**4.4. Grain shape**

It is important for the reader to note that the VPSC model has the capability to account for the grain-shape and its evolution. The influence of different elongated grain-shapes on the limit strains is shown in Fig. 8 for a non-textured material. Generally, the calculated FLDs decrease when increasingly elongated grains are considered. In the negative minor-strain range, no appreciable changes are noted in the FLD's behavior for different aspect ratio grains. To the contrary, in the biaxial-stretching zone, differences are observed, especially when the aspect ratio is more pronounced. For initially equiaxed grains and grains elongated up to an aspect ratios of (3:1:1), the profiles of the simulated limit strains show no significant changes. However, for a (10:1:1) grain aspect ratio, we found a noticeable decrease of the limit-strains moving from plane-strain to equi-biaxial tension. This result can be correlated with the final textures predicted for each grain-shape morphology. The {111} pole figures calculated for the homogeneous deformation zone at the end of the equi-biaxial loading path are shown on the right in Fig. 8.

(*f*0 = 0.99, *m* = 0.01, *h*0 = 1410 MPa, <sup>c</sup> *<sup>s</sup> τ* = 47 MPa).

190 Metal Forming – Process, Tools, Design

strain-rate sensitivity.

**4.3. Hardening coefficients** 

effects are included in the calculations.

**4.4. Grain shape** 

parameter *m* in the viscoplastic law controls the accumulated shear, which in turn drives the hardening. It is also known that as the *m* value increases textural sharpness decreases, though this behavior depends on the imposed strain path, too. The calculated average number of slip systems associated with the main strain paths and several additional material sensitivities are presented in Table 4. It is interesting to note that, as expected, the fewer the number of slip systems and the sharper the texture the lower the limit curve.

ρ = -0.5 4.5 2.8 2.3 ρ = 0.0 2.8 2.2 1.9 ρ = 1.0 3.3 3.1 2.8 **Table 4.** Calculated average number of active slip systems as a function of the deformation path and

As opposed to the results for the limit-strain values, the critical angles at failure are almost insensitive to the strain-rate sensitivity. The predicted final angles rise to between \* Ψ = 34°- 37° for a uniaxial path and decrease to zero degrees for ρ ≥ 0. If we restrict the model interaction to the FC hypothesis, as done by Wu et al. (1997), our calculated values equal theirs.

Slip induced hardening is another important factor influencing the limit strains. From Eqs. (28) and (29) it is easy to see that the parameters *h*0 and *n* govern the strain hardening. We investigated the effect produced by different values of the hardening coefficient *n* (0.16, 0.19 and 0.23) while fixing the other material properties. The calculated FLDs are shown in Fig. 7, where it is clear that the slip hardening coefficient *n* does not affect the shape of the forming limit curves. However, it can be seen that the largest value of *n* produces the highest limit strains. Also, no noticeable dependence with ρ is observed. Because we use isotropic hardening in these calculations, the *n* parameter only guides the stress level, producing these simple behaviors. This will not be the case when latent hardening and other kinematic

It is important for the reader to note that the VPSC model has the capability to account for the grain-shape and its evolution. The influence of different elongated grain-shapes on the limit strains is shown in Fig. 8 for a non-textured material. Generally, the calculated FLDs decrease when increasingly elongated grains are considered. In the negative minor-strain range, no appreciable changes are noted in the FLD's behavior for different aspect ratio grains. To the contrary, in the biaxial-stretching zone, differences are observed, especially when the aspect ratio is more pronounced. For initially equiaxed grains and grains elongated up to an aspect ratios of (3:1:1), the profiles of the simulated limit strains show no significant changes. However, for a (10:1:1) grain aspect ratio, we found a noticeable decrease of the limit-strains moving from plane-strain to equi-biaxial tension. This result can

*m* = 0.05 *m* = 0.02 *m* = 0.01

**Figure 7.** Influence of the slip-induced hardening *n* on the FLD

 (*f*0 = 0.99, *m* = 0.02, *n* = 0.16, *h0* = 4000 MPa, <sup>c</sup> *<sup>s</sup> τ* = 22 MPa).

**Figure 8.** Influence of the initial grain shape on the limit-strains for a non-textured material

#### **4.5. Effects of texture**

A crystallographic texture develops during metal forming and it is a key component of the material's microstructure. It is generally accepted that this microstructural feature

significantly affects forming-limit strains. To investigate these texture effects, we carried out simulations using two initial textures: one a random distribution of orientations (R) and the other a rolling texture (S). Two {111} pole figures illustrating these textures are shown in Fig. 9. To construct the S texture we first fixed the ideal component volume fractions, 10% {001}<100>, 15% {011}<100>, 30% {123}<634>, 10% {112}<111> and 35% {011}<211>, and then spread the distribution by assigning each grain a misorientation angle of θ < 15° with respect to the ideal component.

Self-Consistent Homogenization Methods for Predicting Forming Limits of Sheet Metal 193

R

anisotropy in a more complex way in order to determine the effect of texture updating on the limit strains. This in turn is captured more or less realistically by the different


\* ε22

Such an analysis can explain the opposite trends in limit strains reported by previous researchers, which cannot be understood based only on initial-material textures. In our opinion, the differences in the forming-limit strains are related to material anisotropy and its evolution along the deformation path. This produces an increase or decrease of the FLD profile. Many previous investigations have proven that the VPSC model gives a more realistic description of the anisotropic behavior of polycrystalline materials. We believe that results of the MK-VPSC strategy presented here are a better way to explain and justify the

To assess the influence of the texture evolution on the limit strains, we repeated the calculations shown in Fig. 10 but without texture evolution (Fig. 11). In negative strain space (ρ < 0) the FLDs have practically identical shapes, although the calculated values are slightly lower when the initial texture is not updated. However, in the biaxial zone, the tendency is quite different. The FLDs for both materials now approach each other, and a certain matching is observed. For the R texture, the limit-strain values reflect texture evolution. Texture and hence anisotropy evolution produces greater limit strains. In the case of the S material, when texture updating is off, the limit strains in the biaxial zone increase continuously, showing a different behavior than when the texture is updated. We attribute this behavior to the sharpness of the material yield locus and consequently, to the slip systems selected to accommodate the imposed deformation. The corresponding yield-loci after equi-biaxial stretching for both materials are displayed in Fig. 12 in 11 22 *σ σ* − space. Following Barlat's work (1989), the parameter *p* quantifies the effect of yield-surface shape

*<sup>s</sup> τ* = 22 MPa).

S

homogenization schemes.

0,0

**Figure 10.** Influence of the initial texture on the FLD

different effects of the material parameters.

0,2

0,4

\* ε11

(*f*0 = 0.99, *m* = 0.02, *n* = 0.16, *h0* = 4000 MPa, <sup>c</sup>

0,6

0,8

**Figure 9.** Grain orientation distributions represented by {111} pole figures: material R (left) and material S (right).

These different initial textures strongly affected the forming-limit curves as plotted in Fig. 10. 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. The predictions, however, begin to diverge at ρ = 0, and the differences increase continuously, reaching a maximum for a biaxial deformation path. The S textured material develops a much stronger anisotropy than the R, likely producing the observed results. The S material's forming-limit curve slopes downwards from plane-strain to equi-biaxial tension, and over the whole range ρ > 0 the predicted forming limits for the R case are larger than those for the S. Fig. 10 includes a plot of the final textures of each sample at the end of the equi-biaxial loading path. Clearly, the R and S textures evolve to different states producing the strong effects observed in the FLD behavior.

Our calculations also show that the influence of crystallographic texture evolution is at least as important as effects of the initial grain distributions. Evolution effects have been previously discussed by several authors. Tóth et al. (1996) performed simulations with a rate independent Taylor model, showing that crystal rotations decrease the limit strains. Tang & Tai (2000), using the MK analysis together with continuum damage mechanics (CDM) and the Taylor model, found the same behavior for the limit strains. They claim that the development of texture causes deterioration of the material. On the other hand, Wu et al. (2004b) use a mesoscopic approach and a Taylor homogenization scheme to show that texture evolution increases the limit-strains in the biaxial zone. Finally, Inal et al. (2005) recently analyzed these two studies and their opposite conclusions, adding a study of how texture evolution in BCC materials affects the FLD. In the work of Inal et al., the simulations show that texture development does not have a significant influence on the FLD. Actually based on our simulations, we found it necessary to analyze the development of material anisotropy in a more complex way in order to determine the effect of texture updating on the limit strains. This in turn is captured more or less realistically by the different homogenization schemes.

(*f*0 = 0.99, *m* = 0.02, *n* = 0.16, *h0* = 4000 MPa, <sup>c</sup> *<sup>s</sup> τ* = 22 MPa).

192 Metal Forming – Process, Tools, Design

to the ideal component.

S (right).

behavior.

significantly affects forming-limit strains. To investigate these texture effects, we carried out simulations using two initial textures: one a random distribution of orientations (R) and the other a rolling texture (S). Two {111} pole figures illustrating these textures are shown in Fig. 9. To construct the S texture we first fixed the ideal component volume fractions, 10% {001}<100>, 15% {011}<100>, 30% {123}<634>, 10% {112}<111> and 35% {011}<211>, and then spread the distribution by assigning each grain a misorientation angle of θ < 15° with respect

**Figure 9.** Grain orientation distributions represented by {111} pole figures: material R (left) and material

These different initial textures strongly affected the forming-limit curves as plotted in Fig. 10. 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. The predictions, however, begin to diverge at ρ = 0, and the differences increase continuously, reaching a maximum for a biaxial deformation path. The S textured material develops a much stronger anisotropy than the R, likely producing the observed results. The S material's forming-limit curve slopes downwards from plane-strain to equi-biaxial tension, and over the whole range ρ > 0 the predicted forming limits for the R case are larger than those for the S. Fig. 10 includes a plot of the final textures of each sample at the end of the equi-biaxial loading path. Clearly, the R and S textures evolve to different states producing the strong effects observed in the FLD

Our calculations also show that the influence of crystallographic texture evolution is at least as important as effects of the initial grain distributions. Evolution effects have been previously discussed by several authors. Tóth et al. (1996) performed simulations with a rate independent Taylor model, showing that crystal rotations decrease the limit strains. Tang & Tai (2000), using the MK analysis together with continuum damage mechanics (CDM) and the Taylor model, found the same behavior for the limit strains. They claim that the development of texture causes deterioration of the material. On the other hand, Wu et al. (2004b) use a mesoscopic approach and a Taylor homogenization scheme to show that texture evolution increases the limit-strains in the biaxial zone. Finally, Inal et al. (2005) recently analyzed these two studies and their opposite conclusions, adding a study of how texture evolution in BCC materials affects the FLD. In the work of Inal et al., the simulations show that texture development does not have a significant influence on the FLD. Actually based on our simulations, we found it necessary to analyze the development of material

TD

RD

**Figure 10.** Influence of the initial texture on the FLD

Such an analysis can explain the opposite trends in limit strains reported by previous researchers, which cannot be understood based only on initial-material textures. In our opinion, the differences in the forming-limit strains are related to material anisotropy and its evolution along the deformation path. This produces an increase or decrease of the FLD profile. Many previous investigations have proven that the VPSC model gives a more realistic description of the anisotropic behavior of polycrystalline materials. We believe that results of the MK-VPSC strategy presented here are a better way to explain and justify the different effects of the material parameters.

To assess the influence of the texture evolution on the limit strains, we repeated the calculations shown in Fig. 10 but without texture evolution (Fig. 11). In negative strain space (ρ < 0) the FLDs have practically identical shapes, although the calculated values are slightly lower when the initial texture is not updated. However, in the biaxial zone, the tendency is quite different. The FLDs for both materials now approach each other, and a certain matching is observed. For the R texture, the limit-strain values reflect texture evolution. Texture and hence anisotropy evolution produces greater limit strains. In the case of the S material, when texture updating is off, the limit strains in the biaxial zone increase continuously, showing a different behavior than when the texture is updated. We attribute this behavior to the sharpness of the material yield locus and consequently, to the slip systems selected to accommodate the imposed deformation. The corresponding yield-loci after equi-biaxial stretching for both materials are displayed in Fig. 12 in 11 22 *σ σ* − space. Following Barlat's work (1989), the parameter *p* quantifies the effect of yield-surface shape on limit strains for the last four cases (Table 5). It should be noted that the trends of *p* and the predicted limit-strain values are consistent, as Hiwatashi et al. (1998) and Friedman & Pan (2000) have noted.

Self-Consistent Homogenization Methods for Predicting Forming Limits of Sheet Metal 195

Material Texture evolution *p*  R Yes 1.159 S No 1.072 R Yes 1.131 S No 1.087

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

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

**Material FCC-FC FCC-SC BCC-FC BCC-SC**  *h*<sup>0</sup> (MPa) 1950 2720 1850 3100 *n* 0.250 0.224 0.250 0.265

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 in-

*<sup>s</sup> τ* (MPa) 31.5 47.0 37.0 45.0

*<sup>s</sup> τ* of all slip systems are taken equal, the rate sensitivity is *m* = 0.02, and a

**Table 5.** *p*-parameter for the R and S equi-biaxial stretching cases.

hardening.

slip resistances <sup>c</sup>

c

reference slip rate of <sup>0</sup> *γ* = 0.001 s-1 is assumed.

**Table 6.** Material parameters used in the simulations.

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

(*f*0 = 0.99, *m* = 0.02, *n* = 0.16, *h0* = 4000 MPa, <sup>c</sup> *<sup>s</sup> τ* = 22 MPa).

**Figure 11.** Influence of texture evolution on the FLD

**Figure 12.** Calculated yield loci for materials R and S with (solid lines) and without (dotted lines) texture evolution. The equi work-rate surface is normalized to the work rate for uniaxial stretching, as calculated with the FC model.


**Table 5.** *p*-parameter for the R and S equi-biaxial stretching cases.

194 Metal Forming – Process, Tools, Design

Pan (2000) have noted.

0,0

**Figure 11.** Influence of texture evolution on the FLD

0,2

0,4

\* ε11

(*f*0 = 0.99, *m* = 0.02, *n* = 0.16, *h0* = 4000 MPa, <sup>c</sup>


calculated with the FC model.



0,0

σ22

0,1

σp

σb

0,2

0,3

0,6

0,8

on limit strains for the last four cases (Table 5). It should be noted that the trends of *p* and the predicted limit-strain values are consistent, as Hiwatashi et al. (1998) and Friedman &

> Texture evolution with without

*<sup>s</sup> τ* = 22 MPa).

plane strain

R material S material

0,0 0,1 0,2 0,3

σ11

 with without


\* ε22


**Figure 12.** Calculated yield loci for materials R and S with (solid lines) and without (dotted lines) texture evolution. The equi work-rate surface is normalized to the work rate for uniaxial stretching, as



0,0

0,1

0,2

0,3

0,0 0,1 0,2 0,3

σ11

 with without

plane strain

S

R
