- Email: [email protected]
A model for strain-induced martensitic transformation of TRIP steel with strain rate
Computational Materials Science 40 (2007) 101–107 www.elsevier.com/locate/commatsci
A model for strain-induced martensitic transformation of TRIP steel with strain rate W.J. Dan *, W.G. Zhang, S.H. Li, Z.Q. Lin School of Mechanical Engineering, Shanghai JiaoTong University, Shanghai 200240, PR China Received 20 September 2006; accepted 9 November 2006
Abstract Strain rate is one of the key factors which induce martensitic transformation. In this study, a constitutive model, which can describe the transformation-induced plasticity (TRIP) accompanying the strain-induced martensitic transformation in TRIP steel, is developed. The increase of nucleation site in the austenite due to the plastic deformation is formulated as the increase of the shear band intersection. The nucleation site probability is derived not only by stress state, plasticity strain and constant environment temperature, but also by strain rate, where shear band intersection decrease through strain rate adiabatic thermal to simulate the transformation-induced plasticity characteristic for TRIP steel. Anisotropic yield function is used to describe the sheet anisotropic property. A mixture hardening law with four phases is developed instead of the mixture hardening law with two phases used commonly. The constitutive model is implemented into ABAQUS/UMAT for the analysis of the material deforming processes. The martensitic volume fraction is tested by X-ray to describe the comparison between experimental data and simulation. Stress–strain curve is measured under strain rate from 0.001/s to 0.1/s to identify the simulation results based on the new constitutive model. All the results are agreeable. Ó 2006 Elsevier B.V. All rights reserved. PACS: 81.30.Kf Keywords: Martensite transformation; TRIP steel; Strain rate; Finite element analysis
1. Introduction High strength TRIP steels have received increasing attention to improve manufacturability, safety and to reduce weight, for these steels have a high ductility, which results from the transformation of unmetastable austenite to martensite under plastic strain. Then a localized increasing of the strain hardening coefficient during deformation with TRIP steels can delay the onset of necking and ultimately leads to a higher uniform and total elongation. The two main effects usually invoked in TRIP steels are (1) the Greenwood and Johnson effect [1] corresponding to the plastic strain generated in the parent austenitic phase by the martensitic transformation, and (2) the Magee effect
*
Corresponding author. E-mail address: [email protected] (W.J. Dan).
0927-0256/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2006.11.006
[2], associated with the selection of martensite variants arising from the applied and internal stress states. Several micromechanical models have been proposed to account for the TRIP effect. Most of the models are based on the Greenwood and Johnson model. Such as Leblond [3,4], Taleb [5,6] and Fischer [7–9] are the representatives. Taleb was devoted to an investigation of some discrepancies from a reevaluation of the micromechanical model as originally used by Leblond et al. [4]. A more complete formulation taking into account the elasticity in both phases was developed and solved resulting in an improved model enabling a better description of the experimental results and removing the singularity. Leblond’s model [3,4] took into account the interaction between classical plasticity and TRIP, but its experimental validation has not yet been performed [6]. Fischer et al. [8,9] investigated shape memory alloys orientation effect, and quantifies the orientation effect based on Greenwood–Johnson model and Magee model.
102
W.J. Dan et al. / Computational Materials Science 40 (2007) 101–107
The kinetics phase transformation is the key model to describe martensite transformation, which was modeled by Olson and Cohen [10], Stringfellow et al. [11], Sumigoto et al. [12–14], Angel et al. [15], Ludwisgson and Berger [16]. Angel et al. and Ludwisgson and Berger studied the stainless steel transformation induced plasticity behaviors and constituted kinetics phase transformation of martensitic; Olson and Cohen (OC model) modeled the kinetics phase transformation of martensitic and Stringfellow et al. developed the model based on shear band intersection. Sumigoto et al. obtained the retained austenitic formability and stability of stainless steel. Based on the [10,11] Tomita and Iwanmoto (TI) [17–20] have developed a phenomenological constitutive model based on Stringfellow‘s model (S Model) and Serri et al. [21] used the TI model with ABAQUS/ VUMAT code to simulate the TRIP sheet steel forming. Strain rate can be divided into standard strain rate (below 1021/s), middle strain rate (1021021/s) and high strain rate (above 1021/s). Surely, TRIP steel is a material sensitivity to strain rate. Some authors have studied the effects of strain rate on TRIP steel [25–29]. Kenji Saito et al. studied the strain rate and temperature influence on the law-carbon TRIP steel. The enhancement of strain rate and temperature restrain martensite transformation. Choi et al. analyzed the effects of strain rate and retained austenite volume fraction on tensile properties. Increasing the retained austenite volume fraction increases UTS, total elongation, uniform strain and total absorbed energy, but decreases yield strength and absorbed energy below 10% engineering strain. Based on the Green function and selfconsistent approximation for homogenization, Takeshi Iwamoto et al. derived strain rate field in the microregion and a macroscopic constitutive equation for TRIP steels. Then, the tensile test of the plane strain block of SUS304 is simulated by FEM along with the proposed constitutive equation. Vuoristo et al. compared the high strain rate behavior of dual phase steel with that of TRIP steel. The test results showed that the energy absorption capability of the dual phase steels is higher than that of the studied TRIP steel at strains less than 0.1. The HSLA steel and carbon–manganese steels showed a higher strain rate sensitivity but a lower energy absorption capability and a lower uniform elongation at high strain rates than the multiphase steels. Ludovic samek et al. focused on the effect of the strain rate and temperature on the mechanical behaviors of the low alloy high strength TRIP steel, and argued that the adiabatic conditions present have a beneficial effect on the behaviors of TRIP steel during high strain rate deformations. All the contents above were focused on the experiments to study martensite transformation and the mechanical properties during the tensile of TRIP steel, until TI model has constituted the constitutive equation based on the shear-band theory and according to the strain rate, constant environment temperature during deformation. However, the strain rate can lead to temperature enhancement during forming processes for the adiabatic effective.
In this study, TRIP steel sheet, composed of ferrite, bainite, retained austenite and martensite four phases, is analyzed to learn about martensite transformation during deformation with ABAQUS/UMAT Code according to strain rate adiabatic effective. A mixture hardening law with four phases has been used. Anisotropic property of sheet has been shown to replace the isotropic yield function in the TI model. 2. Transformation kinetic The TI model only shows that martensitic transformation driving force is influenced by stress state, strain rate and permanent temperature. However, temperature is changing during sheet forming process especially involving the strain rate. In this study, material deformation is not an isothermal process. The increasing temperature is considered as a driving force to influence on shear band intersection. The rate of increase in the extended volume fraction of the martensitic variant, f_ m , is proportional to the rate of increase in the number of the martensitic variant embryos per unit austenite volume, N_ m ð1Þ f_ m ¼ ð1 f Þvm N_ m where vm is the average volume per martensitic variant unit, N_ m can be expressed in terms of the number of nucleation sites per unit austenite volume, N_ I , and the probability, P, that a shear band intersection will act as a nucleation site. N_ m is given as N_ m ¼ P N_ I þ N I P_ H ðP_ Þ
ð2Þ
where H is the Heaviside step function, reflecting the fact that the martensitic transformation is irreversible. Based on the observation that the strain-induced nucleation occurs predominantly at shear-band intersections [11,17], N_ I can be defined as fI ð3Þ NI ¼ vI where fI and vI are the volume fraction of shear-band intersection and the average volume of shear-band intersection, respectively. The relation between the parameter, fI, and the volume fraction of shear-band, fsb, is assumed to be a power-law expression of the form [11] r
fI ¼ Cðfsb Þ
ð4Þ
where C and r are geometric constants. The volume fraction of the shear-band,fsb, is also related to the plastic strain accumulated in austenite, e_ pa . It can be expressed as follows: f_ sb ¼ að1 fsb Þ_epa
ð5Þ
where a is the parameter on the rate of shear-band formation. The probability parameter, P, is determined assuming that there exits a Gaussian distribution of shear band intersection potencies [11]. P is cast in the form of cumulative probability distribution function,
W.J. Dan et al. / Computational Materials Science 40 (2007) 101–107
1 P ¼ pffiffiffiffiffiffi 2prg
Z
g 1
2 ! 1 g0 g exp dg0 2 rg
ð6Þ
g ¼ g0 þ g 1 T þ g2 R
ð7Þ
where g0, g1, g2 are material parameters. Temperature T and stress state R are defined as T ¼ T 0 þ DT Z ep v DT ¼ rðep Þdep qC p 0 r1 þ r2 þ r3 rii ¼ R¼ 3 r 3 r
ð8Þ ð9Þ ð10Þ
where T0 is the environment temperature and DT is the increase temperature arisen by strain rate. q is the material density. Cp is the specific heat, and v is the Taylor–Quinney coefficient that represents the proportion of plastic work converted into heat. In this paper, q is taken to be 7870 kg/m3, Cp is taken to be 452 J/ (kg K), and v is taken to be 0–0.1. The rate of change of the probability function, P_ , is thus defined as 2 ! 1 1 g g _ P_ ¼ pffiffiffiffiffiffi exp ð11Þ ðg1 DT_ þ g2 RÞ 2 rg 2prg where it can be easily shown that v DT_ ¼ rðep Þ_ep qC q _ _R ¼ R r_ ii r rii r
ð12Þ ð13Þ
From the equations above, we obtain f_ m as following: _ f_ m ¼ ð1 fm ÞðA_ep þ BRÞ
Jaumann derivative of stress tensor can be expressed as r
T ¼ C e ½D Dp ¼ T_ WT þ TW
where g is the dimensionless mean of a given probability distribution function and rg is its standard deviation. And the driving force g is defined as
ð14Þ
103
ð17Þ
The spin tensor is given as 1 W ¼ ðL LT Þ 2
ð18Þ
where L is velocity gradient. The plastic strain rates are split into three parts [17–20], such as Dpslip slip deformation rate, Dpshape shape change rate and Dpdilat volumetric change rate. Then plastic strain rate is given as Dp ¼ Dpslip þ Dpshape þ Dpdilat Miller et al. presented the effective yield stress of the biphasic material, which involves the second and the third invariant of the deviatoric stress tensor, after observation of an asymmetrical behavior in tension and compression [22]. Because, the sheet has an anisotropic property, the second invariant of the deviatoric stress tensor in the yield function is replaced by Hill’48 yield function. It can be expressed as H2 2 ¼ 0 f ¼ 3H 1 3k pffiffiffiffiffiffi r H1 H 1 ¼ hri½Mfrg; H 2 ¼ det jrj
ð19Þ ð20Þ
is the equivalent stress, H1 is the Hill’48 anisowhere r tropic yield function, H2 is the third invariant of stress tensor and k is the parameter which reflects the development of microstructure and texture inside the materials. The simplified evolution equation for k proposed as k_ ¼ C k ð1 kÞe_ pslip
ð21Þ
where Ck = 0.49. The anisotropic coefficients of Hill’48 in Eq. (19) are F = 0.332, G = 0.47, H = 0.531, M = 1.0, N = 1.312, L = 1.0. The shape change plastic strain rate is co-axial with the normal of yield surface [11]. then
r1
g1 v where A ¼ rabP ð1 fsb Þðfsb Þ Bf qC rðep Þ, B = g2Bf q ! 2 b 1 g g Cvm r Bf ¼ pffiffiffiffiffiffi ðfsb Þ exp H ðP_ Þ; b ¼ vI 2 rg 2prg
Parameter a is given as a ¼ ða0 þ a1 T þ a2 T 2 þ a3 RÞ
p a4 e_ e_ p0
ð15Þ
where a0, a1, a2, a3, a4 are material constant. Where a0 = 2.05 E 4, a1 = 4.52 E 2, a2 = 11.8, a3 = 7.4, a4 = 0.0083, g0 = 0.05, g1 = 1.2, g0 = 78.3, b = 4.5. 3. Constitutive equation The strain tensor increment D of an elastoplastic material in a large deformation can be decomposed into elastic and plastic components as D ¼ De þ Dp
ð16Þ
Dpshape ¼ Rf_ m
of of =rya Þf_ m ¼ ðR0 þ R1 r or or
ð22Þ
where R0, R1 is the material parameter. rya is the initial is the effective stress. yield stress of austenite and r R0 = 0.02, R1 = 0.02, rya = 450 MPa. The dilatation plastic strain rate can be expressed in terms of the volume change Dv due to transformation as 1 Dpdilat ¼ Dvf_ m d 3
ð23Þ
where Dv = 0.02–0.05 and d is the unit tensor. Assuming that the stress–strain curve of each single phase can be expressed by the Swift equation which has successfully been used in continuum plastic theory [23,24], the modified swift equation is expressed as p m _ p n e r ¼ aðb þ e Þ p ð1 kDT Þ ð24Þ e_ 0
104
W.J. Dan et al. / Computational Materials Science 40 (2007) 101–107
where, a,b are the material prosperities. r,ep refer to the equivalent stress and strain. k is the thermal softening parameter. It is noted that the selection of Swift equation is just a choice here and you can select any other appropriate stress–strain model for individual constituent phases. The relationship between the strain of the constituent phases and the total stress is then determined as r ¼ fa ra þ fb rb þ ff rf þ fm rm
ð25Þ
where fm = 1 fa fb ff, fa is the retained austenitic volume fraction, fb is the bainitic volume fraction, and ff is the ferrite volume fraction, and fm is the martensitic volume fraction. ra is the retained austenitic effective stress. rb is the bainitic effective stress. rf is the ferrite effective stress and rm is the martensitic effective stress. During calculating process, strain-hardening coefficient, H 0 , can be defined as H0 ¼
of experiments have been performed with solid elements with a reduced integration, called C3D8R. 4.1. Mixture hardening law The material investigated in this study is an uncoated cold-rolled Si–Mn TRIP800 sheet steel. Its chemical composition is listed in Table 1 and the thickness is 1.0 mm. The specimens tested in uniaxial tension are prepared according to ISO6892 with the gauge length 40 mm. The uniaxial tests an carried out in Zwick Roell tensile testing
Table 1 Chemical composition of TRIP steel (mass %) C
Mn
Si
P
S
Al
Fe
0.11
1.65
0.62
0.008
0.0069
0.031
Balance
d r a Þ ¼ fm H m þ ð1 fm ÞH a þ A0 ð1 fm Þð rm r dep þ fb H b þ ff H f
ð26Þ
4. Numerical simulation and results All the simulations discussed below have been performed using the ABAQUS/UMAT code. The simulations
Table 2 Materials constants of Swift equation
Ferrite Bainite Austenite Martensite
a (MPa)
b
N
631 1421 1490 2498
0.002 0.002 0.02 1E 7
0.39 0.121 0.44 0.29
Fig. 1. Stress–strain curves of TRIP steel. (a) Stress–strain curves in experiment. (b) Mixture hardening law with 0.001/s. (c) Mixture hardening law with 0.01/s. (d) Mixture hardening law with 0.1/s.
W.J. Dan et al. / Computational Materials Science 40 (2007) 101–107
105
The volume fraction of martensite is measured on DMAX-III X-ray diffraction machine. In order to obtain enough data to verify the relation between the volume fraction of martensite and strain, several identical specimens are strengthened to different displacement level such as 2 mm, 4 mm, 6 mm, 8 mm for 0.001/s, 3 mm, 6 mm, 9 mm, 12 mm for 0.01/s and 0.1/s, and until to fracture. The final result is shown in Fig. 2. So the fitting of the data
is precise enough to get the relation between the volume fraction and strain for the material tested in this study. In order to identify the developed model in this study, the martensite transformation of 204 M stainless steel during deformation process is shown in Fig. 3, where the martensite volume fraction in simulations under 0.00013/s, 0.0013/s 0.013/s and 0.13/s agrees with the data tested in Ref. [30]. Fig. 4 shows the martensite transformation during the deformation of 304-stainless steel in [17]. Fig. 4a is the comparison of martensite transformation between simulations based on the developed model in this study and experiment in [17]. Fig. 4b is the comparison of martensite transformation between simulations based on TI model and experiment in [17]. The results show that the new model with strain rate is more agreeable with the experiment than does the TI model. In order to clarify the martensite transformation during the deformation process, Fig. 5 shows shear-band, driving force and parameter-a transformation with strain rate 0.001/s. Fig. 5a shows that the martensite transformation in the new model is lower than the TI model, which results from the temperature increment during specimens deformation process. Surely, the increasing temperature also
Fig. 2. Martensite transformation in experiment.
Fig. 3. Martensite transformation comparison with Ref. [30].
machine. The material property parameters are shown in Table 2. During the simulation, the strain rate sensitive coefficient (in Eq. (24)) m ¼ 0:083; k ¼ 0:02; e_ p0 ¼ 0:0005). All the stress–strain curves are shown in Fig. 1. The results show that the increment of strain rate can increase the material stress shown in Fig. 1a. The simulation results (b–d in Fig. 1) present that the developed model can describe the stress–strain relationship clearly, except that Eq. (24) cannot describe the Lu¨ders strain in the hardening law of the TRIP steel.
4.2. Martensite transformation
Fig. 4. Martensite transformation comparison with Ref. [17] (a) New model. (b) TI model (from Ref. [17]).
106
W.J. Dan et al. / Computational Materials Science 40 (2007) 101–107
Fig. 5. Shear-band, driving force and parameter-a transformation during deformation process with strain rate 0.001/s. (a) Martensite transformation. (b) Shear-band transformation. (c) Driving force. (d) Parameter-a.
has an influence on the shear-band, driving force and parameter-a in Fig. 5b–c. In conclusion, martensite transformation in the forming process is not only depended on stress state, environment temperature, strain rate, but also on the temperature increment produced by material deformation.
Fig. 6. Temperature increment in Ref. [30].
4.3. Temperature increment In order to simulate and analyze the response of the structure under different loading rate conditions, the use of rate-dependent and temperature-dependent constitutive description is required in finite element codes. Typically,
Fig. 7. Temperature increment simulation in experiment.
W.J. Dan et al. / Computational Materials Science 40 (2007) 101–107
the stress is expressed as a function of strain, strain rate, and temperature, which is shown in Eqs. (9), (24) and (25). Temperature increment is shown in Fig. 6 with different strain rate (0.0013/s, 0.013/s and 0.13/s). And the temperature profile of the specimen during the tensile test was measured by a J-type thermocouple, which was attached to the gauge part of specimen. The specimen temperature and the tensile data were obtained concurrently in [30]. The results show that the increasing strain rate deduces the higher temperature increment, and that temperature increment calculated by the new model is suitable with the experiments. Fig. 7 shows the temperature increment of TRIP steel during tensile with the new model under 0.001/s, 0.01/s and 0.1/s. 5. Conclusion In this study, the constitutive equation that models for martensitic transformation depending on strain rate, temperature and applied stress system, is developed as an extension of those proposed in Refs. [11,17]. All the simulations discussed below are performed using the ABAQUS/ UMAT code. Subsequently, the computational strategy was applied to clarify the deformation behavior of the axial tension of TRIP steels with different strain rates. During the deformation process, the work rate deduced by stress, strain and strain rate results in temperature increment. The simulation results show that the new model with temperature increment is more feasible to describe martensite transformation during tensile process than does the TI model with constant temperature. During the analysis of the tensile process with strain rate 0.001/s, the results present that the temperature increment leads to the martensite transformed slowly. The shear-band, driving force and parameter-a transformation also decrease when the deformation temperature is higher. The results reveal that the temperature increment in the forming process agrees with the experiment in Ref. [30]. In other words, the new model can describe the temperature increment of material during the deformation process with the strain rate, and the new model can simulate the martensite transformation of TRIP steels during forming process with different strain rate. Acknowledgements This work was together supported by the National Natural Science Foundation of China (No. 50,405,015), the National Basic Research Program of China (973) (No. 2005CB724,103) and the Key Foundation Research Project of Shanghai City (No. 06DJ14005).
107
References [1] G.W. Greenwood, R.H. Johnson, Proc. Roy. Soc., Lond. A 283 (1965) 403. [2] C.L. Magee, Transformation Kinetics, Microplasticity and ageing of martensite in Fe–3l–Ni, Ph.D. Thesis, Carnegie Institute of Technologie University, Pittsburgh, PA, 1966. [3] J.B. Leblond, Int. J. Plasticity 5 (1989) 573–591. [4] J.B. Leblond, G. Mottet, J.C. Devaux, J. Mech. Phys. Solid 34 (1986) 411–432. [5] L. Taleb, F. Sidoroff, Int. J. Plasticity 19 (2003) 1821–1842. [6] L. Taleb, S. Petit, Int. J. Plasticity 22 (2006) 110–130. [7] F.D. Fischer, M. Berveiller, K. Tanaka, E.R. Oberaigner, Arch. Appl. Mech. 64 (1994) 54. [8] F.D. Fischer, Q.-P. Sun, K. Tanaka, ASME Appl. Mech. Rev. 49 (1996) 317. [9] F.D. Fischer, G. Reisner, E. Werner, K. Tanaka, G. Cailletaud, T. Antretter, Int. J. Plasticity 16 (2000) 723–748. [10] G.B. Olson, M. Cohen, Metall. Trans. 6A (1975) 791–795. [11] R.G. Stringfellow, D.M. Parks, G.B. Olson, Acta Metall. Mater. 40 (7) (1992) 1703–1716. [12] K. Sugimoto, A. Kanda, R. Kikuchi, et al., ISIJ Int. 42 (8) (2002) 910–915. [13] K. Sugimoto, K. Iida, J. Sakaguchi, et al., ISIJ Int. 40 (9) (2000) 902–908. [14] K. Sugimoto, K. Nakano, S. Song, et al., ISIJ Int. 42 (4) (2002) 450–456. [15] T. Angle, J. Iron Steel Inst. 177 (1954) 165–174. [16] D.C. Ludwigson, J.A. Berger, J. Iron Steel Inst. 192 (1969) 63–69. [17] Y. Tomita, T. Iwamoto, Int. J. Mech. Sci. 43 (9) (2001) 2017–2034. [18] Y. Tomita, T. Iwamoto, Int. J. Mech. Sci. 37 (12) (1995) 1295. [19] Y. Tomita, K. Morioka, Mater. Charact. 38 (4–5) (1997) 243–250. [20] T. Iwamoto, T. Tsuta, Int. J. Plasticity 18 (2002) 1583–1606. [21] J. Serri, M. Martiny, G. Ferron, Int. J. Mech. Sci. 47 (6) (2005) 884– 901. [22] M.P. Miller, D.L. McDowell, Int. J. Plasticity 12 (1996) 875–902. [23] M. Umemoto, Z.G. Liu, S. Sugimoto, et al., Metallurg. Mater. Trans. A 31 (2000) 1785. [24] HaiYan Yu, Investigation on transformation-induced plasticity behavior in complex strain state for TRIP steels and its application into formability prediction autobody panels. Ph.D. Thesis, ShangHai JiaoTong University, ShangHai, China, 2005 (in chinese). [25] Kenji Saito, Shushi Ikeda, Koichi Makii, et al., A Hen/Transactions of the Japan Society of Mechanical Engineers, Part A 71 (3) (2005) 465–471. [26] I.D. Choi, D.M. Bruce, S.J. Kim, et al., ISIJ Int. 42 (12) (2002) 1483– 1489. [27] Takeshi Iwamoto, Toshiyuki Sawa, Masahiro Sasaki, Zairyo/J. Soc. Mater. Sci. Japan 53 (12) (2004) 1365–1370. [28] T. Vuoristo, V-T. Kuokkala, M. Apostol et al., Deformation and work hardening of cold-formed high strength steels at high rates of strain, Materials Science and Technology, Materials Science and Technology, MS and T 2004; Volume 1: AIST Process Metallurgy, Product Quality and Applications Proceedings vol. 1, 2004, pp. 471– 480. [29] Ludovic Samek, Bruno C Cooman, Joost Van Slycken, et al., Steel Res. Int. 75 (11) (2004) 716–723. [30] H.N. Han, Ch.G. Lee, Chang-Seok Oh, et al., Acta Mater. 52 (2004) 5203–5214.
A model for strain-induced martensitic transformation of TRIP steel with strain rate W.J. Dan *, W.G. Zhang, S.H. Li, Z.Q. Lin School of Mechanical Engineering, Shanghai JiaoTong University, Shanghai 200240, PR China Received 20 September 2006; accepted 9 November 2006
Abstract Strain rate is one of the key factors which induce martensitic transformation. In this study, a constitutive model, which can describe the transformation-induced plasticity (TRIP) accompanying the strain-induced martensitic transformation in TRIP steel, is developed. The increase of nucleation site in the austenite due to the plastic deformation is formulated as the increase of the shear band intersection. The nucleation site probability is derived not only by stress state, plasticity strain and constant environment temperature, but also by strain rate, where shear band intersection decrease through strain rate adiabatic thermal to simulate the transformation-induced plasticity characteristic for TRIP steel. Anisotropic yield function is used to describe the sheet anisotropic property. A mixture hardening law with four phases is developed instead of the mixture hardening law with two phases used commonly. The constitutive model is implemented into ABAQUS/UMAT for the analysis of the material deforming processes. The martensitic volume fraction is tested by X-ray to describe the comparison between experimental data and simulation. Stress–strain curve is measured under strain rate from 0.001/s to 0.1/s to identify the simulation results based on the new constitutive model. All the results are agreeable. Ó 2006 Elsevier B.V. All rights reserved. PACS: 81.30.Kf Keywords: Martensite transformation; TRIP steel; Strain rate; Finite element analysis
1. Introduction High strength TRIP steels have received increasing attention to improve manufacturability, safety and to reduce weight, for these steels have a high ductility, which results from the transformation of unmetastable austenite to martensite under plastic strain. Then a localized increasing of the strain hardening coefficient during deformation with TRIP steels can delay the onset of necking and ultimately leads to a higher uniform and total elongation. The two main effects usually invoked in TRIP steels are (1) the Greenwood and Johnson effect [1] corresponding to the plastic strain generated in the parent austenitic phase by the martensitic transformation, and (2) the Magee effect
*
Corresponding author. E-mail address: [email protected] (W.J. Dan).
0927-0256/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2006.11.006
[2], associated with the selection of martensite variants arising from the applied and internal stress states. Several micromechanical models have been proposed to account for the TRIP effect. Most of the models are based on the Greenwood and Johnson model. Such as Leblond [3,4], Taleb [5,6] and Fischer [7–9] are the representatives. Taleb was devoted to an investigation of some discrepancies from a reevaluation of the micromechanical model as originally used by Leblond et al. [4]. A more complete formulation taking into account the elasticity in both phases was developed and solved resulting in an improved model enabling a better description of the experimental results and removing the singularity. Leblond’s model [3,4] took into account the interaction between classical plasticity and TRIP, but its experimental validation has not yet been performed [6]. Fischer et al. [8,9] investigated shape memory alloys orientation effect, and quantifies the orientation effect based on Greenwood–Johnson model and Magee model.
102
W.J. Dan et al. / Computational Materials Science 40 (2007) 101–107
The kinetics phase transformation is the key model to describe martensite transformation, which was modeled by Olson and Cohen [10], Stringfellow et al. [11], Sumigoto et al. [12–14], Angel et al. [15], Ludwisgson and Berger [16]. Angel et al. and Ludwisgson and Berger studied the stainless steel transformation induced plasticity behaviors and constituted kinetics phase transformation of martensitic; Olson and Cohen (OC model) modeled the kinetics phase transformation of martensitic and Stringfellow et al. developed the model based on shear band intersection. Sumigoto et al. obtained the retained austenitic formability and stability of stainless steel. Based on the [10,11] Tomita and Iwanmoto (TI) [17–20] have developed a phenomenological constitutive model based on Stringfellow‘s model (S Model) and Serri et al. [21] used the TI model with ABAQUS/ VUMAT code to simulate the TRIP sheet steel forming. Strain rate can be divided into standard strain rate (below 1021/s), middle strain rate (1021021/s) and high strain rate (above 1021/s). Surely, TRIP steel is a material sensitivity to strain rate. Some authors have studied the effects of strain rate on TRIP steel [25–29]. Kenji Saito et al. studied the strain rate and temperature influence on the law-carbon TRIP steel. The enhancement of strain rate and temperature restrain martensite transformation. Choi et al. analyzed the effects of strain rate and retained austenite volume fraction on tensile properties. Increasing the retained austenite volume fraction increases UTS, total elongation, uniform strain and total absorbed energy, but decreases yield strength and absorbed energy below 10% engineering strain. Based on the Green function and selfconsistent approximation for homogenization, Takeshi Iwamoto et al. derived strain rate field in the microregion and a macroscopic constitutive equation for TRIP steels. Then, the tensile test of the plane strain block of SUS304 is simulated by FEM along with the proposed constitutive equation. Vuoristo et al. compared the high strain rate behavior of dual phase steel with that of TRIP steel. The test results showed that the energy absorption capability of the dual phase steels is higher than that of the studied TRIP steel at strains less than 0.1. The HSLA steel and carbon–manganese steels showed a higher strain rate sensitivity but a lower energy absorption capability and a lower uniform elongation at high strain rates than the multiphase steels. Ludovic samek et al. focused on the effect of the strain rate and temperature on the mechanical behaviors of the low alloy high strength TRIP steel, and argued that the adiabatic conditions present have a beneficial effect on the behaviors of TRIP steel during high strain rate deformations. All the contents above were focused on the experiments to study martensite transformation and the mechanical properties during the tensile of TRIP steel, until TI model has constituted the constitutive equation based on the shear-band theory and according to the strain rate, constant environment temperature during deformation. However, the strain rate can lead to temperature enhancement during forming processes for the adiabatic effective.
In this study, TRIP steel sheet, composed of ferrite, bainite, retained austenite and martensite four phases, is analyzed to learn about martensite transformation during deformation with ABAQUS/UMAT Code according to strain rate adiabatic effective. A mixture hardening law with four phases has been used. Anisotropic property of sheet has been shown to replace the isotropic yield function in the TI model. 2. Transformation kinetic The TI model only shows that martensitic transformation driving force is influenced by stress state, strain rate and permanent temperature. However, temperature is changing during sheet forming process especially involving the strain rate. In this study, material deformation is not an isothermal process. The increasing temperature is considered as a driving force to influence on shear band intersection. The rate of increase in the extended volume fraction of the martensitic variant, f_ m , is proportional to the rate of increase in the number of the martensitic variant embryos per unit austenite volume, N_ m ð1Þ f_ m ¼ ð1 f Þvm N_ m where vm is the average volume per martensitic variant unit, N_ m can be expressed in terms of the number of nucleation sites per unit austenite volume, N_ I , and the probability, P, that a shear band intersection will act as a nucleation site. N_ m is given as N_ m ¼ P N_ I þ N I P_ H ðP_ Þ
ð2Þ
where H is the Heaviside step function, reflecting the fact that the martensitic transformation is irreversible. Based on the observation that the strain-induced nucleation occurs predominantly at shear-band intersections [11,17], N_ I can be defined as fI ð3Þ NI ¼ vI where fI and vI are the volume fraction of shear-band intersection and the average volume of shear-band intersection, respectively. The relation between the parameter, fI, and the volume fraction of shear-band, fsb, is assumed to be a power-law expression of the form [11] r
fI ¼ Cðfsb Þ
ð4Þ
where C and r are geometric constants. The volume fraction of the shear-band,fsb, is also related to the plastic strain accumulated in austenite, e_ pa . It can be expressed as follows: f_ sb ¼ að1 fsb Þ_epa
ð5Þ
where a is the parameter on the rate of shear-band formation. The probability parameter, P, is determined assuming that there exits a Gaussian distribution of shear band intersection potencies [11]. P is cast in the form of cumulative probability distribution function,
W.J. Dan et al. / Computational Materials Science 40 (2007) 101–107
1 P ¼ pffiffiffiffiffiffi 2prg
Z
g 1
2 ! 1 g0 g exp dg0 2 rg
ð6Þ
g ¼ g0 þ g 1 T þ g2 R
ð7Þ
where g0, g1, g2 are material parameters. Temperature T and stress state R are defined as T ¼ T 0 þ DT Z ep v DT ¼ rðep Þdep qC p 0 r1 þ r2 þ r3 rii ¼ R¼ 3 r 3 r
ð8Þ ð9Þ ð10Þ
where T0 is the environment temperature and DT is the increase temperature arisen by strain rate. q is the material density. Cp is the specific heat, and v is the Taylor–Quinney coefficient that represents the proportion of plastic work converted into heat. In this paper, q is taken to be 7870 kg/m3, Cp is taken to be 452 J/ (kg K), and v is taken to be 0–0.1. The rate of change of the probability function, P_ , is thus defined as 2 ! 1 1 g g _ P_ ¼ pffiffiffiffiffiffi exp ð11Þ ðg1 DT_ þ g2 RÞ 2 rg 2prg where it can be easily shown that v DT_ ¼ rðep Þ_ep qC q _ _R ¼ R r_ ii r rii r
ð12Þ ð13Þ
From the equations above, we obtain f_ m as following: _ f_ m ¼ ð1 fm ÞðA_ep þ BRÞ
Jaumann derivative of stress tensor can be expressed as r
T ¼ C e ½D Dp ¼ T_ WT þ TW
where g is the dimensionless mean of a given probability distribution function and rg is its standard deviation. And the driving force g is defined as
ð14Þ
103
ð17Þ
The spin tensor is given as 1 W ¼ ðL LT Þ 2
ð18Þ
where L is velocity gradient. The plastic strain rates are split into three parts [17–20], such as Dpslip slip deformation rate, Dpshape shape change rate and Dpdilat volumetric change rate. Then plastic strain rate is given as Dp ¼ Dpslip þ Dpshape þ Dpdilat Miller et al. presented the effective yield stress of the biphasic material, which involves the second and the third invariant of the deviatoric stress tensor, after observation of an asymmetrical behavior in tension and compression [22]. Because, the sheet has an anisotropic property, the second invariant of the deviatoric stress tensor in the yield function is replaced by Hill’48 yield function. It can be expressed as H2 2 ¼ 0 f ¼ 3H 1 3k pffiffiffiffiffiffi r H1 H 1 ¼ hri½Mfrg; H 2 ¼ det jrj
ð19Þ ð20Þ
is the equivalent stress, H1 is the Hill’48 anisowhere r tropic yield function, H2 is the third invariant of stress tensor and k is the parameter which reflects the development of microstructure and texture inside the materials. The simplified evolution equation for k proposed as k_ ¼ C k ð1 kÞe_ pslip
ð21Þ
where Ck = 0.49. The anisotropic coefficients of Hill’48 in Eq. (19) are F = 0.332, G = 0.47, H = 0.531, M = 1.0, N = 1.312, L = 1.0. The shape change plastic strain rate is co-axial with the normal of yield surface [11]. then
r1
g1 v where A ¼ rabP ð1 fsb Þðfsb Þ Bf qC rðep Þ, B = g2Bf q ! 2 b 1 g g Cvm r Bf ¼ pffiffiffiffiffiffi ðfsb Þ exp H ðP_ Þ; b ¼ vI 2 rg 2prg
Parameter a is given as a ¼ ða0 þ a1 T þ a2 T 2 þ a3 RÞ
p a4 e_ e_ p0
ð15Þ
where a0, a1, a2, a3, a4 are material constant. Where a0 = 2.05 E 4, a1 = 4.52 E 2, a2 = 11.8, a3 = 7.4, a4 = 0.0083, g0 = 0.05, g1 = 1.2, g0 = 78.3, b = 4.5. 3. Constitutive equation The strain tensor increment D of an elastoplastic material in a large deformation can be decomposed into elastic and plastic components as D ¼ De þ Dp
ð16Þ
Dpshape ¼ Rf_ m
of of =rya Þf_ m ¼ ðR0 þ R1 r or or
ð22Þ
where R0, R1 is the material parameter. rya is the initial is the effective stress. yield stress of austenite and r R0 = 0.02, R1 = 0.02, rya = 450 MPa. The dilatation plastic strain rate can be expressed in terms of the volume change Dv due to transformation as 1 Dpdilat ¼ Dvf_ m d 3
ð23Þ
where Dv = 0.02–0.05 and d is the unit tensor. Assuming that the stress–strain curve of each single phase can be expressed by the Swift equation which has successfully been used in continuum plastic theory [23,24], the modified swift equation is expressed as p m _ p n e r ¼ aðb þ e Þ p ð1 kDT Þ ð24Þ e_ 0
104
W.J. Dan et al. / Computational Materials Science 40 (2007) 101–107
where, a,b are the material prosperities. r,ep refer to the equivalent stress and strain. k is the thermal softening parameter. It is noted that the selection of Swift equation is just a choice here and you can select any other appropriate stress–strain model for individual constituent phases. The relationship between the strain of the constituent phases and the total stress is then determined as r ¼ fa ra þ fb rb þ ff rf þ fm rm
ð25Þ
where fm = 1 fa fb ff, fa is the retained austenitic volume fraction, fb is the bainitic volume fraction, and ff is the ferrite volume fraction, and fm is the martensitic volume fraction. ra is the retained austenitic effective stress. rb is the bainitic effective stress. rf is the ferrite effective stress and rm is the martensitic effective stress. During calculating process, strain-hardening coefficient, H 0 , can be defined as H0 ¼
of experiments have been performed with solid elements with a reduced integration, called C3D8R. 4.1. Mixture hardening law The material investigated in this study is an uncoated cold-rolled Si–Mn TRIP800 sheet steel. Its chemical composition is listed in Table 1 and the thickness is 1.0 mm. The specimens tested in uniaxial tension are prepared according to ISO6892 with the gauge length 40 mm. The uniaxial tests an carried out in Zwick Roell tensile testing
Table 1 Chemical composition of TRIP steel (mass %) C
Mn
Si
P
S
Al
Fe
0.11
1.65
0.62
0.008
0.0069
0.031
Balance
d r a Þ ¼ fm H m þ ð1 fm ÞH a þ A0 ð1 fm Þð rm r dep þ fb H b þ ff H f
ð26Þ
4. Numerical simulation and results All the simulations discussed below have been performed using the ABAQUS/UMAT code. The simulations
Table 2 Materials constants of Swift equation
Ferrite Bainite Austenite Martensite
a (MPa)
b
N
631 1421 1490 2498
0.002 0.002 0.02 1E 7
0.39 0.121 0.44 0.29
Fig. 1. Stress–strain curves of TRIP steel. (a) Stress–strain curves in experiment. (b) Mixture hardening law with 0.001/s. (c) Mixture hardening law with 0.01/s. (d) Mixture hardening law with 0.1/s.
W.J. Dan et al. / Computational Materials Science 40 (2007) 101–107
105
The volume fraction of martensite is measured on DMAX-III X-ray diffraction machine. In order to obtain enough data to verify the relation between the volume fraction of martensite and strain, several identical specimens are strengthened to different displacement level such as 2 mm, 4 mm, 6 mm, 8 mm for 0.001/s, 3 mm, 6 mm, 9 mm, 12 mm for 0.01/s and 0.1/s, and until to fracture. The final result is shown in Fig. 2. So the fitting of the data
is precise enough to get the relation between the volume fraction and strain for the material tested in this study. In order to identify the developed model in this study, the martensite transformation of 204 M stainless steel during deformation process is shown in Fig. 3, where the martensite volume fraction in simulations under 0.00013/s, 0.0013/s 0.013/s and 0.13/s agrees with the data tested in Ref. [30]. Fig. 4 shows the martensite transformation during the deformation of 304-stainless steel in [17]. Fig. 4a is the comparison of martensite transformation between simulations based on the developed model in this study and experiment in [17]. Fig. 4b is the comparison of martensite transformation between simulations based on TI model and experiment in [17]. The results show that the new model with strain rate is more agreeable with the experiment than does the TI model. In order to clarify the martensite transformation during the deformation process, Fig. 5 shows shear-band, driving force and parameter-a transformation with strain rate 0.001/s. Fig. 5a shows that the martensite transformation in the new model is lower than the TI model, which results from the temperature increment during specimens deformation process. Surely, the increasing temperature also
Fig. 2. Martensite transformation in experiment.
Fig. 3. Martensite transformation comparison with Ref. [30].
machine. The material property parameters are shown in Table 2. During the simulation, the strain rate sensitive coefficient (in Eq. (24)) m ¼ 0:083; k ¼ 0:02; e_ p0 ¼ 0:0005). All the stress–strain curves are shown in Fig. 1. The results show that the increment of strain rate can increase the material stress shown in Fig. 1a. The simulation results (b–d in Fig. 1) present that the developed model can describe the stress–strain relationship clearly, except that Eq. (24) cannot describe the Lu¨ders strain in the hardening law of the TRIP steel.
4.2. Martensite transformation
Fig. 4. Martensite transformation comparison with Ref. [17] (a) New model. (b) TI model (from Ref. [17]).
106
W.J. Dan et al. / Computational Materials Science 40 (2007) 101–107
Fig. 5. Shear-band, driving force and parameter-a transformation during deformation process with strain rate 0.001/s. (a) Martensite transformation. (b) Shear-band transformation. (c) Driving force. (d) Parameter-a.
has an influence on the shear-band, driving force and parameter-a in Fig. 5b–c. In conclusion, martensite transformation in the forming process is not only depended on stress state, environment temperature, strain rate, but also on the temperature increment produced by material deformation.
Fig. 6. Temperature increment in Ref. [30].
4.3. Temperature increment In order to simulate and analyze the response of the structure under different loading rate conditions, the use of rate-dependent and temperature-dependent constitutive description is required in finite element codes. Typically,
Fig. 7. Temperature increment simulation in experiment.
W.J. Dan et al. / Computational Materials Science 40 (2007) 101–107
the stress is expressed as a function of strain, strain rate, and temperature, which is shown in Eqs. (9), (24) and (25). Temperature increment is shown in Fig. 6 with different strain rate (0.0013/s, 0.013/s and 0.13/s). And the temperature profile of the specimen during the tensile test was measured by a J-type thermocouple, which was attached to the gauge part of specimen. The specimen temperature and the tensile data were obtained concurrently in [30]. The results show that the increasing strain rate deduces the higher temperature increment, and that temperature increment calculated by the new model is suitable with the experiments. Fig. 7 shows the temperature increment of TRIP steel during tensile with the new model under 0.001/s, 0.01/s and 0.1/s. 5. Conclusion In this study, the constitutive equation that models for martensitic transformation depending on strain rate, temperature and applied stress system, is developed as an extension of those proposed in Refs. [11,17]. All the simulations discussed below are performed using the ABAQUS/ UMAT code. Subsequently, the computational strategy was applied to clarify the deformation behavior of the axial tension of TRIP steels with different strain rates. During the deformation process, the work rate deduced by stress, strain and strain rate results in temperature increment. The simulation results show that the new model with temperature increment is more feasible to describe martensite transformation during tensile process than does the TI model with constant temperature. During the analysis of the tensile process with strain rate 0.001/s, the results present that the temperature increment leads to the martensite transformed slowly. The shear-band, driving force and parameter-a transformation also decrease when the deformation temperature is higher. The results reveal that the temperature increment in the forming process agrees with the experiment in Ref. [30]. In other words, the new model can describe the temperature increment of material during the deformation process with the strain rate, and the new model can simulate the martensite transformation of TRIP steels during forming process with different strain rate. Acknowledgements This work was together supported by the National Natural Science Foundation of China (No. 50,405,015), the National Basic Research Program of China (973) (No. 2005CB724,103) and the Key Foundation Research Project of Shanghai City (No. 06DJ14005).
107
References [1] G.W. Greenwood, R.H. Johnson, Proc. Roy. Soc., Lond. A 283 (1965) 403. [2] C.L. Magee, Transformation Kinetics, Microplasticity and ageing of martensite in Fe–3l–Ni, Ph.D. Thesis, Carnegie Institute of Technologie University, Pittsburgh, PA, 1966. [3] J.B. Leblond, Int. J. Plasticity 5 (1989) 573–591. [4] J.B. Leblond, G. Mottet, J.C. Devaux, J. Mech. Phys. Solid 34 (1986) 411–432. [5] L. Taleb, F. Sidoroff, Int. J. Plasticity 19 (2003) 1821–1842. [6] L. Taleb, S. Petit, Int. J. Plasticity 22 (2006) 110–130. [7] F.D. Fischer, M. Berveiller, K. Tanaka, E.R. Oberaigner, Arch. Appl. Mech. 64 (1994) 54. [8] F.D. Fischer, Q.-P. Sun, K. Tanaka, ASME Appl. Mech. Rev. 49 (1996) 317. [9] F.D. Fischer, G. Reisner, E. Werner, K. Tanaka, G. Cailletaud, T. Antretter, Int. J. Plasticity 16 (2000) 723–748. [10] G.B. Olson, M. Cohen, Metall. Trans. 6A (1975) 791–795. [11] R.G. Stringfellow, D.M. Parks, G.B. Olson, Acta Metall. Mater. 40 (7) (1992) 1703–1716. [12] K. Sugimoto, A. Kanda, R. Kikuchi, et al., ISIJ Int. 42 (8) (2002) 910–915. [13] K. Sugimoto, K. Iida, J. Sakaguchi, et al., ISIJ Int. 40 (9) (2000) 902–908. [14] K. Sugimoto, K. Nakano, S. Song, et al., ISIJ Int. 42 (4) (2002) 450–456. [15] T. Angle, J. Iron Steel Inst. 177 (1954) 165–174. [16] D.C. Ludwigson, J.A. Berger, J. Iron Steel Inst. 192 (1969) 63–69. [17] Y. Tomita, T. Iwamoto, Int. J. Mech. Sci. 43 (9) (2001) 2017–2034. [18] Y. Tomita, T. Iwamoto, Int. J. Mech. Sci. 37 (12) (1995) 1295. [19] Y. Tomita, K. Morioka, Mater. Charact. 38 (4–5) (1997) 243–250. [20] T. Iwamoto, T. Tsuta, Int. J. Plasticity 18 (2002) 1583–1606. [21] J. Serri, M. Martiny, G. Ferron, Int. J. Mech. Sci. 47 (6) (2005) 884– 901. [22] M.P. Miller, D.L. McDowell, Int. J. Plasticity 12 (1996) 875–902. [23] M. Umemoto, Z.G. Liu, S. Sugimoto, et al., Metallurg. Mater. Trans. A 31 (2000) 1785. [24] HaiYan Yu, Investigation on transformation-induced plasticity behavior in complex strain state for TRIP steels and its application into formability prediction autobody panels. Ph.D. Thesis, ShangHai JiaoTong University, ShangHai, China, 2005 (in chinese). [25] Kenji Saito, Shushi Ikeda, Koichi Makii, et al., A Hen/Transactions of the Japan Society of Mechanical Engineers, Part A 71 (3) (2005) 465–471. [26] I.D. Choi, D.M. Bruce, S.J. Kim, et al., ISIJ Int. 42 (12) (2002) 1483– 1489. [27] Takeshi Iwamoto, Toshiyuki Sawa, Masahiro Sasaki, Zairyo/J. Soc. Mater. Sci. Japan 53 (12) (2004) 1365–1370. [28] T. Vuoristo, V-T. Kuokkala, M. Apostol et al., Deformation and work hardening of cold-formed high strength steels at high rates of strain, Materials Science and Technology, Materials Science and Technology, MS and T 2004; Volume 1: AIST Process Metallurgy, Product Quality and Applications Proceedings vol. 1, 2004, pp. 471– 480. [29] Ludovic Samek, Bruno C Cooman, Joost Van Slycken, et al., Steel Res. Int. 75 (11) (2004) 716–723. [30] H.N. Han, Ch.G. Lee, Chang-Seok Oh, et al., Acta Mater. 52 (2004) 5203–5214.