**5. Discussions about MK–FC and MK–VPSC approaches**

A comparison between the FC and the VPSC interaction models is the subject of this section. For this purpose, we calculate forming-limit strains using both homogenization schemes together with the MK approach. The numerical procedure, previously applied to a FCC structure, is extended to include the slip-system families of BCC polycrystals. The consequences of the FCC and BCC crystallographic-slip assumptions, coupled with the selection of either FC or SC type grain-interactions, are investigated in detail. Then, we focus on the effect of the cube texture on the forming-limit behavior, and seek to explain why a spread about cube exhibits unexpectedly high limit strains close to equi-biaxial stretching when the MK-FC is used. Finally, we explore the right-hand side of the FLD for a BCC material considering either 24 or 48 active slip systems for crystal-plasticity simulations. The advantages of using the VPSC material model in the MK approach are discussed at the end of this section. For all calculations in this section, and as was pointed out in Section 4, the strain hardening between slip systems is taken into account by adopting isotropic hardening.

In what follows, we apply the MK-VPSC and MK-FC approaches for predicting FLDs to both FCC and BCC materials. We assume that plastic deformation occurs by 12 crystallographic slip systems of the type {111} <110> for the FCC material and 48 slip systems of the type {110} <111>, {112} <111> and {123} <111> for the BCC. The crystal level properties listed in Table 6 are determined by imposing the same uniaxial behaviors for all simulations. An initially random texture, described by 1000 equiaxed grains is assumed. The slip resistances <sup>c</sup> *<sup>s</sup> τ* of all slip systems are taken equal, the rate sensitivity is *m* = 0.02, and a reference slip rate of <sup>0</sup> *γ* = 0.001 s-1 is assumed.



To analyze the development of deformation localization during proportional straining, the calculations are performed assuming an initial imperfection of *f*0 = 0.99 over the different strain paths. The predicted limit strains are presented in Fig. 13. For each homogenization method, both materials have about the same profile from uniaxial tension (ρ = -0.5) to inplane plane-strain tension (ρ = 0). Over this entire range, the major limit strains decrease with increasing ρ. It can be seen that, for the BCC-SC material, the largest value of the parameter *n* results in the highest limit values. Also, it is interesting to note that MK-FC simulations using the same *n* value predict similar limit strains for the in-plane plane strain condition.

Self-Consistent Homogenization Methods for Predicting Forming Limits of Sheet Metal 197

in the prediction of a relatively low limit strain. The yield potentials of the materials were calculated by imposing different plastic strain-rate tensors under a state of plane stress in 11 22 *σ σ* − space. With the simulation, we deformed the material in equi-biaxial stretching up to a given plastic strain and then performed the yield-locus calculations. For the purpose of comparing polycrystal yield surfaces, all work-rates were normalized to that of FCC-FC uniaxial stretching. We calculated the yield loci corresponding to each of the necking limit strains, in order to highlight the link between the yield-surface shape and the forming-limit behavior. Shapes and curvatures predicted by the FC and VPSC models are shown in Fig. 14.

*p* = 1.14

*p* = 1.05

 FCC-FC FCC-SC BCC-FC BCC-SC

0,0 0,2 0,4 0,6 0,8

σ11

As expected, the curvature of the yield locus of the FCC-FC material in the equi-biaxial stretching zone is much sharper than those of the other materials, which is consistent with its limit-strain value, lowest among the cases considered, shown in Fig. 13. Similarly, the curvature of the BCC-FC yield locus is rounder than the others, which is again in agreement with the predicted limit strains. We also calculated the parameter *p* for our materials finding that the values of *p* for the FCC-FC and BCC-FC materials are the lowest and the highest,

The FLD results depend on the homogenization scheme, and the differences are explained in terms of the sharpness of the yield-loci and texture development. The MK-FC framework predicts both the highest and lowest limit strains, for the BCC and FCC materials

As mentioned in section 2.2, in rolled FCC sheets, crystallographic textures are frequently classified in terms of the ideal rolling and recrystallization components. Such classifications are well suited for theoretical modeling where mathematical descriptions of particular

0,0

**Figure 14.** Calculated yield loci at equi-biaxial failure for all tested materials.

**5.1. Influence of cube texture on sheet-metal formability** 

0,2

0,4

σ22

respectively.

respectively.

0,6

0,8

Predictions begin to diverge in the biaxial-stretching range. Here, results clearly illustrate large differences between the homogenization schemes and between materials. These differences reach a maximum for the equi-biaxial deformation path. The MK-FC framework predicts both the highest and lowest limit strains, for the BCC and FCC materials respectively. The MK-FC FCC material calculation leads to a remarkably low limit curve. Completely the opposite behavior is observed within the MK-VPSC scheme. In this case, the FCC material shows better formability than the BCC for ρ ≥ 0.3, and for both materials, the calculated limit-strain curves remain between those calculated with the MK-FC scheme near the equi-biaxial zone. In the case of the FCC material, the MK-VPSC approach predicts a noticeable increase of the limit strains over the whole right side of the diagram, while the BCC material only shows that behavior in the region 0 ≤ ρ ≤ 0.6. For ρ ≥ 0.6, the MK-VPSC limit-strain values are nearly constant. Fig. 13 also includes the {100} stereographic pole figures of each material at the end of the equi-biaxial stretching path. As can be seen, FCC and BCC material textures evolve differently depending on the model assumption. For a FCC material, the FC model develops a weaker texture than that produced by the VPSC calculation. For the BCC case, the final textures are qualitatively similar but quantitatively different in their degree of intensity.

**Figure 13.** Influence of the slip microstructure and interaction model on the FLD.

According to Lian et al. (1989), the yield-surface shape has a tremendous effect on the FLD, and Neale & Chater (1980) demonstrated that a decrease in the sharpness of the stress potentials in equi-biaxial stretching promotes larger limit strains. A sharp curvature allows the material to quickly select a deformation path approaching plane strain, and this results in the prediction of a relatively low limit strain. The yield potentials of the materials were calculated by imposing different plastic strain-rate tensors under a state of plane stress in 11 22 *σ σ* − space. With the simulation, we deformed the material in equi-biaxial stretching up to a given plastic strain and then performed the yield-locus calculations. For the purpose of comparing polycrystal yield surfaces, all work-rates were normalized to that of FCC-FC uniaxial stretching. We calculated the yield loci corresponding to each of the necking limit strains, in order to highlight the link between the yield-surface shape and the forming-limit behavior. Shapes and curvatures predicted by the FC and VPSC models are shown in Fig. 14.

196 Metal Forming – Process, Tools, Design

different in their degree of intensity.

0,0

0,2

0,4

\* ε11 0,6

0,8

condition.

plane plane-strain tension (ρ = 0). Over this entire range, the major limit strains decrease with increasing ρ. It can be seen that, for the BCC-SC material, the largest value of the parameter *n* results in the highest limit values. Also, it is interesting to note that MK-FC simulations using the same *n* value predict similar limit strains for the in-plane plane strain

Predictions begin to diverge in the biaxial-stretching range. Here, results clearly illustrate large differences between the homogenization schemes and between materials. These differences reach a maximum for the equi-biaxial deformation path. The MK-FC framework predicts both the highest and lowest limit strains, for the BCC and FCC materials respectively. The MK-FC FCC material calculation leads to a remarkably low limit curve. Completely the opposite behavior is observed within the MK-VPSC scheme. In this case, the FCC material shows better formability than the BCC for ρ ≥ 0.3, and for both materials, the calculated limit-strain curves remain between those calculated with the MK-FC scheme near the equi-biaxial zone. In the case of the FCC material, the MK-VPSC approach predicts a noticeable increase of the limit strains over the whole right side of the diagram, while the BCC material only shows that behavior in the region 0 ≤ ρ ≤ 0.6. For ρ ≥ 0.6, the MK-VPSC limit-strain values are nearly constant. Fig. 13 also includes the {100} stereographic pole figures of each material at the end of the equi-biaxial stretching path. As can be seen, FCC and BCC material textures evolve differently depending on the model assumption. For a FCC material, the FC model develops a weaker texture than that produced by the VPSC calculation. For the BCC case, the final textures are qualitatively similar but quantitatively


\* ε22

According to Lian et al. (1989), the yield-surface shape has a tremendous effect on the FLD, and Neale & Chater (1980) demonstrated that a decrease in the sharpness of the stress potentials in equi-biaxial stretching promotes larger limit strains. A sharp curvature allows the material to quickly select a deformation path approaching plane strain, and this results

**Figure 13.** Influence of the slip microstructure and interaction model on the FLD.

 FCC-FC FCC-SC BCC-FC BCC-SC

**Figure 14.** Calculated yield loci at equi-biaxial failure for all tested materials.

As expected, the curvature of the yield locus of the FCC-FC material in the equi-biaxial stretching zone is much sharper than those of the other materials, which is consistent with its limit-strain value, lowest among the cases considered, shown in Fig. 13. Similarly, the curvature of the BCC-FC yield locus is rounder than the others, which is again in agreement with the predicted limit strains. We also calculated the parameter *p* for our materials finding that the values of *p* for the FCC-FC and BCC-FC materials are the lowest and the highest, respectively.

The FLD results depend on the homogenization scheme, and the differences are explained in terms of the sharpness of the yield-loci and texture development. The MK-FC framework predicts both the highest and lowest limit strains, for the BCC and FCC materials respectively.

#### **5.1. Influence of cube texture on sheet-metal formability**

As mentioned in section 2.2, in rolled FCC sheets, crystallographic textures are frequently classified in terms of the ideal rolling and recrystallization components. Such classifications are well suited for theoretical modeling where mathematical descriptions of particular components can be input into simulations. In particular, we focus on how the strength of the cube texture affects localized necking. To investigate this effect, we modeled variations of the cube texture. The variations were constructed with different spreads of grain orientations around the ideal cube component. The procedure for modeling textures is the same as that used in Signorelli & Bertinetti (2009). As an example, the cube-15° texture is one whose grains have a misorientation with respect to the ideal cube orientation {100}<001> of less than 15°, uniformly distributed over that area. Fig. 15 shows the {111} stereographic pole figures for cube-3°, cube-7°, cube-11° and cube-15° distributions. For the cube set of textures, the number of individual orientations was set in order to obtain an adequate representation of a uniform distribution.

Self-Consistent Homogenization Methods for Predicting Forming Limits of Sheet Metal 199


\* ε22


**Figure 16.** Calculated FLDs for MK-FC (left) and MK-VPSC (right) models

 Ideal cube-3<sup>o</sup> cube-7<sup>o</sup> cube-11<sup>o</sup> cube-15<sup>o</sup> Random

*<sup>s</sup> τ* = 42 MPa).

In order to assess the effect of the yield-surface shape on the forming-limit behavior close to the balanced-biaxial stretching zone, ρ = 1.0, we prestrained the texture sets along the equibiaxial path. The amounts of equi-biaxial strain corresponded to the necking-limit strains. Then, we calculate the yield-loci for cube-11°, cube-15° and random cases, using FC and VPSC models. The corresponding 11 22 *σ σ* − projections are shown in Fig. 17. The equi workrate surfaces are normalized to the work rate for uniaxial stretching as calculated with the FC model. As expected, the yield loci are quite different. The curvatures of the VPSC yield loci are blunter than those of FC model, particularly for the random texture. This explains the higher limit-strains predicted by the MK-VPSC model as shown in Fig. 16. For the cube-11° and cube-15° initial textures, the FC yield loci are sharper and larger. As other researchers concluded and our simulations confirm, regions of reduced yield-locus

Fig. 18 shows the initial and final (at failure) inverse pole figures of the cube-15º for both constitutive-model approaches at ρ = 1.0. We found that the behavior of certain crystallographic orientations depends on the interaction model used. Particularly, near the <100> orientation, results of the models diverge. Using the VPSC approach, no grains remain close to <100> (Θ < 5º), but for the FC simulations this is not the case, and the grains rotate in widely different directions. In both cases, one can trace an imaginary line that delineates a zone containing a high density of orientations and one vacant of orientations. The grain orientations tend to rotate and accumulate in the region approximately defined by <115> - <114> and <104> - <102> for FC and by <116> - <115> and <104> - <305> for VPSC, respectively. In addition, we found that the FC final orientations are distributed rather uniformly in the inhabited region. Interestingly, for the VPSC calculations, there is a preference to rotate half-way up the <104> - <305> segment line. Similar behavior can be

0,0

(*f*0 = 0.99, *m* = 0.02, *n* = 0.24, *h0* = 1218 MPa, <sup>c</sup>

curvature correspond to lower FLD values.

found for the cube-11º distribution.

0,1

0,2

0,3

\* ε11

0,4

0,5

0,6

**Figure 15.** {111} pole figures used in the simulations for cube-3°, cube-7°, cube-11° and cube-15° distributions.

In the following FLD simulations, in both the homogeneous and MK band zones, standard FCC {111}<110> crystal slip is used, and the initial textures are assumed to be the same. Fig. 16 shows the predicted limit strains for the cube set of texture distributions using the MK-FC and MK-VPSC approaches. The simulations clearly show that there are differences between the VPSC and FC homogenization schemes, although the shapes and levels of the predicted FLDs are similar in the uniaxial range. No significant differences were found between the ideal cube and the cube-3° textures, since for this case the HEM nearly corresponds that of a single crystal. In these cases, both models closely predict shape and tendency. In the negative minor-strain range (ρ < 0) of the FLD, the shapes are nearly straight lines with the maximum values at ρ = - 0.5. For both textures, the FLD curves slope downwards from plane-strain tension to equi-biaxial stretching, with the minimum limitstrain values at ρ = 1. These values are far below that of the random texture.

We would expect that a spread around the ideal orientation would give greater formability and higher limit strains. The FLD calculated from the random texture should be above all others, with the FLDs of particular spreads lying between those of the ideal cube and the random cases. Wu et al.'s (2004a) results do not show this expected behavior; their calculated forming-limit curves for cube-11° and cube-15° are significantly higher than that of the random texture near equi-biaxial stretching. These results were confirmed by Yoshida et al. (2007) using the same modeling hypothesis. Our simulations are similar to those reported by Wu et al. when the MK-FC approach is used. To the contrary, FLDs calculated with the MK-VPSC approach behave as expected. For 0 < ρ ≤ 1, the limit strains move upwards with an increase in the dispersion cut-off angle, and the strain-limit values never reach those of the random case.

 (*f*0 = 0.99, *m* = 0.02, *n* = 0.24, *h0* = 1218 MPa, <sup>c</sup> *<sup>s</sup> τ* = 42 MPa).

of a uniform distribution.

reach those of the random case.

distributions.

components can be input into simulations. In particular, we focus on how the strength of the cube texture affects localized necking. To investigate this effect, we modeled variations of the cube texture. The variations were constructed with different spreads of grain orientations around the ideal cube component. The procedure for modeling textures is the same as that used in Signorelli & Bertinetti (2009). As an example, the cube-15° texture is one whose grains have a misorientation with respect to the ideal cube orientation {100}<001> of less than 15°, uniformly distributed over that area. Fig. 15 shows the {111} stereographic pole figures for cube-3°, cube-7°, cube-11° and cube-15° distributions. For the cube set of textures, the number of individual orientations was set in order to obtain an adequate representation

**Figure 15.** {111} pole figures used in the simulations for cube-3°, cube-7°, cube-11° and cube-15°

strain values at ρ = 1. These values are far below that of the random texture.

In the following FLD simulations, in both the homogeneous and MK band zones, standard FCC {111}<110> crystal slip is used, and the initial textures are assumed to be the same. Fig. 16 shows the predicted limit strains for the cube set of texture distributions using the MK-FC and MK-VPSC approaches. The simulations clearly show that there are differences between the VPSC and FC homogenization schemes, although the shapes and levels of the predicted FLDs are similar in the uniaxial range. No significant differences were found between the ideal cube and the cube-3° textures, since for this case the HEM nearly corresponds that of a single crystal. In these cases, both models closely predict shape and tendency. In the negative minor-strain range (ρ < 0) of the FLD, the shapes are nearly straight lines with the maximum values at ρ = - 0.5. For both textures, the FLD curves slope downwards from plane-strain tension to equi-biaxial stretching, with the minimum limit-

We would expect that a spread around the ideal orientation would give greater formability and higher limit strains. The FLD calculated from the random texture should be above all others, with the FLDs of particular spreads lying between those of the ideal cube and the random cases. Wu et al.'s (2004a) results do not show this expected behavior; their calculated forming-limit curves for cube-11° and cube-15° are significantly higher than that of the random texture near equi-biaxial stretching. These results were confirmed by Yoshida et al. (2007) using the same modeling hypothesis. Our simulations are similar to those reported by Wu et al. when the MK-FC approach is used. To the contrary, FLDs calculated with the MK-VPSC approach behave as expected. For 0 < ρ ≤ 1, the limit strains move upwards with an increase in the dispersion cut-off angle, and the strain-limit values never

**Figure 16.** Calculated FLDs for MK-FC (left) and MK-VPSC (right) models

In order to assess the effect of the yield-surface shape on the forming-limit behavior close to the balanced-biaxial stretching zone, ρ = 1.0, we prestrained the texture sets along the equibiaxial path. The amounts of equi-biaxial strain corresponded to the necking-limit strains. Then, we calculate the yield-loci for cube-11°, cube-15° and random cases, using FC and VPSC models. The corresponding 11 22 *σ σ* − projections are shown in Fig. 17. The equi workrate surfaces are normalized to the work rate for uniaxial stretching as calculated with the FC model. As expected, the yield loci are quite different. The curvatures of the VPSC yield loci are blunter than those of FC model, particularly for the random texture. This explains the higher limit-strains predicted by the MK-VPSC model as shown in Fig. 16. For the cube-11° and cube-15° initial textures, the FC yield loci are sharper and larger. As other researchers concluded and our simulations confirm, regions of reduced yield-locus curvature correspond to lower FLD values.

Fig. 18 shows the initial and final (at failure) inverse pole figures of the cube-15º for both constitutive-model approaches at ρ = 1.0. We found that the behavior of certain crystallographic orientations depends on the interaction model used. Particularly, near the <100> orientation, results of the models diverge. Using the VPSC approach, no grains remain close to <100> (Θ < 5º), but for the FC simulations this is not the case, and the grains rotate in widely different directions. In both cases, one can trace an imaginary line that delineates a zone containing a high density of orientations and one vacant of orientations. The grain orientations tend to rotate and accumulate in the region approximately defined by <115> - <114> and <104> - <102> for FC and by <116> - <115> and <104> - <305> for VPSC, respectively. In addition, we found that the FC final orientations are distributed rather uniformly in the inhabited region. Interestingly, for the VPSC calculations, there is a preference to rotate half-way up the <104> - <305> segment line. Similar behavior can be found for the cube-11º distribution.

Self-Consistent Homogenization Methods for Predicting Forming Limits of Sheet Metal 201

The crystal level properties are determined, by imposing same uniaxial behaviors for all the cases: BCC48-FC, BCC24-FC, BCC48-SC and BCC24-SC. Accordingly, the hardening parameters are chosen to give an identical uniaxial-stress response. They are listed in Table 7. The initial texture, the reference plastic shearing rate and shear strain-rate sensitivity are

**Material BCC48-FC BCC48-SC BCC24-FC BCC24-SC** 

The predicted limit strains are shown in Fig. 19. Large differences are between the MK-FC and MK-VPSC results, particularly near equi-biaxial stretching, regardless of the material. For each homogenization method, both materials have about the same forming limit in plane strain. At ρ = 0, we found no difference in the predicted limit-strains values given by either the BCC24 or BCC48 approach, but a discrepancy appears between the FC and SC polycrystal models. Within the MK-VPSC framework, the profiles of the BCC48 and BCC24 simulations are very close for the strain ratios ρ ≤ 0.6, and differences can be seen near equibiaxial stretching. The MK-FC limit strains are similar as ρ increases up to 0.8, but for ρ > 0.8, the critical values calculated with the BCC24 deformation model increase to an unrealistic high value at ρ = 1. Our simulations clearly show that in equi-biaxial stretching the exclusion of {123}<111> crystallographic slip as a potential active deformation mode promotes higher limit strains for both models. For MK-VPSC this gap is only an increase in the limit strain

0,0 0,2 0,4 0,6 0,8

ε \* 22  BCC48-FC BCC24-FC BCC48-SC BCC24-SC

*h*<sup>0</sup> (MPa) 808 808 795 980 *n* 0.23 0.26 0.23 0.26

*<sup>s</sup> τ* (MPa) 30.5 40.0 30.0 37.0

from 0.37 to 0.39; whereas for MK-FC the value changes from 0.53 to 0.97.

0,0

**Figure 19.** Influence of the slip microstructure and interaction model on the FLD.

0,2

0,4

\* ε11 0,6

0,8

the same as used in the previous section.

**Table 7.** Material parameters used in the simulations.

c

**Figure 17.** Calculated yield loci for VPSC and FC models. The equi work-rate surfaces are normalized to the work rate for uniaxial stretching, as calculated with the FC model.

**Figure 18.** Stereographic triangles showing the initial cube-15° texture (left) and predicted final orientations after equi-biaxial stretching to failure by FC (center) and VPSC (right) models.

Simulations of FLDs show that the MK-FC strategy leads to unrealistic results, since an increasing spread about the cube texture produces unexpectedly high limit strains. However, results with the MK-VPSC approach successfully predict a smooth transition in the limit strains from the ideal-cube texture, through dispersions around the cube texture with increasing cut-off angles, ending with a random texture.
