INSTITUTE OF PHYSICS PUBLISHING MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING

Modelling Simul. Mater. Sci. Eng. 12 (2004) 511527 PII: S0965-0393(04)76087-8

A mesoscopic approach for predicting sheet metalformability

P D Wu1, S R MacEwen1, D J Lloyd1 and K W Neale2

1 Alcan International Limited, Kingston Research and Development Centre, Kingston,Ontario K7L 5L9, Canada2 University of Sherbrooke, Faculty of Engineering, Sherbrooke, Quebec, J1K 2R1, Canada

Received 4 December 2003Published 12 March 2004Online at stacks.iop.org/MSMSE/12/511 (DOI: 10.1088/0965-0393/12/3/011)

AbstractA mesoscopic approach for constructing a forming limit diagram (FLD) isdeveloped. The approach is based on the concept of a unit cell. The unitcell is macroscopically infinitely small and thus represents a material point inthe sheet, and is microscopically finitely large and thus contains a sufficientlylarge number of grains. The responses of the unit cell under biaxial tensionare calculated using the finite element method. Each element of a mesh/unitcell represents an orientation and the constitutive response at an integrationpoint is described by the single crystal plasticity theory. It is demonstrated thatthe limit strains are the natural outcomes of the mesoscopic approach, and theartificial initial imperfection necessitated by the macroscopic MK approach isnot relevant in the mesoscopic approach. The effects of strain-rate sensitivity,single slip hardening and latent hardening, texture evolution, crystal elasticityand spatial orientation distribution on necking are discussed. Numerical resultsbased on the mesoscopic approach are compared with experimental data.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

The concept of the forming limit diagram (FLD) has proved to be very useful for representingconditions for the onset of sheet necking (see, e.g. Hecker (1975)), and is now a standardtool for characterizing materials in terms of their overall forming behaviour. However,experimentally measuring the FLD is a very time consuming procedure. Furthermore, thescatter in experimental data for a given sheet is usually so large that researchers often questionthe accuracy and precision with which the FLD is determined (see, e.g. Janssens et al (2001)).As a result, a significant effort has been spent on developing more accurate and reliablenumerical procedures to construct FLDs, while experimental procedures for measuring FLDsconstantly improves.

0965-0393/04/030511+17$30.00 2004 IOP Publishing Ltd Printed in the UK 511

512 P D Wu et al

Most theoretical and numerical studies on FLD analysis have been based on the so-calledMK approach developed by Marciniak and Kuczynski (1967). The basic assumption of thisapproach is the existence of a material imperfection, in the form of a groove, on the surface of thesheet. They showed that a slight intrinsic inhomogeneity in load bearing capacity throughouta deforming sheet can lead to unstable growth of strain in the region of the imperfection, andsubsequently cause localized necking and failure. Within the MK framework, the influenceof various constitutive features on FLDs has been explored using phenomenological plasticitymodels (see, e.g. Neale and Chater (1980), Wu et al (2003a)) and crystal plasticity (see, e.g. Wuet al (1997)). Using the MK approach, the predicted FLDs based on crystal plasticity werein good agreement with measured FLDs for rolled aluminium alloy sheets (Wu et al 1998).However, the MK approach needs the value of an artificial initial imperfection parameter f0,which cannot be directly measured by physical experiments. Instead, the value of f0 has to beestimated by fitting the FLD prediction of a strain path (often the in-plane plane strain tension)to the corresponding experimental limit strain. Therefore, in order to calculate the FLD ameasured necking point is required, in addition to a measured initial texture and a stressstrain curve for characterizing hardening behaviour. As mentioned previously, measuringFLDs is a very time consuming procedure that is not very reliable. Making things evenworse is the fact that a significant number of universities and companies have no capabilityof measuring FLDs. The challenge then arises: knowing the initial state and a stressstraincurve up to large strains of a sheet, is it possible to determine the limit strains/FLD for thesheet?

The purpose of this paper is to develop a mesoscopic approach for predicting FLDs. Theapproach is based on the concept of a unit cell. The unit cell is macroscopically infinitelysmall and thus represents a material point in the sheet, but it is microscopically finitely largeand thus contains a sufficiently large number of grains. The response of the unit cell underbiaxial tension is calculated using the finite element method. Each element of a mesh/unitcell represents a grain and the constitutive response at an integration point is described by thesingle crystal plasticity theory. The initiation of localized necking is defined as the instantwhen the unit cell goes to an in-plane plane strain state, i.e. the minor strain reaches itsmaximum and the applied major strain is almost completely compromised by accelerating thereduction of sheet thickness. The corresponding in-plane strain of the unit cell is the limit strain,which produces a point on the FLD. The entire FLD of the sheet is determined by repeatingthe procedure for different deformation paths. It is demonstrated that the limit strains are thenatural outcomes of the mesoscopic approach, and the artificial initial imperfection necessitatedby the macroscopic MK approach is not relevant in the mesoscopic approach. The effectsof strain-rate sensitivity, single slip hardening and latent hardening, texture evolution, crystalelasticity and spatial orientation distribution on the FLD are discussed. Numerical resultsbased on the mesoscopic approach are compared with the experimental data.

2. Constitutive equations

In this section, we very briefly recapitulate the constitutive framework adopted in this paper.For details we refer to Asaro and Needleman (1985) and Wu et al (1996). In the rate-dependentcrystal plasticity model employed, the elastic constitutive equation for each grain is specified by

= LD 0 tr D, (1)

where denotes the Jaumann rate of the Cauchy stress tensor, L is the tensor of elastic moduli,D represents the strain-rate tensor, and 0 accounts for the viscoplastic type stress rate that is

Sheet metal formability 513

Figure 1. Schematic representation of a unit cell under a remote biaxial stress state.

determined by the slip rates on the various slip systems in the crystal. The slip rates are takento be governed by the power-law expression

() = (0) sgn () ()g()

1/m

(2)

where (0) is a reference shear rate taken to be the same for all slip systems, () is the resolvedshear stress on slip system , and m is the strain-rate sensitivity. g() characterize the currentstrain-hardened state of all slip systems. For multiple slip, the evolution of the hardening isgoverned by

g() =

h()|()| (3)

where g()(0) is the initial hardness, taken to be a constant 0 for each slip system, and h()are the hardening moduli. The form of these moduli is

h() = q()h() (no sum on ) (4)where h() is a single slip hardening rate and q() is the matrix describing the latent hardeningbehaviour of the crystallite. The latter is determined by a latent hardening parameter q 1;if q = 1, hardening is isotropic.

The single slip hardening law employed in this paper takes the following power-law formof the constitutive function h():

h() = h0(

h0a

0n+ 1)n1

(5)

where h0 is the systems initial hardening rate, n is the hardening exponent and a is theaccumulated slip.

3. Problem formulation and method of solution

We define a unit cell as a small region of sheet under a remote biaxial stress state: 33 = 0and 22 = 11, as illustrated in figure 1. The unit cell is defined as a globally small regionof the sheet that contains all the essential micro-structural and textural features that characterize

514 P D Wu et al

the sheet. Orientations in the measured texture data are randomly assigned to elements in themesh/unit cell. In other words, each element of the mesh represents an orientation fromthe measured texture. The overall response of the unit cell is described by macroscopic strainsEij and stresses ij , which are obtained by averaging their respective values of stresses (I)ijover the total number of integration points N :

ij = 1N

I

(I)ij , (6)

where the grain stresses (I)ij are computed according to the single crystal plasticity modelpresented in section 2.

The loading is assumed to be strain controlled by prescribing the strain rate E11. Thetransverse strain rate E22 is determined at each instant such that the stress ratio 22/11 = is kept at a constant value during the deformation process (see figure 1). Uniaxial tensionalong the RD corresponds to = 0; while = 0.5 describes a nearly in-plane plane straintension in the RD, = 1 gives a balanced biaxial tension. The initiation of localized neckingis defined as the instant when the unit cell goes to an in-plane plane strain state, i.e. the minorstrain reaches its maximum and the applied major strain is almost completely compromisedby accelerating reduction of sheet thickness. The corresponding in-plane biaxial strain state ofthe unit cell represents the limit strains, which produces a point on the FLD. The entire FLD ofthe sheet is determined by repeating the procedure for different deformation paths prescribedby the stress ratio .

When the major straining direction is in the TD (such as uniaxial tension along thetransverse direction), it is more convenient numerically to control the deformation byprescribing the strain rate E22. In this case, the vertical strain rate E11 is calculated at eachinstant such that the stress ratio 11/22 = is kept at a constant value during the deformationprocess.

The finite element scheme used in this paper is similar to that described by Wu and Van derGiessen (1994), and used recently by Wu et al (2001, 2003b). We use quadrilateral elements,each built up of four linear velocity, triangular sub-elements arranged in a crossed triangleconfiguration. An equilibrium correction procedure is employed to avoid drift away from thetrue equilibrium path during the incremental procedure.

4. Results

The initial sheet had the texture shown in figure 2 in terms of the {111} pole figure with500 grains/orientations, and the rolling direction is aligned with the major stress direction (x1).

The mesoscopic approach developed in this paper determines the limit strains for sheetmetals with a homogeneous spatial distribution of grain orientations. In other words, thesheet has a randomly distributed texture. However, when measured orientations are assignedto elements in a mesh, an inevitable inhomogeneous spatial orientation distribution will beintroduced numerically. To reduce the effect of spatial distribution of texture components innumerical simulations, we use a regular mesh consisting of 50 50 elements (50 elementsin the RD and 50 elements in the TD). The measured 500 crystal orientations are randomlyassigned five times to elements in the mesh. The global textures used in this case is exactlythe same as the experimental texture shown in figure 2.

The values for the material parameters in the crystal plasticity analysis are C11 = 236 GPa,C12 = 135 GPa and C44 = 62 GPa, 0 = 0.001 s1, m = 0.01, 0 = 48 MPa, h0/0 = 28,n = 0.285 and q = 1.0.

Sheet metal formability 515

Figure 2. Initial texture represented in terms of the {111} pole figure.

Figure 3. Calculated uniaxial stressstrain curve for the unit cell with the initial texture shown infigure 2.

These values are in the range of those for rolled aluminium sheets, and will be used in allsimulations reported in this paper except where noted otherwise. The predicted stressstraincurve under uniaxial tension in the RD is shown in figure 3. It is found from figure 3 that thesimulated stress reaches its maximum at a strain of about 0.24 and then gradually decreaseswith further straining. The calculated strain path is presented in figure 4. It is observed that atrelatively small strains up to the maximum stress point there is a linear relationship betweenthe minor strain E22 and the applied major strain E11. At large strains, |E22| increases moreslowly until it reaches its saturation value 0.12 at E11 = 0.34. It is clear that the unit cell(or the representative small region of the sheet) enters an in-plane plane strain tension state at(0.34,0.12). Therefore, E11 = 0.34 and E22 = 0.12 can be defined, respectively, as themajor and minor limit strains under uniaxial tension. Figure 5 shows contour plots of the vonMises type effective plastic strain at different stages of the deformation. It is observed that thedeformation within the unit cell is quite inhomogeneous and the inhomogeneity starts when

516 P D Wu et al

Figure 4. Calculated in-plane strain path for the unit cell under uniaxial tension.

Figure 5. Predicted distribution of the effective plastic strain for the unit cell of figure 3 at differentvalues of strain E11.

the applied deformation is small. The inhomogeneous deformation gradually evolves withincreasing applied strain and forms a few localized deformation bands at around the maximumstress. At necking, localized deformation bands have crossed the entire width of the unit cell.As a result, the unit cell ceases to deform in the minor stress direction (x2) and material withinthe unit cell is, on the average, under in-plane plane strain state.

The entire FLD of the sheet is determined by repeating the procedure for differentdeformation paths as prescribed by the stress ratio : from uniaxial tension = 0, through anearly in-plane plane strain tension with = 0.5, to equi-biaxial tension for = 1. Figure 6

Sheet metal formability 517

Figure 6. Predicted FLD for a sheet with the initial texture shown in figure 2.

Figure 7. Influence of the strain-rate sensitivity m on the predicted FLDs.

presents the predicted FLD. Also included in figure 6 are strain paths under a few importantstress ratios. It is noted that the necking point for the strain path associated with = 1 is onthe left-hand side of the necking point for strain path for = 0.95 in the predicted FLD. Thisis due to the anisotropy of the material. Previous studies have indicated that FLDs are usuallysensitive to effects of material properties such as strain hardening and material rate sensitivity(e.g. Hutchinson and Neale (1978), Wu et al (1997)).

Figure 7 shows the change in the predicted FLD when the value of the material ratesensitivity m is decreased by a factor of 10 to m = 0.001. Decreasing the rate sensitivity tendsto degrade the hardening at large strains. Consistent with this, figure 7 shows that the limitstrain is decreased relative to that in figure 6. The effect ofm on FLDs predicted by the proposedmesoscopic approach is similar to that based on the MK approach in conjunction withphenomenological plasticity (Neale and Chater 1980) and crystal plasticity (Wu et al 1997).

518 P D Wu et al

Figure 8. Influence of the hardening parameter n on the predicted FLDs.

Figure 9. Influence of the latent hardening parameter q on the predicted FLDs.

In this paper, the strain hardening is described by the power-law expression in terms ofthe parameters n, q and h0. It is known that the effect of parameter h0 on material responseis noticeable only when the applied strains are very small, and has no significant influenceon FLDs. In figure 8, the effect of hardening is first considered using the same value of q(q = 1.0) but different values of n, n = 0.285 and 0.245. It is clear that a larger value of nincreases the limit strain, which can be attributed simply to the fact that hardening increaseswith increasing n. The effect of the hardening characteristics by using the same value of n(n = 0.285) but different values of q, q = 1.0 and 1.4 are shown in figure 9. It is apparentthat the latent hardening (q > 1.0) has no significant effect on FLDs: it only slightly increaseslimit strains near the in-plane plane strain tension ( 0.5).

Sheet metal formability 519

Figure 10. Influence of crystal elastic properties on the predicted FLDs.

Previous works based on phenomenological plasticity models suggested that the predictedFLDs are not sensitive to elastic properties of the materials. Consequently, elasticity has beenneglected in most FLD analyses; for instance, in Graf and Hosford (1990) and Zhou andNeale (1995). However, Wu et al (1997) have revealed, according to their work on the crystalplasticity based MK approach, that crystal elasticity has an important effect on the FLD. Toexamine the effect of crystal elasticity on the FLD based on the mesoscopic approach, weconsider a rigid-plastic material with very large elastic constants C 11, C 12 and C 44, whichare all a factor 100 larger than the corresponding values of C11, C12 and C44, but with allother parameters, such as hardening parameters, kept unchanged. The term rigid plasticis used here because the stressstrain curve of this material is almost identical to that of thecorresponding idea of rigid-plastic material. The predicted FLDs are presented in figure 10. Itis found that increasing the elastic modulus of a sheet metal can improve its formability. Thisobservation is similar to that from the crystal plasticity based MK approach.

It is generally accepted that texture evolution has a significant effect on the initiation andpropagation of shear bands in FCC polycrystalline metals (see, e.g. Wu et al (2001) and Inalet al (2002)). In this paper, repeating calculations reported in figure 6 but turning off thetexture evolution assesses the influence of the texture evolution on FLDs. Numerical resultsare presented in figure 11, and it is observed that texture evolution has a negligible effect onlimit strains for stress paths 0.0 0.3. However, texture evolution increases the limitstrains significantly when 0.4 1.0.

The results shown previously are based on a mesh with 50 50 elements or 5 500orientations, and the same spatial orientation distribution: 500 measured crystal orientationsare randomly assigned five times to elements in the mesh. These 5 500 orientations are nowre-assigned randomly into elements in the mesh to have an additional three different spatialorientation distributions without changing the measured texture. Results based on all fourspatial orientation distributions are presented in figures 12 and 13 for uniaxial tension andfigure 14 for the FLD. (The spatial orientation distribution with symbol C is the same as usedin figures 310.) In all simulations the values of the material parameters are the same. It isobserved from figures 12 and 13 that the four distributions give virtually the same responseup to about E11 = 0.24, where B and C reach their maximum stress points. The in-plane

520 P D Wu et al

Figure 11. Influence of the texture evolution on the predicted FLDs.

Figure 12. Calculated uniaxial stressstrain curves for the unit cell with four different spatialorientation distributions of the texture shown in figure 2.

strain components (E11, E22) at maximum stress and localized necking are

at max. stressA: (0.293,0.129)B: (0.244,0.108)C: (0.238,0.107)D: (0.297,0.129)

at neckingA: (0.323,0.136)B: (0.306,0.123)C: (0.337,0.124)D: (0.337,0.139)

It is found that the predicted in-plane strain components at maximum stress are quitedifferent for the four different spatial orientation distributions with the same measured globaltexture. However, it is also observed from figure 12 that for distributions B and C, inwhich the maximum stress is reached early, the stress decreases slowly after the maximum

Sheet metal formability 521

Figure 13. Calculated in-plane strain paths for the unit cell with four different spatial orientationdistributions of the texture shown in figure 2.

Figure 14. Predicted FLDs for the unit cell with four different spatial orientation distributions ofthe texture shown in figure 2.

stress point. However, for distributions A and D, in which the maximum stress is achievedat relatively larger strains, the stresses decrease more dramatically. As a result, the predictedlimit strains under uniaxial tension for all four distributions are within the typical range ofexperimental error. The predicted FLDs based on the different spatial orientation distributionsare shown in figure 14, and it is clearly seen that the predicted necking points form a relativelynarrow band, indicating a relatively small spatial orientation distribution effect. The deformedmeshes at necking for the four different spatial orientation distributions are presented foruniaxial tension ( = 0), in-plane plane strain tension ( = 0.5) and balanced biaxial tension( = 1) in figures 1517, respectively. As expected, the pattern of localized deformationis sensitive to the spatial distribution of the measured orientations. However, it could be

522 P D Wu et al

Figure 15. Deformed mesh at necking under uniaxial tension for the unit cell with four differentspatial orientation distributions of the texture shown in figure 2.

Figure 16. Deformed mesh at necking under in-plane plane strain tension for the unit cell withfour different spatial orientation distributions of the texture shown in figure 2.

Sheet metal formability 523

Figure 17. Deformed mesh at necking under balanced biaxial tension for the unit cell with fourdifferent spatial orientation distributions of the texture shown in figure 2.

concluded statistically, from figures 15 to 17 and deformed meshes for other deformationpaths not shown in this paper, that at necking at least one main localized deformation band hascrossed the width of the unit cell. As a result, the unit cell ceases to deform in the minor straindirection and materials within the unit cell are under in-plane plane strain.

At this stage, it is of interest to compare simulated and measured FLDs. As an example,we consider a commercial aluminium sheet with the measured initial texture shown in figure 2.The crystal elastic constants, the strain-rate sensitivity m and the slip system reference plasticshearing rate 0 are taken as identical to the values used previously. We assume isotropichardening (q = 1.0) and estimate the hardening parameters in the crystal plasticity constitutivemodel by curve-fitting numerical simulations of uniaxial tension in the RD to the correspondingexperimental data. From this procedure and based on the mesh with spatial orientationdistribution C, we find that 0 = 48 MPa, h0/0 = 28 and n = 0.285, which are alsoidentical to the values used previously. Figure 18 shows that there is quite good agreementbetween the simulated and experimental curves.

It is necessary to point out that the experimental curve is the averaged response of thewhole sample under macroscopic uniaxial tension, while the simulation is intended to modelthe local material response of a small region represented by the unit cell. Only when thedeformation is homogeneous over the whole sample can such a simulation be considered as areal curve-fitting. So the determined hardening parameters, together with the unit cell model,can be used to describe the hardening behaviour of the sheet. Therefore, the ideal experimentused for determining hardening parameters should be the one in which the deformation in thegauge section remains essentially homogeneous and no macroscopic localized shear band isdetected at large to very large strains. On the other hand, in order to validate the predictivecapability of the proposed mesoscopic approach, the ideal experiment used for determining

524 P D Wu et al

Figure 18. Tensile stress response in uniaxial tension for an aluminium sheet with its initial texturegiven in figure 2.

Figure 19. Predicted and measured FLDs for the aluminium sheet of figure 18.

hardening parameters should not be the one that itself is a strain path included in the FLD. Ithas been noted that a planar simple shear is quite different compared to any strain path fromthe biaxial states 0 1. Furthermore, Gasperini et al (1996) have indicated that theplanar simple shear can be used to characterize the plastic behaviour of thin sheet metals atlarge strains (see also Wu et al (2001)). Unfortunately, a shear test is not available for the sheetconsidered. In this paper, therefore, the hardening parameters are calculated by curve-fittingthe numerical simulation of uniaxial tension in the RD to the corresponding experimental data.

Figure 19 shows the simulated and measured FLDs. It is seen that the agreement betweenthe simulated and measured FLDs is reasonable, with the simulation tending to underestimatethe limit strains for the stretch region of the diagram. For a comparison, the predicted FLD basedon the MK approach in conjunction with the Taylor polycrystal plasticity is also included in

Sheet metal formability 525

figure 19. It should be pointed out that in the MK analysis the value of the initial imperfectionparameter f0 was estimated by fitting the limit strain of in-plane plane strain tension (Wu et al1998), while f0 was not relevant for the mesoscopic approach.

5. Discussion and conclusions

In this paper, a mesoscopic approach for predicting sheet metal formability has been developed.The approach is based on the concept of the unit cell, which is macroscopically infinitely smalland thus represents a material point in the sheet, but is microscopically finitely large and thuscontains a large number of grains. The response of the unit cell under a range of stress pathsis calculated using the finite element method. Each element of a mesh/unit cell representsan orientation and the constitutive response at an integration point is described by the singlecrystal plasticity theory.

It is demonstrated that the limit strains are the natural outcomes of the mesoscopicapproach, and the artificial initial imperfection necessitated by the macroscopic MK approachis not needed in the mesoscopic approach. The effects of strain-rate sensitivity, single sliphardening and latent hardening, texture evolution, crystal elasticity and spatial orientationdistribution on the FLD have been discussed.

There are three major difficulties involved in the development and application of theproposed mesoscopic approach to predict sheet metal formability.

The first is how to define the necking point for a given deformation path. In this paper, theinitiation of localized necking is defined as the instant when the unit cell goes to an in-planeplane strain state, i.e. the minor strain reaches its maximum or saturation value, and the appliedmajor strain is almost completely compromised by accelerating reduction of sheet thickness.The corresponding in-plane strain components of the unit cell are the limit strains, whichform a point on the FLD. The entire FLD of a sheet is determined by repeating the procedurefor different deformation paths as prescribed by the stress ratio . This definition appearsphysically appropriate. However, it should be pointed out that this mesoscopic approach maytend to overestimate the limit strains because the major or minor strain direction is constrained tobe only possible in either vertical or horizontal direction. A more conservative estimate ofthe forming limit strain could be obtained by calculating the limit strain for various values ofthe chosen angle by which the initial texture rotated around the sheet normal and selecting theminimum limit strain as the forming limit strain. However, in reality, the simulation is in goodagreement with experiments in the drawn region of the FLD and underestimates the behaviourin the stretch region, figure 19.

The second difficulty is how to reduce the numerical effect of spatial orientationdistribution of a measured texture. In our approach, Ng measured grain orientations arerandomly assigned Na times to Na Ng elements in the mesh. Our detailed numerical tests,not shown in this paper, have indicated that the appropriate value of Na depends on the texture.Usually, the stronger the texture, the smaller the value of Na. In general, we first fix a value ofNa based on personal experience on the texture. The resulting Na Ng orientations are thenassigned randomly into elements in the mesh to have different spatial orientation distributionswithout changing the measured texture. The value of Na is considered large enough if thepredicted necking points based on the different distributions form a relatively narrow band.Otherwise, the value of Na is increased until a relatively small spatial orientation distributioneffect is reached. For the texture with 500 grains shown in figure 2, the results presented infigure 13 clearly demonstrated that Na = 5 was large enough to give relatively actuate results.

The third major difficulty involved in the proposed mesoscopic approach is related to howto characterize the hardening behaviour of sheet at large strains. The ideal experiment used

526 P D Wu et al

to determine hardening parameters should be the one in which the deformation in the gaugesection remains essentially homogeneous at large to very large strains, and at same time shouldnot be the one which itself is a strain path included in the FLD. While the uniaxial tension testwas used to estimate values of hardening parameters in this paper, we recommend using theplanar simple shear test.

However, with these essentials in mind, the differences between the predicted andmeasured FLDs could be due to the fact that the simulation considered in-plane stretching,while the experimental data was obtained from hemispherical punch stretching tests. In punchtests there are compressive stresses normal to the sheet, frictional shear stresses, and sheetcurvature. Furthermore, the stress paths are not necessarily proportional. These complicatingfactors have not been accounted for in our analyses. In addition, it has been found that themeasured FLDs from in-plane stretching lie below the corresponding punch stretching FLDs(Ghosh and Hecker 1974). Based on this observation, the agreement between the calculated andexperimental FLDs can be considered to be reasonably good. Nevertheless, both theoreticaland experimental development are required to further validate and improve the mesoscopicapproach.

Finally, it should be emphasized that the mesoscopic approach, together with the numericalscheme for reducing the spatial orientation distribution effect, is developed to estimate FLDsfor sheet metals with homogeneous/random spatial distribution of grain orientations. For asheet with an inhomogeneous distribution of differently oriented grains, plastic deformationroughens a free surface by the development of roping due to the alignment of texturecomponents. For a roping sheet localized necking is often initiated at a valley, and it isexpected that the forming limit strain based on the proposed mesoscopic approach will besensitive to the texture distribution. Research on applying the proposed approach to predictFLDs for roping sheets is in progress and will be reported elsewhere.

Acknowledgments

The authors are grateful to Alcan International Limited for allowing the publication of this work.

References

Asaro R J and Needleman A 1985 Texture development and strain hardening in rate dependent polycrystals ActaMetall. 33 92353

Hecker S S 1975 Formability of aluminum alloy sheet J. Eng. Mater. Tech. 97 6673Gasperini M, Pinna C and Swiatnicki W 1996 Microstructure evolution and strain localization during shear deformation

of an aluminium alloy Acta Mater. 44 4195208Ghosh A K and Hecker S S 1974 Stretching limits in sheet metals: in-plane versus out-of-plane deformation Metall.

Trans. 5 21614Graf A and Hosford W F 1990 Calculation of forming limit diagrams Metall. Trans. 21 A 8794Inal K, Wu P D and Neale K W 2002 Finite element analysis of localization deformation in FCC polycrystalline sheets

under plane stress tension Int. J. Solids Struct. 39 346986Janssens K, Lambert F, Vanrostenberghe S and Vermeulen M 2001 Statistical evaluation of the uncertainty of

experimentally characterised forming limits of sheet steel J. Mater. Process. Technol. 112 17484Marciniak M and Kuczynski K 1967 Limit strains in the processes of stretch-forming sheet metals Int. J. Mech. Sci.

9 60920Neale K W and Chater E 1980 Limit strain predictions for strain-rate sensitive anisotropic sheets Int. J. Mech. Sci. 22

56374Wu P D, Inal K, Neale K W, Kenny L D, Jain M and MacEwen S R 2001 Large strain behaviour of very thin aluminium

sheets under planar simple shear J. Phys. IV 11 22936Wu P D, Jain M, Savoie J, MacEwen S R, Tugcu P and Neale K W 2003a Evaluation of anisotropic yield function for

aluminum sheets Int. J. Plasticity 19 12138

Sheet metal formability 527

Wu P D, Lloyd D J, Bosland A, Jin H and MacEwen S R 2003b Analysis of roping in AA6111 automotive sheet ActaMater. 51 194557

Wu P D, Neale K W and Van der Giessen E 1996 Simulation of the behaviour of FCC polycrystals during reversedtorsion Int. J. Plasticity 12 1199219

Wu P D, Neale K W and Van der Giessen E 1997 On crystal plasticity FLD analysis Proc. R. Soc. Lond. A 4531931848

Wu P D, Neale K W, Van der Giessen E, Jain M, Makinde A and MacEwen S R 1998 Crystal plasticity forming limitdiagram analysis of rolled aluminum sheets Metall. Mater. Trans. A 29 52735

Zhou Y and Neale K W 1995 Predictions of forming limit diagrams using a rate-sensitive crystal plasticity modelJ. Mech. Sci. 37 120