**3.1. Viscoplastic crystal plasticity**

The velocity-gradient tensor, using a dot to indicate the time derivative, is given by

$$\mathbf{L} = \dot{\mathbf{F}} \cdot \mathbf{F}^{-1} = \dot{\mathbf{R}}^{\*} : \mathbf{R}^{\*T} + \mathbf{R}^{\*} : \mathbf{L}^{p} : \mathbf{R}^{\*T}. \tag{4}$$

In this expression, **R**\* represents the crystallographic rotation, **F** corresponds to the effect of dislocation slip on the crystal deformation and p pp <sup>1</sup> : <sup>−</sup> **L FF** = is the plastic velocity gradient resulting from dislocation motion along specific planes and directions in the crystal (all potentially activated slip systems are labeled with the superscript *s*):

$$\mathbf{L}^{\mathcal{P}} = \sum\_{s} \left( \mathbf{n}^{s} \otimes \mathbf{b}^{s} \right) \dot{\gamma}^{s},\tag{5}$$

here *<sup>s</sup> γ* represents the dislocation slip rates, *<sup>s</sup>* **n** and *<sup>s</sup>* **b** are the normal to the system´s or systems´ glide plane and the Burgers' vector, respectively. They define the symmetric *<sup>s</sup>* **m** and the screw-symmetric *<sup>s</sup>* **q** parts of the Schmid orientation tensor:

182 Metal Forming – Process, Tools, Design

**Table 3.** BCC rolling components.

**3. Plasticity framework** 

**3.1. Viscoplastic crystal plasticity** 

In this expression, **R**\*

rate-dependent plastic response at the single-crystal level.

values of 1 ϕ

orientations.

{111}<110> component (Ray et al., 1994). The data are best displayed by sections at constant

(right in Table 3). Table 3 also gives the Miller indices and Euler angles of the typical BCC texture components. The {100} and {110} pole figures best represent the ideal BCC material

We begin this section with the kinematic definitions of crystal-plasticity theory, citing the basic equations. The kinematic development of a single-crystal plasticity model has been well documented by several authors and is the subject of recent works (Kocks et al., 1998; Roters et al., 2010). Here we assume that, during plastic forming operations, it is possible to neglect the elastic contribution to deformation. Consequently, we will restrict ourselves to a

The velocity-gradient tensor, using a dot to indicate the time derivative, is given by

*s*

potentially activated slip systems are labeled with the superscript *s*):

( )<sup>p</sup> , *s ss*

dislocation slip on the crystal deformation and p pp <sup>1</sup> : <sup>−</sup> **L FF** = is the plastic velocity gradient resulting from dislocation motion along specific planes and directions in the crystal (all

1 \* \*T \* \*T <sup>p</sup> : : :: . <sup>−</sup> **L FF R R R L R** == + (4)

**L nb** = ⊗ *<sup>γ</sup>* (5)

represents the crystallographic rotation, **F** corresponds to the effect of

ϕ

= 45º section

, but the most important texture features can all be found in the 2

$$\mathbf{m}^s = \frac{1}{2} \left( \mathbf{n}^s \otimes \mathbf{b}^s + \mathbf{n}^s \otimes \mathbf{b}^s \right) \,\prime \tag{6}$$

$$\mathbf{q}^s = \frac{1}{2} \left( \mathbf{n}^s \otimes \mathbf{b}^s - \mathbf{n}^s \otimes \mathbf{b}^s \right). \tag{7}$$

The dislocation slip rates are derived using a viscoplastic exponential law (Hutchinson, 1976):

$$\dot{\boldsymbol{\gamma}}^{s} = \dot{\boldsymbol{\gamma}}\_{0} \left| \frac{\mathbf{m}^{s} \mathbf{:} \mathbf{S}}{\boldsymbol{\tau}\_{\mathbf{c}}^{s}} \right|^{1/m} \operatorname{sign}(\mathbf{m}^{s} \mathbf{:} \mathbf{S}). \tag{8}$$

where *<sup>0</sup> <sup>γ</sup>* is the reference slip rate, *<sup>s</sup> <sup>c</sup> τ* is the critical resolved shear stress on the slip system labeled *s*, **S** is the deviatoric tensor stress and *m* is the strain-rate sensitivity exponent. The rate sensitivity *m* is typically quite small, a large value of 1/*m* tends to be almost a rate independent case, *~* 50. As 1/*m* ∝, the plastic constitutive formulation becomes formally rate-independent.

The velocity gradient can be additively decomposed into symmetric and skew-symmetric parts

$$\mathbf{L} = \mathbf{D} + \mathbf{W},\tag{9}$$

where **D** is the distortion rate tensor and **W** is the rotation rate tensor. They can be obtained by evaluating the symmetric and screw-symmetric parts of equation (4), respectively:

$$\mathbf{D} = \mathbf{R}^\* : \mathbf{D}^p : \mathbf{R}^{\*T} \tag{10}$$

$$\mathbf{W} = \boldsymbol{\Omega} + \mathbf{R}^\* : \mathbf{W}^p : \mathbf{R}^{\*\mathrm{T}}.\tag{11}$$

The rotation rate contains an extra contribution, the lattice spin tensor, defined as \* \*T **Ω** ≡ **R R**: . Rearranging Eq. (11) allows us to obtain the rate of change of the crystal orientation matrix:

$$\mathbf{Q} = \mathbf{W} - \mathbf{R}^\* : \mathbf{W}^{\mathbb{P}} : \mathbf{R}^{\*\mathbb{T}} \,\tag{12}$$

which is used to determine the re-orientation of the crystal and consequently, to follow the texture evolution. The orientation change during plastic deformation can be described by a list of the Euler angle change rates, ( ) 1 2 ϕ φϕ, , , related to the lattice spin as follows:

$$\begin{aligned} \dot{\varphi}\_1 &= \Omega\_{13} \frac{\sin \varphi\_2}{\sin \phi} - \Omega\_{23} \frac{\cos \varphi\_2}{\sin \phi} \\ \dot{\phi} &= -\Omega\_{23} \cos \varphi\_2 - \Omega\_{13} \sin \varphi\_2 \\ \dot{\varphi}\_2 &= \cos \phi \left( \Omega\_{13} \frac{\cos \varphi\_1}{\sin \phi} - \Omega\_{23} \frac{\sin \varphi\_1}{\sin \phi} \right) + \Omega\_{21} \end{aligned} \tag{13}$$

Self-Consistent Homogenization Methods for Predicting Forming Limits of Sheet Metal 185

**M MB B M M M M** = =+ + (17)

( ) ( ) <sup>1</sup>

The macroscopic compliance can be adjusted iteratively using the following self-consistent

where denotes a weighted average over all the grains in the polycrystal, and **B** is the accommodation tensor defined for each single crystal. The solution is reached using an iterative procedure that involves Eqs. (14), (15) and (17). It gives the stress in each crystal, the local compliance tensor and the corresponding polycrystal tensor, which is consistent

For simulating formability behavior, we implemented the VPSC formulation described above in conjunction with the well-known MK approach. As originally proposed, the analysis assumes the existence of a material imperfection such as a groove or a narrow band across the width of the sheet. In the approach´s modified form, developed by Hutchinson & Neale (1978), an angle ψ0 with respect to the principal axis determines the band's orientation (Fig. 4). Tensor components are taken with respect to the Cartesian X coordinate system, i

The thickness along the minimum section in the band is denoted as ( ) <sup>b</sup> *h* t , with an initial value ( ) <sup>b</sup> *h* 0 , while an imperfection factor *f*0 is given by an initial thickness ratio inside and

> ( ) ( ) b

Equilibrium and compatibility conditions must be fulfilled at the interface with the band. Following the formulation developed by Wu et al. (1997), the compatibility condition at the band interface is given in terms of the differences between the velocity gradients ( ) <sup>b</sup> **L L**,

( ) <sup>b</sup> <sup>1</sup>

( ) <sup>b</sup> <sup>1</sup>

0 , 0 *h*

*<sup>f</sup> <sup>h</sup>* <sup>=</sup> (18)

<sup>b</sup>**L Lc n** = +⊗ . (19)

<sup>2</sup> **D D c nn c** = + ⊗+⊗ , (20)

<sup>2</sup> **W W c nn c** = + ⊗−⊗ (21)

0

Eq. (19) is decomposed into the symmetric, **D** , and screw-symmetric, **W** , parts:

: , : , <sup>−</sup>

equation:

outside the band:

with the impose boundary conditions.

**3.3. Marciniak and Kuczynski technique** 

and quantities inside the band are denoted by the subscript b.

with *h* ( ) 0 being the initial sheet thickness outside the groove.

inside and outside the band respectively:

#### **3.2. The 1-site VPSC-TGT formulation**

For simulating the material response, a rate-dependent polycrystalline model is employed. In what follows, we present some features of the 1-site tangent VPSC-TGT formulation. For a more detailed description, the reader is referred to Lebensohn & Tomé (1993). This model is based on the viscoplastic behavior of a single crystal and uses a SC homogenization scheme for the transition to the polycrystal. Unlike the FC model, for which the local strain in each grain is considered to be equal to the macroscopic strain applied to the polycrystal, the SC formulation allows each grain to deform differently, according to its directional properties and the strength of the interaction between the grain and its surroundings. In this sense, each grain is in turn considered to be an ellipsoidal inclusion surrounded by a homogeneous effective medium, HEM, which has the average properties of the polycrystal. The interaction between the inclusion and the HEM is solved by means of the Eshelby formalism (Mura, 1987). The HEM properties are not known in advance; rather, they have to be calculated as the average of the individual grain behaviors, once a convergence is achieved. In what follows, we will only present the main equations of the VPSC model.

The deviatoric part of the viscoplastic constitutive behavior of the material at a local level is described by means of the non-linear rate-sensitivity equation:

$$\mathbf{D} = \dot{\boldsymbol{\gamma}}\_0 \sum\_{s=1}^{\text{sys}} \mathbf{m}^s \cdot \frac{\mathbf{m}^s : \mathbf{S}}{\boldsymbol{\tau}\_c^s} \left| \frac{\mathbf{m}^s : \mathbf{S}}{\boldsymbol{\tau}\_c^s} \right|^{\dot{\mathcal{N}}\_m^{-1}} = \mathbf{M} : \mathbf{S} \tag{14}$$

where **M** is the visco-plastic grain compliance. The interaction equation, which relates the differences between the micro and the macro strain rates ( ) **D D**, and deviatoric stresses ( ) **S S**, , can be written as follows:

$$\mathbf{D} - \overline{\mathbf{D}} = -\alpha \mathbf{\tilde{M}} \mathbf{:} \left( \mathbf{S} - \overline{\mathbf{S}} \right) \tag{15}$$

The interaction tensor **M** , which is a function of the overall modulus and the shape and orientation of the ellipsoid that represents the embedded grain, is given by:

$$\tilde{\mathbf{M}} = \left(\mathbf{I} - \mathbf{S}^{csh}\right)^{-1} : \mathbf{S}^{csh} : \overline{\mathbf{M}} \ , \tag{16}$$

where *esh* **S** is the Eshelby tensor; **I** is the 4th order identity tensor, and **M** is the macroscopic visco-plastic compliance. The parameter α tunes the strength of the interaction tensor. In the present models, the standard TGT approach is used ( 1 α= ).

The macroscopic compliance can be adjusted iteratively using the following self-consistent equation:

$$\mathbf{\tilde{M}} = \left< \mathbf{M} : \mathbf{B} \right> , \qquad \mathbf{B} = \left( \mathbf{M} + \mathbf{\tilde{M}} \right)^{-1} : \left( \mathbf{\tilde{M}} + \mathbf{\tilde{M}} \right) . \tag{17}$$

where denotes a weighted average over all the grains in the polycrystal, and **B** is the accommodation tensor defined for each single crystal. The solution is reached using an iterative procedure that involves Eqs. (14), (15) and (17). It gives the stress in each crystal, the local compliance tensor and the corresponding polycrystal tensor, which is consistent with the impose boundary conditions.

#### **3.3. Marciniak and Kuczynski technique**

184 Metal Forming – Process, Tools, Design

2 2

ϕϕ

ϕ

2 13 23 21

 = Ω −Ω +Ω 

φ

For simulating the material response, a rate-dependent polycrystalline model is employed. In what follows, we present some features of the 1-site tangent VPSC-TGT formulation. For a more detailed description, the reader is referred to Lebensohn & Tomé (1993). This model is based on the viscoplastic behavior of a single crystal and uses a SC homogenization scheme for the transition to the polycrystal. Unlike the FC model, for which the local strain in each grain is considered to be equal to the macroscopic strain applied to the polycrystal, the SC formulation allows each grain to deform differently, according to its directional properties and the strength of the interaction between the grain and its surroundings. In this sense, each grain is in turn considered to be an ellipsoidal inclusion surrounded by a homogeneous effective medium, HEM, which has the average properties of the polycrystal. The interaction between the inclusion and the HEM is solved by means of the Eshelby formalism (Mura, 1987). The HEM properties are not known in advance; rather, they have to be calculated as the average of the individual grain behaviors, once a convergence is achieved. In what follows, we will only present the main equations of the VPSC model.

The deviatoric part of the viscoplastic constitutive behavior of the material at a local level is

<sup>1</sup> <sup>1</sup> #

 τ

where **M** is the visco-plastic grain compliance. The interaction equation, which relates the differences between the micro and the macro strain rates ( ) **D D**, and deviatoric stresses

> **D D**− =− − α

> > ( ) <sup>1</sup>

α

The interaction tensor **M** , which is a function of the overall modulus and the shape and

where *esh* **S** is the Eshelby tensor; **I** is the 4th order identity tensor, and **M** is the macroscopic

: : *esh esh* <sup>−</sup>

*sys s s <sup>m</sup> <sup>s</sup>*

: : : ,

−

= = **m Sm S D m M S** (14)

**Μ** :( ) **S S** . (15)

tunes the strength of the interaction tensor. In

**M IS S M** = − , (16)

α= ).

s s <sup>1</sup> c c

τ

cos sin

 ϕ

> φ

sin sin

1 1

 ϕ (13)

 φ

23 2 13 2

sin cos sin sin cos sin

1 13 23

= −Ω − Ω

=Ω −Ω

ϕ

φ

cos

described by means of the non-linear rate-sensitivity equation:

*0 s* γ

( ) **S S**, , can be written as follows:

visco-plastic compliance. The parameter

=

orientation of the ellipsoid that represents the embedded grain, is given by:

the present models, the standard TGT approach is used ( 1

 φ

ϕ

φ

ϕ

**3.2. The 1-site VPSC-TGT formulation** 

For simulating formability behavior, we implemented the VPSC formulation described above in conjunction with the well-known MK approach. As originally proposed, the analysis assumes the existence of a material imperfection such as a groove or a narrow band across the width of the sheet. In the approach´s modified form, developed by Hutchinson & Neale (1978), an angle ψ0 with respect to the principal axis determines the band's orientation (Fig. 4). Tensor components are taken with respect to the Cartesian X coordinate system, i and quantities inside the band are denoted by the subscript b.

The thickness along the minimum section in the band is denoted as ( ) <sup>b</sup> *h* t , with an initial value ( ) <sup>b</sup> *h* 0 , while an imperfection factor *f*0 is given by an initial thickness ratio inside and outside the band:

$$f\_0 = \frac{h\_{\text{b}}\left(0\right)}{h\left(0\right)},\tag{18}$$

with *h* ( ) 0 being the initial sheet thickness outside the groove.

Equilibrium and compatibility conditions must be fulfilled at the interface with the band. Following the formulation developed by Wu et al. (1997), the compatibility condition at the band interface is given in terms of the differences between the velocity gradients ( ) <sup>b</sup> **L L**, inside and outside the band respectively:

$$
\overline{\mathbf{L}}^{\rhd} = \overline{\mathbf{L}} + \mathbf{\dot{c}} \otimes \mathbf{n} \tag{19}
$$

Eq. (19) is decomposed into the symmetric, **D** , and screw-symmetric, **W** , parts:

$$\mathbf{\overline{D}}^{\mathbf{b}} = \mathbf{\overline{D}} + \frac{1}{2} (\mathbf{\dot{c}} \otimes \mathbf{n} + \mathbf{n} \otimes \mathbf{\dot{c}}) \,\tag{20}$$

$$\mathbf{\tilde{W}}^{\rm b} = \mathbf{\tilde{W}} + \frac{1}{2} (\dot{\mathbf{c}} \otimes \mathbf{n} - \mathbf{n} \otimes \dot{\mathbf{c}}) \tag{21}$$

Here, **n** is the unit normal to the band, and **c** is a vector to be determined. The equilibrium conditions required at the band interface are given by

$$\mathbf{n} . \overline{\mathbf{o}}^{\text{b}} \ h\_{\text{b}} = \mathbf{n} . \overline{\mathbf{o}}^{\text{b}} \ h\_{\text{\textquotedblleft}} \tag{22}$$

Self-Consistent Homogenization Methods for Predicting Forming Limits of Sheet Metal 187

, , are obtained by solving a mixed boundary-condition in the

prescribed strain path, the non-linear system of two equations is solved (Signorelli et al., 2009). More recently, in Serenelli et al. (2011), Eqs. (25) and (26) were used after obtaining the state ( **L** , **σ** ) in the homogeneous zone, in order to solve the groove state avoiding the 2x2 set of non-linear equations mentioned above. In this case, the remaining unknowns

To analyze the development of deformation localization during proportional straining, the calculations were performed over different strain paths. They were defined in terms of the

of the FLD for each ρ are obtained by performing calculations every 5 degrees of Ψ0 to a

11 ε , \* 22 ε

In this section, we analyze the influences on the limit strains of the initial-imperfection intensity and orientation, the strain-rate sensitivity and the hardening to determine MK-VPSC performance. The sensitivity of the MK-VPSC model to the initial grain-shape and to texture and textural evolution is also addressed, by showing results obtained from sheets with rolling and random initial textures. In this section, the material inside and outside of the groove is taken to be a polycrystal described by 1000 equiaxed grains, except where noted otherwise. Each grain is assumed to be a single crystal with a FCC crystal structure. Plastic deformation occurs on 12 crystallographic slip systems of the type {111}<110>. We constructed the initial texture in both zones to be the same, and assumed a reference plastic shearing rate of <sup>0</sup> *γ* = 0.001 s-1. In order to account for the strain hardening between slip systems, we adopted isotropic hardening. In this case the evolution of the critical shear

= over the range -0.5 ≤ ρ ≤ 1 (step = 0.1). The possible variations

11 ε

outside the band and the critical failure

versus Ψ<sup>0</sup> . In the present work,

b b b

11 12 33 L ,L ,L and <sup>b</sup> b b

strain-rate ratios D /D 22 11 ρ

stresses is given by

maximum of 90 degrees. The failure strains \*

failure is assumed when <sup>b</sup> D 20 D <sup>33</sup> <sup>33</sup> <sup>&</sup>gt; .

angle \* Ψ are obtained after minimizing the curve \*

**4. Effects of the main model´s parameters on the FLDs** 

22 13 23 σσσ

**Figure 4.** A thin sheet in the plane *x*1-*x*2 with an imperfection band.

VPSC module, with the logic time benefits.

where **σ** denotes the Cauchy stress. Noting that ij *δ* is the Kronecker symbol, the boundary condition 33 *σ* = 0 is applied as follows

$$
\overline{\sf T}\_{\overline{\sf i}} = \overline{\sf S}\_{\overline{\sf i}} - \overline{\sf S}\_{33} \,\, \delta\_{\overline{\sf i}} \qquad \text{ (i = 1,2,3)}\,\,.\tag{23}
$$

The integration of the polycrystalline model inside and outside the band is performed in two steps. First, an increment of strain is applied to the material outside the band, **D** Δ*t* , while the imposed strain path on the edges of the sheet is assumed to be

$$\rho = \frac{\overline{\mathbf{L}}\_{22}}{\overline{\mathbf{L}}\_{11}} = \frac{\overline{\mathbf{D}}\_{22}}{\overline{\mathbf{D}}\_{11}} = \text{const.} \tag{24}$$

It is assumed that DD WW0 13 23 13 23 === = outside and inside the band. The instability appears in a narrow zone inclined at an angle ψ0 with respect to the major strain axis. The equilibrium condition, Eq. (22), can be expressed in the set of axes referenced to the groove **n t**, (see Fig. 4):

$$\begin{aligned} \overline{\sigma}\_{\text{nn}}^{\text{b}} h\_{\text{b}} &= \overline{\sigma}\_{\text{nn}} h \\ \overline{\sigma}\_{\text{nt}}^{\text{b}} h\_{\text{b}} &= \overline{\sigma}\_{\text{nt}} h. \end{aligned} \tag{25}$$

The compatibility condition requires equality of elongation in the direction **t** ,

$$
\overline{\mathbf{D}}\_{\text{tt}}^{\text{b}} = \overline{\mathbf{D}}\_{\text{tt}} \; . \tag{26}
$$

Because, we are considering thin sheets with the orthotropic symmetries in the plane of the sheet in this research, in-plane stretching results in a plane-stress state. As discussed by Kuroda & Tveergard (2000), when an orthotropic material is loaded along directions not aligned with the axes of orthotropy, it is necessary to compute the 12 L component by imposing the requirement that 12 *σ* = 0 . After solving each incremental step, the evolution of the groove orientation ψ is given by

$$\mathbf{n}\_{\perp} = \frac{\mathbf{1}}{\sqrt{t\_1^2 + t\_2^2}} \begin{pmatrix} -\mathbf{F}\_{11} \ t\_1^0 - \mathbf{F}\_{12} \ t\_2^0\\ \mathbf{F}\_{21} \ t\_1^0 + \mathbf{F}\_{22} \ t\_2^0 \end{pmatrix} \; ; \tag{27}$$

where **F** is the deformation gradient tensor.

The system of Eqs. (19) and (22, 23) can be solved to obtain **c** . This is done by substituting the macroscopic analogous Eq. (14) into the incremental form of Eq. (22) and using Eq. (20) to eliminate the strain increments in the band. At any increment of strain along the prescribed strain path, the non-linear system of two equations is solved (Signorelli et al., 2009). More recently, in Serenelli et al. (2011), Eqs. (25) and (26) were used after obtaining the state ( **L** , **σ** ) in the homogeneous zone, in order to solve the groove state avoiding the 2x2 set of non-linear equations mentioned above. In this case, the remaining unknowns b b b 11 12 33 L ,L ,L and <sup>b</sup> b b 22 13 23 σσσ , , are obtained by solving a mixed boundary-condition in the VPSC module, with the logic time benefits.

**Figure 4.** A thin sheet in the plane *x*1-*x*2 with an imperfection band.

186 Metal Forming – Process, Tools, Design

**n t**, (see Fig. 4):

conditions required at the band interface are given by

condition 33 *σ* = 0 is applied as follows

the groove orientation ψ is given by

where **F** is the deformation gradient tensor.

Here, **n** is the unit normal to the band, and **c** is a vector to be determined. The equilibrium

where **σ** denotes the Cauchy stress. Noting that ij *δ* is the Kronecker symbol, the boundary

The integration of the polycrystalline model inside and outside the band is performed in two steps. First, an increment of strain is applied to the material outside the band, **D** Δ*t* ,

> 22 22 11 11 L D

It is assumed that DD WW0 13 23 13 23 === = outside and inside the band. The instability appears in a narrow zone inclined at an angle ψ0 with respect to the major strain axis. The equilibrium condition, Eq. (22), can be expressed in the set of axes referenced to the groove

> nt b nt . *σ h σ h σ h σ h* =

Because, we are considering thin sheets with the orthotropic symmetries in the plane of the sheet in this research, in-plane stretching results in a plane-stress state. As discussed by Kuroda & Tveergard (2000), when an orthotropic material is loaded along directions not aligned with the axes of orthotropy, it is necessary to compute the 12 L component by imposing the requirement that 12 *σ* = 0 . After solving each incremental step, the evolution of

> 2 2 0 0 21 1 22 2 1 2

+ +

1 F F ; F F

*t t t t* − − <sup>=</sup>

The system of Eqs. (19) and (22, 23) can be solved to obtain **c** . This is done by substituting the macroscopic analogous Eq. (14) into the incremental form of Eq. (22) and using Eq. (20) to eliminate the strain increments in the band. At any increment of strain along the

0 0 11 1 12 2

**n** (27)

*t t*

L D

b nn b nn

b

The compatibility condition requires equality of elongation in the direction **t** ,

const.

while the imposed strain path on the edges of the sheet is assumed to be

ρ

<sup>b</sup> **n**. ., **σ** *h h* =**n σ** (22)

ij ij 33 ij *σ δ* =− = S S (i 1,2,3) . (23)

== = (24)

= (25)

<sup>b</sup> D D tt tt <sup>=</sup> . (26)

b

To analyze the development of deformation localization during proportional straining, the calculations were performed over different strain paths. They were defined in terms of the strain-rate ratios D /D 22 11 ρ = over the range -0.5 ≤ ρ ≤ 1 (step = 0.1). The possible variations of the FLD for each ρ are obtained by performing calculations every 5 degrees of Ψ0 to a maximum of 90 degrees. The failure strains \* 11 ε , \* 22 ε outside the band and the critical failure angle \* Ψ are obtained after minimizing the curve \* 11 ε versus Ψ<sup>0</sup> . In the present work, failure is assumed when <sup>b</sup> D 20 D <sup>33</sup> <sup>33</sup> <sup>&</sup>gt; .

#### **4. Effects of the main model´s parameters on the FLDs**

In this section, we analyze the influences on the limit strains of the initial-imperfection intensity and orientation, the strain-rate sensitivity and the hardening to determine MK-VPSC performance. The sensitivity of the MK-VPSC model to the initial grain-shape and to texture and textural evolution is also addressed, by showing results obtained from sheets with rolling and random initial textures. In this section, the material inside and outside of the groove is taken to be a polycrystal described by 1000 equiaxed grains, except where noted otherwise. Each grain is assumed to be a single crystal with a FCC crystal structure. Plastic deformation occurs on 12 crystallographic slip systems of the type {111}<110>. We constructed the initial texture in both zones to be the same, and assumed a reference plastic shearing rate of <sup>0</sup> *γ* = 0.001 s-1. In order to account for the strain hardening between slip systems, we adopted isotropic hardening. In this case the evolution of the critical shear stresses is given by

$$\dot{\pi}\_c = \sum\_s h^s \left| \dot{\mathcal{Y}}^s \right| \tag{28}$$

Self-Consistent Homogenization Methods for Predicting Forming Limits of Sheet Metal 189

profiles of the FLD curves remain insensitive to defect severity, except for paths approaching equal-biaxial deformation. In this case, the largest value of *f*0 gives a more pronounced fall in the limit strain. Our results compare favorable to those of Wu et al. (1997), but these authors predict a more noticeable decrease of the limit-strains as the strain path approaches ρ = 1.0. It is possible that the FC homogenization scheme used by Wu et al., rather than the SC scheme of the present work, produced these differences. Also, we assumed a greater strain-rate sensitivity than did Wu and his co-authors. As we will see in the next section, larger rate sensitivities produce higher limit-strain profiles near ρ = 1.0.

The influence of the material's rate sensitivity *m* on the FLDs is addressed. Fig. 6 shows the calculated limit-strains assuming a random initial texture and an initial imperfection of *f*0 = 0.99. Depending on the *m* value, not only are the limit strains different, but the FLD profiles vary as well. The limit-strains decrease with decreasing *m*, for *m*-values of 0.05, 0.02 and 0.01, while the FLD profiles in the negative minor-strain range (ρ < 0) continue to exhibit a nearly linear behavior. For biaxial paths (ρ > 0) near plane strain, the forming-limit strain increases rapidly for low values of *m*, while for high *m*, *m* = 0.05, the major limit-strain is nearly constant. As illustrated by the curves shown in Fig. 6, high values of *m* also displace the ending of the steep-slope profile towards an equi-biaxial path. Analyzing the results for *m* = 0.01 in the neighborhood of ρ = 1.0, we find decreasing limit-strains, indicating that lowering the strain-rate sensitivity produces a similar effect to that found using a Taylor


\* ε22

The effects produced by the different strain-rate sensitivities are consistent with the relationship between the hardening behavior and *m*, described by Eqs. (28) and (29). The

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

0,0

**Figure 6.** Influence of the rate sensitivity *m* on the FLD

*<sup>s</sup> τ* = 47 MPa).

0,2

0,4

\* ε11

(*f*0 = 0.99, *n* = 0.23, *h*0 = 1410 MPa, <sup>c</sup>

0,6

0,8

**4.2. Strain-rate sensitivity** 

model.

where *<sup>s</sup> h* are the hardening moduli behaviors, which depend on Γ (accumulated sum of the single-slip contributions to *<sup>s</sup>* γ ). These moduli can be written using the initial hardening rate, <sup>0</sup> *h* , and the hardening exponent, *n*:

$$h^s = h\_0 \left(\frac{h\_0 \Gamma}{\pi\_c^s n} + 1\right)^{n-1}; \qquad \Gamma = \sum\_s \int\limits\_0^t \dot{\jmath}^s \Big| dt \,\tag{29}$$

The strain-induced hardening law prescribed above is applied to all slip systems.

#### **4.1. Initial imperfection**

The MK approach predicts the FLD based on the growth of an initial imperfection. However, the strength of the imperfection cannot be directly measured by physical experiments. Zhou & Neale (1995) analytically predicted the effect of the initial imperfection parameter *f*0 on the FLD and demonstrated, as expected, that the forming-limit strain decreases with increasing depth of the initial imperfection. Using the MK-VPSC approach, we too determined that the limit strains are greatly affected by the value of *f*0. In addition, we found, as did Zhou & Neale, that the smaller the imperfection the larger the limit strain. The calculations plotted in Fig. 5 were performed using a random initial texture.

(*m* = 0.01, *n* = 0.23, *h*0 = 1410 MPa, <sup>c</sup> *<sup>s</sup> τ* = 47 MPa).

**Figure 5.** Influence of the initial imperfection *f*0 on the FLD

All three curves show the minimum limit strain close to the in-plane plane-strain path. Although the limit strains are different for the different values of the initial imperfection, the profiles of the simulated cases are equivalent over the range -0.5 ≤ ρ ≤ 0.3. For ρ > 0.3, the profiles of the FLD curves remain insensitive to defect severity, except for paths approaching equal-biaxial deformation. In this case, the largest value of *f*0 gives a more pronounced fall in the limit strain. Our results compare favorable to those of Wu et al. (1997), but these authors predict a more noticeable decrease of the limit-strains as the strain path approaches ρ = 1.0. It is possible that the FC homogenization scheme used by Wu et al., rather than the SC scheme of the present work, produced these differences. Also, we assumed a greater strain-rate sensitivity than did Wu and his co-authors. As we will see in the next section, larger rate sensitivities produce higher limit-strain profiles near ρ = 1.0.

#### **4.2. Strain-rate sensitivity**

188 Metal Forming – Process, Tools, Design

the single-slip contributions to *<sup>s</sup>*

**4.1. Initial imperfection** 

rate, <sup>0</sup> *h* , and the hardening exponent, *n*:

c

γ

0 0

<sup>−</sup> <sup>Γ</sup>

s

*τ n*

*s s <sup>τ</sup>* <sup>=</sup> *<sup>h</sup>*

where *<sup>s</sup> h* are the hardening moduli behaviors, which depend on Γ (accumulated sum of

*s*

1

*<sup>t</sup> <sup>h</sup> h h dt*

1 ; *n s s*

= + Γ =

The strain-induced hardening law prescribed above is applied to all slip systems.

The calculations plotted in Fig. 5 were performed using a random initial texture.

0,0

**Figure 5.** Influence of the initial imperfection *f*0 on the FLD

*<sup>s</sup> τ* = 47 MPa).

0,2

0,4

\* ε11

(*m* = 0.01, *n* = 0.23, *h*0 = 1410 MPa, <sup>c</sup>

0,6

0,8

c 0

The MK approach predicts the FLD based on the growth of an initial imperfection. However, the strength of the imperfection cannot be directly measured by physical experiments. Zhou & Neale (1995) analytically predicted the effect of the initial imperfection parameter *f*0 on the FLD and demonstrated, as expected, that the forming-limit strain decreases with increasing depth of the initial imperfection. Using the MK-VPSC approach, we too determined that the limit strains are greatly affected by the value of *f*0. In addition, we found, as did Zhou & Neale, that the smaller the imperfection the larger the limit strain.


\* ε22

All three curves show the minimum limit strain close to the in-plane plane-strain path. Although the limit strains are different for the different values of the initial imperfection, the profiles of the simulated cases are equivalent over the range -0.5 ≤ ρ ≤ 0.3. For ρ > 0.3, the

γ

). These moduli can be written using the initial hardening

γ

 *f* 0 = 0.995

 *f* 0 = 0.990

 *f* 0 = 0.975

. (29)

*s*

(28)

The influence of the material's rate sensitivity *m* on the FLDs is addressed. Fig. 6 shows the calculated limit-strains assuming a random initial texture and an initial imperfection of *f*0 = 0.99. Depending on the *m* value, not only are the limit strains different, but the FLD profiles vary as well. The limit-strains decrease with decreasing *m*, for *m*-values of 0.05, 0.02 and 0.01, while the FLD profiles in the negative minor-strain range (ρ < 0) continue to exhibit a nearly linear behavior. For biaxial paths (ρ > 0) near plane strain, the forming-limit strain increases rapidly for low values of *m*, while for high *m*, *m* = 0.05, the major limit-strain is nearly constant. As illustrated by the curves shown in Fig. 6, high values of *m* also displace the ending of the steep-slope profile towards an equi-biaxial path. Analyzing the results for *m* = 0.01 in the neighborhood of ρ = 1.0, we find decreasing limit-strains, indicating that lowering the strain-rate sensitivity produces a similar effect to that found using a Taylor model.

(*f*0 = 0.99, *n* = 0.23, *h*0 = 1410 MPa, <sup>c</sup> *<sup>s</sup> τ* = 47 MPa).

**Figure 6.** Influence of the rate sensitivity *m* on the FLD

The effects produced by the different strain-rate sensitivities are consistent with the relationship between the hardening behavior and *m*, described by Eqs. (28) and (29). The 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.

Self-Consistent Homogenization Methods for Predicting Forming Limits of Sheet Metal 191

 *n* = 0.23 *n* = 0.19 *n* = 0.16

> RD ND TD

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


\* ε22

 Aspect ratio (1:1:1) Aspect ratio (3:1:1) Aspect ratio (10:1:1)


\* ε22

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

*<sup>s</sup> τ* = 22 MPa).

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

loading path are shown on the right in Fig. 8.

0,0

*<sup>s</sup> τ* = 47 MPa).

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

0,0

0,2

0,4

\* ε11

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

**4.5. Effects of texture** 

0,6

0,8

0,2

0,4

\* ε11

(*f*0 = 0.99, *m* = 0.01, *h*0 = 1410 MPa, <sup>c</sup>

0,6

0,8


**Table 4.** Calculated average number of active slip systems as a function of the deformation path and strain-rate sensitivity.

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.
