**5.2. Influence of the dislocation slip assumption on the formability of BCC sheet metals**

The identification of the active slip systems is a widely discussed issue in the plastic deformation of BCC crystals. The most common deformation mode is {110}<111>, but BCC materials also slip on other planes, {112} and {123}, with the same slip direction. In the literature, it is common for two sets of possible slip systems describing BCC plastic behavior to be considered: {110}<111>, {112}<111> (BCC24); or {110}<111>, {112}<111>, {123}<111> (BCC48). In what follows, we explore the right-hand side of the FLD for a BCC material using the proposed MK-FC and MK-VPSC approaches, and test the crystallographic slip assumption.

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 the same as used in the previous section.


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

200 Metal Forming – Process, Tools, Design

Random texture

Cube-15<sup>o</sup>

**Figure 17.** Calculated yield loci for VPSC and FC models. The equi work-rate surfaces are normalized

104 102 305

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

**5.2. Influence of the dislocation slip assumption on the formability of BCC sheet** 

The identification of the active slip systems is a widely discussed issue in the plastic deformation of BCC crystals. The most common deformation mode is {110}<111>, but BCC materials also slip on other planes, {112} and {123}, with the same slip direction. In the literature, it is common for two sets of possible slip systems describing BCC plastic behavior to be considered: {110}<111>, {112}<111> (BCC24); or {110}<111>, {112}<111>, {123}<111> (BCC48). In what follows, we explore the right-hand side of the FLD for a BCC material using the proposed MK-FC and MK-VPSC approaches, and test the crystallographic slip

**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.

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

σ11

to the work rate for uniaxial stretching, as calculated with the FC model.

with increasing cut-off angles, ending with a random texture.

116115114

 VPSC FC

104

116

**metals** 

assumption.

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

σ<sup>11</sup> σ<sup>11</sup>

116115114

104 102 305

100 110

111

Cube-11<sup>o</sup>

0,0

0,2

0,4

σ22

0,6

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

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

The yield potentials of the materials were calculated by imposing different plastic strain-rate tensors under a state of plane stress in the 11 22 *σ σ* − section. 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 the case of BCC48-FC uniaxial stretching. We calculated the yield loci corresponding to each of the necking limit strains, in order to highlight the effect of the yield-surface shape on the forming-limit behavior. Particular attention must be paid to the surface's curvature near the balanced-biaxial stretching zone.

Self-Consistent Homogenization Methods for Predicting Forming Limits of Sheet Metal 203

θ

ϕ

0 15 30 45 60 75 90

Loading direction ϕ / deg.

 BCC48-FC BCC24-FC BCC48-SC BCC24-SC

the case when we follow the MK-VPSC procedure, at least for a nontextured material. For these calculations, we found that either BCC24 or BCC48 materials give similar FLD curves,


0

Direction of

**Figure 20.** Yield loci at equi-biaxial failure for all tested materials (left); directions of the plastic strain-

Finally, we evaluated the MK-VPSC capability comparing the predictions of our model with recently published experimental and numerical results. Data and predictions for two lowcarbon steel, LCS, sheets are analyzed. The limit-strain calculations performed with MK-FC and MK-VPSC are analyzed and discussed in terms of the crystallographic- slip assumption, and compared with the measured data. All experimental tests were conducted at room

For a first verification, experimental data are taken from Serenelli et al. (2010). The initial texture of the steel sheet was measured by using a Phillips X'Pert X-ray diffractometer. Incomplete pole figures for the {110}, {200} and {112} diffraction peaks were obtained. From these data, the ODF was determined, and the completed {110} and {100} pole figures were calculated (Fig. 21). The texture was discretized into 1000 orientations of equal volume fraction. Due to the annealing process, the texture contains a slight α-fiber ({001}<110> 1.2%, {112}<110> 9.6%), a more intense γ-fiber ({111}<110> 18.7%, {111}<112> 9.0%), and {554}<225> 11.7%. Optical micrographs of the LCS annealed microstructure show oblate spherical

Equi-biaxial bulge tests with different elliptical die rings were conducted in order to obtain three biaxial paths in strain space. The die ring masks had a major diameter of 125 mm and aspect ratios of 0.5, 0.7 and 1.0. The die rings and a corresponding tested circle-gridded specimens are shown in Fig. 22. A grid pattern of 2.5 mm diameter circles was electrochemically etched on the surface of specimens. Details of the experimental procedure can be

**D** (θ / deg.)

45

90

135

over the whole range of deformations.

 BCC48-FC BCC24-FC BCC48-SC BCC24-SC

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

σ11

grains, with an approximate aspect ratio of 1.0:1.0:0.4.

found in Serenelli et al. (2010).

0,0

rate vectors (right).

temperature.

0,2

σ22

0,4

0,6

The shapes and curvatures predicted by the FC and VPSC models are quite different (Fig. 20 left). Within the VPSC framework, the yield loci are sharper, and only small differences can be found between the BCC48 and BCC24 based simulations. This is consistent with the similar limit-strain values predicted by the MK-VPSC model, as shown in Fig. 19. The differences are more pronounced for the MK-FC calculations. The curvature of the BCC24 yield locus is more gradual than that of the BCC48 material, again in agreement with the predicted limit strains. Fig. 20b presents the stress potential in a different but qualitatively similar way, based on the direction of plastic-strain rate and loading direction. In the Cauchy stress reference frame, the directions at different points along the predicted yield surfaces are characterized by two angles, θ and ϕ, as shown in Fig. 20 (right). These angles are taken to be zero along the horizontal axis and assumed positive in a counterclockwise sense. Differences in the sharpness of the stress potentials in equi-biaxial stretching, reflected in the slope of the plots, are clearly illustrated. For the BCC24-FC case, the direction of the plastic-strain rate **D** , or θ, seems nearly invariant in the vicinity of ϕ=45º (equi-biaxial strain-rate states), whereas for BCC48-FC the values of θ increase steadily with ϕ over this range. The curves calculated with the FC theory, in accordance with the observed yield loci, are not steep and clearly different for the two materials. For the BCC48-SC and BCC24-SC materials, the slight changes observed in the predicted critical-strain values correlate to almost identical profiles in Fig. 20b. In the vicinity of ϕ=45º, the sharpness of both yield loci are characterized by an abrupt change of θ. Over the range 42º ≤ ϕ ≤ 47º, θ varies linearly from 23º to 63º. This allows the material to quickly approach a plane-strain state with minor variations of stress state. We verified that this behavior can be mainly attributed to the ability of the VPSC model to distribute the imposed deformation according to the relative hardness of the grains. In addition, we note that a large majority of the grains, approximately 80 percent, shows a similar local-states solution (i.e. stress / strain-rate states, plastic dissipation and accumulated shear), but built from different sets of active slip systems when the simulations were carried out with either BCC48 or BCC24 slip assumptions. Consequently, MK-VPSC predicts very similar limit strains for both microstructural slip assumptions, though this result is specifically dependent on the initial material texture.

In summary, although it is normally accepted that a BCC material can be represented using 24 or 48 slip systems, we found that the MK-FC FLD calculations are sensitive to the material plasticity assumption in the vicinity of equi-biaxial stretching. However, this is not the case when we follow the MK-VPSC procedure, at least for a nontextured material. For these calculations, we found that either BCC24 or BCC48 materials give similar FLD curves, over the whole range of deformations.

202 Metal Forming – Process, Tools, Design

material texture.

The yield potentials of the materials were calculated by imposing different plastic strain-rate tensors under a state of plane stress in the 11 22 *σ σ* − section. 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 the case of BCC48-FC uniaxial stretching. We calculated the yield loci corresponding to each of the necking limit strains, in order to highlight the effect of the yield-surface shape on the forming-limit behavior. Particular attention must be paid to the surface's curvature near the balanced-biaxial stretching zone.

The shapes and curvatures predicted by the FC and VPSC models are quite different (Fig. 20 left). Within the VPSC framework, the yield loci are sharper, and only small differences can be found between the BCC48 and BCC24 based simulations. This is consistent with the similar limit-strain values predicted by the MK-VPSC model, as shown in Fig. 19. The differences are more pronounced for the MK-FC calculations. The curvature of the BCC24 yield locus is more gradual than that of the BCC48 material, again in agreement with the predicted limit strains. Fig. 20b presents the stress potential in a different but qualitatively similar way, based on the direction of plastic-strain rate and loading direction. In the Cauchy stress reference frame, the directions at different points along the predicted yield surfaces are characterized by two angles, θ and ϕ, as shown in Fig. 20 (right). These angles are taken to be zero along the horizontal axis and assumed positive in a counterclockwise sense. Differences in the sharpness of the stress potentials in equi-biaxial stretching, reflected in the slope of the plots, are clearly illustrated. For the BCC24-FC case, the direction of the plastic-strain rate **D** , or θ, seems nearly invariant in the vicinity of ϕ=45º (equi-biaxial strain-rate states), whereas for BCC48-FC the values of θ increase steadily with ϕ over this range. The curves calculated with the FC theory, in accordance with the observed yield loci, are not steep and clearly different for the two materials. For the BCC48-SC and BCC24-SC materials, the slight changes observed in the predicted critical-strain values correlate to almost identical profiles in Fig. 20b. In the vicinity of ϕ=45º, the sharpness of both yield loci are characterized by an abrupt change of θ. Over the range 42º ≤ ϕ ≤ 47º, θ varies linearly from 23º to 63º. This allows the material to quickly approach a plane-strain state with minor variations of stress state. We verified that this behavior can be mainly attributed to the ability of the VPSC model to distribute the imposed deformation according to the relative hardness of the grains. In addition, we note that a large majority of the grains, approximately 80 percent, shows a similar local-states solution (i.e. stress / strain-rate states, plastic dissipation and accumulated shear), but built from different sets of active slip systems when the simulations were carried out with either BCC48 or BCC24 slip assumptions. Consequently, MK-VPSC predicts very similar limit strains for both microstructural slip assumptions, though this result is specifically dependent on the initial

In summary, although it is normally accepted that a BCC material can be represented using 24 or 48 slip systems, we found that the MK-FC FLD calculations are sensitive to the material plasticity assumption in the vicinity of equi-biaxial stretching. However, this is not

**Figure 20.** Yield loci at equi-biaxial failure for all tested materials (left); directions of the plastic strainrate vectors (right).

Finally, we evaluated the MK-VPSC capability comparing the predictions of our model with recently published experimental and numerical results. Data and predictions for two lowcarbon steel, LCS, sheets are analyzed. The limit-strain calculations performed with MK-FC and MK-VPSC are analyzed and discussed in terms of the crystallographic- slip assumption, and compared with the measured data. All experimental tests were conducted at room temperature.

For a first verification, experimental data are taken from Serenelli et al. (2010). The initial texture of the steel sheet was measured by using a Phillips X'Pert X-ray diffractometer. Incomplete pole figures for the {110}, {200} and {112} diffraction peaks were obtained. From these data, the ODF was determined, and the completed {110} and {100} pole figures were calculated (Fig. 21). The texture was discretized into 1000 orientations of equal volume fraction. Due to the annealing process, the texture contains a slight α-fiber ({001}<110> 1.2%, {112}<110> 9.6%), a more intense γ-fiber ({111}<110> 18.7%, {111}<112> 9.0%), and {554}<225> 11.7%. Optical micrographs of the LCS annealed microstructure show oblate spherical grains, with an approximate aspect ratio of 1.0:1.0:0.4.

Equi-biaxial bulge tests with different elliptical die rings were conducted in order to obtain three biaxial paths in strain space. The die ring masks had a major diameter of 125 mm and aspect ratios of 0.5, 0.7 and 1.0. The die rings and a corresponding tested circle-gridded specimens are shown in Fig. 22. A grid pattern of 2.5 mm diameter circles was electrochemically etched on the surface of specimens. Details of the experimental procedure can be found in Serenelli et al. (2010).

Self-Consistent Homogenization Methods for Predicting Forming Limits of Sheet Metal 205

 Experimental BCC48-FC BCC24-FC BCC48-SC BCC24-SC

well with the measured points. The plane strain behavior is similar to that predicted for an initial non-textured material, and no influence from the crystallographic slip assumption is found. However, the differences between the interaction models remain. In these calculations, the MK-VPSC results showed a sensibility to the addition of {123}<111> slip as a potentially

0,0 0,2 0,4 0,6 0,8 1,0

ε22 \*

**Figure 23.** Influence of the slip microstructure and interaction model on limit strains for the LCS rolled

For a second verification, we consider data – experimental FLD and material´s properties – from Signorelli et al. (2012) for an electro-galvanized DQ-type steel sheet 0.67 mm thick. Texture measurements were conducted using X-Ray diffraction in a Phillips X´Pert Pro-MPD system equipped with a texture goniometer, Cu*K* alpha radiation and an X-ray lens. The initial pole figures obtained for the {110}, {112} and {100} diffraction peaks are shown in Fig. 24 (left). From these data the ODF was calculated. The measured texture represented by the ϕ2 = 45° section is also presented in Fig. 24 (right). It shows a high concentration of orientations with {111} planes lying parallel to the sample (sheet) surface together with the

**Figure 24.** Experimental equal-area pole figures {110}, {112} and {100} (left); ϕ2 = 45° section of the ODF

TD

RD

{554}<225> orientations. This is typical of a cold-rolled and annealed steel.

active deformation mode; only BCC48-SC predicts a realistic strain-limit profile.

0,0

0,2

0,4

0,6

ε \* 11

sheet.

(right).

0,8

1,0

**Figure 21.** Experimental {110} and {100} stereographic pole figures (lines are multiples of a half-random distribution). Reference frame: X1 top (RD), X2 right (TD), X3 center (ND).

**Figure 22.** Die-ring masks (top) and photograph of the gridded specimens after the tests (bottom) with ratios of 0.5 (a), 0.7 (b) and 1.0 (c) between the major and minor diameters. The marked points where the grid was measured can be observed in the photographs.

The alloy's hardening parameters were estimated in order to fit tensile test data. The coefficients for the BCC48-FC, BCC24-FC, BCC48-SC and BCC24-SC materials are listed in Table 8.


**Table 8.** Material parameters used in the simulations for the LCS sheet.

The initial value of the imperfection factor, *f*0, was taken to be 0.996. The FLD predictions are shown in Fig. 23 together with the bulge-test data. The BCC48-SC predicted limit-strains agree well with the measured points. The plane strain behavior is similar to that predicted for an initial non-textured material, and no influence from the crystallographic slip assumption is found. However, the differences between the interaction models remain. In these calculations, the MK-VPSC results showed a sensibility to the addition of {123}<111> slip as a potentially active deformation mode; only BCC48-SC predicts a realistic strain-limit profile.

204 Metal Forming – Process, Tools, Design

**Figure 21.** Experimental {110} and {100} stereographic pole figures (lines are multiples of a half-random

**Figure 22.** Die-ring masks (top) and photograph of the gridded specimens after the tests (bottom) with ratios of 0.5 (a), 0.7 (b) and 1.0 (c) between the major and minor diameters. The marked points where the

The alloy's hardening parameters were estimated in order to fit tensile test data. The coefficients for the BCC48-FC, BCC24-FC, BCC48-SC and BCC24-SC materials are listed in

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

The initial value of the imperfection factor, *f*0, was taken to be 0.996. The FLD predictions are shown in Fig. 23 together with the bulge-test data. The BCC48-SC predicted limit-strains agree

*h*<sup>0</sup> (MPa) 1680 2900 1770 2900 *n* 0.203 0.212 0.201 0.212

*<sup>s</sup> τ* (MPa) 49.0 60.0 47.5 59.0

**Table 8.** Material parameters used in the simulations for the LCS sheet.

distribution). Reference frame: X1 top (RD), X2 right (TD), X3 center (ND).

grid was measured can be observed in the photographs.

Table 8.

c

**Figure 23.** Influence of the slip microstructure and interaction model on limit strains for the LCS rolled sheet.

For a second verification, we consider data – experimental FLD and material´s properties – from Signorelli et al. (2012) for an electro-galvanized DQ-type steel sheet 0.67 mm thick. Texture measurements were conducted using X-Ray diffraction in a Phillips X´Pert Pro-MPD system equipped with a texture goniometer, Cu*K* alpha radiation and an X-ray lens. The initial pole figures obtained for the {110}, {112} and {100} diffraction peaks are shown in Fig. 24 (left). From these data the ODF was calculated. The measured texture represented by the ϕ2 = 45° section is also presented in Fig. 24 (right). It shows a high concentration of orientations with {111} planes lying parallel to the sample (sheet) surface together with the {554}<225> orientations. This is typical of a cold-rolled and annealed steel.

**Figure 24.** Experimental equal-area pole figures {110}, {112} and {100} (left); ϕ2 = 45° section of the ODF (right).

The forming-limit diagrams were determined by following an experimental procedure involving three stages: applying a circle grid to the samples, punch stretching to maximum load, and measuring strains. As we are not focused on the experimental methods and techniques, we will not present experimental details here. Readers are referred to Signorelli et al. (2012) for a completed description of the specific techniques for measuring the FLDs.

Self-Consistent Homogenization Methods for Predicting Forming Limits of Sheet Metal 207

The shapes and levels of the predicted FLDs for both models are similar in the tensioncompression range, and the trends between measured and simulated limit strains are close, except near uniaxial tension. In this region, simulations show that there are differences between the MK-VPSC and MK-FC schemes. MK-FC predictions are more conservative and this curve lies below the region of localized flow. Examining the calculated FLDs in the biaxial quadrant of strain space, we found that the limit values predicted by MK-VPSC model accurately separate the regions of safe (uniform) and insecure (localized) deformation. On the other hand, the critical values calculated with the MK-FC approach, are only accurate for strain-path values to 0.3. These differences reach a maximum for the equibiaxial deformation path. Our simulations clearly show that limit values calculated with the MK-FC approach increase unrealistically as ρ increases. It is clear that the VPSC scheme together with the MK approach provides accurate predictions of the DQ-type steel behavior


Experimental Data Safe Unsafe

> MK-VPSC MK-FC

\*

ε22

**Figure 26.** Experimental data and simulated FLDs for both MK-FC and MK-VPSC schemes for the DQ-

From the results presented in this chapter it can generally be concluded that the calculation of the FLD is strongly influenced by the selected constitutive description. In the present work we highlight the important role that the assumed homogenization scheme plays, which cannot be omitted in the discussion of the simulation's results. The predicative capability of a particular crystal-plasticity model is then assessed by comparing its predictions with those experimental data not used for the fitting. In our case, the discussion is framed in terms of the predicted FLD, texture evolution and polycrystal

over the entire biaxial range.

0.0

0.2

0.4

0.6

\* ε11

type steel.

**5.3. Summary** 

yield surface.

0.8

1.0

Simulations were performed following the methodology described in previous sections. The measured initial texture was discretized into 1000 orientations of equal volume fraction. In this case, we assumed that plastic deformation occurred by slip on the {110}, {112} and {123} planes with a <111> slip direction for each case (BCC48). The hardening parameters were established by numerically fitting the uniaxial tensile data taken parallel to the rolling direction with the following results: <sup>c</sup> <sup>s</sup> *τ* = 62 MPa, *h*0 = 2275 MPa and *n* = 0.222 for VPSC simulations; and <sup>c</sup> <sup>s</sup> *τ* = 55 MPa, *h*0 = 1100 MPa and *n* = 0.209 for the FC calculations. For the calculations, the initial slip resistances, <sup>c</sup> <sup>s</sup> *τ* , of all slip systems are assumed equal, the strain-rate sensitivity and the reference slip rate at the crystal level were taken to be *m* = 0.02 and 0 *s* γ = 1 s-1, respectively. The simulated and the experimental curves are shown in Fig. 25.

Before performing simulations, we adjusted *f*0 such that the predicted limit strains matched the experimental results in plane strain. For the MK-FC and MK-VPSC simulations these values of *f*0 were 0.999 and 0.996 respectively. Together with the experimental data, the simulated FLDs for both the MK-FC and MK-VPSC schemes are shown in Fig. 26. The open symbols define a safe zone of uniform deformation for metal forming. The solid symbols correspond to measured circles that experienced local necking or fracture, specifying an insecure zone for metal forming.

**Figure 25.** Experimental and simulated uniaxial tests parallel to the rolling direction.

The shapes and levels of the predicted FLDs for both models are similar in the tensioncompression range, and the trends between measured and simulated limit strains are close, except near uniaxial tension. In this region, simulations show that there are differences between the MK-VPSC and MK-FC schemes. MK-FC predictions are more conservative and this curve lies below the region of localized flow. Examining the calculated FLDs in the biaxial quadrant of strain space, we found that the limit values predicted by MK-VPSC model accurately separate the regions of safe (uniform) and insecure (localized) deformation. On the other hand, the critical values calculated with the MK-FC approach, are only accurate for strain-path values to 0.3. These differences reach a maximum for the equibiaxial deformation path. Our simulations clearly show that limit values calculated with the MK-FC approach increase unrealistically as ρ increases. It is clear that the VPSC scheme together with the MK approach provides accurate predictions of the DQ-type steel behavior over the entire biaxial range.

**Figure 26.** Experimental data and simulated FLDs for both MK-FC and MK-VPSC schemes for the DQtype steel.

#### **5.3. Summary**

206 Metal Forming – Process, Tools, Design

direction with the following results: <sup>c</sup>

calculations, the initial slip resistances, <sup>c</sup>

the FLDs.

simulations; and <sup>c</sup>

*s* γ

insecure zone for metal forming.

0,0

0,1

0,2

0,3

σ11 (GPa)

0,4

0,5

0.02 and 0

Fig. 25.

The forming-limit diagrams were determined by following an experimental procedure involving three stages: applying a circle grid to the samples, punch stretching to maximum load, and measuring strains. As we are not focused on the experimental methods and techniques, we will not present experimental details here. Readers are referred to Signorelli et al. (2012) for a completed description of the specific techniques for measuring

Simulations were performed following the methodology described in previous sections. The measured initial texture was discretized into 1000 orientations of equal volume fraction. In this case, we assumed that plastic deformation occurred by slip on the {110}, {112} and {123} planes with a <111> slip direction for each case (BCC48). The hardening parameters were established by numerically fitting the uniaxial tensile data taken parallel to the rolling

strain-rate sensitivity and the reference slip rate at the crystal level were taken to be *m* =

Before performing simulations, we adjusted *f*0 such that the predicted limit strains matched the experimental results in plane strain. For the MK-FC and MK-VPSC simulations these values of *f*0 were 0.999 and 0.996 respectively. Together with the experimental data, the simulated FLDs for both the MK-FC and MK-VPSC schemes are shown in Fig. 26. The open symbols define a safe zone of uniform deformation for metal forming. The solid symbols correspond to measured circles that experienced local necking or fracture, specifying an

= 1 s-1, respectively. The simulated and the experimental curves are shown in

0,0 0,1 0,2 0,3 0,4

Uniaxial fitting RD DQ-type steel

> VPSC FC

ε11

**Figure 25.** Experimental and simulated uniaxial tests parallel to the rolling direction.

<sup>s</sup> *τ* = 62 MPa, *h*0 = 2275 MPa and *n* = 0.222 for VPSC

<sup>s</sup> *τ* , of all slip systems are assumed equal, the

<sup>s</sup> *τ* = 55 MPa, *h*0 = 1100 MPa and *n* = 0.209 for the FC calculations. For the

From the results presented in this chapter it can generally be concluded that the calculation of the FLD is strongly influenced by the selected constitutive description. In the present work we highlight the important role that the assumed homogenization scheme plays, which cannot be omitted in the discussion of the simulation's results. The predicative capability of a particular crystal-plasticity model is then assessed by comparing its predictions with those experimental data not used for the fitting. In our case, the discussion is framed in terms of the predicted FLD, texture evolution and polycrystal yield surface.

The emphasis in this chapter has been on cubic metals, and all calculations were carried out using either Full-Contraint or Self-Consistent models. All simulations clearly show that there are differences between the MK-VPSC and MK-FC assumptions, although the shapes and levels of the predicted FLDs are similar in the tension-compression range. Some examples were analyzed in order to highlight these differences:

Self-Consistent Homogenization Methods for Predicting Forming Limits of Sheet Metal 209

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