Microstructural inhomogeneity and biaxial stretching limits in aluminium alloy AA6016

Microstructural inhomogeneity and biaxial stretching limitsin aluminium alloy AA6016

Mark Moore, Pete Bate*

Manchester Materials Science Centre, The University of Manchester, Grosvenor Street, Manchester M1 7HS, UK

Received 3 December 2001; accepted 15 January 2002

Abstract

The ductility of sheet metals in the regime where both principal in-plane strains are positive is strongly dependent on macroscopic

inhomogeneity of the sheet. This can be clearly seen when the grain size becomes large compared with the sheet thickness. However, even with

quite small grain sizes there is a tendency in aluminium alloys for grains of similar orientation to occur in colonies, and this complicates any

simple correlation between grain size and limit strains in stretching. Sheets of the alloy AA6016 Al–Mg–Si were heat treated to give a wide

difference in grain size, and the microstructure characterised using electron back scattered diffraction (EBSD) electron microscopy, which can

reveal the presence of orientation colonies. Marciniak driving blank tests were used to reveal the development of dimensional inhomogeneity and

the limit strains in balanced biaxial tension. An approach using the Marciniak–Kuczynski model with yield functions derived from texture data

and the ‘defect’ associated with the measured inhomogeneity of crystallographic texture gives first-order predictions of the limit strains in this

material, although issues of texture evolution and adequate quantification of inhomogeneity need further consideration.

# 2002 Elsevier Science B.V. All rights reserved.

Keywords: Sheet forming; Biaxial stretching; Grain size; Limit strains; Grain colonies

1. Introduction

Although of limited practical value—most failures in

sheet forming occur where plane strain tension is imposed

by the geometry of the process—the prediction of limit

strains under conditions of biaxial stretching has attracted

much academic interest. This is because the simple, classic,

analysis of strain localisation failure appropriate to the

uniaxial tension of sheet [1] does not apply when both in-

plane principal strains are positive. The fact that ductile

failure still occurs under those conditions means that a more

general way of treating such failure is required.

In uniaxial tension, or any straining state where the minor

in-plane principal strain is less than or equal to zero, the

process of strain localisation failure is relatively simple to

deal with. In a ‘perfect’ specimen, instability begins at

maximum load. When the load is decreasing, two solutions

are available. The material can continue to deform plasti-

cally with reducing cross-section or it can unload elastically.

Both occur, and continuing plastic deformation becomes

confined to a ‘neck’ (this is for a strain rate insensitive

material: the same overall result occurs when there is rate

sensitivity). This neck develops to a stage where it occupies a

narrow band in the sheet plane. For further localisation to

occur, continuity must be maintained which means that the

band of the localised necking needs to be oriented in the

direction of zero stretch. The local neck continues to deform

under conditions of plane strain tension until fracture occurs.

The ‘useful’ deformation possible is determined by the

formation of the local neck—unless fracture occurs before

it forms, which is unusual in ductile metals—and the limit

strains in sheet forming operations are equivalent to the

strain required to form a local neck.

The problem with biaxial stretching is that there is no zero

stretch direction in the sheet plane. The first, and still most

useful, way of dealing with this difficulty is to assume the

existence of a ‘defect’, in the form of a pre-existing band, in

the sheet. This was introduced by Marciniak and Kuczynski

[2] and the overall method is named after them; it will be

referred to here as MK. In most cases, the pre-existing defect

is assumed to have a reduced initial thickness, but differ-

ences in material properties have also been invoked [3,4]. A

major overall result of the many analyses made using MK

models is the high sensitivity of limit strains in biaxial

tension to the severity of the initial defect. Details of the

model of plasticity involved also have a large effect on MK

Journal of Materials Processing Technology 125–126 (2002) 258–266

* Corresponding author. Tel.: þ44-161-200-8842;

fax: þ44-161-200-3586.

E-mail address: [email protected] (P. Bate).

0924-0136/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 4 - 0 1 3 6 ( 0 2 ) 0 0 3 0 4 - 7

limit strain predictions, and the choice of yield function is

very important [5,6]. Of course, the strain hardening beha-

viour and strain rate sensitivity have major effects on limit

strains in biaxial tension, as for other deformation states.

The most significant aspect, for present purposes, of MK

model predictions is the high sensitivity of limit strains to the

initial defect severity. There has been criticism of the MK

model because of the nature of the initial defect assumed, but

in reality all metals are inhomogeneous and the problem is

really one of relating the actual inhomogeneity in a material

to the idealised defect assumed in the MK model. One

unavoidable source of inhomogeneity in polycrystals is

the spatial distribution of grains with different orientations,

and so plastic responses. This leads to surface roughening in

stretching: the well-known phenomenon of ‘orange peel’ in

coarse grained materials is an example. This is a local

phenomenon, and the formation of a local neck is clearly

a much longer range one, but both are related to the grain-

scale inhomogeneity.

In the work presented here, sheets of an aluminium alloy

used for autobody manufacture have been prepared with

three different grain sizes. These have been deformed in

equibiaxial stretching, and the limit strains obtained are

compared with predictions made by using data derived from

microstructural examination and a MK model.

2. Experimental work

2.1. Materials

The material used was AA6016 Al-0.5%Mg-1.2%Si alloy

sheet, provided by Alcan International. The sheet thickness,

as received, was 1.05 mm. To give different grain sizes,

pieces of that sheet were rolled to 0.94 and 0.84 mm, at room

temperature. All materials for the experiments described

here were then solution treated at 535 8C for 600 s and

quenched by placing between cold, flat aluminium plates.

They were then naturally aged for 7 days at ambient

temperature (�20 8C) to give a T4 temper, and subsequently

stored at �35 8C until tested, at ambient temperature. The

three conditions are referred to by their approximate final

cold rolling reductions in percent: r0, r10 and r20.

The grain sizes were measured as mean linear intercepts

from images obtained by backscattered electron imaging in a

scanning electron microscope. The values are given in Table 1.

To give both the mean crystallographic texture and informa-

tion about the disposition of orientations, specimens were

examined using electron back scattered diffraction (EBSD).

The data were analysed using VMAP software [7]. Informa-

tion was gathered over sample areas of about 25 mm2 at the

mid-planes of the sheets, involving about 40,000 orientations.

Average crystallographic textures were derived using harmo-

nic series, with no additional Gaussian spread of the orienta-

tions. All the textures were dominated by the ‘cube’

component, j2 ¼ 0 sections are given in Fig. 1, and the

degree of preferred orientation decreased with increased

rolling reduction.

Samples of maps showing the distribution of material

misoriented less than 158 from the ‘cube’ orientation are

shown in Fig. 2. As well as the changes in the volume of

material near to the ideal orientation, the nature and degree

of clustering is apparent, especially in the r0 material where

bands of similar orientations aligned with the rolling direc-

tion are clearly visible. The significance of these textures

and the spatial distribution of orientations will be dealt with

below.

2.2. Equibiaxial stretching

The Marciniak driving blank technique [2] was used to

give equibiaxial stretching. The driving blanks were

annealed 70–30 brass sheet, 1.2 mm thick with a 15 mm

diameter central hole. The punch was 50 mm diameter with

a 10 mm profile radius, and that punch was driven at

0.5 mm s�1. The outer surface of the aluminium test pieces

were marked with a photoresist pattern of 2 mm diameter

circles to allow strain measurement after testing. These grid

circles were measured via digital imaging.

An example of a failed piece is shown in Fig. 3 and the

limit strains are given, as functions of grain size and, more

meaningfully, the initial sheet thickness divided by the grain

size is shown in Fig. 4. There was a marked effect of grain

size on limit strain, but it is notable that the difference in

limit strain between the r0 and r20 materials is much less

than between r20 and r10 cases, despite the differences in

grain size relative to the initial sheet thickness.

2.3. Tensile testing

Tensile tests were carried out on specimens with a 12 mm

wide parallel gauge, 63.5 mm long. A cross head speed of

0.083 mm s�1 was used and extensions were measured using

an electronic extensometer with a 50 mm gauge length. Only

specimens in the rolling direction were tested, and these

were also marked with a photoresist grid to allow uniaxial

limit strains, as well as the strain hardening characteristics,

to be measured.

Stress–strain curves for the three conditions are shown in

Fig. 5. Although there are minor differences, these are within

the scatter of repeat tests for material in the same conditions

and so can be considered as being the same. As well as the

material condition having negligible effect on the stress–

strain behaviour, it also had relatively little effect on the limit

Table 1

The MLI grain sizes of the material conditions used

Condition MLI grain size (mm)

r0 29.9

r10 75.2

r20 35.1

M. Moore, P. Bate / Journal of Materials Processing Technology 125–126 (2002) 258–266 259

Fig. 1. Sections of the orientation distribution functions for the three material conditions. In all cases, the ‘cube’ texture {0 0 1}h1 0 0i is the main component

and occurs at the corners of these sections. The degree of preferred orientation decreases with the rolling strain. The contour levels are in multiples of random

density.

Fig. 2. Maps showing the spatial distribution of orientations within 158 of {0 0 1}h1 0 0i in sheet plane sections for the three material conditions.

strain in uniaxial tension, values of about 0.25 were obtained

for all conditions.

It is known that this alloy has very little strain rate

sensitivity in the T4 condition, so that was ignored. The

strain hardening was fitted by a modified Voce relationship

[8], which in terms of plastic strain hardening rate, Y is

Y ¼ Y0 1 � s� s0

ass

� �a

(1)

A good fit to the general behaviour in uniaxial tension was

given by using Y0 ¼ 1325 MPa, s0 ¼ 160 MPa, ss ¼170 MPa and a ¼ 1:4.

3. Modelling

3.1. Quantifying the inhomogeneity

Unlike surface roughening, only inhomogeneities over

length scales greater than about half the sheet thickness will

have a significant effect on macroscopic strain localisation

[9,10]. The EBSD data can be used to estimate this long-

wavelength variability. We make a supposition: the material

which will form a band-like defect will have an enhanced

fraction of material close to the ideal ‘cube’ orientation.

These orientations will certainly be ‘weaker’ than average,

Fig. 3. The appearance of an r20 specimen tested to failure, showing the grid circles and the localised necking failure.

Fig. 4. The measured limit strains plotted against the grain size, d, and against the initial thickness, h, divided by the grain size.


but in fact many complicating factors are ignored in this

assumption.

It might be thought that one could assume a random

spatial disposition of such ‘soft’ grains and calculate the

probability of their forming a defect of some given severity

with the required spatial size. Unfortunately, the spatial

distribution of orientations is not random and there is a

tendency for grains of similar orientations to cluster in the

sheet plane. This is a result of the thermomechanical pro-

cessing history of the sheet and is not well understood. To

quantify the inhomogeneity, Fourier transforms were used.

The transform of a function Z is given by

ZðoÞ ¼Z

zðxÞ e�2piox dx (2)

where x is a position vector, o the frequency vector, Z the

complex and the discrete fast Fourier transform was used for

calculation. The parameter z in the sample planes of EBSD

measurements was defined as 1 if the orientation was less

than 158 misorientation from the ideal ‘cube’ orientation,

otherwise z ¼ 0. It is clearly easy to filter the Fourier

transform, and each term was multiplied by a window

function which was unity when |o| corresponded to wave-

lengths greater than the sheet thickness, ramping to zero for

wavelengths less than half the thickness.

The (statistical) variance of z is represented by the sum of

terms Z(o)Z(o), o 6¼ 0, where Z is the complex conjugate

of Z. Sums taken along lines from the origin in frequency

space give the variance associated with ‘band-like’ features

normal, in real space, to those lines. However, this is only

true for the single section plane, and a through-thickness

average is required.

It is known from other work [11] that there is no sig-

nificant correlation through the sheet thickness, and that the

whole sheet can be considered as a series of layers, one grain

thick, with uncorrelated but similar property variations in

their planes. Under those conditions, the overall through-

thickness averaged in-plane variance—at least for ‘long’

wavelengths—is reduced in proportion to thickness, h,

divided by the grain size, l. There is no filtering effect in

this averaging, and so the through-thickness average var-

iances will be the same as that measured for one plane, but

with their magnitude reduced by l/h. The magnitude of

‘likely’ inhomogeneities depends on the square root of the

variance, and so onffiffiffiffiffiffiffiffiffiffiffiðl=hÞ

p; a result obtained previously by

numerical trials [12] but which can be obtained more

rigorously using Fourier transforms.

Estimates for the severity of ‘enhanced cube’ bands in the

three material conditions were made using the square roots

of the variances (i.e. the standard deviations (S.D.)) follow-

ing multiplication by l/h. Expectation values were obtained

by multiplying these standard deviations by three. The

resulting values are given in Table 2.

Fig. 5. Stress–strain curves from uniaxial tensile tests of the three material condition, with tensile axes along the sheet rolling direction.

Table 2

The standard deviations given by the square roots of the line sums of

filtered Fourier coefficients divided by the number of grains through the

sheet thickness, multiplied by three to give expectation valuesa

Condition S.D. 08 S.D. 458 S.D. 908

r0 0.002 0.003 0.006

r10 0.008 0.008 0.010

r20 0.004 0.003 0.002

a The angles correspond to deviations of the band normal from the

rolling direction.

262 M. Moore, P. Bate / Journal of Materials Processing Technology 125–126 (2002) 258–266

The effect of orientation on these values is quite weak in

the r10 and r20 cases, and average values of 0.009 and 0.003,

respectively, can be assumed. This is clearly not so for r0, as

a result of the strong colony effect for that condition, and the

three values were used in the appropriate MK calculations.

3.2. MK modelling

An iterative method was used to solve load equilibrium

between material inside and outside the band in small

deformation increments. Isotropic plasticity was assumed

and evolution of the yield locus shape was ignored. The work

hardening behaviour derived from the uniaxial tensile tests

was used throughout this modelling.

The yield surface shapes were calculated from the crystal-

lographic texture using the Taylor model, via harmonic

coefficients. The important fitting process used a ‘dual

potential’ approach [13]. In the present case the Taylor

factor, in terms of the major principal strain rate and scaled

by the uniaxial tension value, was fitted to the square root of

a low order polynomial for the important region of straining

states between plane strain tension and equibiaxial tension.

This method gives good fitting characteristics: exact for both

the von Mises and Tresca yield criteria.

Material in the ‘band’ was assumed to have a texture given

by the normalised sum of the mean texture plus a fraction

of ‘cube’ material, given by random orientations within a

158 spread about the ideal {0 0 1}h1 0 0i orientation. The

fraction of that ideal ‘cube’ texture was determined by the

standard deviations derived from the Fourier analysis of

the EBSD data.

The lowest limit strains, in all cases, were predicted for

bands at 458 to the rolling direction. This is due to increased

curvature of the predicted yield loci near balanced biaxial

deformation at that angle. This even overwhelms the mark-

edly greater degree of inhomogeneity for bands in the rolling

direction of this material. The predicted limit strains are

shown in Fig. 6, where they are plotted against the experi-

mental values. The predicted limit strains are significantly

higher than the measured ones. For the r10 and r20 cases, the

prediction is consistently 75% greater than experiment, but

for r0 it is less, at about 40%.

4. Discussion

It is clear that the material condition is having a significant

effect on the balanced biaxial limit strain, whilst having little

effect on either the work hardening behaviour or the limit

strains in uniaxial tension. There are strong indications that

this is due to the inhomogeneous distribution of crystal

orientation. Ultimately this inhomogeneity must exist at

the length scale of the grain size, but the clustering of grains

with similar orientations in, especially, the r0 material means

that we cannot simply use the grain size to characterise the

likely material inhomogeneity. In fact, the MK modelling

has overestimated this effect, as well as predicting limit

strains considerably greater than those measured. Before

considering possible reasons for this, we shall discuss some

general aspects of the effect of grain structure on limit strain

which are relevant to the experiments presented above.

There have been several attempts at correlating grain size

and textural inhomogeneity with limit strains in sheet

stretching. As far as grain size is concerned, it is interesting

to compare the present results with the comprehensive set of

results given by Wilson et al. [14] for copper-based alloys

and steels, processed to give weak crystallographic textures.

This is presented in Fig. 7, using a logarithmic scale for the

ratio of initial thickness to grain size. Wilson et al. results

give near-straight lines in that space, in contrast to the results

for our AA6016 materials. They also, incidentally, showed

that there is no correlation between actual grain size and

limit strain.

A linear relationship also exists between defect severity

and balanced biaxial limit strains predicted by MK model-

ling. This is shown in Fig. 8, and used the work hardening

parameters of the aluminium alloy, in fact, the linearity is

rather insensitive to the hardening characteristics. Although

defects of reduced initial thickness were used for the results

of Fig. 8, this linearity extends to a more generalised notion

of ‘defectiveness’, or inhomogeneity. It is tempting, then, to

relate inhomogeneity to grain size. The near-linearity of

Fig. 7 infers some simple power-law relationship between

the effective MK defect size and the grain size relative to the

sheet thickness. This is reasonable if there is no orientation

clustering, or at least if the clustering does not change with

changing grain size. If that were the case, we might expect

the r0 condition to have given a balanced biaxial limit strain

about 12% higher than was actually obtained.Fig. 6. Predicted balanced biaxial limit strains as functions of the

experimental values.


The presence of orientation colonies in aluminium alloys

has attracted much attention, mainly because it can lead to a

characteristic form of surface roughness known as ‘roping’

[15,16]. In the context of limit strains, this phenomenon was

dealt with by Wilson and Rodrigues [17] and, using the MK

approach, by Rodrigues et al. [18]. Subsequent work showed

that the level of long-wavelength textural inhomogeneity

required to give adequate MK predictions could be derived

Fig. 7. The data fitting lines given by Wilson et al. [14], replotted with a logarithmic scale for the initial thickness divided by the grain size. The results from

the present work have been included.

Fig. 8. Graphs of major in-plane limit strains predicted by the MK model as functions of the inverse of the severity of reduced thickness defects. Unless

stated, these results are for balanced biaxial tension.


from local texture measurements [12]. It is clear from that

work that such colonies can be very significant in determin-

ing the limit strains of aluminium alloys.

The shape of the yield locus has a great influence on

biaxial limit strains. A plethora of yield functions have been

proposed, and used in MK modelling. Many of the yield

functions invoked have little or no physical basis, but

those based on crystal plasticity, specifically using Taylor’s

assumptions, have been reasonably useful [6,19,20]. The

gradient of the balanced biaxial limit strain vs. log defect

severity relationships shown in Fig. 8 depends on the yield

function. It decreases as the curvature of yield locus around

balanced biaxial tension increases. This could contribute to

the increased gradient of Wilson et al. steel results compared

to their copper alloy results, and also our r10 and r20 results.

This is because pencil glide, assumed to occur in BCC

ferrite, gives much more rounded yield loci than those

expected for the octahedral glide which occurs in FCC

metals. While there is clear influence of yield locus shape,

this is not a significant factor in explaining the difference

between the three material conditions examined in this work.

Although they had different textures, the predicted yield loci

were all rather similar.

It cannot be assumed that crystallographically based yield

functions will be perfect. Quite large deviations between

crystallographic predictions of anisotropy and yield func-

tions, and experimental results are often found in aluminium

alloys, particularly those containing significant amounts of

second phase particles. However, without measuring plastic

behaviour in the biaxial regime, they remain our best

estimate of plastic response at this time.

One potentially important factor which we have ignored is

the development of yield locus shape during stretching. One

cause of this is the change in crystallographic texture. This

will certainly occur, and the texture will become dominated

by orientations with h1 1 0i directions close to the sheet

plane normal. This type of texture can have a deleterious

effect on stretching limits [21] and this could contribute to

the curvature of the line for the copper alloy results (Fig. 7).

In our work, so far, we have ignored the influence of texture

changes, but it has been shown that the evolution of texture

has a significant effect on limit strains predicted by MK

models using reduced thickness defects [22].

Although it might seem intuitively correct, associating

regions of enhanced ‘cube’ texture with the MK defects may

not be appropriate. For example, work on surface rough-

ening [23] has failed to reveal any simple relationship

between orientation and local deformation. It is necessary

to use techniques such as finite element modelling, with

constitutive laws based on crystal plasticity [24,25], to

properly elucidate the effect of the disposition of orienta-

tions on inhomogeneous deformation in stretching. The use

of such modelling would also help to resolve the effect of

texture development in stretching. It is clear that correct

identification of textural inhomogeneity when the texture is

evolving will be a major challenge.

It may be that assuming a strain localisation limit is not

appropriate. Stress-based fracture criteria for stretching have

been proposed, and there is a possibility that the r0 condition

simply exceeded such a limit. This is a difficult area: there is

nothing in the MK model to prevent the band being narrow

and, apart from a modification of the terminal straining state

from plane strain to an inclined simple shear, so include

macroscopic shear failure. Further work is required to

resolve this issue.

Despite rather poor quantitative predictions, the approach

used here, with an MK model having a crystallographically

based yield function and a defect defined by an inhomo-

geneity in crystallographic texture estimated from orienta-

tion mapping, is potentially useful. It certainly reinforces the

practical requirement to avoid orientation clustering, for a

given grain size, even though balanced biaxial stretching is

rarely associated with failure in real pressings.

5. Conclusion

The balanced biaxial limit strains in AA6016 are strongly

dependant on the grain size of the material relative to the

sheet thickness and the degree of clustering of similarly

orientated grains. Despite several assumptions and limita-

tions, an approach using the Marciniak–Kuczynski model

with yield functions derived from texture data and the

‘defect’ associated with measured inhomogeneity of crys-

tallographic texture gives a first-order prediction of these

effects.

Acknowledgements

This work was supported in many ways by Alcan Inter-

national. One of us (MM) received additional financial

support from the ORS scheme and Manchester Materials

Science Centre. Thanks also to the many staff of Alcan

International and the MMSC who helped with practical

aspects of this work.

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Microstructural inhomogeneity and biaxial stretching limits in aluminium alloy AA6016

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