Mechanical behavior prediction of TWIP steel in plastic deformation

Computational Materials Science xxx (2014) xxx–xxx

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Mechanical behavior prediction of TWIP steel in plastic deformation W.J. Dan ⇑, F. Liu, W.G. Zhang Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200240, PR China

a r t i c l e

i n f o

Article history: Received 2 October 2013 Received in revised form 4 February 2014 Accepted 4 March 2014 Available online xxxx Keywords: Twinning steel Twinning-induced plasticity Mechanical behavior FEM

a b s t r a c t This study aims to predict the influence of twinning volume fraction and grain size on the mechanical behavior of TWIP steel in plastic deformation with a dislocation-based model considering austenite twinning. The preceding model was implemented into FEM codes based on the conventional elastoplasticity increment theory, and verified by the experimental data of Allain et al. (2002). Then the influence of twinning volume fraction and grain size on the work-hardening of TWIP steel in tensile deformation was investigated. A damage factor was proposed to evaluate the failure trend in plastic deformation based on the Freudenthal fracture criterion, and the effect of twinning volume fraction and grain size on the proposed factor was analyzed. The results indicate that higher twinning volume fraction improves strength and ductility, delays the failure of TWIP steel. However, finer grain improves strength and ductility, but may deteriorate fracture resistance of TWIP steel. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction TWIP (Twinning-induced plasticity) steels have high strength and ductility simultaneously due to austenite twinning in plastic deformation process. These desirable properties meet the demand of energy economy, environmental protection and safety in modern automobile industry. This ideal material was named by Grassel et al. [1] in 1997 while studying the mechanical behavior of Fe– Mn–Si–Al TRIP steels. These authors found its tensile strength up to 600 MPa and total elongation higher than 80% due to twining of austenite. Subsequently many investigations have been carried out to understand the mechanical behavior of this material. Allain et al. [2] found that twinning nucleation in TWIP steels was related to local deformation and residual stress. Bracke et al. [3] considered that the nucleation and growth of the recrystallized state were without preferred orientation in a relatively energetic homogeneous microstructure. Jin and Lee [4] reported the twinning volume fraction increased with tensile strain due to the increase in the number of deformation twins rather than the lateral growth of each deformation twin. Vercammen et al. [5] concluded that the strain increased the induced volume fraction of twins at low strain, and that non-homogeneous deformation mechanisms were active at high levels of strain. Dai et al. [6] indicated that the true stress–strain curve of TWIP steels from tension tests exhibited the repeated serrations, the deformed microstructure displayed typical planar glide characteristics, such as no cell formation, dislocation pile-ups on a single slip plane, mechanical twins ⇑ Corresponding author. Tel./fax: +86 86 21 34203084. E-mail address: [email protected] (W.J. Dan).

and stacking faults. Dini et al. [7] indicated the strength induced by mechanical twinning was related to the decreased dislocation mean free path. Lebedkina et al. [8] confirmed that the plastic flow instability, associated with fluctuation of the flow stress, resulted from quasi-continuous propagation of deformation band along the specimen axis. Barbier et al. [9] reported the development of the main h1 1 1i//TD fiber was related to the increased deformation. The deformation twins developed in grains having orientations close to the two main texture components- h1 1 1i//TD and h1 0 0i//TD fibers generated orientations close to those orientations. Ahn et al. [10] investigated the anisotropic property, Bauschinger, transient, and permanent softening behavior of TWIP steel during reverse loading in forming process. Chung et al. [11] developed the forming limit diagram of TWIP steel and implemented it into the simulation program using Hill’s and M-K theories as well as Dorn’s and Swift’s diffusion theories. Considering grain size and Bauschinger effect, Bouaziz et al. [12] established a physical model to describe the isotropic and kinematic hardening behavior of high-manganese austenitic twinning-induced plasticity steel. Based on the scale transition method, Shiekhelsouk et al. [13] developed a micromechanical model to describe the behavior of austenitic steels that displayed TWIP effect. These authors proposed a physically based constitutive equation at the grain scale considering crystallographic slip and twinning to simulate material behavior at both macro and micro scales. The research results above show that Fe–Mn–C steels exhibit high strength and ductility due to austenite phase twinning during forming. However, it is necessary to explain how the twinning volume fraction and grain size influences the mechanical behavior in plastic deformation process.

http://dx.doi.org/10.1016/j.commatsci.2014.03.007 0927-0256/Ó 2014 Elsevier B.V. All rights reserved.

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Fig. 1. Numerical implementation procedure.

Based on the physical model of Allain et al. [2], a simple strain hardening rule is proposed in this study. A constitutive model is presented based on the elastoplasticity increment theory, and implemented into FEM codes. The distribution of the stress–strain curve, dislocation density, hardening rate and strain hardening exponent are calculated to analyze the influence of twinning volume fraction and grain size on the mechanical behavior of TWIP steel in tensile deformation. A damage factor based on the Freudenthal fracture criterion is proposed to investigate the failure trend in plastic deformation. 2. Theory model The crystallographic slip and twinning are modeled to describe the deformation process of Fe–Mn–C TWIP steels [2,6,7,9,13,14].

Table 1 Material parameters. Parameters

Value

r0 (MPa) a

120 0.4 3.0 72 30 30 25 1.95 1.5 0.018

M

l (GPa) d (lm) e (lm) b (lm) m k1 k2

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Fig. 2. Sketch of tensile specimen.

The effective stress is related to the evolution of the dislocations densities [15].

pffiffiffiffi

r ¼ r0 þ aMlb q

ð1Þ

 is effective stress, r  0 is yield stress, M is Taylor factor; a is where r material constant, l is shear modulus, b is Burgers vector, q is dislocation density. The dislocation is accumulated and annihilated in plastic deformation due to dynamic recovery accompanied with austenite twinning [2]. The evolution of the dislocation density with gliding dislocation effective plastic strain and austenite twinning can be described as

q_ ¼

1 d

  pffiffiffiffi þ 1t þ k1 q M m  k2  q  pffiffiffi e_ p 1F b 2

ð2Þ

(1)

where d is grain size, t is average twins spacing, k1, k2 are material parameters. F is twinning volume fraction, related to parameter m and effective plastic strain ep : The increasing of m-value and ep induces the twinning volume fraction increasing [16]. It is supposed as the following function:

F_ ¼ ð1  FÞme_ p

ð3Þ

The average twins spacing t is related to the austenite twinning volume fraction [17,18], can be shown as

t ¼ 2e

1F F

ð4Þ

where e is average twin thickness. Combining Eqs. (2)–(4) and substituting into Eq. (1), a dislocation-based stress–strain model can be built to describe the mechanical behavior of TWIP steel. The hardening rate H and strain hardening exponent n are defined as follows:

H¼

 dr q_ ¼ aMlb pffiffiffiffi p  de 2 q

ð5Þ

n¼

ep q_ Þ d ln ðr ¼ aM lb pffiffiffiffi p d ln ðe Þ r 2 q

ð6Þ

(2)

In this study, the associated flow rule with Hill’s 1948 anisotropic yield criterion [19] is written as

2 ¼ 0 U ¼ hri½Mfrg  r

ð7Þ

where r is stress tensor, [M] is the matrix of anisotropic parameters,

r is equivalent stress. The anisotropic coefficients of Hill’48 can be calculated by R0 = 0.816, R45 = 1.188, R90 = 1.339 [11]. The strain tensor increment e_ of an elastoplastic material in large deformation can be decomposed into elastic and plastic components as

(3) Fig. 3. Verification of the model, (1) stress–strain curve, (2) hardening rate, (3) strain hardening exponent.

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(2)

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(5) Fig. 4. Mechanical behavior of TWIP steel at different m-value, (1) twinning volume fraction at different m-value, (2) stress–strain curves at different m-value, (3) dislocation density at different m-value, (4) H-value at different m-value, (5) n-value at different m-value.

e_ ¼ e_ e þ e_ p

ð8Þ

e_ p ¼ k_

@U @r

ð10Þ

ð9Þ

@U e_ p ¼ k_  k_ ¼ 2r  @r

ð11Þ

Stress tensor increment can be expressed as

r_ ¼ De ðe_  e_ p Þ

According to the plastic flow rule, the incremental plastic strain and effective plastic strain are

Basing on the consistency condition, the following relationship is obtained

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W.J. Dan et al. / Computational Materials Science xxx (2014) xxx–xxx

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Fig. 5. Mechanical behavior of TWIP steel with different grain size, (1) stress–strain curves with different grain size, (2) dislocation density with different grain size, (3) hardening rate with different grain size, (4) strain hardening exponent with different grain size.

U_ ¼

@U r_  2r r_ ¼ 0 @r

ð12Þ

From Eqs. (8)–(12), the increment in plastic strain can be expressed as

e_ p ¼ @UT @r

@ UT e @ U D @r @r e @U D @r þ 4  2 H

r

e_

ð13Þ

All the simulations discussed below are performed using the ABAQUS/UMAT codes with element type is solid element with reduced integration C3D8R. Numerical implementation procedure for constitutive model coupled with ABAQUS/UMAT is shown in Fig. 1. 3. Material and model verification A TWIP steel (Fe–Mn 27 wt%–C 2 wt%) is adopted to verify the preceding model. Its yield stress is 120 MPa, and shear modulus is 72 GPa. Tensile tests are performed at a constant strain rate of 0.005/s [2]. The material parameters are listed in Table.1. The corresponding unaxial tensile deformation simulation is carried out. The specimen has a parallel length of 7.9 mm, a width of 5 mm and a thickness of 2.3 mm (in Fig. 2). The stress–strain curve determined from the preceding model is shown in Fig. 3(1). The hardening rate and strain hardening exponent are shown in Fig. 3(2) and (3), respectively. The determined

stress–strain curve agrees well with the experimental data, the hardening rate and strain hardening exponent from experimental data are well-distributed along the corresponding determined curves. The above results comparison between simulation and experiment illuminate that the proposed model can describe the mechanical behavior of TWIP steel in plastic deformation. 4. Prediction of mechanical behavior Austenite twinning can improve the mechanical behavior of TWIP steels (Fe–Mn–C), and some models and experimental results are reported recently [6–18]. However, it is not investigated specifically how the TWIP effect affects the distribution of stress–strain fields and improves the mechanical behavior in plastic deformation. Therefore, the influence of twinning volume fraction on mechanical behavior is investigated in this section. According to Olson and Cohen’s model [16], equation for twinning in plastic deformation is given in Allain et al. [2]. It is assumed that the parameter m takes the values of 1.0, 1.95 and 2.5 in Eq. (3) to control the twinning volume fraction (as shown in Fig. 3(1)). It can be seen that the volume fraction of twinning increases as m-value increases. The stress–strain curves, dislocation density and H-value and n-value are also calculated considering the variation of m-value (in Fig. 4(2)–(5)). The calculated results indicate that: (1) stress decreases as m increases when strain is less than 0.3, whereas

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stress increases with m once strain exceeds 0.3 (in Fig. 4(2)). (2) The change of the dislocation density is similar to the stress–strain curve (in Fig. 4(3)). When strain is no more than 0.3, more twinning volume fraction decelerates dislocation accumulation, leads to a stress decrease. However, when strain is more than 0.3, more twinning volume fraction accelerates dislocation accumulation, leads to a stress increase. (3) As m-value increases, H-value decreases at initial deformation stage, and increases at large deformation stage, respectively. H-value increasing with m-value at large deformation stage indicates that austenite twinning can improve the work-hardening performance of TWIP steel. When m equals 1.0, H-value decreases continuously with strain. At middle deformation stage, H-value keeps stable in the case of m = 1.95, and increases in the case of m = 2.5. (in Fig. 4(4)). (4) The relationship between n-value and strain seems to be a quadratic function. The n-value increasing with m-value will retard material failure and improve material ductility. The failure strain ef is 0.51, 0.56 and 0.56 in the case of m = 1.0, 1.95 and 2.5, respectively. (in Fig. 4(5)). Grain size is also known to drastically influence the mechanical behavior of TWIP steels. The grain refinement affects the strength properties of the steels [12,20–22], benefits the strain hardening and fatigue resistance [23,24] by improving deformation twinning [24], phase transformation [25]. In this study, the effect of grain size on the strain hardening of TWIP steel is also investigated by the proposed dislocation-based model. Three different values, 30 lm, 15 lm and 10 lm, are assumed for grain size, and the corresponding stress–strain curves, dislocation density, hardening rate (H-value) and strain hardening exponent (n-value) are shown in Fig. 5(1)–(5). The calculated results are concluded that: finer grain results in more grain boundary, improves the dislocation density [23–25], and promotes the strength of material [12,20–22]. The effect of grain refinement within the micrometer range on the twinning stress and, hence, on twinning inhibition, is obvious and deteriorates the ductility of this material, leads to the decrease of the hardening rate finally [25]. However, strain hardening exponent increasing with finer grain, by which the ef is 0.52, 0.54 and 0.56 for grain size 10 lm, 15 lm and 30 lm respectively (in Fig. 5(4)).

Table 2 Material parameter C1 (MPa). Parameters (m = 1.95)

C1

Parameters (d = 30 lm)

C1

d = 10 lm d = 15 lm d = 30 lm

316.07 357.92 363.10

m = 1.0 m = 1.95 m = 2.5

358.61 360.90 357.92

(1)

(2) Fig. 6. The failure trends prediction in tensile loading, (1) m = 1.0, m = 1.95 and m = 2.5, (2) d = 10 lm, d = 15 lm and d = 3 0 lm.

5. Failure trends evaluation in plastic deformation Ductile fracture has been extensively investigated in large plastic deformation of sheet metal forming. A diffuse necking criterion was developed to predict the plastic at the maximum load in the one principal stress direction [26] or two principal stress directions [27]. However, actual failure is quite often initiated only when a localized necking forms. Postulated that the localized necking was formed along the zero-extension direction, a criterion for the localized necking under the plane stress condition was presented [28]. Assumed that necking occurred in local regions of initial imperfections, another criterion was developed [29]. Furthermore, the forming limit diagram (FLD) was proposed to experimentally assess the amount of deformation prior to necking failure of sheet metals under various stretch modes [30]. Based on the Freudenthal Fracture criterion [31,32], a damage factor D is proposed to evaluate the failure trends in plastic deformation.

D¼

1 C1

Z

e

r de

ð14Þ

0

Re  de is material parameter, its value is listed in where C 1 ¼ 0 f r Table 2 with different m-value and grain size. In the unaxial tension condition, one end of specimen is fixed and the other end is loaded with a displacment of 8 mm. The

Table 3 Damage factor D in tensile loading. Parameters (d = 30 lm)

Dmax

Parameters (m = 1.95)

Dmax

m = 1.0 m = 1.95 m = 2.5

>1.00 0.80 0.89

d = 10 lm d = 15 lm d = 30 lm

0.95 0.97 0.97

damage factor of TWIP steel with different m-value and grain size is illustrated in Fig. 6, and the maximum damage factor (Dmax) is summarized in Table 3. It can be concluded that higher twinning volume fraction can suppress failure [24], and the failure may occur in the material with coarse grain [25]. The cup drawing simulation has been carried out with geometry shown in Fig. 7. The diameter of the blank is 200 mm, thickness is 1.0 mm, and the punch stroke is 50 mm. The anisotropic coefficients of Hill’48 in Eq. (7) are F = 0.336, G = 0.551, H = 0.449, M = 1.0, N = 1.496, L = 1.0. The initial blank-holder force is 25 kN, friction coefficient is assumed to be 0.15 for blank-punch contact, and 0.10 blank-die contact and blank-holder contact. The damage factor for sheet forming process with different m-value and grain size is shown in Fig. 8. The maximum, minimum damage factor (Dmax, Dmin) and DD = Dmax  Dmin are summarized

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W.J. Dan et al. / Computational Materials Science xxx (2014) xxx–xxx Table 5 Damage factor in cup drawing with different grain size. Parameters (m = 1.95)

Dmax

Dmin

DD = Dmax  Dmin

d = 10 lm d = 15 lm d = 30 lm

0.70 0.66 0.61

0.09 0.09 0.09

0.61 0.57 0.52

in Tables 4 and 5, where Dmax is to describe the failure trend and DD reflects the uniform deformation in sheet forming. The results indicate that higher twinning volume fraction retards damage accumulation, delays failure. However, finer grain accelerates damage accumulation and fracture. 6. Conclusions

Fig. 7. The dimension of cup drawing tools.

This paper reported the influence of twinning volume fraction and grain size on the mechanical behavior of TWIP steel during plastic deformation. The main conclusions from this study are: (1) The stress–strain relationship is developed based on a physical model and twinning empirical rule. The constitutive model is proposed based on the elastoplasticity increment theory, and implemented into FEM codes. The stress–strain curves, hardening rate and strain hardening exponent are verified by experimental data. (2) The tensile deformation is performed using ABAQUS/UMAT codes with the constitutive equation, and the strain hardening behavior is evaluated with different grain size and mvalue. The results have shown that higher twinning volume fraction and finer grain size improve strength and ductility of TWIP steel. (3) A damage factor D is proposed to evaluate the failure trend in plastic deformation based on the Freudenthal fracture criterion. In the tensile loading case, more twinning can suppress failure and failure may occur in the material with coarse grain. In the cup drawing case, higher volume fraction retards the damage accumulation, delays failure. However, finer grain accelerates damage accumulation and fracture.

(1)

Acknowledgements This work was supported by the National Natural Science Foundation of China (Nos. 51275296, 51222505, 51375307 and 51075266). References

(2) Fig. 8. The failure trends prediction in cup drawing, (1) m = 1.0, m = 1.95 and m = 2.5, (2) d = 10 lm, d = 15 lm and d = 30 lm.

Table 4 Damage factor in cup drawing with different m-value. Parameters (d = 30 lm)

Dmax

Dmin

DD = Dmax  Dmin

m = 1.0 m = 1.95 m = 2.5

0.73 0.61 0.59

0.08 0.12 0.1

0.65 0.49 0.49

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