Deep drawing simulation of a metastable austenitic stainless steel using a two-phase model

Journal of Materials Processing Technology 210 (2010) 835–843

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Journal of Materials Processing Technology journal homepage: www.elsevier.com/locate/jmatprotec

Deep drawing simulation of a metastable austenitic stainless steel using a two-phase model S. Gallée ∗ , P. Pilvin Laboratoire d’Ingénierie des MATériaux de Bretagne, Université de Bretagne-Sud, Rue de Saint-Maudé, BP 92116, F-56321 Lorient, France

a r t i c l e

i n f o

Article history: Received 8 September 2009 Received in revised form 22 December 2009 Accepted 15 January 2010

Keywords: Stainless steel Martensitic transformation Two-phase model Numerical simulation Deep drawing Eddy Current measurements

a b s t r a c t This paper deals with the numerical simulation of the deep drawing of a metastable austenitic stainless steel of AISI 304 type. A two-phase model of elastoviscoplastic type has been developed by Gallée et al. (2007) to take into account the martensitic transformation induced by plastic deformation during cold forming that can exhibit this stainless steel. This two-phase model is used to describe the behavior of the steel during the numerical simulation of the deep drawing of a cylinder cup using a commercial finite element code. The numerical results of the simulations are compared with experimental results of the same deep drawing test in order to validate the model. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Due to their high corrosion resistivity and nice looking, stainless steels are increasingly used for industrial applications. Among them, type AISI 304 austenitic stainless steel sheets are most commonly used for forming products since they are superior in formability. Austenite in these stainless steels is unstable and can undergo a martensitic transformation induced by the plastic deformation at room temperature. This martensitic transformation has been studied by Mangonon and Thomas (1970) for a type 304 austenitic stainless steel and Choi and Jin (1997) investigated the influence of the strain induced martensite on the strain hardening behavior of a 304 austenitic stainless steel. This martensitic transformation depends on the temperature, stress state and strain rate. Hecker et al. (1982) and Murr et al. (1982) investigated the influence of strain rate and strain state on deformation-induced tranformation in 304 stainless steel and Dan et al. (2007) developed a model for TRIP steel depending on the strain rate. The gradual transformation of austenite to strain induced martensite increases the work-hardening of these steels, which is desirable for the high formability because the onset of the necking is delayed, as explained by Shan et al. (2008) and by Serri et al. (2005) for TRIP steels forming. As a consequence, these steels are commonly used

∗ Corresponding author. Tel.: +33 297880532; fax: +33 297874572. E-mail address: [email protected] (S. Gallée). 0924-0136/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2010.01.008

for deep drawing processes. However, the strain induced martensitic transformation can generate several problems in deep drawing processes and particulary delayed cracking as observed by Kim et al. (1999) and Nagumo et al. (2003). Berrahmoune et al. (2006) also studied the delayed cracking phenomenon of 301LN austenitic stainless steel during the deep drawing of cylindar cups. It is then necesssary to quantify the growth of martensite in the austenitic phase during a forming process and to compare results with a mechanical approach that is able to take into account the constitutive behavior of the induced martensite. As a consequence, a two-phase micromechanical model has been identified by Gallée et al. (2007) to describe the behavior of a AISI 304 austenitic stainless steel. This model takes into account the behavior of both austenitic and martensitic phases. Different models have been proposed to describe the kinetics of the strain-induced martensitic transformation. Olson and Cohen (1975) proposed a law depending on the plastic strain and temperature to describe the martensitic volume fraction evolution. Stringfellow et al. (1992) modified the Olson and Cohen model to take into account the influence of the stress state. Powell et al. (1958) investigated the effects of temperature and strain rate on the martensitic transformation of austenitic stainless steels and also the influence of the stress state (tension, torsion and compression). They clarified the influence of the stress state on the martensitic transformation (with uniaxial tension test involving more martensite than torsion or compression tests). Okutani et al. (1995) also studied the influence of the stress state by performing tension, compression, equi-biaxial compression and deep

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Table 1 Ponderal chemical composition and supplier measured mechanical properties of the studied AISI 304 austenitic stainless steel. Nuance AISI 304

Cr 18.3%

Ni 9.2%

C 0.04%

Mn 1.5%

Si 0.5%

Mo 0.18%

Md30 ◦

40 C

Rm

Rp0.2

A%

610 MPa

270 MPa

55

drawing tests on a 304 austenitic stainless steel and found that the induced volume fraction of martensite is higher in compression than tension. Iwamoto et al. (1998) clarified the evolution of the martensitic transformation. They performed uniaxial tension and compression tests on a 304 austenitic stainless steel at different temperatures with a constant strain rate. At 20 ◦ C, they showed that the martensitic transformation is higher and faster for low plastic strain level and then, for more important plastic strain, the deformation is higher and faster in tension. Uniaxial tension, compression and torsion tests performed by Lebedev and Kosarchuk (2000) have shown that the induced volume fraction of martensite is higher in tension. In this paper, the martensitic transformation induced by the plastic deformation is described by using a modification of the Shin et al. law (Shin et al., 2001). To take into account the influence of the stress state on the phase transformation, a criteria based on the triaxiality stress ratio is introduced in the Shin et al. law. This criteria allows to have different kinetics of transformation in tension, shear, compression and biaxial stress state. The following study presents the numerical simulation of the deep drawing of a AISI 304 austenitic stainless steel. The deep drawing process has been carefully captured experimentally and so provides an ideal basis from which to characterize the twophase model. In a first part, the material properties and the two-phase model are detailled. The parameters identification of the model is also presented. The second part of this study deals with the numerical simulation of the deep drawing of a cylindar cup using the two-phase model. The experimental process is first presented with a special attention to get the martensite distribution by measuring the Eddy Currents on the drawn cup. The numerical simulation using the finite element code ABAQUS® is then described. Two simulations are performed: the first one with an axisymmetric two-dimensional model and the second one in three dimensions. Numerical results are compared with experimental data, with special attention to force–displacement curves, thickness distribution on the drawn cup and martensitic phase evolution.

Rectangular samples of dimension 20 mm × 180 mm × 0.78 mm are cut on the initial cold rolled sheets at different orientations to the RD (0◦ , 45◦ and 90◦ ). In order to eliminate the hardened area induced by the cutting, the free edges are machined. The strain rate imposed during tensile tests is of ε˙  2.4 × 10−3 s−1 . Logarithmic strain and Cauchy stress are used throughout the paper to plot stress–strain data. Stress–strain curves in Fig. 1 highlight the anisotropy of the steel and show that the material characteristics are higher in the RD. The anisotropy can be evaluated by the calculation of the plastic p p anisotropy coefficients r˛ . These coefficients r˛ = dε22 /dε33 , where 1 corresponds to the tensile direction, 2 the transverse direction and 3 the normal direction, are calculated from the assumption of volume conservation in the plastic area (see Table 2) and are plotted in Fig. 2.

2. Material characterization

2.2. Shear tests

The studied material is an AISI 304 austenitic stainless steel (X4CrNi18-9) supplied by Arcelor Company. The material is cold rolled sheets of 500 mm × 500 mm × 0.78 mm in a shinning annealed final state. The chemical composition of this steel as well as the mechanical properties measured by the supplier are given in Table 1. A complete experimental characterization has been carried out in order to be able to identify the material behavior. This experimental database has already been detailed in Gallée et al. (2007) and only major aspects are presented here. Uniaxial tensile tests, simple and cyclic shear tests are performed out on a tensile test machine of 100 kN maximum load capacity. Strains measurement are carried out by using a high resolution video camera (Thuillier and Manach, 2009). The use of this video camera (resolution of 10−4 ) allows the investigation of a wide strain range (compared to the use of an extensometer) and the measurement of transverse strains. Both tensile and shear tests are performed at 0◦ , 45◦ and 90◦ to the rolling direction (RD) to investigate the anisotropy of the sheet. Three tests are performed each time to check the reproductibility of the experiments.

In addition to tensile tests, shear tests are perfomed on a specific device developed by Manach and Couty (2001) at the laboratory. Cyclic shear tests can be carried out using the symmetry of the device by reversing the direction of the test. Furthermore, in the case of thin sheets, this test can be considered as homogeneous in the central part of the sample, as explained by Rauch and G’Sell (1989). The dimensions of the shear sample are 17 mm × 50 mm × 0.78 mm and the width of the gauge area is 4.5 mm. Monotonic shear tests are performed on samples at 0◦ , 45◦ and 90◦ to the RD and the strain rate is fixed at ˙ = 2 × 10−3 s−1 where  is the shear strain. Cyclic shear tests are also performed in order to highlight the Bauschinger effect (see Fig. 3).

Fig. 1. Uniaxial tensile tests performed at 0◦ , 45◦ and 90◦ to the RD.

2.1. Tensile tests

Table 2 Plastic anisotropy coefficients of the AISI 304 austenitic stainless steel. Material

r0

r45

r90

AISI 304

0.90

1.28

0.86

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p

837

p

Fig. 2. Evolution of transverse strain (ε22 ) vs normal strain (ε33 ) for calculation of plastic anisotropy coefficients.

Fig. 4. General framework of two-phase micromechanical model.

In the following paragraphs, the three steps of the homogenization technique are described, as well as the kinetics of the strain-induced martensitic transformation. 3.1. Representative step The formulation at the phase level is of the elastoviscoplastic type and the plasticity criterion takes into account the orthotropic symmetry of sheet in its natural reference frame. It is assumed that the elastic part of the deformation is homogeneous and isotropic. The constitutive law of each phase is a modification of the elastoplastic model of Hill (1950) in which the viscous behavior and kinematic work-hardening are taken into account.

Fig. 3. Monotonic shear tests at 0◦ , 45◦ and 90◦ to the RD and Bauschinger tests in the RD.

3.1.1. Austenitic phase behavior At the initial state, the steel is 100% austenitic. Due to the orthotropic symmetry of the sheet, the austenite behavior is modeled using on orthotropic criterion, whereas the martensite behavior is described using an isotropic behavior. The yield function for austenite is expressed in the form:



3. Two-phase model a

The two-phase model developed has already been detailled (Gallée et al., 2003) and is only summarized here. The aim of this model is to characterize the behavior of the steel at a macroscopic scale by taking into account the individual behavior and the volume fraction of each phase (austenite and martensite). An homogenization technique is used to describe the steel behavior. In this work, a self-consistent approach is used and it assumes that each phase is embedded in a homogeneous equivalent medium (HEM). This technique is based on three steps (Fig. 4). The first step is called representative step and its aim is to identify the phases of the representative elementary volume (REV) and to describe their behavior. The REV is constituted here of two phases, i.e. austenite and martensite, and it is assumed that both phases have the same isotropic elastic behavior. The second step is the scale transition. The aim of this step is to write relationship between macroscopic stress () and strain (E) tensors and microscopic stress ( r ) and strain (εr ) tensors in each phase r. The homogenization step is the last one. It consists in calculating macroscopic quantities by performing stress and strain averaging. AISI 304 austenitic stainless steel undergoes a martensitic phase transformation induced by plastic deformation. The description of this transformation must be incorporated in the model in addition to austenitic and martensitic phases behavior.

a

a

f ( , X , R ) =

3 a ( − X a ):H:( ad − X a ) − R0a − Ra 2 d

(1)

where  ad is the deviatoric part of the stresses tensor  a , X a is the second-order tensor of kinematic work-hardening and Ra is the term of isotropic work-hardening. The initial yield stress R0a is equal to the elastic limit in tension along the RD and H is the fourthorder Hill’s tensor. The 6 coefficients of H can be deduced from the 6 parameters F, G, H, L, M, N of the quadratic Hill’s criterion. The condition on the initial elastic limit along the RD allows to impose the relation G + H = 2. The strain viscoplastic component follows a flow rule derived from a viscoplastic potential ˝ that is a power function of the yield function (Lemaitre and Chaboche, 1985): K ˝(f ) = n+1



f+ K

n+1 (2)

where n is the sensitivity coefficient of strain rate, K a weighting coefficient of the viscous part of the stress and f + the positive part of f. The behavior is elastic if f ≤ 0 and if f > 0 the viscoplastic strain rate is written as: vp

εa =

∂˝ ∂f = ˝ (f ) a ∂ a ∂

(3)

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The evolution of the isotropic work-hardening is associated to the cumulated plastic strain pa in the austenite and its evolution follows a Voce law: Ra = Qa [1 − exp(−ba pa )]

(4)

where Qa represents the amplitude of isotropic work-hardening and ba the coefficient controling its saturation rate. The kinematic work-hardening evolution law is similar to the one proposed by Armstrong and Frederick (1966) with the addition of a linear component of Prager type: Xa =

3 a 3 vp C ˛a + Ha εa 2 2

vp

with ˛˙ = ε˙ a − a ˛a p˙

and a control the intensity of work-hardening and Ha is where the slope of the linear component of the kinematic work-hardening. 3.1.2. Martensitic phase behavior The martensite behavior is described using an isotropic criterion with only an isotropic component for the work-hardening. The yield funcion for martensite is expressed in the form: (6)

is the deviatoric part of the stresses tensor  m and Rm is where  m d the term of isotropic work-hardening. The evolution of the isotropic work-hardening is similar to the austenite evolution law and is associated to the cumulated plastic strain pm in the martensite: Rm = R0m + Qm [1 − exp(−bm pm )]

(7)

where Qm represents the amplitude of isotropic work-hardening and bm the coefficient controling its saturation rate. 3.2. Description of the austenite–martensite transformation In this study, the evolution of the volume fraction of martensite is given by a Shin et al. law (Shin et al., 2001), in which the martensitic transformation is considered as a continual relaxation of the internal strain energy accumulated during the plastic deformation. This volume fraction depends on the equivalent strain in eq the austenite εa and not on the macroscopic strain. Furthermore, it is clearly established that the kinetic of the martensitic transformation depends on the temperature and strain path (Lebedev and Kosarchuk, 2000). In order to take into account the transformation dependance on the strain path, the threshold of the austenite–martensite transformation depends on the stress triaxiality ratio in austenite ( a ). This ratio is equal to 0 in shear, 1/3 in tension and −1/3 in compression. The evolution law of the martensite volume fraction fm is vp expresssed as a function of a criterion g(εa ,  a ) as follows: f˙m = nM ˇM [max(0, g)]nM −1 (fs − fm )p˙ a

(8)

with vp

(11)

ˇ=

2 (4 − 5) 15 (1 − )

(12)

B = fa ˇa + fm ˇm

(13)

and  vp ˇ˙ r = ε˙ r − D(ˇr − ıˇr )p˙ r

and

  vp ˇ˙ r = ε˙ r − D (ˇr )p˙ r

(14)

where  is the shear modulus, fr the volume fraction of each phase and pr is the cumulated plastic strain of each phase r given by the following equation: dpr = dt

  3 2

vp

dεr dt

  :

vp

dεr dt

 (15)

The parameters D, ı and D allow to fulfill the self-consistent condition and also describe the evolution law of the accomodation  variables ˇr and ˇr . The adjustement of these three parameters is performed by the comparison of the response of this model with the response of the Berveiller–Zaoui model in the case of a monotonous tensile test. The aim of the comparison is to identify the set of parameters giving the same local behavior for the two phases with both models (using an inverse method). The values of these three parameters are: D = 660, ı = 0.04 and D = 1.7. 3.4. Homogenization The final stage of the methodology consists in obtaining the average behavior at the scale of the representative elementary volume. With the condition fa + fm = 1, the macromechanical behavior is approximated by the volume average of the micromechanical behavior: E = fa εa + fm εm

and

 = fa  a + fa  a

(16)

With the hypothesis of isotropic elasticity and homogenous elastic proporties for both phases, the macroscopic plastic strain rate can be obtained with the following expression: vp vp vp vp vp E˙ = (1 − fm )ε˙ a + fm ε˙ m + f˙m (εm − εa )

(17)

4. Identification of material parameters eq

g(εa ,  a ) = εa − [ε0 + ε1 ( a )]

(9)

and ( a ) =

 r =  + 2(1 − ˇ)(B − ˇr ) with

(5)

Ca

f ( m , Rm ) = J2 ( m ) − Rm

variables on the macroscopic scale (, E). The method proposed by Pilvin (1990) is used here. This method is a modification of the Berveiller–Zaoui rule (Berveiller and Zaoui, 1979) and has the advantage to be able to take into account cyclic loading by intro ducing accomodation variables ˇr and ˇr , which take in charge the inelastic accomodation between the HEM and each phase. The scale transition for each phase r is described by the following rules:

1 tr( a ) 3 J2 ( a )

(10)

where fs is the saturation value of the martensite volume fraction, ˇM the evolution rate of the transformation and ε0 and ε1 describe the dependance of the transformation threshold as a function of the stress triaxiality ratio. 3.3. Scale transition The aim of the scale transition is to connect the mechanical variables of the two phases, i.e. ( a , εa ) and ( m , εm ) with the

The complexity of the model and the important number of parameters introduced in the equations need to compare this model with an important experimental database for the identification of the parameters. It appears necessary to use an inverse approach. The software SiDoLo (Pilvin, 1988) is used for the identification of the parameters. The identification of the optimum set of parameters (represented by a vector A) is based on the minimization of a cost function L(A). This function measures the difference between experimental and simulated data and is defined by: L(A) =

N  n=1

Ln (A)

(18)

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Table 3 Values of material coefficients of the two-phase model for the austenitic phase (R0a , Qa , Ca and Ha are expressed in MPa). Coefficient

F

G

H

N

R0a

Qa

ba

Ca

a

Ha

Value

1.16

1.07

0.93

5.02

193

1002

1.69

11360

123

344

Table 4 Values of material coefficients of the two-phase model for the martensitic phase (R0m and Qm are expressed in MPa). Coefficient

R0m

Qm

bm

Value

1130

209

1.0

Table 5 Values of the coefficients of the volume fraction of martensite evolution law for the two-phase model. Coefficient

fs

ˇM

ε0

ε1

nM

Value

0.51

2.80

0.051

−0.15

2.2

with N the number of tests in th experimental database and: 1  Mn Mn

Ln (A) =

i=1



Zexp (ti ) − Zsim (ti , A) Zn

2 (19)

Mn is the number of points of test n, ti is the observation time, Z is the observable variable and Zn is the weighting coefficient. The experimental database is composed of monotonous tensile and shear tests at 0◦ , 45◦ and 90◦ to the RD and of a cyclic shear test in the RD. The weighting coefficient affected to an observable variable is chosen according to the uncertainty on the experimental measures. The value of the weighting coefficient is 12 = 3 MPa for shear tests and for the tensile tests, 11 = 5 MPa and ε11 = 0.0005. Furthermore, several coefficients of the models are fixed during the identification: L = M = 3, E = 190 GPa,  = 0.29, K = 20 MPa s1/n , n = 15. The value of the identified coefficients are given in Tables 3–5. Figs. 5 and 6 present a comparison between experimental data and results obtain with the set of parameters identified. The results highlight a good description of the experimental data in tension, as well as in shear, especially for the cyclic shear tests thanks to the use of a kinematic hardening.

Fig. 6. Comparison between experimental and simulation for uniaxial tensile tests (UT), monotonous shear test (MS) and cyclic shear tests (Bauschinger).

5. Deep drawing process An experimental device has been developed by Thuillier et al. (2002) to perform the deep drawing of cylindar cups (see Fig. 7). The tool geometry of the experimental device is given in Table 6. The function of this device is to draw a blank sheet of initial diameter of 170 mm in a cylindrical cup with a 100 mm diameter (deep drawing ratio DR = 1.7). Experiments are performed at room temperature on a tensile test machine of 500 kN maximum load capacity. The blank-holder force is obtained by calibrating the tightening of 8 screws with a torque wrench. A small amount of lubricant (Fuchs 4107S) is applied to both sides of the blank and cups are drawn down to a punch displacement of 60 mm. Particular attention was paid to the alignement between the punch, the die and the blankholder. The operating speed was 30 mm/min. Fig. 14 presents the experimental force–displacement curve of the punch during the deep drawing test. Maximum punch load obtained during the test is about 109 MPa for a punch displacement of 32 mm. The thickness distributions in the cup wall have been measured at 0◦ , 45◦ and 90◦ to the RD using a 3D measuring machine and the results are similar to the ones obtained by Dan et al. (2007) for a TRIP steel. The results, presented in Fig. 8, correspond to the average values of the measurements performed on 5 cups. The thickness evolution is similar in the 3 orientations. At the beginning of the drawing, the process involves a stretching of the sheet that induces a thinning in the cup’s bottom. After this step, a circumferential compression stress state appears under the blankholder and a thickening occurs in the cup’s top. This phenomenon is also observed by Laurent et al. (2009) during the deep drawing of aluminium sheet. A thinning is observed on the bottom of the cup wall (near the punch noze). This thinning can be explained by the fact that no martensitic transformation occurs in this region. Due to the initial anisotropy of the sheet, the thickness distribution is different following the orientation to the RD. The thickness Table 6 Dimensions of the deep drawing device (in mm).

Fig. 5. Comparison between experimental and simulation for the description of anisotropy coefficients.

dp

rp

dm

rm

dblank

100

5.5

104.5

8

170

840

S. Gallée, P. Pilvin / Journal of Materials Processing Technology 210 (2010) 835–843

Fig. 9. Volume fraction of martensite and Eddy Current real signal vs distance from the cup center.

Fig. 7. Deep drawing experimental device.

distributions are similar at 0◦ and 90◦ to the RD but lower at 45◦ . These results are in good agreement with the values of the plastic anisotropy coefficients (Table 2). Due to the anisotropy of the sheet, an earing profil is observed on the top of the cup wall. Similar earing profile has been observed by Moreira et al. (2000) on IF steel. The martensitic transformation as well as the position of each phase in the material (austenite and martensite) can be estimated by the change of magnetic properties in the drawn cup. The initial

Fig. 10. Numerical volume fraction of martensite vs punch displacement at the top (A) and bottom (B) of the cup wall.

blank is an austenitic stainless steel, known to be non-magnetic. On the contrary, the induced martensite is magnetic. A method to investigate the martensite distribution in the drawn cup consists in measuring the Eddy Current (EC) in the material. This method has been used by Lois and Ruch (2006) to estimate the amount of induced martensite in different specimens of austenitic stainless steels (AISI 304, 316 and 347). This method relies on the change of impedance of a sensing coil whose flux is partially filled by a specimen. An Eddy Current evaluation is carried out on the drawn cup with an experimental device consisting of a probe Förster and an EC instrument Defectoscop AF. Before measurement, calibration of the probe has been made at the bottom of the cups where no martensitic transformation occurs. All measurements have been performed at room temperature. An exciting frequency of 20 Hz is chosen for EC evaluation in order to obtain EC of suitable skin depth compare to the specimens thickness. The results of the measurement (Figs. 9 and 10) show that the volume fraction of martensite increases from the bottom to the top of the cup wall. As the martensitic transformation is induced by the deformation, the results are in good correlation with what can be expected. 6. Numerical simulation

Fig. 8. Experimental cup thickness evolution in the wall at 0◦ , 45◦ and 90◦ to the RD.

The two-phase model is used to simulate the above-presented deep drawing process. The numerical simulations are performed

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841

using the finite element code ABAQUS® and the user subroutine UMAT. The aim of the simulation is to compare numerical results with experimental data to evaluate the performances of the two-phase model. The comparisons can be performed in term of thickness distribution on the final shape of the cylindrical cup, induced martensitic phase distribution, earing profil due to the anisotropy of the steel or strain distribution. The simulations are performed using a quasi-static implicit method. The Coulomb’s friction coefficients are assumed to 0.11 for the contact between the tools (die, punch and blank-holder) and the blank. Only the blank is meshed, the tools are considered as rigid bodies. Two simulations are performed. Firstly, an axisymmetric twodimensional simulation is performed. Then, a three-dimensional model is used to simulate the deep drawing process. 6.1. 2D simulation In a first step, calculation is performed using an axisymmetric numerical simulation and anisotropy is not taken into account. This first simulation allows to mainly study the volume fraction of martensite using a fine mesh. The mesh is composed of 4 elements in the thickness and 200 elements in the radius, elements being CAX4R. In order to simulate a realistic blank-holder force, this force is maintained constant at 93 kN until a punch displacement of 44 mm is reached and then removed. After a punch displacement of 44 mm, the blank is no more under the blank-holder. The final displacement of the punch is 60 mm. Fig. 9 presents the distribution of the martensite volume fraction as a function of the distance from the cup center on the deformed blank. The left axis (FvM) refers to the numerical volume fraction of martensite distribution on inner and outer skins of the cup while the right axis (Vx ) is related to the real signal given by the Eddy Current device. Numerical results show a gradient of the volume fraction of martensite between inner and outer skins. A peak is noticed near the punch nose on the outer skin (tension stress state) while on the inner skin (compression stress state), no martensite is produced. Similarly, on the top of the cup, the volume fraction of marteniste is higher on the inner skin (traction stress state) than on the outer skin (compression stress state). This difference of martensite distribution between inner and outer skin has also been observed by Hallberg et al. (2007) during the deep drawing simulation of a cylindrical cup. On cup wall, the gradient is weak between inner and outer skins that are on the same stress state. These results highlight the influence of the stress state on the kinetic of the martensitic transformation. Takuda et al. (2003) obtain similar results on a AISI 304 type austenitic stainless steel with a 2.1 deep drawing ratio. The influence of the stress state on the volume fraction of martensite distribution has also been highlighted by Serri et al. (2005) during the deep drawing simulation of a cylindar cup (304 type austenitic stainless steel, 2.0 deep drawing ratio) using the Iwamoto and Tsuta numerical model (Iwamoto and Tsuta, 2002). Finally, the numerical results are in good agreement with the experimental measurements performed on the cup after the drawing using the Eddy Current device. Fig. 10 presents the evolution of the volume fraction of martensite during the process at the top (A) and the bottom (B) of the cup wall, both on the inner and outer skin. The martensitic transformation occurs firstly near the punch nose, i.e. area B, and the transformation is faster and higher on the inner skin. In this area, the inner skin is in a tension stress state at the beginning of the deep drawing. At the end of the process, the volume fraction of martensite is higher on the outer skin that is on a tension stress state compared to the inner skin that is on a compression stress state (for area B). Concerning area A, the martensitic transformation begins later but the volume fraction is higher, the strain being more important in this area. The gap between inner and outer skins on

Fig. 11. Blank mesh used for 3D numerical simulations.

the area A at the end of the process is smaller than the one observed on the area B. 6.2. 3D simulation In addition to the axisymmetric simulation, a three-dimensional simulation is performed. One-fourth of the circular blank is meshed (see Fig. 11) in order to investigate the influence of the anisotropy of the steel. The blank is meshed using 8-node finite elements with linear interpolation, with 3 elements in the thickness for a total of 5184 elements. The tools are considerd are rigid bodies (see Fig. 12). The punch load–displacement curve (see Fig. 13) shows that the two-phase model (2P) gives a good description of the experimental results. Until a punch displacement of 30 mm, the experimental and numerical curves are identical and after this value of displacement,

Fig. 12. 3D modelisation of the tools and the blank.

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S. Gallée, P. Pilvin / Journal of Materials Processing Technology 210 (2010) 835–843

Fig. 13. Comparison between experimental (exp) and numerical (2P) results for the evolution of the punch load during deep drawing process.

the two-phase model underestimates the experimental values. At the end of the drawing, a stagnation of the punch load is observed experimentaly (related to the friction of the top of the cup on the die). This stagnation is well represented by the two-phase model, which validates the value of the friction coefficient used in the simulation. Fig. 14 shows a comparison between ewperimental and numerical data concerning the thickness distribution (at 45◦ and 90◦ to the RD). The two-phase model gives a good description of the thickness on the top of the wall but underestimates the thickness on the bottom of the wall. The anisotropy of the sheet induces an earing profil on the final shape of the drawn cup (see Fig. 15). Due to the higher value of the anisotropy coefficient at 45◦ to the RD, an ear appears in this direction. The influence of the anisotropy on the earing profil has also been investigated by Moreira et al. (2000). Due to the sheet anisotropy, the cumulated plastic strain distribution differs in function of the orientation to the RD. Fig. 16 shows that the cumulated plastic strain is higher at 45◦ to the RD, orientation with the higher anisotropy coefficient. The distribution of the cumulated plastic strain are similar in the RD and at 90◦ to the RD, orientation with similar anisotropy coefficients. The results also highlight a difference of distribution between inner and outer skin. These results are in concordance with the distribution of the volume fraction of martensite (Fig. 17). The martensitic trans-

Fig. 14. Comparison between experimental (exp) and numerical (2P) results for the thickness distributions on the final shape of the cup.

Fig. 15. Comparison between experimental (exp) and numerical (num) earing profile.

Fig. 16. Evolution of the cumulated plastic strain vs distance from the cup center.

formation being induced by the plastic deformation, the volume fraction is higher at 45◦ to the RD, where the cumulated plastic strain is higher. The distribution is similar to the one obtain with the two-dimensional simulation (see Fig. 10). Finally, Fig. 18 gives the distribution of the martensite volume fraction on the drawn cup.

Fig. 17. Evolution of the martensite fraction volume vs distance from the cup center.

S. Gallée, P. Pilvin / Journal of Materials Processing Technology 210 (2010) 835–843

Fig. 18. Distribution of the martensite volume fraction on the drawn cup (SDV1 is the volume fraction of martensite).

7. Conclusions The numerical simulation of the deep drawing of an austenitic stainless steel is proposed in this paper. The simulation of this process is used to validate the performance of the two-phase model developed by Gallée et al. (2007). The experimental deep drawing device, the numerical simulation of the process as well as the twophase micromechanical model used are described. The two-phase model takes into account the behavior of the austenitic phase and also the behavior of the martensitic phase induced by the deformation in such stainless steel. The use of a modified Shin et al. law in this model allows to take into account the evolution of the martensite volume fraction. The numerical simulation of the deep drawing is performed using the software ABAQUS® and the two-phase model. The model gives a good description of the punch load–displacement curve but underestimates the thickness distribution. The main advantages of this model are the prediction of the martensite volume fraction distribution on the drawn cup and also the prediction of the earing induced by the anisotropy of the steel. Acknowledgements The authors thank Pr. P.-Y. Manach and Pr. S. Thuillier for their help on this work and G. Lovato from Arcelor Group for supplying the material. References Armstrong, P.J., Frederick, C.O., p. 731 1966. CEGB Report RB/B/N. Berrahmoune, M.R., Berveiller, S., Inal, K., Patoor, E., 2006. Delayed cracking in 301LN austenitic steel after deep drawing: Martensitic transformation and residual stress analysis. Mater. Sci. Eng. A 438–440, 262–266. Berveiller, M., Zaoui, A., 1979. An extension of the self-consistent scheme to plasticity flowing polycristals. J. Mech. Phys. Solids 26, 325–344. Choi, J.Y., Jin, W., 1997. Strain induced martensite formation and its effect on strain hardening behavior in the cold drawn 304 austenitic stainless steels. Scr. Mater. 36, 99–104. Dan, W.J., Zhang, W.G., Li, S.H., Lin, Z.Q., 2007. A model for strain-induced martensitic transformation of TRIP steel with strain rate. Comput. Mater. Sci. 40, 101–107. Dan, W.J., Zhang, W.G., Li, S.H., Lin, Z.Q., 2007. Finite element simulation on straininduced martensitic transformation effects in TRIP steel sheet forming. Comput. Mater. Sci. 39, 593–599.

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