Topographic analysis and Weibull regression

for prediction of strain localisation in an

automotive aluminium alloy

J. B. Hubbard1, M. R. Stoudt*1 and A. M. Possolo2

We fit a two-parameter Weibull regression model by maximum likelihood estimation (MLE) to

filtered surface roughness data. These data were acquired with scanning confocal laser

microscopy performed on aluminium alloy AA 5754-O surfaces that were subjected to a range of

plastic strain intensities in three different in-plane strain modes. Noting that one of the two Weibull

regression parameters is, to a good approximation, invariant with strain intensity for a given strain

mode, the authors find that the variation of the second parameter with strain intensity conforms to

a simple quadratic function. These functions may then be used to generate accurate, statistically

significant, single parameter predictors of both strain intensity and strain mode up to and

including the onset of critical strain localisation and/or failure.

Keywords: Strain localisation modelling, Strain mode, Scanning laser confocal microscopy, 3D characterisation, Weibull statistics, Sheet metal forming

Introduction

The need to reduce gross vehicle weight and improve fuel

economy has escalated the demand for new lightweight

materials such as aluminium alloys, magnesium alloys and

advanced high strength steel throughout the transportation

industry. However, the available property data and

constitutive laws for these new materials are often limited

or inadequate for use in the numerical models that are used

to predict the behaviour during sheet metal forming. The

ensuing inability to reliably model the mechanical behaviour creates significant obstacles that impede widespread

incorporation of new alloys. Inaccurate prediction of the

manner in which surface inhomogeneities evolve during

sheet metal forming is a prime example of the problem. In

addition to creating a surface that is cosmetically unacceptable, surface inhomogeneities have profoundly

detrimental effects on the formability of the metal sheet.

Specifically, they generate unpredictable variability in the

friction between the metal sheet and the die surfaces during

stretch forming, and they promote strain localisation, both

of which lead to unexpected failures in the sheet metal

component.25 For this reason, the character of the deformed surface can essentially determine whether an alloy is

suitable for a particular application.

During sheet metal forming, macroscopic deformation

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

biaxial, uniaxial and plane strain). Since each strain mode

imposes specific constraints on the material flow during

stretch forming, each mode imparts distinctly different

surface morphologies for the same overall level of plastic

strain.23 Thus, it is essential that the deformation produced

by each component of the macroscopic strain be properly

identified and accurately characterised. Such characterisations tend to be expensive and complicated processes that

require numerous high resolution measurements of the

deformation under each strain mode.13 Despite all of the

many improvements in the numerical models currently

adopted by the US automotive industry to predict the

evolution of the deformed surface,1,6–8,15,19,21,27 inconsistencies still exist between the results predicted by these models

and what is observed experimentally. This creates a

situation where a numerical model that predicts the material

behaviour during forming may correctly indicate the overall

trends of the deformation, but fails to reliably predict the

actual magnitudes of the strain at which critical localisation

(failure) occurs.9

It was recently noted23 that a two-parameter Weibull

probability distribution, in which a is the scale parameter and b is the shape parameter, provides good

overall fits to filtered surface roughness data. These

roughness data were acquired from the surfaces of

aluminium alloy AA 5754-O specimens that were subjected to a range of plastic strain intensities in three

different in-plane strain modes. The authors refer to this

data as filtered because they analysed raw topographic

data with a technique that extends the profile based peak

to valley surface roughness parameter Rt to a matrix

form. This process generated sets of local intensity maps

composed of coarse grained (n6n) arrays of Rt ‘cells’

that were constructed from the raw topographic data.

The maximum height of a profile, Rt, is defined in the

surface roughness literature as the vertical distance

between the highest and the lowest points within a

profile of any given length3

1

NIST Centre for Metal Forming, Materials Science and Engineering

Laboratory, Gaithersburg, MD 20899, USA

2

Statistical Engineering Division, Information Technology Laboratory,

National Institute of Standards and Technology, 100 Bureau Drive,

Gaithersburg, MD 20899, USA

*Corresponding author, email stoudt@nist.gov

1206

2011 Institute of Materials, Minerals and Mining

Published by Maney on behalf of the Institute

This article was written and prepared by an officer and/or employee of the

US Government as part of their official duties and is not copyrightable

Received 22 March 2010; accepted 29 March 2010

DOI 10.1179/026708310X12701149768179 Materials Science and Technology 2011 VOL 27 NO 7
Rt~RpzRv (1)

In this equation, Rp is the vertical distance between the

highest point of the profile and the mean line, and Rv is

the vertical distance between the lowest point of the

profile and the mean line within the same evaluation

length. Note that the data located below the mean line of

the profile are traditionally given a negative sign, so by

this convention, all of the Rt values are positive.

The Rt parameter demonstrates several attractive properties that make it ideal to characterise the surface conditions

that promote critical strain localisation, and to quantify the

relative magnitudes of those surface conditions:

(i) it is highly sensitive to local changes in surface

height

(ii) it is based on a straightforward calculation

(iii) the data over which Rt is determined can range

from a short profile segment to the entire

profile. Consequently, the Rt parameter can

track the changes in the surface height on a

local or on an overall basis

(iv) the magnitudes of the local surface extremes will

change with strain to the point where they

reflect the onset of critical strain localisation

(v) the Rt parameter can be easily determined

between any two points or nodes during a

numerical simulation, thereby making changes

in the local surface conditions directly integrable into formability models.

Mathematically, an Rt matrix is analogous to a Euclidean

distance matrix.10 Construction of an Rt matrix required

subdividing the original 512 row6512 column matrix of

height data into a 128 row6128 column array of Rt

‘cells.’ Each element in the 16 384 cell Rt matrix, Rt(i,j), is

the maximum difference for the set of 16 height values

contained within each (4 row64 column) cell. While

similar in concept to the profile form, the matrix format

preserves the spatial coordinates for each Rt(i,j) cell by

construction. Therefore, this approach enables direct

quantification of the local Rt magnitudes as well as the

specific location of that Rt value within the original

topography. That is, this technique directly links any

discernible feature in the source topography to the

corresponding change in magnitude of the local surface

height. Obviously, the level of resolution can easily be

adjusted by simply changing the dimensions of the

submatrix cells. For this reason, this process was repeated

using larger (64 element 868 cells) to evaluate the

strength of the resolution dependence in the Rt data.

An example of this construction exhibiting the relationships between the original topography (Fig. 1A), the

distribution of surface heights in the topography (Fig. 1B),

the corresponding Rt map (Fig. 1C) and the distribution

of filtered surface heights (Fig. 1D) is shown as Fig. 1.

Each cell in the Rt map (Fig. 1C) is derived from the

topography (Fig. 1A) and it accentuates the height

disparities between topographical features. As such, this

approach generates a direct three-dimensional quantification of the local intensities produced by a change in the

surface morphology. Since maps were constructed for

multiple strain levels in three different strain modes, one

can evaluate the morphological conditions that promote

critical strain localiation for each strain condition.

It was noted in Refs. 22 and 23 that the Weibull a

parameter seems to vary systematically with strain mode

and strain intensity, and that the Weibull b parameter was

essentially constant for each strain mode. For this reason,

the Weibull b parameter could possibly be regarded as a

characteristic of the AA 5754 alloy in the O condition. In

the present paper, the authors attempt to quantify these

general observations and to ascertain the extent to which

the Weibull scale parameter a can serve as a predictor of

the strain intensity up to and including the onset of critical strain localisation and/or failure. The existence of an

isomorphism (i.e. a one to one relationship) between the

strain mode, the strain intensity and a Weibull regression

scale parameter, would strongly suggest an empirical

correlation between these Rt maps and the active

deformation mechanisms for this particular alloy.

Experimental

Sets of 30630 cm (12612 in.) blanks were sheared from

commercially available 1 mm thick AA 5754-O sheet

stock for testing. This aluminium alloy was developed

primarily for automotive applications, and like most alloys

in the 5xxx series, AA 5754 is substitutionally strengthened, and it demonstrates good overall formability. The

literature indicates that AA 5754 typically contains

2?8 mass%Mg for solid solution strengthening, and

y0?5 mass%Mn for grain refinement and stability.4 The

grain structure was relatively equiaxed in the rolling plane

and slightly elongated along the rolling direction of the

sheet, which is indicative of the recrystallised microstructure normally associated with the O temper. The grain size

for this alloy was 40¡20 mm.5 All of the specimens used

for this evaluation were polished to a 6 mm diamond finish

using standard metallographic 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 inplane proportional strain modes, which are traditionally

defined in terms of the ratio of in-plane principal strains:

r5e2/e1.

14 The first strain mode was equibiaxial (r51).

The second strain mode was uniaxial (r520?5), and the

third was plane strain (r50). Both the equibiaxial and

plane strain deformations were imposed using an

augmented20 Marciniak flat bottom ram test.16 Samples

were strained in equibiaxial tension using a typical strain

rate of 561024 s

21 to 5, 10, 15 and 20% nominal true

strain. The uniaxial samples were all 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 methods2 with a

constant displacement rate resulting in an average strain

rate of 661024 s

21

. A third set of samples was strained to

similar true strain levels in the plane strain condition.

One sample was taken to ‘failure’ in each strain mode. In

this case, the term ‘failure’ denotes the maximum uniform

strain outside of the region where critical localisation

occurred. Note that the actual fracture event is not

included in any of the topographic measurements at the

failure strain level. For this reason, the surface roughness

data acquired from these samples are the maximum

roughness values that can be generated uniformly in the

AA 5754-O alloy by each strain mode. These data also

reflect the surface conditions that produced the failure

event. Coupons were cut from the centre of each deformed

specimen for topographic analysis.

The surfaces were quantified for each strain condition

with scanning laser confocal microscopy (SLCM). Each

Hubbard et al. Topographic analysis for prediction of strain localisation

Materials Science and Technology 2011 VOL 27 NO 7 1207
topographic measurement consisted of acquiring five

well separated SLCM images of the specimen surface

using a 610 objective lens and a nominal z-scan depth

of y40 mm. These imaging conditions generated sets of

6406512 pixel intensity images acquired with 12 bit

resolution and were consistent for all measurements.

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

was fixed by the objective lens at 1?56 mm, and the

spacing between the individual focal planes within each

image was y100 nm. These conditions yielded nominal

physical dimensions (x,y,z) of 10006800640 mm for

each image. As described in Ref. 24, the SLCM stores

each topographic image data as a raw depth map in

tagged image file format (TIFF), which contains the

complete set of imaging parameters and binary pixel

values. A computer code utilising the format standards

within these TIFF images was developed to convert the

raw depth map into a simple numerical matrix of surface

heights. The resulting matrix was then trimmed to a

square 512 row6512 column array to facilitate the

matrix based mathematical operations. Using the aforementioned x–y pixel spacing, the values in each matrix

correspond to an 8006800 mm area of the surface.

After conversion, the extreme values (defined as the

values in the height data greater than ¡6s, where s is

the standard deviation for all the heights in that matrix)

were screened from the datasets. This step was required

because some of the statistical parameters used to

A topography of AA 5754-O at maximum uniform strain condition in uniaxial strain mode; B corresponding distribution of

surface heights for each of five areas sampled in this strain condition; C Rt map constructed from topography shown in

A; D corresponding distribution of Rt values for each of five Rt maps

1 Series of figures illustrating relationships between topography, Rt maps and Weibull distribution used to evaluate and

model Rt data

Hubbard et al. Topographic analysis for prediction of strain localisation

1208 Materials Science and Technology 2011 VOL 27 NO 7
interpret the surface data are highly sensitive to outlier

data points. Any individual Rt value that exceeded the

¡6s threshold was reset to the mean value for that

particular surface. Note that the number of affected data

points for a given surface was typically less than 20 points

(or ,0?008% of the total number of data points within the

matrix). The resulting sets of residual matrices were used

as the source for all subsequent assessments of the surface

character. Note that 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.24

Results and discussion

Weibull regression

The authors modelled the dispersion of the Rt values

using Weibull probability distributions in which the

scale parameter is expressed as an explicit function of

the strain intensity. This model is often called a Weibull

Regression, a special case of survival regression, which is

widely used in studies of reliability. Such a model is

appropriate when one wishes to describe how the

lifetime of a mechanical part depends on one or several

attributes (either qualitative or quantitative).

The Weibull probability density pa,b is defined as

pa,bð Þr ~ b

a

 r

a

 b{1

exp { r

a

 b   (2)

for any Rt value r.0, and fixed values of the scale (a.0)

and the shape (b.0) parameters. Given a sample of Rt

values (r1, ..., rm), the maximum likelihood estimates

(MLEs) of the scale and shape parameters^a and ^b (the hats

denote MLE values), maximise the product pa,b(r1), pa,b(r2),

..., pa,b(rm) of the likelihood function L with respect to the

Weibull a and b parameters for a given strain mode.17

The method of maximum likelihood is one of several

statistical methods that can be used for estimating

parameter values from empirical data. However, when it

is applicable, MLE is generally preferred over other

methods because MLEs have attractive statistical

properties. In addition, the maximum likelihood estimation process also produces approximate assessments of

the parameter uncertainties. When the empirical data

are similar outcomes of independent Weibull random

variables, all with the same values of a and b, the MLEs

can be found by solving a system of two non-linear

equations, LL/La50, and LL/Lb50, where L, the

logarithm of the likelihood function, is defined as

Lð Þ a,b ~m log b

a

zð Þ b{1

Xm

i~1

log ri

a

 {Xm

i~1

ri

a

 

b

(3)

Alternatively, the MLEs can be found by maximising L

directly, which is what was carried out in this case, using

Nelder–Mead’s18 algorithm as implemented in function

‘optim’ of the R programming environment for statistical analysis and graphics.26

The Rt values in this evaluation are not necessarily

statistically independent because the Rt values in

neighbouring cells tend to be positively correlated.

That is, if an individual cell has an Rt value larger than

the median Rt value, then it is highly likely that a

neighbouring cell will also have a similarly large Rt

value. Therefore, even though the parameter estimates

will be optimised by the MLE technique, the associated

uncertainty assessments, which are based on the

assumption of statistical independence, will tend to

underestimate the magnitude of the actual uncertainties.

The measured strain values are expressed as nominal

true strain values, as equivalent scalar modal true strain

values emodal (where modal denotes either equibiaxial,

uniaxial or plane strain and is equivalent to e11 in the

table), and as effective true strain values eeff (where the

effective strain is the magnitude of a modal strain

expressed in terms of the equivalent uniaxial strain) in

Table 1. The effective strain is typically determined from

the principal components of the modal strain tensor (i.e.

e11, e22 and e33 in Table 1 and the von Mises criterion11

eef f~ð Þ2 1=2

3 ð Þ e11{e22

2

zð Þ e22{e33

2

zð Þ e11{e33

2 h i1=2

(4)

The authors have found that the maximum likelihood

estimate of the Weibull scale parameter ^a varies

markedly and systematically with strain intensity within

each strain mode. That is, while the magnitude of a^ is

proportional to the strain intensity, the manner in which

a^ varies depends on the macroscopic constant volume

constraints imposed by the individual strain modes. In

contrast, the overall magnitude of the maximum likelihood estimate for the Weibull shape parameter ^

b

Table 1 Strain levels achieved during in-plane deformation: effective strains were calculated using von Mises criterion

(equation (4))11

Nominal true strain emodal;e11 e22 e33 eeff

Equibiaxial

0.05 0.049 0.051 20.106 0.104

0.10 0.097 0.097 20.216 0.210

0.15 0.140 0.139 20.278 0.279

0.20 0.200 0.199 20.412 0.408

0.25 0.258 0.254 20.638 0.512

Uniaxial

0.05 0.051 ... ... 0.051

0.10 0.116 ... ... 0.116

0.15 0.166 ... ... 0.166

0.20 0.228 ... ... 0.228

Plane strain

0.05 0.051 20.

002 20.064 0.066

0.10 0.100 20.008 20.089 0.109

0.15 0.147 20.001 20.145 0.169

0.20 0.158 20.003 20.207 0.181

Hubbard et al. Topographic analysis for prediction of strain localisation

Materials Science and Technology 2011 VOL 27 NO 7 1209
remains essentially constant with strain intensity for a

given strain mode. However, the relative magnitudes of

the ^b values do exhibit slight differences between the

three strain modes. Because of the relative insensitivity

of ^b to changes in strain, and given the approximately

quadratic trend of the values of a

_

as a function of e, the

authors model these as

logð Þa ~c0zc1ezc2e

2 (5)

In these circumstances, the likelihood function L

becomes a function of four parameters, b, c0, c1 and

c2. The values of these parameters that maximise the

likelihood function are the maximum likelihood estimates ^

b, c

^

0, c

^

1 andc

^.

Figures 2 and 3 exhibit the results of the Weibull

regression model fits to the Rt data, using the modal

strain intensity, separately for each particular strain

mode. The plots shown in Fig. 2 are derived from the set

of Rt maps produced by the 64664 cell format. Similar

plots, as shown in Fig. 3, are derived from the set of Rt

maps produced by the 1286128 cell format.

The lines in Figs. 2A and 3A exhibit how a varies with

the set of strain intensities determined from each set of

modal strain values (for the 64664 and the 1286128

cases respectively). While the a magnitudes are somewhat

greater in Fig. 2A, the patterns of the relationships are

strikingly similar to those shown in Fig. 3A. Each plot

symbol represents the value of a^ derived from all five

regions sampled on an individual coupon. Data were

pooled across all five regions because each topographic

sample has the same nominal strain intensity. The

uncertainty in the vertical placement of each plotted

symbol in Figs. 2A and 3A (and also in both panels of

Fig. 4) is the uncertainty of a^ that derives from the

dispersion of Rt values within and between regions in the

same coupon. The magnitudes of these uncertainties are

roughly comparable to the sizes of the plot symbols and

are not visible in these figures. Note that the curved lines

shown in these figures are not fitted to the points. The

lines result directly from the Weibull regression. The

points, which represent the MLEs derived for each value

of strain intensity (and strain mode) separately, are

included to emphasise that the model is not a perfect fit.

Figure 4 shows the behaviour of a as a function of the

effective true strain. Since the effective strain is an

estimate of the uniaxial strain component in each strain

mode, this figure enables a direct comparison of the a

behaviour with respect to a common strain intensity.

Each point in Figs. 2B and 3B illustrates how b varies

with the same set of modal strain intensities. The

corresponding plots for the b estimates as a function of

the effective strain are qualitatively comparable to those

shown in Figs. 2B and 3B. The superimposed thin vertical

lines shown in these plots represent the 95% confidence

intervals for b. These confidence intervals are based on

standard uncertainties derived from a simulation study

that was performed to account for the fact that the Rt

values are not statistically independent. More precisely,

the authors took the following steps to estimate the

standard uncertainty of the maximum likelihood estimate

of b for each modal strain intensity value. First, the

authors modelled the logarithms of the Rt values as a

Gaussian random field on a regular lattice (based on

either the 64664 or 1286128 maps), with Matern’s

covariance function.12 Second, the authors used the

parameters of the fitted model to simulate 1000 realisations of arrays of Rt values, to which the authors fitted

the Weibull probability distributions described above.

Third, the authors computed the standard deviation of

the resulting ^b values. This approach provides a more

conservative and accurate assessment of the true uncertainties in the b parameter estimates. The results revealed

that the standard deviations of the ^b values produced by

the simulation are approximately twice as large as the

standard deviations that were based on the assumption

that the Rt values are statistically independent.

When the confidence intervals of all the b values

within a particular strain mode are projected onto the

vertical axis in either Fig. 2B or Fig. 3B, an overlap

between two or more intervals indicates that the values

of b corresponding to those strain intensities are

A Weibull scale parameter a as function of modal true strain; B Weibull shape parameter b as function of modal true

strain

2 Magnitudes of Weibull parameters estimates calculated for set of Rt values based on 64664 maps

Hubbard et al. Topographic analysis for prediction of strain localisation

1210 Materials Science and Technology 2011 VOL 27 NO 7
statistically indistinguishable. Even when the b values do

not overlap (i.e. they are significantly different statistically), the relative magnitude of the differences between

these b values is quite small with respect to the

magnitude of the changes in the a values for the same

strain intensities. For this reason, the variability in the b

values is of little consequence from an engineering

perspective.

Strain prediction

The Weibull regression models fitted to the Rt data are

all highly statistically significant, in that the Weibull

parameters and the coefficients in the quadratic regression fit in equation (5) are statistically non-negligible.

This implies that the fitted Rt values are a relevant

predictor of strain intensity for the three strain modes

considered in this study. Furthermore, the Rt values

retain their predictive value whether they are expressed

as either modal or effective strains. To illustrate how the

models used in this evaluation can be used to derive

predictions of strain intensity information from a set of

roughness data, consider the specimen that was strained

in uniaxial tension to produce a strain intensity of

emodal50?116 as an example. The Weibull MLEs

calculated for one of the five sets of Rt values from

this coupon were ^a56?526 and ^b52?896 (note that in

Figs. 2 and 3, there is only one point representing the

MLE for each combination of strain intensity and strain

mode because, as indicated above, the Rt data from all

five coupons were pooled to estimate the MLEs shown).

The Weibull regression produced the following fit for

this strain intensity for this particular strain mode

logð Þa ~0:8487z6:633emodalz9:

321e

2

modal

A Weibull scale parameter a as function of modal true strain; B Weibull shape parameter b as function of modal true

strain

3 Magnitudes of Weibull parameters estimates calculated for set of Rt values based on 1286128 maps

A a values calculated from set of 64664 maps; B a values calculated from set of 1286128 maps

4 Magnitudes of Weibull scale parameters shown as function of effective true strain

Hubbard et al. Topographic analysis for prediction of strain localisation

Materials Science and Technology 2011 VOL 27 NO 7 1211
Substituting the ^

a value of 6?526 into this expression and

solving for emodal produces a result of emodal50?13.

Similar calculations produce estimates of emodal for the

remaining four surface measurements for this particular

strain intensity: 0?11, 0?12, 0?11 and 0?12. These results

suggest that in this example, there is little variability in

the strain intensity between regions of the same coupon.

It should be re-emphasised that the following

simplifying assumptions were made to enable the

application of Weibull regression analysis to these

data:17 the shape parameter b remains invariant with

strain intensity for each strain mode and the a parameter

and the strain intensity are related as described by

equation (3). This model is advantageous because it

allows for integration of the information from each

surface measurement, irrespective of the strain intensity,

which corresponds to a particular strain mode.

The differences in magnitudes of a and b produced

from the two sets of Rt maps (64664 and 1286128)

demonstrate that the filtering and the binning of the data

have some influence on the behaviour of both a and b.

That is, the values of a and b both exhibit some

resolution dependence. However, the similarities in the

overall trends of these two parameters across the two

different resolutions suggest that they reflect characteristics of the material as a whole. Therefore, the key

question raised by these results is whether this predictive

capability only holds for this specific alloy and heat

treatment, or whether it might be more generally

applicable. If the latter possibility turns out to be valid,

this approach could provide a basic methodology for

constructing a fundamental constitutive relationship

between surface roughness and strain. Such a relationship could have substantial practical relevance. To

determine the answer, it is necessary to ascertain

whether the Weibull shape parameter b remains

approximately invariant for any or all of the strain

modes after implementing a change in a material

characteristic of this alloy that has a strong influence

on the mechanical properties, such as the grain size, the

workhardening behaviour or the crystallographic texture. It is also of particular interest to determine whether

the b parameter demonstrates a similar approximate

invariance in a different alloy system. If b were to

demonstrate approximate invariance and a were to scale

with strain intensity monotonically under either of these

conditions, it may then be possible to use surface

roughness measurements straightforwardly to determine

the principal components of the strain tensor (i.e. e11, e22

and e33) in each strain mode. Such a constitutive

relationship could prove to be quite useful as a

predictive tool in sheet metal forming.

Conclusions

In the present paper, the authors have shown that a

Weibull regression, fitted using the method of maximum

likelihood, is an appropriate model for the Rt values

that were derived from topographic measurements on

strained coupons of AA 5754. The results also indicated

that the Weibull model could be used to develop an

accurate parametric predictor of strain intensity for a

given set of Rt data. Even though the differences

observed between the 64664 Rt maps and the

1286128 Rt maps revealed that the behaviour of the

Weibull scale and shape parameters are somewhat

dependent on the Rt cell size, the similarities in the

general trends suggest that these parameters are likely to

be characteristics of the material as a whole.

The critical issue raised with this analysis is whether

the Weibull parameters will demonstrate the same

characteristics that were observed here in a different

alloy. That is, shape parameter is approximately

invariant, and the scale parameter scales monotonically

with strain intensity. If so, it may then be possible to use

surface roughness measurements straightforwardly to

determine the principal components of the strain tensor

(i.e. e11, e22 and e33) in each strain mode. A constitutive

relationship such as this could be quite useful as a

predictive tool in sheet metal forming. For this reason,

further studies are in progress to determine the universal

nature of these characteristics.

References

1. H. A. Al-Quershi, A. N. Klein and M. C. Fredel: J. Mater. Process.

Technol., 2005, 170, 204–210.

2. ‘Standard test methods of tension testing of metallic materials’, E08, ASTM, Philadelphia, PA, USA, 1993.

3. ‘Surface texture (surface roughness, waviness and lay)’, B46?1,

ASME, New York, USA, 2002.

4. The Aluminum Association: ‘Aluminum standards and data 2003’,

235; 2003, Washington, DC, The Aluminum Association.

5. S. W. Banovic, M. A. Iadicola and T. Foecke: Metall. Mater.

Trans. A, 2008, 39A, 2246–2258.

6. F. Barlat, R. C. Becker, Y. Hayashida, Y. Maeda, M. Yanagawa,

K. Chung, J. C. Brem, D. J. Lege, K. Matsui, S. J. Murtha and S.

Hattori: Int. J. Plast., 1997, 13, 385–401.

7. A. J. Beaudoin, A. Acharya, S. R. Chen, D. A. Korzekwa and

M. G. Stout: Acta Mater., 2000, 48, 3409–3423.

8. R. Becker and O. Richmond: Modell. Simul. Mater. Sci. Eng.,

1994, 2, 439–454.

9. J. E. Bird and J. L. Duncan: Metall. Trans. A, 1981, 12A, 235–241.

10. J. Dattorro: ‘Euclidean distance matrix’, in ‘Convex optimization

and euclidean distance geometry’, Chapter 4, 219–314; 2005, Palo

Alto, CA, MeBoo Publishing.

11. G. E. Dieter: ‘Mechanical metallurgy’, 751; 1986, New York,

McGraw-Hill.

12. P. J. Diggle and P. J. Ribeiro: ‘Model-based geostatistics’; 2007,

New York, Springer.

13. T. Foecke, M. A. Iadicola, A. Lin and S. W. Banovic: Metall.

Mater. Trans. A, 2007, 38A, 306–313.

14. M. A. Iadicola, T. Foecke and S. W. Banovic: Int. J. Plast., 2008,

24, 2084–2101.

15. W. B. Lievers, A. K. Pilkey and D. J. Lloyd: Acta Mater., 2004, 52,

3001–3007.

16. Z. Marciniak and K. Kuczynski: Int. J. Mech. Sci., 1967, 9, 609–

620.

17. W. Meeker and L. A. Escobar: ‘Statistical methods for reliability

data’; 1998, New York, John Wiley & Sons.

18. J. A. Nelder and R. Mead: Comput. J., 1965, 7, 308–313.

19. D. Rabbe, M. Sachtleber, H. Weiland, G. Scheele and Z. Zhao:

Acta Mater., 2003, 51, 1539–1560.

20. K. S. Raghavan: Metall. Mater. Trans. A, 1995, 26A, 2075–2084.

21. J. Savoie, M. Jain, A. R. Carr, P. D. Wu, K. W. Neale, Y. Zhou

and J. J. Jonas: Mater. Sci. Eng. A, 1998, A257, 128–133.

22. M. R. Stoudt and J. B. Hubbard: Philos. Mag., 2009, 89, 2403–

2425.

23. M. R. Stoudt, J. B. Hubbard, M. A. Iadicola and S. W. Banovic:

Metall. Mater. Trans. A, 2009, 40A, 1611–1622.

24. M. R. Stoudt, J. B. Hubbard and S. A. Janet: Mater. Sci. Technol.,

2008, 24, 253–260.

25. M. R. Stoudt, J. B. Hubbard, S. P. Mates and D. E. Green: SAE

Trans., J. Mater. Manuf., 2006, 114, (5), 183–190.

26. R Development Core Team: ‘R: A language and environment for

statistical computing’, R Foundation for Statistical Computing,

Vienna, Austria, 2009.

27. P. D. Wu, D. J. Lloyd, M. Jain, K. W. Neale and Y. Huang: Int. J.

Plast., 2007, 23, 1084–1104.

Hubbard et al. Topographic analysis for prediction of strain localisation

1212 Materials Science and Technology 2011 VOL 27 NO 7