations reported in a previous

study performed on AA5052 [23].

3.2. Strain localization analysis

Quantification of the relationship between surface roughness

and strain localization requires a measure that describes the local

changes in the magnitudes of the peaks and valleys on the surface

[11]. The parameter chosen for this analysis is the maximum profile

height, Rt, which is defined as the total vertical distance between

the highest and the lowest points of a profile within a particular

evaluation length [24]. That is:

Rt = Rp + Rv (1)

where Rp is the absolute distance between the highest point of the

profile and the mean line, and Rv is the absolute 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 shown as negative). Because of the high level of sensitivity to changes in surface height, the Rt parameter is an ideal

tool to characterize the surface conditions that promote critical

strain localization and to quantify the magnitudes of those surface

conditions.

Extending the Rt parameter to a matrix form minimizes the

inherent statistical uncertainty associated with profile-based analysis [10,11]. This is accomplished by constructing a Euclidean

distance matrix [25], which divides the 512-row × 512-column

source matrix into smaller sub-matrices, or cells, each consisting

of 2 pixels/row × 2 pixels/column. The result is a differential matrix

consisting of 65,536 elements (i.e., 256 cells × 256 cells). The maximum difference in the surface height for any given cell, Rt(i,j), was

determined from the set of 4 height values contained within that

cell. Thus, each Rt(i,j) value is the absolute value of the maximum

displacement normal to themean plane at thematrix coordinates of

that cell. This construction is equivalent to plotting the maximum

measured vertical slope as a function of position on the surface.

Since the matrix format preserves the spatial coordinates for each

Rt cell by construction, this technique directly links any feature(s)

in the topography to the corresponding change in magnitude of the

local surface height.

An example showing the Rt matrix construction as well as the

relationship of this matrix to the original topography is presented

for the TD orientation at 26% strain in Fig. 5. Note that the topography, Fig. 5a, is the same image shown in Fig. 3d except that the 8-bit

grayscale is now based on the range of measured surface heights

instead of the local intensity of the reflected light. Fig. 5b is the Rt

map that was constructed from the topography (Fig. 5a).
Fig. 4. A scanning laser confocal micrograph showing the surface morphology evolution as a function of uniaxial strain for the AA6022 alloy in the T43 heat treatment: 

at 10% strain, (b) at 15% strain, (c) at 22% strain and, (d) at 26% strain (just prior to fracture). The rolling direction is parallel to the tensile direction (RD orientation) in t

figure.

A previous study showed that the Rt data are well characterized with a Weibull distribution [12]. Weibull statistics are often

used to estimate failure because they emphasize the incidence of

statistically rare events (i.e., those that lie in the tail regions of

a distribution). The Weibull distribution is commonly expressed

in terms of the cumulative distribution function (CDF) because in

this form, the expression enables a straightforward estimate of the

failure probability [26]:

fCDF (x; ̨, ˇ) = 1 − e−(x/ ̨)

ˇ

. (2)

In this equation, x ≥ 0 and fCDF(x; ̨,ˇ) = 0 for x < 0. The 2 parameters,

a scale parameter ( ̨), and a shape parameter (ˇ), must be real, and

>0. Thus, for any given strain level, the CDF is the probability, P,

of the occurrence of an Rt magnitude that is less than or equal to

a given Rt i.e., fCDF(Rt given) = P(Rt ≤ Rt given). In addition, the CDF is

bounded by the following conditions: fCDF(0) = 0, and fCDF(∞) = 1.

The statistically rare events in the tail of the Rt distribution are

particularly germane to this analysis because the largest magnitudes in the Rt distribution reflect the highest probability that the

surface conditions will promote strain localization and/or failure

[10]. For this analysis, the tail was defined as the segment of the

Rt distribution where the Rt magnitudes have a probability ≥0.95

based on the CDF constructed from the Rt data set with Eq. (2). 

this case, the 0.95 threshold corresponds to an Rt value of 3.0 

By filtering the values <3.0 m from the Rt map, one can gain insig

about how these ‘extreme’ Rt values are distributed with respect 

the surface morphology at failure. As such, Fig. 5c shows the loc

tions of the Rt values after the values <3.0 m have been remove

This figure was generated by mapping the range of Rt magnitud

into a standard 8-bit color scale, where blue (For interpretation 

the references to color in this text, the reader is referred to the w

version of the article.) is low and red is high, so that the color 

an individual cell directly indicates the magnitude of Rt. Fig. 5d i

composite image created by overlaying the filtered Rt map (Fig. 5

onto Fig. 5a. As expected, the remaining Rt values in this figu

mostly surround the largest surface features.While some of these 

values correlate with slip-based (i.e., intra-grain) surface roughe

ing, the figure is consistent with previous observations in that mo

of the surface displacements tend to occur along grain bounda

regions [23,27–29].

The results from the EBSD-based grain orientation analysis a

shown for the TD and RD specimens in the as-polished condition 

Fig. 6a and b, respectively. Each map displays the grain orientati

of the sample normal direction in a typical inverse pole figure t

angle. The color key indicates that while the grain orientations a
Fig. 5. An example of the method used to construct the Rt overlay from the TD topography data: (a) The surface from Fig. 3d shown as a topography, (b) the Rt map constructed

from (a), (c) the Rt map shown in (b) after all values < 3.0 (a 0.95 probability threshold) have been eliminated and, (d) the composite figure resulting from overlaying (c) on

(a).

for the most part, uniformly distributed, there is a slight tendency

toward the (0 0 1) orientation (red). Additional grain size analysis performed with EBSD indicated that the variability in the grain

size was on the order of ±20 m from the nominal grain size of

30 m. This relatively wide dispersion is observable in the EBSD

maps from both specimens. No further textural analysis was performed on either specimen because of the small number of grains in

the measurements (1080 grains for the TD sample, and 870 grains

for the RD sample).

Fig. 7 is an overlay of the EBSD map of the TD orientation

(Fig. 6a) over the composite topography/Rt map (Fig. 5d). Finding

the congruent regions in these two images presented two significant challenges. First, the EBSD scans contain substantial distortion

due to the strain applied to the sample (EBSD was performed prior

to straining), the cumulative misalignment produced by stitching

the individual images together, the distortion resulting from tilting the surface to 70◦ with respect to the incident electron beam,

and the aberrations present in the magnetic lenses. Thus, the shape

and size of each grain shown in Fig. 6 was considerably different

from the plan view exhibited in Figs. 3 and 5. Compensating for

this difference required a stretch, shear, and rotation operation on

the EBSD image to place the two images in phase. A set of distinct grains common to both images (labeled in the figure as  ̨, ˇ,

etc.) was used to facilitate this correction. The second challenge

was compensating for the difference in scan areas. While the scan

areas were nominally in the exact center of the sample, there was

a slight offset, and this is reflected in Fig. 7. The overlapped region,

highlighted with a black line, is the area where the topography and

the EBSD images are congruent, and based on this figure, the offset

between the EBSD y-axis (bounding line to the left) and the tensile

axis of the sample is 4◦. As before, the Rt map overlay indicates the

location of the largest Rt magnitudes with respect to the topography; but with the addition of the EBSD overlay, one can now directly

correlate a specific topographic feature with the local microstructural conditions that produced it. Fig. 8 is a similar construction

for the RD orientation; however, the offset between the EBSD and

tensile axes is 8◦. Also the congruent region shown in this figure is

noticeably smaller than that shown for the TD orientation.

It is well known that the differences in crystal orientation within

a polycrystalline material produce variations in the local yield
Fig. 6. The sample normal orientation maps for the (a) TD and (b) RD specimens

in the as-polished, undeformed condition. The grains (labeled  ̨, ˇ, , etc.,) in each

figure were used to facilitate construction of Rt-orientation overlays. A standard

tri-color inverse pole figure for cubic materials is also shown in the figure.

criteria. One common approach to assess these variations is to

determine the differences in the local plastic flow conditions by

computing the individual Taylor factors. Taylor [30] made the simplifying assumption that all of the grains in a polycrystal would

deform identically, thus maintaining compatibility. Mises [31] had

earlier pointed out that at least five independent strain components

were required to plastically deform a (constant volume) solid without geometric restrictions and Taylor hypothesized that five slip

systems would be active in each grain.2 Since there are more than

five combinations of slip systems available in fcc materials, additional criteria were required. Taylor [30] further hypothesized that

the slip systems that are active during plastic deformation are those

that minimize the internal work. Taylor then made the assumptions that the critical shear stress, 0, for all slip systems is the

same and that work hardening does not vary between grains. Thus,

2 The assumption that exactly five slip systems would be active only applies under

the assumptions made by the full constraints Taylor model. If this assumption is

relaxed, then different numbers of slip systems can be active.

Fig. 7. The composite image produced by overlaying Fig. 6a on top of Fig. 3d. T

area bounded by the black line is the region where both images are congruent.

minimizing the internal work becomes equivalent to minimizi

the total incremental crystallographic shear for a given grain:

da =

5

i=1

dai (

Fig. 8. The composite image produced by overlaying Fig. 6b on top of Fig. 4d. T

area bounded by the black line is the region where both images are congruent.
Fig. 9. The congruent region of the image produced by overlaying the Rt composite

image in Fig. 3d on top of a Taylor map that was constructed from the orientation

data shown in Fig. 6a.

where dai is the incremental shear strain on slip system i. The

increment of internal work then becomes dW= 0da, which can

be equated with the external work, dW= dε, where is the

macroscopic tensile stress and dε is the corresponding incremental

macroscopic strain. The result is



0

= da

dε = M, (4)

where M is the Taylor factor. For fccmetals, the average M is approximately 3.1. For a more detailed treatment of Taylor factors, see [32]

or [33]. Note that the increment of work is proportional to da. Thus,

from Eq. (4) we see that for a given increment of applied tensile

strain, a large Taylor factor corresponds to a grain that requires a

large amount of plastic work to deform and is therefore “stronger”.

Of course, this calculation is based on the assumption that all grains

deform identically in simple uniaxial tension with no regard to

complex neighbor interactions.

Figs. 9 and 10 are maps of the Taylor factors for the TD and

RD orientations, respectively. Both maps were computed from the

EBSD data using the TaylorCubicY.dll toolbox within the HKL crystallographic software suite. This routine assumes that the tensile

axis is parallel to the y-axis of the EBSD scan, and as noted previously, this is only approximately correct (4◦ and 8◦ offsets for TD and

RD respectively). The corresponding Rt maps are overlaid upon the

Taylor factor maps. As in the case of the Rt-EBSD overlays shown

in Figs. 7 and 8, the areas shown in both figures are the regions

where the Taylor factor maps and the Rt-overlays are congruent.

One distinct difference between Figs. 9 and 10 is the higher density of Rt values in Fig. 9 (the TD orientation). The Rt values appear

to be more concentrated in the grain boundary regions, which is

consistent with the behavior shown in Figs. 3 and 4.

The relationship between the grain orientation and the morphology of the deformed surface is shown at a higher magnification

for the TD orientation in Fig. 11. Fig. 11a shows a SLCM image of

this region where some of the more distinctive grains are again

indicated by ( ̨, ˇ, ı, ε) to match those shown in Fig. 10. The grain

Fig. 10. The congruent region of the image produced by overlaying the Rt composite

image in Fig. 4d on top of a Taylor map that was constructed from the orientation

data shown in Fig. 6b.

marked  in Fig. 10 is not included in the figure. EBSD scans were

also performed on the samples after deformation. As such, Fig. 11b

and c are the TD orientation maps before and after 26% tensile

strain, respectively.

We also calculated the local misorientation (also called the kernel average misorientation, or KAM) by calculating the difference in

orientation from pixels three steps away in each direction (square)

up to an orientation cut off of 5◦. These calculations were carried

out for the TD sample to investigate the intergranular deformation;

the results are presented in Fig. 12.

4. Discussion of results

As described above, the Taylor and Rt maps that are overlaid

in Figs. 9 and 10 are not completely contiguous largely due to

deformation-induced changes in the grain shapes and cumulative

errors in assembling the 25 EBSD maps. Nevertheless, several features are apparent that are common to both the TD and RD samples.

First, most of the highest (red) Rt locations occur at triple junctions

where there is a large difference between the Taylor factors of the

individual grains. This is not surprising since triple junctions are

widely recognized as possible stress concentration points and intergrain deformation is expected to be largest between grains with a

large difference in the Taylor factors [34]. Another common feature is that pronounced slip band behavior occurs predominately

in grains with low Taylor factor. This is particularly evident in the TD

sample (Fig. 9), where the intra-grain slip appears in the upper and

lower regions of the figure and the Taylor factor is low. Conversely,

the central region of this figure has a collection of grains with much

higher Taylor factors, none of which exhibit enough intra-grain slip

to appear on the Rt map.

It is tempting to conclude that slip is minimal in grains with high

Taylor factor, but this is too simplistic. For example, Fig. 11a shows a

high-resolution SLCM image of this same high-Taylor-factor region

of the deformed TD sample. Finely distributed slip is clearly evident

in the grains that did not show activity on the Rt map. This slip does
Fig. 11. The surface of the TD specimen shown at a higher magnification to better

reveal the relationship between grain orientation and the deformed surface morphology: (a) a SLCM image of the region exhibiting some of the distinctive grains,

(b) the TD orientation map before deformation and, (c) the TD orientation map after

the application of 26% strain.

not appear because of the criteria used in producing the Rt map:

a 100× overall magnification and a 0.95 threshold. That is, if the

higher magnification image in Fig. 11b was used as the source for

an Rt/Taylor map overlay construction, it is likely that some of the

fine slip exhibited in the figure would be observable. In contrast,

the lower Taylor factor grains often have slip confined to gross slip

bands which have large enough local height changes to exceed the

Rt map threshold [35]. In the following, “fine slip” refers to graininterior slip structures that do not show up on the Rt map.

Another related factor that directly influences the surface roughening is the change in the local grain orientation [15]. Fig. 11b and

Fig. 12. A map showing the local misorientation (also called the kernel average m

orientation or KAM) performed by calculating the average difference in orientat

from pixels 3 steps away in each direction (square) up to an orientation cut off of 

c are orientation maps of the same region of the TD sample tak

before and after deformation, respectively. The two high-Taylo

factor grains just below the center of this figure (labeled as ı and 

show a broad array of colors, indicative of fine-scale crystal break

into numerous sub-grains. The fine slip in these grains varies co

siderably with position, which is consistent with this result. T

large grain labeled  ̨ in Fig. 11a also displays fine slip; but unli

ı and ε, all of the slip lines in  ̨ appear to be parallel. This wou

explain why the grain orientation remained uniform during defo

mation.

The above observations are consistent with the idea that fi

slip is favored in grains with high Taylor factor. Such grains a

observed to deform more “smoothly” than those with lower Tay

factor and thus often exhibit larger local intra-grain misorient

tion, but little intra-grain surface roughness (as defined by the 

criteria). However, this same ability to deform around obstac

can lead to substantial surface roughening at the grain boundari

Thus, a grain with high Taylor factor may exhibit a plastic stra

“discontinuity” when adjacent to a grain with different prima

slip directions or substantially lower Taylor factor. This is cons

tent with the general pattern of large grain-boundary roughne

delineating many of the central high-Taylor factor grains in Fig. 

Although many of these active grain boundaries occur betwe

grains of similar Taylor factor, inspection of the EBSD map in Fig

shows that the corresponding crystallographic orientations a

very different.

Another factor that seems to play some role is the grain si

Smaller grains are generally expected to be stronger than larg

grains (Hall–Petch relationship [36]) and inspection of the Rt m

overlays in Figs. 7 and 8 demonstrates that the sample regio

with the largest Rt values are frequently located on grain boun

aries adjacent to the smallest grains (with one dimension belo

≈15 m).
Finally, the correspondence between changes in local orientation and distributed fine slip in highly deformed grains suggests

that an EBSD map showing local misorientation may provide useful

information comparable to the Rt map construction. Fig. 12 shows

an EBSD-derived map of the local misorientation in the TD sample after 26% plastic strain. As described above, this map shows the

local difference in orientation from pixels three steps away in each

direction (square) up to an orientation cut off of 5◦ (which is considered a grain boundary). The brightest red regions in this figure

exhibit the largest local misorientation. These regions are primarily

restricted to the grain boundaries and they correspond remarkably

well with the regions of largest Rt value in Fig. 7. As such, these

results contradict the findings of Raabe et al. [15]. They concluded

that strain localization is primarily dependent on the local orientation, and therefore, can initiate at grain boundaries and within

the interior of grains with similar probability. While Fig. 12 indicates that appreciable changes in the local orientation occurred in

most of the grains in the figure, the largest Rt values tended to

be concentrated along the grain boundary regions and not in the

grain interiors. Considering that the largest Rt values reflect the

greatest surface displacements, strain localization is more likely in

the grain boundary regions than in the interior. This would agree

with the conclusions by Bieler et al. [37] that failure was more

likely to initiate along grain boundary regions where unfavorable

slip interactions reduced the local strength of the boundary and

not in regions with the highest local strain conditions. Based on

these results, the overlay mapping approach demonstrated here

can be used to evaluate the local conditions in the grain boundary

regions to assess the strength of the relationship between the highest surface roughening, the degree of grain misorientation, and the

location of failure.

5. Conclusions

The general rules described above are only true in a rough statistical sense, since many exceptions exist. Considering the numerous

assumptions made in the Taylor factor analysis, it is quite surprising

how well this criterion worked. Other factors, such as grain boundary misorientation, locations of coincident site lattice orientations,

and highest Schmid factors were also investigated. These analyses

were not presented since the agreement between these parameters

and the Rt maps was relatively poor with respect to the Taylor and

misorientation maps. Even the Taylor map does not provide adequate information to completely predict the local deformation and

roughening behavior of the samples.While the correlation between

the surface roughness and the EBSD-derived maps reported here

is remarkable, the only microstructural information included in

the analysis was the orientation of each grain. The primary factors

that are missing from all of these approaches are: (1) information

about the subsurface grains, (2) the activity of individual slip systems in each grain and how this affects local inter-grain stresses

and strains, and (3) a consideration of how “clusters” of grains

affect deformation at the multiple-grain length scale. However,

many of these shortcomings can be overcome by combining the

unique multiple-technique surface analysis approach described

in this paper with crystal-plasticity-based finite element modeling of the deformation experiment. Some attempts have already

been made to relate subgrain microstructure with slip activity

[38,39] through polycrystalline modeling efforts. Nevertheless,

obtaining three-dimensional information about the shapes and orientations of the subsurface grains would require either destructive

measurements (such as serial sectioning) or synchrotron-based Xray diffraction methods.

Acknowledgement

The authors would like to acknowledge the contributions of S.

M. Olson of the Materials Science and Engineering Department at

Lehigh University who performed the mechanical tests, as well as

the initial SLCM and EBSD measurements. Mr. Olson was supported

in part by the National Science Foundation’s Research Experience

for Undergraduates (REU) program, Division of Materials Research.

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