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Fundamental relationships between deformation-induced surface roughness,

critical strain localisation and failure in AA5754-O

Mark R. Stoudt a; Joseph B. Hubbard a

a

 Materials Science and Engineering Laboratory, National Institute of Standards and Technology,

Gaithersburg 20899, MD, USA

Online Publication Date: 01 September 2009

To cite this Article Stoudt, Mark R. and Hubbard, Joseph B.(2009)'Fundamental relationships between deformation-induced surface

roughness, critical strain localisation and failure in AA5754-O',Philosophical Magazine,89:27,2403 — 2425

To link to this Article: DOI: 10.1080/14786430903120343

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Philosophical Magazine

Vol. 89, No. 27, 21 September 2009, 2403–2425

Fundamental relationships between deformation-induced surface

roughness, critical strain localisation and failure in AA5754-O

Mark R. Stoudt* and Joseph B. Hubbard

Materials Science and Engineering Laboratory, National Institute of Standards and

Technology, Gaithersburg 20899, MD, USA

(Received 6 March 2009; final version received 7 June 2009)

This research employs two approaches to characterise the apparent

structure observed in localised strain maps constructed from surface

topography data acquired from AA5754-O sheet stock that was deformed

in three in-plane stretching modes. The first uses a conventional two-point

autocorrelation function (ACF), while the second uses the combination of

the eigenvalue spectrum associated with each map and information theory.

The results from the ACF analysis are inconclusive, implying that this

technique lacks the sensitivity necessary to quantify the relationships

between multi-point clustering and strain localisation. The information

theory-based approach reveals that the relative spectral entropy increases

monotonically, attains a maximum and then decreases sharply to the failure

strain. This behaviour occurs in all three strain modes and results from two

competing processes: one where the formation of structure is favourable

and one where it is not. The crossover point is a clear indicator of the onset

of critical strain localisation and, therefore, can be regarded as a precursor

to failure because once the dominant process shifts, additional strain results

in the precipitate formation of a critical strain localisation event.

Keywords: information theory methods; aluminium alloys; mechanical

behaviour; plastic deformation; polycrystalline metals; sheet metal forming;

topological analysis

1. Introduction

The inability to reliably model the evolution of the surface inhomogeneities produced

during sheet metal forming is a significant obstacle impeding the widespread

incorporation of new alloys designed to reduce gross vehicle weight and increase

overall vehicle fuel economy. The macroscopic deformation in a typical metal

stamping occurs through a complex combination of strain modes (e.g. biaxial,

uniaxial, plane strain). Since each strain mode imposes distinctly different constraints

on the material flow during stretch forming, the strain mode strongly influences both

the magnitudes of the limiting strains achievable through stretch forming, and the

overall character of the deformed surface. The topographies shown in Figure 1

provide an example of how the constraints on material flow result in considerably

*Corresponding author. Email: stoudt@nist.gov

ISSN 1478–6435 print/ISSN 1478–6443 online

This work was authored as part of Contributors’ official duties as employees of the United States Government and is

therefore a work of the United States Government. In accordance with 17 U.S.C. 105, no copyright protection is

available for such works under U.S. law.

DOI: 10.1080/14786430903120343

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different surface morphologies for the same level of modal plastic strain.

The problem arises from the fact that a combination of strain modes normally

produces failure at a lower strain compared to that achieved in a single strain mode.

Owing to the complexity of the automotive component shapes produced with metal

stamping, the limiting strains for these components are generally determined through

Figure 1. (Colour online). A representative set of scanning laser confocal topographies

showing the surface structure of AA5754-O at three strain levels in three strain modes. Images

A, B and C are in equi-biaxial strain, D, E, and F are in uniaxial strain and G, H, and I are in

plane strain. Images A, D, and G show the surfaces after 5% nominal true strain, B, E, and H

show the surfaces after 10% nominal true strain and C, F, and I show the surfaces at the

maximum uniform strain.

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numerical simulations specifically designed to predict the onset of critical strain

localisation (i.e. one that is a direct precursor to failure such as the formation of

cracks or splits, necks, etc.). Clearly, the accuracy of these models is heavily

dependent on the quality of the materials data, and a correct prediction of the

forming behaviour requires that the surfaces produced by each component of

the macroscopic strain be properly identified and correctly characterised.

Such characterisations are expensive and complicated processes that require

numerous high-resolution measurements of the surface morphology produced by

each strain mode [1].

The incorporation of revised plasticity and kinematic hardening models, as well

as the results from numerous studies of the influence of various material parameters,

such as grain size, grain orientation effects, surface roughening effects and other

damage mechanisms, on strain localisation have greatly enhanced the reliability of

the numerical models used to predict the limiting strains [2–9]. Despite all the

improvements, inconsistencies still exist between the mechanical behaviour predicted

numerically and that observed experimentally. This creates a situation where the

numerical models often correctly indicate the general trends, but fail to consistently

predict the actual strains at which localisation occurs [10].

This paper is one in a series documenting analyses of strain localisation

behaviour in commercial aluminium sheet [11–13]. The inspiration for this research is

derived from the work of Frost and Ashby [14], and others [15,16], in that changes in

the microstructure or, in this case, surface morphology, can be ‘mapped’ with respect

to the level of plastic deformation. Previously [11], rigorous matrix-based statistical

analysis methods were developed and integrated with high-resolution topographical

imaging (Figure 1) to assess how plastic deformation and microstructure evolution

influence strain localisation in aluminium sheet deformed in three in-plane strain

modes. A key component of this approach involves extending the peak-to-valley

surface roughness (Rt) to a matrix form, which is done for three reasons: (1) the

magnitudes of the local surface extremes will change with strain to the point where

they reach a value that directly reflects the onset of critical strain localisation, (2) the

Rt parameter is highly sensitive to those localised changes and it is acquired through

a straightforward calculation, and (3) the Rt parameter is based on a simple sum of

two values, so it can be easily determined between any two nodes within a particular

region in a finite element simulation, allowing for direct integration of changes in the

intensity of the local surface conditions into formability models. The result is a series

of local intensity maps consisting of discretised arrays of Rt ‘cells’ constructed from

the raw topographic data (an example is shown in Figure 2). Each cell accentuates

the height disparities between topographical features, enabling a direct threedimensional quantification of the microstructural conditions that promote critical

strain localisation. So far, this study has established that (1) the morphological

conditions that promote strain localisation can be assessed directly through

topographic analysis, (2) an accurate and straightforward probabilistic expression

that captures the subtle changes produced in the morphology can be developed, and

(3) strain localisation is controlled by a stochastic process that can be reliably

predicted with Weibull statistical methods.

The objective of the present investigation is to ascertain whether critical strain

localisation requires the formation of ‘structure’ within the Rt maps (i.e. several Rt

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cells exceeding a threshold magnitude that are in close proximity and act in a

cooperative manner). The motivation for this study comes from the numerous cells

present in the Rt maps that exhibit magnitudes that exceed the level at which critical

strain localisation was predicted to occur by the Weibull analysis, yet no critical

localisation was observed. As a result, the goal of this research is to determine

(a) whether structure exists within the Rt data, and (b) the strength of any spatial

correlations that may be present within this structure. (Note that quantification of

spatial correlations in three dimensions presents a substantial challenge in itself, as

Equi-Biaxial Uniaxial Plane Strain

5 % Strain Maximum Strain

100 mm

15 % Strain

B

C

D

F

E

I

G

H

A

Rt Magnitude (mm)

0 5 10 15 20 ≥ 25 0 4 8 12 16 ≥ 20 0 4 8 12 >16

Figure 2. (Colour online). The set of localisation maps constructed from the peak-to-valley

distance (Rt) data calculated for the topographies shown in Figure 1. Each map exhibits the

propensity for strain localisation as a function of strain level according to the 8-bit colour scale

shown for each strain mode.

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the available tools are designed for and, generally, limited to the analysis of twodimensional data.) If the observable structure has no influence, then critical strain

localisation is an uncorrelated stochastic process that can only be estimated through

probability analyses. However, if structure is required, then where critical strain

localisation occurs on the surface will depend on the local magnitudes of the surface

heights as well as on the size, shape and density of that structure.

Two approaches were adopted to quantify the spatial correlations in the Rt data.

The first is based on the conventional two-point autocorrelation function (ACF).

The second is based on spectra derived from sets of complex eigenvalues that were

calculated for the high-resolution topographies [17]. The second approach is similar

to that employed in pattern recognition algorithms [18] and incorporates analysis

concepts that are derived from information theory. The motivation for introducing

information theory methods into the analysis of deformation-induced surface

roughness is two-fold. First, given that plastic deformation is a volume conserving

process, the topography of a deformed polycrystalline material contains substantial

characteristic information regarding all of the physical mechanisms involved.

Second, the relevant literature shows that information theory methods are rigorous,

quantitative, flexible and minimally biased [19–24]. As a result, these methods are

ideally suited to analyse large, complex data sets such as those produced by

topographical analysis.

2. Experimental

2.1. Generation and evaluation of surface roughness

This section is a synopsis of the procedures presented in [11] that were used to

generate surface data in the various strain conditions. Sets of 30 30 cm (12 12 in)

blanks were sheared from 1-mm thick commercially available AA5754-O sheet stock

for testing. This aluminium alloy was primarily developed for automotive

applications and, like most alloys in the 5xxx series, AA5754 is substitutionally

strengthened, and demonstrates good overall formability. Aluminium alloy AA5754

typically contains 2.8% mass fraction Mg for solid solution strengthening and

approximately 0.5% mass fraction Mn for grain refinement and stability [25].

The grain structure was relatively equiaxed in the rolling plane and slightly elongated

along the rolling direction of the sheet (RD), which is indicative of the recrystallised

microstructure normally associated with the O-temper. The grain size for this alloy

was 40  20 mm [12]. All the specimens in this evaluation were polished to a 6 mm

diamond finish using standard metallographic polishing practice to better reveal the

surface character at low strains and to produce more consistent surface roughness

measurements.

After polishing, the blanks were deformed in three in-plane, proportional strain

modes defined in terms of the in-plane principal strain ratio 1⁄4 "2="1 [13]. The first

strain ratio was 1⁄4 1 (hereafter referred to as equi-biaxial). The second strain ratio

was p 1⁄4 –0.5 (hereafter referred to as uniaxial), and the third strain ratio was 1⁄4 0

(hereafter referred to as plane strain). The uniaxial samples were machined to an

ASTM E08-91 standard sheet-type tensile specimen geometry and then strained to

nominal true strain values of 5, 10 and 15% using standard test methods [26] with a

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constant displacement rate resulting in an initial strain rate of 6 104 s

1

. Both the

equi-biaxial and plane strain deformations were imposed using an augmented [27]

Marciniak flat bottom ram test [28]. Samples were strained in equi-biaxial tension

using a typical strain rate of 5 104 s

1 to 5, 10, 15 and 20% nominal true strain.

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 for this investigation, ‘failure strain’ refers to the uniform strain measured after

a localisation event occurred. Therefore, the surface roughness data acquired in this

condition establish: (1) the maximum uniform surface roughness obtainable for the

AA5754-O alloy in each strain mode, and (2) the surface conditions that gave rise to

the formation of the failure event. Coupons were cut from the centre of each

deformed specimen for topographic analysis.

The surface topography was quantified for each strain level in each strain mode

with scanning laser confocal microscopy (SLCM). Note that the morphology within

a localised region is specific to a unique set of surface conditions and no longer

represents the character of the surface as a whole. In other words, due to the high

degree of heterogeneity in the surface data in the proximity of a localised region,

these data are non-stationary in a statistical sense. For this reason, neither the critical

localisations, nor the regions immediately surrounding them, were included in any of

the SLCM images acquired from the surfaces in the failure condition. Each

measurement in this evaluation consisted of five, well-spaced SLCM images taken

from different locations on the specimen surface. 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. The SLCM images were

converted from the image format to simple matrices of topographic data according

the procedure described in reference [29].

2.2. Generation of strain localisation maps

The relationship between surface roughness and strain localisation was quantified

via the Rt roughness parameter. This parameter is formally defined as the vertical

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

evaluation length [30]. That is,

Rt 1⁄4 Rp þ Rv: ð1Þ

In this equation, 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 below the mean plane are

traditionally considered negative). Constructing a Euclidean distance matrix based on

this measure [31], extended the traditionally linear profile-based Rt parameter to a

matrix format. Construction of this matrix required sub-dividing a 512-row 512-

column source matrix (e.g. those shown in Figure 1) into smaller matrices each

consisting of 16 elements (i.e. 4 heights/row 4 heights/column). The difference in the

surface height for a given cell, Rt(i, j), was determined from the set of 16 height values

contained in that cell. That is, each Rt(i, j ) value represents the largest displacement

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normal to the mean plane at the coordinates of the source with the matrix index (i, j).

The result is a differential matrix that contains 16,384 Rt values.

The maps in Figure 2 are examples of this construction. Note that both the

magnitude and the physical location of each Rt(i,j) value are quantified by this

method, since the matrix format retains the spatial coordinates for an Rt cell by

construction. That is, this technique directly links any feature in the source image to

the corresponding change in magnitude of the local surface height. Obviously, cell

size has a strong influence on the information contained within the Rt map, and the

level of resolution can easily be affected by a simple change in the dimensions of the

sub-matrix cells. The cell size used in the previous evaluation was (8 8). Decreasing

the number total of elements in a cell from 64 to 16 increased the resolution of the Rt

maps by a factor of four. The surfaces were re-evaluated at the higher resolution to

differentiate between grain boundary roughening and within-grain roughening

events. Assuming a uniform distribution of grains with an approximate area of

1600 mm2

, the smaller 6.25 6.25 mm cells would yield a minimum of 40 samples per

grain. This sample density provided sufficient sensitivity to clearly discern the

roughening that occurs within a grain (e.g. slip) and that located along the grain

boundaries. Moreover, the improved resolution enabled a direct assessment of the

topographical features produced by the plastic strains in each strain mode (Figure 1)

and the corresponding change in the relative potential for critical strain localisation

produced by those topographical features (Figure 2). Figures 1F and 2F were

extracted, enlarged and presented as Figure 3A and B, respectively, to illustrate the

good correlation between the surface topographies and the Rt maps.

Maps of the Rt data were constructed for the five surfaces acquired in each strain

condition. As shown in Figure 2, the range of Rt magnitudes is mapped into a

standard 8-bit colour scale, where blue is low and red is high, so that the colour of an

individual cell reflects the local magnitude of Rt. From the previous findings, the

maximum (the dark red cells online) in Figure 2 reflects a threshold Rt magnitude

where the probability of a critical strain localisation event occurring is 95% [11].

3. Analysis of Rt Data

As noted earlier, several cells were observed that exceeded the Rt magnitude predicted

for the onset of critical strain localisation at strain levels where failure did not occur.

This behaviour implies that the occurrence of an isolated, or single, Rt cell with a

critical magnitude is not sufficient to produce failure. Further examination of the Rt

maps for each strain condition (Figure 2) revealed that, as the strain level increases,

some structure (i.e. clustering), or groups of cells with similar relative probabilities,

appears to form in the Rt images. The sizes, locations, and densities of these Rt cell

clusters also appear to vary with strain mode. It should be emphasised that in large

part, this ‘structure’ is a consequence of the threshold values used for the upper and

lower bounds of the colour scales in each map. However, it can also be linked to

specific morphological features produced by the plastic deformation (Figure 3).

Therefore, the presence of this structure raises the following question: Does the

structure in the Rt maps play a significant role in the localisation, or is it merely an

artefact arising from the methods used to construct and colour the Rt maps?

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Figure 3. (Colour online). Figure 1F and Figure 2F enlarged and presented together to exhibit

the correlation between the strain localisation (Rt map) and surface morphology. In addition,

areas with a large number of cells that have localisation probabilities exceeding 95% and the

corresponding locations on the topography have been isolated to illustrate the notion of

clustering that may exist within the Rt maps.

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3.1. Autocorrelation function approach

The auto-correlation function (ACF), defined as the correlation of a source with

itself, is one of the more familiar techniques used to characterise the spatial

arrangement in linear profile-based roughness data [30,32,33]. This technique can be

easily extended to a three-dimensional form for evaluating the spatial characteristics

of surface data [34]. The three-dimensional form of the ACF used for this analysis is

derived from the cross correlation function for any (m n) matrix, A, with itself [35]:

P Að Þ i,j

 1⁄4

1

MN

PM1

m1⁄40

PN1

n1⁄40 Að Þ m,n Að Þ mþi,nþj

 j k

Sq2 : ð2Þ

In this equation, M and N are the dimensions of the matrix, Sq2 is the square of

the root mean squared roughness (i.e. the variance of the matrix), and the values in

the ACF matrix, hP(A(i,j))i, are the expectation values produced by lag (offset)

vectors of a given magnitude, sð Þ i,j



  1⁄4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

i

2 þ j2 ð Þ p , evaluated over the entire matrix.

Since these values have been normalised, they range between (–1.0) and (þ1.0). Thus,

the physical meaning of an individual ACF value is as follows: (a) a value of (þ1.0)

indicates that the matrix is perfectly correlated at all values of the lag vector s(i, j );

(b) a value of (0) indicates that there is no correlation at all values of the lag vector

s(i, j) (i.e. random); (c) a value of (1.0) indicates that the matrix is perfectly anticorrelated at all values of the lag vector s(i, j ) (i.e. 180 out of phase) [33]. By

construction, the ACF values returned by a lag vector with positive and negative

values for (i, j ) are identical, so the resulting ACF surfaces possess a quadrantal

symmetry.

ACF surfaces were constructed for each Rt map using Equation (2) and the signal

processing toolbox in MATLAB1 [36]. The set of ACF surfaces derived from the Rt

maps in Figure 2 are shown in Figure 4. Each plot in this figure depicts the

magnitude of the ACF at any point in the Cartesian plane with each contour

corresponding to a constant ACF value. Note that the Rt values used to construct

these figures are all positive, so no anti-correlated regions are present in the ACF

surfaces. Thus, the range of the corresponding 8-bit colour scale lies between

(0, blue) and (þ1, dark red) and is common to all the plots in the figure.

As shown by Brochard et al. [37], analyses of ACF surfaces can reveal a great

deal of information pertaining to the spatial arrangement of the surface features.

Unfortunately, the boundary conditions used in the ACF calculation have a strong

influence on the results. For this reason, analyses of surface data with the ACF can

be subjective and difficult to interpret. The most apparent feature in these plots is the

area of high correlation, which appears as a diamond-shaped region distributed

about the origin and extends more than 200 mm in both the x and y directions.

Outside this central diamond region, the degree of correlation rapidly decays to zero.

The correlation length, or the transition from the correlated to the random state, is

the most common characteristic derived from the ACF and it is based on the decay

of the ACF [32]. While this may be suitable to assess the characteristics of an ACF

generated from a linear profile, extending this approach to three-dimensional data is

not as straightforward, and does not readily produce meaningful results. Since the

ACF data decay in more than one direction, the correlation length becomes a vector

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and must be determined with vector-based tools. One of the more common

approaches is the fastest decay autocorrelation length, Sal [30]. This approach is

normally used to generate a polar spectrum that can be used to assess the anisotropy

of a surface [38]. While small differences are observable between the individual plots

in Figure 4, the overall characters of the plots are strikingly similar to one another.

This suggests that any differences are likely to occur at longer lag lengths where the

statistical reliability may be severely degraded; thus, the fastest decay autocorrelation

length is not the appropriate tool to evaluate the strength of the correlations in the Rt

data. Methods that are more statistically robust, and exhibit better sensitivity to

Equi-Biaxial Uniaxial Plane Strain

5 % Maximum Strain 15 % 

100 μm

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 4. (Colour online). Two-point autocorrelation (ACF) surfaces constructed from

Rt data sets shown in Figure 2. Each surface is a contour plot that reflects the magnitude of

the ACF at any point in the Cartesian plane. Each contour line corresponds to a constant

ACF value.

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correlations at long lag lengths must be employed to evaluate whatever differences

may be present among the ACF surfaces. Two methods were adopted for this

purpose.

The first method evaluated the overall volume of the ACF surface (i.e. the space

enveloped by the ACF surface and the mean plane), which was estimated by a simple

sum of all of the ACF magnitudes. Unlike the topographies, where the zero mean of

the surface heights would produce a zero overall volume, the means of the individual

Rt data sets were all non-zero and positive, producing ACF surface volumes that are

also non-zero and positive. As shown in Table 1, the largest volume was observed in

the as-polished data. This is not a surprising result in that little variation was

observed in the overall surface structure for this particular condition, which

produced large areas of similar height (i.e. high correlation). While the mean ACF

volumes exhibit some variability in magnitude, the uncertainties associated with

these variations indicate that the ACF volume data are relatively insensitive to strain

level or to strain mode.

The second method employed to discriminate among the ACF surfaces was the

moment of inertia. The quadrantal symmetry of the ACF surface allows for the

construction of a centroid at the origin about which a moment of inertia can be

calculated. The trace of the moment of inertia tensor [I(i,j)] is a scalar quantity (I) that

is determined from the sum of the products of the masses and the square of their

distance from the axis of rotation [39]. In the case of an ACF surface, the ‘masses’ are

taken to be the magnitudes of the ACF values at any given location in the ACF

matrix P(i,j), and the distances are the matrix coordinates (i,j) of that ACF value.

Thus, the moment of inertia for an (n n) matrix is expressed as

I 1⁄4 Xn

i1⁄40

Xn

j1⁄40

Pð Þ i,j R2

ð Þ i,j

  where R2

ð Þ i,j 1⁄4 i

2 þ j2 : ð3Þ

Table 1. Analysis of the two-point autocorrelation function (ACF) data.

Strain

path

Nominal true

strain

ACF volume

(mm2

)

Uncert.

 (mm3)

ACF

inertia (mm2)

Uncert.

 (mm3)

As-Polished 0.00 1.41E þ 04 1.31E þ 02 1.21E þ 08 1.61E þ 06

Equi-biaxial 0.05 1.32E þ 04 2.13E þ 02 1.15E þ 08 1.89E þ 06

– 0.10 1.33E þ 04 1.31E þ 02 1.17E þ 08 1.47E þ 06

– 0.15 1.35E þ 04 9.75E þ 01 1.18E þ 08 9.95E þ 05

– 0.20 1.34E þ 04 9.44E þ 01 1.17E þ 08 9.44E þ 05

– 0.26 1.40E þ 04 1.74E þ 02 1.22E þ 08 1.23E þ 06

Uniaxial (RD ?) 0.05 1.35E þ 04 2.46E þ 02 1.19E þ 08 1.96E þ 06

– 0.10 1.33E þ 04 3.36E þ 02 1.17E þ 08 2.22E þ 06

– 0.15 1.26E þ 04 1.69E þ 02 1.10E þ 08 1.94E þ 06

– 0.23 1.32E þ 04 3.90E þ 02 1.15E þ 08 3.05E þ 06

Plane strain (RD ?) 0.05 1.25E þ 04 2.71E þ 02 1.10E þ 08 2.71E þ 06

– 0.10 1.27E þ 04 1.05E þ 02 1.11E þ 08 1.59E þ 06

– 0.15 1.28E þ 04 1.61E þ 02 1.12E þ 08 7.98E þ 05

– 0.16 1.33E þ 04 2.40E þ 02 1.15E þ 08 2.26E+ 06

Note: ‘Uncert.’ refers to one standard deviation of the mean, .

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The main advantage of this quantity over the ACF volume is that the moment of

inertia is a measure that directly links the magnitude of an individual ACF value to

the exact location in the Cartesian plane. Since the moment of inertia weights the

values in the ACF as a function of their distance from the origin, it would seem to be

a more proper metric to gauge the relative influences of the variations in the size of

the diamond region and the small or near-zero values in Figure 4. However, as

shown in Table 1, the moment of inertia data are similar to the ACF volume data in

that they are relatively insensitive to changes in strain level or strain mode. The

relative uncertainties associated with the ACF volume and moment of inertia

estimates indicate that these quantities are credible measures of the strength of

the spatial correlations present in the Rt data. Based on the relative magnitudes

of the uncertainties shown in Table 1, the results of this analysis indicate that neither

the strain level nor the strain mode produced statistically significant changes in the

degree of correlation in the Rt data for this particular alloy.

3.2. Information theory approach

The spatial relationships within the Rt data were also analysed by constructing a set

of normalised complex eigenvalues for all the Rt maps acquired in each strain

condition. As shown in [17], the eigenvalue spectrum, [1,2, ... , n], for a matrix A,

i.e. ðð 1ffiffi

n p AÞÞ serves as a minimally biased ‘fingerprint’ that represents all of the

magnitude information as well as all of the spatial correlations latent within A at all

levels of complexity up to and including the rank of the matrix, n. In other words, the

individual complex eigenvalue spectra directly reflect all of the deformation effects at

a given strain condition for any Rt map. This deformation information is reduced to

a set of n complex eigenvalues each containing both a magnitude and a phase

component. Extracting the magnitudes and disregarding the phase information

contained in the eigenvalue spectrum (i.e. setting each phase angle equal to zero)

allows for the construction of a scalar quantity that distils the information regarding

the magnitudes and the spatial correlations existing in the Rt matrix into a

convenient form. This is accomplished by taking the absolute value of each

eigenvalue, i.e. multiplying each eigenvalue by its complex conjugate and taking the

square root: i j j 1⁄4 ði

i Þ

1=2

.

Information theory allows one to compare the information contained in a given

distribution with respect to some baseline, or reference state [20,21,24,40–47]. For

this evaluation, a set of reference states was created by randomly permuting [48] all

of the elements contained within each Rt matrix, thereby completely destroying all of

the correlations in each matrix while preserving all of the characteristics of the

original Rt distribution [49]. Since the first order statistical properties of each

randomised Rt matrix are identical to those of the original Rt matrix, the eigenvalues

of the randomised matrices reflect only the characteristics of the Rt magnitudes

generated by the individual deformation conditions.

The eigenvalues for the five Rt matrices in each strain condition were combined

to form a single ‘strip’ and then sorted in order of decreasing magnitude. The

resulting superposition of eigenvalues represents the character of the entire surface in

that strain condition. A similar superposition was constructed for the eigenvalues of

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the randomised matrices. An example of the differences between the superposed

correlated and superposed random eigenvalue distributions is shown in Figure 5.

A pair of discrete probability measures, or ratios, was constructed from the

eigenvalue distributions for each strain condition. One probability measure, P(jij)

(hereafter referred to as P), was based on the correlated eigenvalue distributions. A

second, Q(jij) (hereafter referred to as Q), was based on the random eigenvalue

distributions. The probability measure, A(jij), was defined in terms of the eigenvalue

distribution (i), the total number of eigenvalues (n), and the ratio associated with

the ith eigenvalue magnitude as follows [17]2

:

A i ð Þ1⁄4 j j

i j j

Pn

i1⁄4i i j j  	, where A 1⁄4 ðP, QÞ: ð4Þ

The information content in A can be quantified directly by constructing another

scalar measure known as the ‘spectral’ or Shannon entropy SS. [41]. This spectral

entropy is expressed as the negative of the expectation value of the logarithm of the

probability measure, A:

Ss 1⁄4 X

i

Aið Þ ln Ai where X

i

Ai 1⁄4 1: ð5Þ

0

1

2

3

4

5

6

7

8

0 100 200 300 400 500 600 700

Correlated

Random

Eigenvalue Number

Eigenvalue Magnitude, μm

Figure 5. (Colour online). Two superposed eigenvalue distributions that represent the surface

of AA5754-O at the maximum uniform strain condition in the uniaxial strain mode in two

different conditions are shown. One contains the structural information (correlated) and the

other has been permuted so that all of the structural information is destroyed (random).

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The differential spectral entropy DSS, or the global difference between the

random and correlated conditions for a pair of eigenvalue distributions, defined as

DSs  ðÞ ð Ssð Þ Q Ssð Þ P 6Þ

is a measure of the complexity of the spatial correlations within a given Rt map. The

differential spectral entropy is shown as a function of true strain and strain mode in

Figure 6. In this figure, the DSS behaviour for the uniaxial and plane strain modes are

quite similar, showing a pronounced peak value at 15% nominal true strain followed

by a sharp decrease to the failure strain. In contrast, the peak exhibited in the

equi-biaxial mode is not as distinct as that shown in the other two modes and the

spectral entropy decreases more slowly to failure.

If the two distributions have similar overall shapes, such as those shown in

Figure 5, the differential spectral entropy may not have sufficient sensitivity to

clearly distinguish the differences between the two. In this situation, one can define a

‘relative spectral entropy’, Sr, based on the Kullback–Leibler distance (KLD), which

measures the ‘local’ discrepancy between any two pairs of probability distributions

(P, Q) [40,46,50,51]. The symmetric form of the KLD has been adopted for this

evaluation and is expressed as

Sr 1⁄4 X

i

Pi ln Pi

Qi


  
  þX

i

Qi ln Qi

Pi


  
  where X

i

Pi 1⁄4 1 and X

i

Qi 1⁄4 1: ð7Þ

0

0.02

0.04

0.06

0.08

0.1

Equi-biaxial

Uniaxial

Plane Strain

Nominal True Strain

Differential Spectral Entropy, 

Δ

Ss

0 0.05 0.10 0.15 0.20 0.25 0.30

Figure 6. (Colour online). The differential spectral entropy, DSs, is shown as a function of the

nominal true strain and the strain mode. Note that DSs is a dimensionless quantity.

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It should be noted that Sr40 for all cases except where Pi 1⁄4 Qi for all values of i.

In this case, Sr 1⁄4 0, which signifies that the two distributions are indeed identical.

Even though the KLD is not a true metric or a physical ‘distance’ in a strict

measurement sense, it can be regarded as an optimally sensitive discriminator

between any pair of distributions, provided the two distributions are based on a

common measure.

Relative spectral entropy data were plotted against the nominal true strain level

and shown as Figure 7. The format for this figure is identical to Figure 6 enabling a

direct comparison between the two entropies. Two trends are apparent. First, while

only the uniaxial and plane strain modes exhibited similar shapes in Figure 6, the

overall shapes of the curves in Figure 7 are more similar in that each strain mode

exhibits a distinct peak that is immediately followed by a sharp decrease to the failure

strain. This suggests that the decrease in relative spectral entropy might be a

signature for the onset of critical localisation. Second, the magnitude ranges of the Sr

data are considerably greater than the DSS data for the same strain conditions. This

amplitude increase indicates that the element-by-element comparison between the

correlated and random distributions used in the relative spectral entropy calculation

substantially enhances the signal-to-noise ratio in the Rt data compared to the

differential spectral entropy, DSS.

Figures 6 and 7 also illustrate how strain mode influences the general

deformation behaviour of the aluminium sheet. That is, when presented as a

0

10

20

30

40

50

60

70

0 0.05 0.10 0.15 0.20 0.25 0.30

Relative Spectral Entropy, 

Sr

Nominal True Strain

Equi-biaxial

Uniaxial

Plane Strain

Figure 7. (Colour online). The relative spectral entropy, Sr, is shown as a function of the

nominal true strain level and the strain mode. Note that Sr is a dimensionless quantity.

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function of the strain mode, the failure strain is considerably higher for the equibiaxial mode compared to the uniaxial and plane strain modes. This is primarily due

to the manner in which the macroscopic constant volume conditions are imposed by

the individual strain mode during the deformation process [52]. Since the intent is to

assess the entropy change as a function of strain intensity and strain mode, the

results are all expressed in terms of a true strain in each strain mode. However, the

entropy changes can also be expressed in terms of a normalised strain intensity,

which can be calculated using the von Mises relationship for the effective uniaxial

strain. Note if the entropy change is expressed in terms of the effective strain, the

trends may not necessarily be the same as those shown in Figures 6 and 7.

The Kullback–Leibler distance [51] analysis can also provide a direct assessment

of the statistical variability for the surface in general. This is accomplished by

evaluating the KLD for each of the possible combinations of the five datasets

according to the following:

KLDð Þ M,N 1⁄4 X

i

Mi ln Mi

Ni


  
  þX

i

Ni ln Ni

Mi


  
  where X

i

Mi 1⁄4 1 and X

i

Ni 1⁄4 1:

ð8Þ

In this equation, M and N represent the five data sets. Due to the form of the

KLD adopted for this evaluation, this equation produces a 25-element symmetric

matrix. A KLD analysis was performed on the data in each strain condition for both

the correlated and randomised conditions. The KLD values are presented in Tables 2

and 3, respectively. Note that the magnitudes of the KLDs are relatively small and

that the maximum KLD is approximately twice that of the mean value. This

indicates that the five Rt maps for a given strain condition are remarkably similar in

a general statistical sense.

Table 2. Analysis of the divergence between individual surface measurements: Correlated.

Strain path

Nominal true

strain

Mean K–L

distance Uncert. 

Maximum

K–L distance

As-polished 0.00 0.0097 0.0054 0.0173

Equi-biaxial 0.05 0.0073 0.0045 0.0135

– 0.10 0.0089 0.0064 0.0220

– 0.15 0.0142 0.0099 0.0347

– 0.20 0.0077 0.0043 0.0135

– 0.26 0.0077 0.0045 0.0142

Uniaxial (RD ?) 0.05 0.0087 0.0055 0.0188

– 0.10 0.0104 0.0059 0.0182

– 0.15 0.0068 0.0042 0.0133

– 0.23 0.0099 0.0061 0.0178

Plane strain (RD ?) 0.05 0.0054 0.0031 0.0091

– 0.10 0.0057 0.0033 0.0102

– 0.15 0.0089 0.0049 0.0156

– 0.16 0.0065 0.0042 0.0129

Note: ‘Uncert.’ refers to one standard deviation of the mean, .

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4. Discussion of results

Throughout this evaluation, clusters have been referred to, but, in fact, the only

statement resembling a definition of a cluster in this paper has been ‘several Rt cells

exceeding a threshold magnitude that are in close proximity and act in a cooperative

manner’. The rationale behind this is that defining a cluster in terms of a specific metric

biases the analysis used to evaluate the spatial relationships that may, or may not, exist

within a given dataset [53]. The definition of a cluster is not the only source of bias in

these analyses. As noted previously, the colour scales used in Figures 2 and 3 are based

on a 95% failure probability as predicted by a Weibull analysis. As a result, the 8-bit

colour scales compress the tail regions of the Rt distributions to accentuate the areas of

the Rt map where failure is most probable. Using this format to present the Rt data

introduces a visual bias by artificially creating regions of cells with similar colour.

However, the apparent structure that is present in these figures is simply an artefact

introduced by the presentation. It is also possible to accentuate the segment of the Rt

distribution that lies above the 95% probability level. Figure 8 exhibits how simply

redefining the bounds of the 8-bit colour scale can influence the perception of clusters.

While it is impossible to conduct any analysis without introducing preconceptual

bias in some form, the results of a spatial analysis are especially sensitive to the

influence of such bias. Avoiding an a priori definition of a cluster (e.g. some

predetermined shape consisting of n or more Rt elements, or n Rt elements in contact

with one another in some arbitrary arrangement) and quantifying the multi-point

spatial correlations directly produces a robust statistical analysis of any and all

spatial relationships that are present, while simultaneously increasing the confidence

in the assumption of independent conclusions drawn from that analysis.

The remarkable similarity between the ACF surfaces shown in Figure 4 and the

ACF data in Table 1 demonstrates that the differences between the Rt maps are

virtually indistinguishable at the two-point ACF level. One implication of this

Table 3. Analysis of the divergence between individual surface measurements: Random.

Strain path

Nominal true

strain

Mean K–L

distance Uncert. 

Maximum K–L

distance

As-polished 0.00 0.0065 0.0037 0.0112

Equi-biaxial 0.05 0.0066 0.0037 0.0113

– 0.10 0.0057 0.0030 0.0089

– 0.15 0.0096 0.0068 0.0221

– 0.20 0.0094 0.0062 0.0191

– 0.26 0.0100 0.0064 0.0221

Uniaxial (RD ?) 0.05 0.0075 0.0043 0.0127

– 0.10 0.0090 0.0053 0.0173

– 0.15 0.0055 0.0032 0.0093

– 0.23 0.0099 0.0065 0.0235

Plane strain (RD ^) 0.05 0.0090 0.0062 0.0213

– 0.10 0.0078 0.0057 0.0195

– 0.15 0.0086 0.0058 0.0176

– 0.16 0.0058 0.0045 0.0129

Note: ‘Uncert.’ refers to one standard deviation of the mean, .

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Figure 8. (Colour online). The influence of the colour scale on the perception of ‘‘clusters’’

that have formed within the Rt maps. Both maps present the Rt data shown in Figure 2F. A is

shown using the original 8-bit colour scale (i.e. derived from the Weibull localisation

probability analysis outlined in reference [11]), and B is shown using an 8-bit colour scale that

has been recomputed to accentuate the range of probabilities that lie between the 95% and

99.9%.

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statistical degeneracy is clear. The unexpected levels of similarity in the correlation

results indicate that a two-point analysis lacks sufficient sensitivity to discriminate

the complex spatial relationships that exist in the Rt maps. The presence of diamondshaped regions of high-correlation in Figure 4 denote that correlations are indeed

present within the Rt data; however, complex clustering, such as that exhibited in

Figure 3, implies that these correlations are of a higher order and cannot be

differentiated with a two-point analysis technique. While a two-point analysis may

be entirely appropriate to assess the spatial relationships that exist within other

datasets (such as those based on the measurement of a power spectrum), the ACF

technique does not provide any reliable information that elucidates the nature of the

relationship between complex clustering and the onset of critical strain localisation in

the AA5754 alloy. This result also indicates that whatever influence the formation of

Rt clusters may have on critical strain localisation can only be extracted with

analytical methods that are based on multi-point spatial statistics [54].

The eigenvalue analysis is an attractive and powerful approach to quantify the

existence of clusters within the Rt data for two important reasons. First, the complex

eigenvalue spectrum calculated for each Rt map distils the spatial information

present in that map into a compact scalar form. Second, it allows for the

establishment of a well-defined reference state for comparison (e.g. the randomised

condition). Analysis of the spatial relationships in the eigenvalue distributions by

information theory-based methods, which offer high flexibility and minimal

preconceptual bias by design, yields a systematic evaluation of any structure that

may be present in any Rt map. Quantifying the comparative degrees of disorder

(entropy) for two normalised distributions that correspond to a deformed surface in

two different conditions (i.e. one that contains the correlation information, and one

that does not) reveals the change in entropy produced by the formation of that

structure. Both the spectral entropy and the relative spectral entropy are capable of

quantifying this entropy change; however, the considerably higher signal-to-noise

ratio in the relative spectral entropy results clearly indicates that this method is the

more appropriate tool for this particular application.

The initial monotonic increase exhibited in each of the three relative spectral

entropy curves shown in Figure 7 reflects the divergence of the correlated and

random surface conditions at each strain level. Recall that all of the surface effects

produced by plastic strain are directly reflected in the shape of the Rt distribution

and that this distribution can be completely rearranged (randomised) with no change

in the first-order statistical properties (i.e. mean, standard deviation, skew and

kurtosis). This isolates the overall effects produced by an increase in plastic strain

from the changes that occur in the surface structure generated by that increase. While

it may appear relatively homogeneous on a macroscopic level, the deformation in a

polycrystalline material occurs by highly complex and non-uniform processes. Each

grain in a polycrystalline network deforms by different amounts as determined by the

individual orientation of the grain, the local Schmid factor and the constraints

imposed by neighbouring grains at or below the surface [55–57] and, for this reason,

the local or grain level strain conditions have a strong influence on the overall

character of the ensuing surface roughness. Increasing the level of plastic strain

intensifies the crystallographic nature of the surface roughness and adds to the

overall structure of the surface (see Figure 1). This increase in surface structure

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results in a complementary change in the degree of disorder associated with the

surface, and that is precisely what is exhibited in Figure 7.

Given that Rt is effectively the scalar projection of the height gradient, Rt maps

such as those shown in Figure 2, serve as measures of the change in the local peak-tovalley height for a surface topography (Figure 1). However, the relationship between

a topography and the corresponding Rt map is not necessarily isomorphic. Because it

is derived from a set of surface height values, an Rt cell is a coarse-grained measure

of the differential topography on a local level, which accentuates the microstructural

features that promote critical strain localisation (e.g. cracks, splits and tears and

necking). For example, consider a group of adjacent grains with orientations that do

not favour primary slip. As the strain increases, these grains will rotate relative to

one another in response to the applied strain and, in this situation, the deformation

tends to concentrate in the grain boundary regions resulting in relatively large local

displacements. This will produce a cluster of Rt cells that emphasises the

displacement in the grain boundary regions rather than in the grain interiors.

Therefore, the shape of this cluster is determined directly by the microstructure

(Figure 3).

However, as shown in our previous analysis [11], the range of values (i.e. Rt

magnitudes) in the distribution broadens markedly with increasing plastic strain,

thereby augmenting and accentuating the tail region of the Rt distribution. This

dispersion, caused by statistically rare events, makes it progressively less probable for

similar Rt values to lie within close proximity of one another. In other words, since

the range of potential Rt values expands with increasing plastic strain, it becomes

increasingly more difficult to form groups of cells with similar Rt magnitude

(clusters) as the strain level increases. This results in two competing processes: one

where the formation of clusters is favourable and one where it is not. The maximum

exhibited in the three curves in Figure 7 is produced by this competition. Both

processes are dependent on changes in the surface morphology. At strains less than

approximately 15% in the uniaxial and plane strain modes, and approximately 22%

in the equi-biaxial mode, the first process, which primarily reflects the coalescence of

Rt cells of similar magnitude, dominates the behaviour. At strains larger than these,

the second process, which is more strongly influenced by rare probabilistic events

that occur in the disproportionately long tail of the Rt distribution (such as the

isolated cells present in Figure 8), dominates. Since the range of variability within the

individual data sets for each strain condition, in both the randomised and correlated

conditions is small, as shown in Tables 2 and 3, the maxima in Figure 7 cannot be an

artefact caused by large statistical fluctuations in the surface data (i.e. the breakdown

of stationarity). For this reason, the crossover can be considered a signature of the

onset of critical strain localisation and, thus, a precursor to failure.

More explicitly, recalling that the data acquired in the failure condition establish

the maximum obtainable surface roughness for the AA5754-O alloy as well as the

morphological conditions that gave rise to the localisation event in each strain mode,

the fact that exceptionally high Rt values were observed (Figure 8) and no failure

occurred indicates that the surfaces in this condition are, in fact, metastable. It is,

therefore, reasonable to hypothesise that the onset of metastability coincides with the

occurrence of the maximum relative spectral entropy.

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The results of these analyses prove that critical strain localisation requires the

formation of structure within the Rt data. This structure cannot be distinguished by

statistical methods that only describe correlations at the two-point level such as the

autocorrelation function. While higher order correlation analysis clearly demonstrates that structure is present and that it is produced by the microstructure, the

crossover from the process where the formation of clusters is favourable to the

probabilistic process where it is not signifies the onset of critical strain localisation

and failure. After this shift in the dominant process occurs, the surfaces become

metastable and the application of additional strain produces perturbations that

trigger the failure event. Because this process is more dependent on statistically rare

events, predicting the exact location of that failure event with a deterministic model

becomes significantly more difficult.

5. Conclusions

The spatial correlations present in surface data produced by three in-plane

deformation modes (equi-biaxial, uniaxial and plane strain) were characterised

with two different analytical approaches. The first was based on a conventional twopoint autocorrelation function. The second was based on the complex eigenvalue

spectra associated with each surface and information theory methods.

The data from the ACF approach were assessed by calculating the overall volume

and the moment of inertia for each ACF surface. While the small uncertainties

associated with the ACF volume and moment of inertia data confirmed that these

quantities were statistically reliable measures of the spatial correlations, the results

indicated that neither the strain level nor the strain mode produced a statistically

significant change in the degree of correlation in the Rt data. Therefore, the

conclusion drawn from the ACF analysis was that the ACF technique does not

contain sufficient sensitivity to elucidate the nature of the relationship between

complex clustering and critical strain localisation in the AA5754 alloy.

The data from the information theory-based approach revealed that quantifying

the comparative degrees of disorder (i.e. information–theoretic entropy) for two

normalised distributions that represent a deformed surface under two different

conditions (one that contains the correlation information, and one that does not)

directly reveals the change in the degree of disorder associated with the surface

produced by the plastic strain. While both the spectral entropy and the relative

spectral entropy are capable of revealing this change, the relative spectral entropy

calculation was the more appropriate tool for this particular application. Each strain

mode exhibited a monotonic increase in the relative spectral entropy as a function of

true strain that attained a maximum, and then decreased sharply to the failure strain.

This behaviour resulted from two competing processes: one where the formation of

structure is favourable and one where it is not. The first process dominated the

behaviour at strains up to and including the maximum Sr. At higher strains, the

second process, driven by statistically rare events in the disproportionately long tail

of the Rt distribution, dominated. Therefore, the crossover point can be considered

as a precursor to failure because, once the dominant process shifts, additional strain

results in the precipitate formation of a critical strain localisation event.

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Notes

1. Certain commercial equipment, instruments, 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.

2. Note this measure is equivalent to a normalised eigenvalue distribution.

References

[1] T. Foecke, M.A. Iadicola, A. Lin and S.W. Banovic, Metall. Mater. Trans. A 38 (2007)

p.306.

[2] F. Barlat, R.C. Becker, Y. Hayashida, Y. Maeda, M. Yanagawa, K. Chung, J.C. Brem,

D.J. Lege, K. Matsui and S.J. Murtha, Int. J. Plasticity 13 (1997) p.385.

[3] A.J. Beaudoin, A. Acharya, S.R. Chen, D.A. Korzekwa and M.G. Stout, Acta Mater. 48

(2000) p.3409.

[4] D. Raabe, M. Sachtleber, H. Weiland, G. Scheele and Z. Zhao, Acta Mater. 51 (2003)

p.1539.

[5] R. Becker and O. Richmond, Model. Simul. Mater. Sci. Eng. A 2 (1994) p.439.

[6] H.A. Al-Quershi, A.N. Klein and M.C. Fredel, J. Mater. Proc. Tech. 170 (2005) p.204.

[7] J. Savoie, M. Jain, A.R. Carr, P.D. Wu, K.W. Neale, Y. Zhou and J.J. Jonas, Mater. Sci.

Eng. A 257 (1998) p.128.

[8] P.D. Wu, D.J. Lloyd, M. Jain, K.W. Neale and Y. Huang, Int. J. Plasticity 23 (2007)

p.1084.

[9] W.B. Lievers, A.K. Pilkey and D.J. Lloyd, Acta Mater. 52 (2004) p.3001.

[10] J.E. Bird and J.L. Duncan, Metall. Trans. A 12 (1981) p.235.

[11] M.R. Stoudt, J.B. Hubbard, M.A. Iadicola, S.W. Banovic, Metall. Mater. Trans. A 40

(2009) p.1611.

[12] S.W. Banovic, M.A. Iadicola and T. Foecke, Metall. Mater. Trans. A 39 (2008) p.2246.

[13] M.A. Iadicola, T. Foecke and S.W. Banovic, Int. J. Plasticity 24 (2008) p.2084.

[14] H.J. Frost and M.F. Ashby, Deformation-Mechanism Maps: The Plasticity and Creep of

Metals and Ceramics, Pergamon Press, Oxford, 1982.

[15] G.W. Greenwood, Mater. Sci. Eng. A 410/411 (2005) p.12.

[16] F.A. Mohamed and T.G. Langdon, Metall. Trans. 5 (1974) p.2339.

[17] J.B. Hubbard, M.R. Stoudt and L.E. Levine, J. Appl. Phys. 102 (2007) p.1.

[18] M. Turk and A. Pentland, J. Cogn. Neurosci. 3 (1991) p.71.

[19] C. Arndt, Information Measures Information and Its Description in Science and

Engineering, Springer, New York, 2001.

[20] R.B. Ash, Information Theory, Dover, New York, 1990.

[21] M.R. Fazlollah, An Introduction to Information Theory, Dover, New York, 1994.

[22] R.M. Gray, Entropy and Information Theory, Springer, New York, 1990.

[23] G. Jumarie, Relative Information Theories and Applications, Springer, New York, 1990.

[24] A.I. Khinchin, Mathematical Foundations of Information Theory, Dover, New York, 1957.

[25] Anon, Aluminium Standards and Data 2003, The Aluminium Association, Washington,

DC, 2003.

[26] Anon, in 1993 Annual Book of ASTM Standards, Section 3, Metals Test Methods and

Analytical Procedures, ASTM, Philadelphia, PA, 1993.

[27] K.S. Raghavan, Metall. Mater. Trans. A 26 (1995) p.2075.

[28] Z. Marciniak and K. Kuczynski, Int. J. Mech. Sci. 9 (1967) p.609.

[29] M.R. Stoudt, J.B. Hubbard and S.A. Janet, Mater. Sci. Tech. 24 (2008) p.253.

2424 M.R. Stoudt and J.B. Hubbard

Downloaded By: [NIST National Institiutes of Standards & Technology] At: 18:53 2 September 2009
[30] Anon, ASME Designation B46.1-2002: Surface Texture (Surface Roughness, Waviness,

and Lay), The American Society of Mechanical Engineers, New York, 2002.

[31] J. Dattorro, in Convex Optimization and Euclidean Distance Geometry, MeBoo

Publishing, Palo Alto, CA, 2005.

[32] T.R. Thomas, Rough Surfaces, Imperial College Press, London, 1999.

[33] M.R. Stoudt and J.B. Hubbard, Acta Mater. 53 (2005) p.4293.

[34] W.P. Dong, P.J. Sullivan and K.J. Stout, Wear 178 (1994) p.45.

[35] S.J. Orfanidis, Optimum Signal Processing, Prentice-Hall, Englewood Cliffs, NJ, 1996.

[36] Anon, MatlabR2007a, The MathWorks. Inc, Natick, MA, 2007.

[37] J. Brochard, M. Khoudeir and B. Augereau, Pattern Recogn. Lett. 22 (2001) p.759.

[38] F. Blateyron, New 3D Parameters and Filtration Techniques for Surface Metrology,

Quality Magazine (2006). Available at: www.qualitymag.com/.../New3DParameters

andFiltrationTechniquesforSurfaceMetrology.pdf

[39] D. Halliday and R. Resnick, Physics, Wiley, New York, 1978.

[40] Anon. Quantum Relative Entropy, Wikipedia, The Free Encyclopedia, 2006. Available at:

en.wikipedia.org/wiki/Quantum_relative_entropy

[41] C.E. Shannon and W. Weaver, The Mathematical Theory of Communication, University

of Illinois Press, Urbana, 1949.

[42] P. Diaconis, Bull. Am. Math. Soc. 40 (2003) p.155.

[43] A. Edelman, J. Multivar. Anal. 60 (1997) p.203.

[44] S. Geman, Ann. Prob. 14 (1986) p.1318.

[45] J. Ginibre, J. Math. Phys. 6 (1965) p.440.

[46] S. Kullback, Information Theory and Statistics, Wiley, New York, 1959.

[47] N. Lehmann and H.J. Sommers, Phys. Rev. Lett. 67 (1991) p.941.

[48] G.J. Chaitin, Sci. Am. 259 (1988) p.80.

[49] L.E. Levine, A Permutation Algorithm for Randomizing the Elements within Square

Matrices, NIST, Gaithersburg, MD, 2006.

[50] Anon., Kullback–Leibler Divergence, Wikipedia, The Free Encyclopedia (2006). Available

at: en.wikipedia.org/wiki/Kullback-Leibler_divergence

[51] S. Kullback and R.A. Leibler, Ann. Math. Stat. 22 (1951) p.79.

[52] G.E. Dieter, Mechanical Metallurgy, McGraw-Hill, New York, 1986.

[53] A. Kraskov, H. Sto ̈gbauer, R.G. Andrzejak and P. Grassberger, Europhys. Lett. 70

(2005) p.278.

[54] B.D. Ripley, Spatial Statistics, Wiley-Interscience, Hoboken, NJ, 2004.

[55] G.I. Taylor, J. Inst. Metals 62 (1938) p.307.

[56] J.D. Eshelby, Proc. R. Soc. Lond. A 241 (1957) p.376.

[57] M.F. Ashby, Phil. Mag. 21 (1970) p.399.

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