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Numerical modeling and simulation of carburized and nitrided quenching process
Journal of Materials Processing Technology 143–144 (2003) 880–885
Numerical modeling and simulation of carburized and nitrided quenching process Dong-Ying Ju a,∗ , Chuncheng Liu b , Tatsuo Inoue c a
Department of Mechanical Engineering, Saitama Institute of Technology, Saitama, Japan b High Technology Research Center, Saitama Institute of Technology, Saitama, Japan c Department of Energy Conversion Science, Kyoto University, Kyoto 606-8501, Japan
Abstract In order to study metallo-thermo-mechanical behavior during carburized and nitrided quenching, a numerical model when considering quantitative effects of diffused carbon and nitrogen gradients and kinetics of phase transformation is proposed. Coupled calculations of diffusion, phase transformation and stress/strain give the final distributions of carbon and nitrogen contents as well as residual stress and distortion. Within the numerical simulation method, the effects of both transformation and lattice expansion induced from carbon and nitrogen absorption are considered when calculating the evolution of the internal stress and strain. In order to verify the method of numerical simulation, the simulated distributions of carbon and nitrogen content and residual stress/strain of a cylinder during carburized and nitrided quenching are compared with the measured results. © 2003 Published by Elsevier Science B.V. Keywords: Carburizing; Nitriding; Quenching; Metallo-thermo-mechanical behavior; Numerical simulation
1. Introduction Nitriding and carburizing are thermochemical surface treatments of large importance in engineering practice [1–4]. They are widely used in machine parts such as automotive gears or shafts due to their beneficial effects on fatigue resistance, tribological and anti-corrosion properties. The reasons for the property improvement are ascribed to the compound layers composed of fine nitrides or carbides of iron or alloy elements, and residual surface stresses are also attributed to this process. As a known fact, residual stresses are always accompanied by the changes in shape and size of workpiece, which often results in significant distortion of machine parts. This also leads to undesired effects such as increasing noise from transmission gears. In order to improve such kind of defects, manufacturing engineers need to have a thorough understanding how various process parameters influence the final profiles of carbon/nitrogen, stress/distortion and microstructures of heat-treated components. To do this, in recent years, there have been a great number of experimental and theoretical interests in mathematical description of carburizing and/or nitriding processes [5–17]. Numerical simulation by the finite element method (FEM), due to its benefit in reducing
∗
Corresponding author.
0924-0136/$ – see front matter © 2003 Published by Elsevier Science B.V. doi:10.1016/S0924-0136(03)00378-9
time-consuming and trial-and-error experiments that plague engineers, is looking forward to being used widely in engineering problems. Some FEM softwares have realized the simulation of carburizing, like HEARTS [6], ABAQUS [17], SYSWELD [18] and DEFORM-HT [19]. However, in these commercialized programs, some problems such as effect of diffusion of nitrogen or simultaneous diffusion of carbon and nitrogen on the following thermo-mechanical simulation still remain imperfect. With respect to this, the numerical modeling for simulating the metallo-thermo-mechanical behavior during carburized and nitrided quenching is developed in this paper. The purpose of the modeling aims to make it capable to calculate not only diffusion of multi-element during chemical treatment but also the evolution of microstructure, stress and distortion during carburized and nitrided quenching by considering the effect of C/N diffusion on distortion and phase transformation kinetics. This paper motivates to propose a numerical modeling of coupled diffusion of carbon/nitrogen, heat conduction, phase transformation and stress/strain. Within the numerical simulation method, the effects of both transformation and lattice expansion relevant to describing carbon and nitrogen absorption are considered when calculating the evolution of the internal stress and strain. In order to verify the numerical method, the simulated distributions of carbon and nitrogen contents, temperature and variation of microstructure as well as
D.-Y. Ju et al. / Journal of Materials Processing Technology 143–144 (2003) 880–885
residual stress/strain of a cylinder during carburized and nitrided quenching are compared with the measured results.
2. Numerical simulation models The modeling developed by the authors can be used to predict the evolution of temperature, microstructure, stress/distortion and profiles of chemical component such as carbon and nitrogen and so on during some processes of thermochemical treatment and heat treatment. The models involved in the FEM procedure can be divided into four categories of analysis: heat conduction, diffusion of C/N, transformation kinetics and stress/strain analysis by use of elasto-plastic constitutive equations. The following depictions are based on 3D coordinate, though it is also capable to axisymmetric problems in the program. When the boundary conditions are involved, the prescribed flux boundary conditions are employed. 2.1. Heat conduction The heat conduction equation governing temperature field T is ∂ ∂T ˙ ρcT − k − rv = 0 (1) ∂xi ∂xi ∂T ni = h(T − Tw ), ∂xi
of each constituent DNI (N), which is designed to be calculated by the following formula: QN ˆ DNI (N) = DNI,0 f(C, N)h(grad C, grad N) exp − RT (6) ˆ with the functions f(C, N) and h(grad C, grad N) of carbon and nitrogen concentrations and the gradients, respectively. And QN in Eq. (6) is the activation energy for diffusing nitrogen, and DNI,0 and R the material constants. The coefficient of surface reaction rate βN in Eq. (4) of nitriding process under partial pressure pH2 of hydrogen in the mixed gas is given by QN βN = β0 pH2 exp − , (7) RT where β0 is a constant [20]. The equation governing carbon diffusion kinetics can be simply obtainable by exchanging N and C in Eqs. (3)–(6). When the coefficient of surface reaction rate βC for carbon is decided by using experimental method [21], the carbon diffusion in steel can be expressed by the following boundary condition: ∂C −DC ni = βC (C − Cw ). (8) ∂xi 2.3. Phase transformation kinetics
with heat transfer boundary condition −k
881
(2)
where ρ, c and k denote density, specific heat and heat conductivity, respectively; rv the internal heat source, h and Tw the mean heat transfer coefficient and environmental temperature, respectively, on the boundary and ni the unit normal. The value of ρ, c and k should be calculated according to the mixture rule X = XI ξI , where X is a material property and ξ I is volume fraction of the Ith phase.
The microstructures involved in the present simulation are assumed to be composed of three constituents: austenite, pearlite or bainite and martensite. The diffusion type transformation from austenite to pearlite is calculated based on the modified Johnson–Mehl relation [22]. In this case of carbon steel S45C, the volume fraction of pearlite ξ p is expressed as [23]: t ξp = 1 − exp − f1 (T)f2 (C)f3 (σ)(t − τ)3 dτ , (9)
0
T − a1 a2
a3
a4 − T a5
a6
2.2. Diffusion of nitrogen and carbon
f1 (T) = a0
The following equations are the related diffusion formulas used for nitriding: ∂ ∂N ˙ − N DN − rN = 0 (3) ∂xi ∂xi
f2 (C) = exp{−a7 (C − C0 )},
(11)
f3 (σ) = exp{a8 σm },
(12)
with boundary condition −DN
∂N ni = βN (N − Nw ). ∂xi
(4)
Here, N, Nw and rN , respectively, denote diffused mass percent, environment potential and internal source of nitrogen. Global diffusivity DN is also assumed to be given by the mixture law DN = DNI (N)ξI (5)
,
(10)
where σ m is mean stress, C and C0 the current and initial carbon contents, respectively, and ai (i = 0, 1, 2, . . . , 8) are some transformation kinetic parameters. Diffusionless transformation is controlled by ξM = 1 − exp{φ1 T + φ2 (C − C0 ) + φ3 (N − N0 ) + φ4 σm +φ5 σe + φ6 },
(13)
by modified Magee’s rule [24], where ξ M is the volume fraction of martensite, φi (i = 1, 2, . . . , 6) are material parameters. N and N0 mean the current and initial nitrogen contents (wt.%) in the nodes, respectively.
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D.-Y. Ju et al. / Journal of Materials Processing Technology 143–144 (2003) 880–885
In the above equations, the effects of C/N profile on transformation kinetics are considered. Due to the small penetrating depth of N atoms and high contents of C in the nitrogen diffusion layer, little pearlite transformation occurs in this layer. It should not matter greatly even though no effect of nitrogen is taken into account in diffusion type transformation kinetics. 2.4. Constitutive equations Total strain rate ε˙ ij is assumed to be sum of rates of elastic strain ε˙ eij , plastic strain ε˙ pij , thermal strain ε˙ th , transformation ij tr df strain ε˙ ij and diffusion induced strain ε˙ ij of carbon and nitrogen which is newly introduced in this paper. ε˙ ij =
ε˙ eij
p + ε˙ ij
+ ε˙ th ij
+ ε˙ trij
+ ε˙ df ij .
(14)
Here the four terms are given by 1+ν ν σij − σkk δij , E E K ∂F ∂F ∂F ∂F p ˆ ξ˙I ε˙ ij = G σkl + T˙ + ∂σkl ∂T ∂ξI ∂σij εeij =
(15)
(16)
I=1
with yield function p
F = F(σij , εij , κ, T, ξI ), ε˙ th ij =
K
(17)
αI T˙ δij ,
(18)
βJ→I ξ˙I δij .
(19)
I=1
ε˙ trij
=
K
If no transformation occurs during diffusion of carbon and nitrogen, the microstructure is only composed of austenite. Thus, the strain rate ε˙ df ij simply depends on the diffusion rate of carbon and nitrogen in the form: 1 a3 (Ct1 , Nt1 ) ε˙ df = − 1 ij 3 a3 (Ct0 , Nt0 ) 1 1 ˙ δij , × (21) C˙ + N Ct1 − Ct0 Nt1 − Nt0 where a is the lattice parameter of austenite, which is a function of current carbon and nitrogen contents. 3. Materials and condition of experiments and simulation The steel used in this paper is the carbon steel (JIS-S45C). The heat pattern is composed of five steps, as shown in Fig. 1. In the first step, the specimen is heated up to 860 ◦ C from room temperature and then mixed C/N gas is provided with certain carbon and nitrogen potentials, which are both 0.9 wt.% in the present case. The second step is for carburizing and nitriding for 270 min. After that, the specimen is cooled to 850 ◦ C for 10 min and then is held at this temperature for 30 min as diffusion period to decrease the gradients of carbon and nitrogen. The final step is for quenching of the specimen into oil quenchant of 60 ◦ C. The sizes of the cylindrical specimen are shown in Fig. 2, in which the holes (nos. 1 and 2) are drilled to insert thermocouples for temperature measurement during quenching. The mesh division for FEM calculation is also depicted in Fig. 2. Due to the symmetry, the one-fourth of the cross-section is
I=1
Here, E and ν are the Young’s modulus and Poisson’s ratio, respectively, T the temperature, and βJ→I the dilatation due to the microstructure change from the Jth to Ith phase. αI is thermal expansion coefficient of the Ith phase. Compared with the previous constitutive equations [16,17] of the authors, a new term ε˙ df ij in (14) due to carbon or nitrogen diffusion is introduced in this paper. When carbon or nitrogen is absorbed as interstitial atoms, the lattice parameters change to induce the volumetric dilatation. Some researchers proposed formulas of the lattice parameters depending on carbon and nitrogen contents [25]. If transformation (α → β) occurs between time interval t0 and t due to the diffusion of carbon and nitrogen, the strain ε˙ df ij is
T 1 Vβ (Ct1 , Nt1 )/nβ,Fe ε˙ df − 1 ξ˙α→β δij , (20) ij = 3 VαT (Ct0 , Nt0 )/nα,Fe where nI ,Fe means the number of Fe atoms in single unit cell of certain microstructure. VαT and VβT denote the aαT and aβT lattice parameter before and after α → β transformation, while Ct and Nt are the carbon and nitrogen contents at time t, respectively.
Fig. 1. Heat treatment process.
Fig. 2. Specimen and FE mesh.
D.-Y. Ju et al. / Journal of Materials Processing Technology 143–144 (2003) 880–885
Fig. 3. Carbon profiles at different times.
883
Fig. 5. Carbon contents vs. time.
used for meshing by 160 nodes and 133 elements by use of quadrilateral iso-parameter elements with four nodes. The thermal and diffusion boundary conditions are both the flux type, and hardness and residual stresses are measured after experiments, the results of which are used to compare with the calculated ones.
4. Results and discussions Figs. 3 and 4 show the carbon and nitrogen profiles in the surface layer on the middle section after carburizing and nitriding, respectively. Because of the high surface reaction rate from nitrogen gas to surface and low diffusion rate of nitrogen in austenite, the depth of diffused nitrogen is only 500 m or so after 6 h. From the time 270–310 min, while gas is not provided and the environment is kept at high temperature, carbon and nitrogen atoms still diffuse into the inner side from surface layer, such that the carbon content at the surface drops after 270 min, and that the same situations occur in the diffused nitrogen. The variations of carbon and nitrogen contents versus time for certain special positions (corner and certain positions on the middle section with 0, 0.5 and 1.0 mm beneath the middle surface) are also represented in Figs. 5 and 6. Temperatures calculated by the program are compared with measured ones, which are shown in Fig. 7. The cal-
Fig. 4. Nitrogen profiles at different times.
Fig. 6. Nitrogen contents vs. time.
culated temperatures agree well with the measured ones. A small discrepancy between both data could be decreased by improving the heat exchange coefficient on the surface. Fig. 8 shows the calculated microstructure evolution during quenching with temperature change, which follows that the volume fraction of martensite is higher at certain depth (0.5 mm) than that on the surface in spite of the decrease in cooling rate from surface to center. This is due to the different carbon and nitrogen contents at the corresponding positions. As we know, diffused carbon and nitrogen prohibit martensite transformation, so positions with high carbon and nitrogen contents form smaller volume fraction of martensite. Thus, the distributions of microstructures in the outer layer have the profiles as shown in Fig. 9.
Fig. 7. Comparison of cooling curves.
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D.-Y. Ju et al. / Journal of Materials Processing Technology 143–144 (2003) 880–885
Fig. 8. Microstructures at 0.5 mm from the surface.
Fig. 11. Stress profiles on the middle section.
Fig. 9. Microstructure in the surface layer.
The measured hardness distribution on the middle section of the cylinder is depicted in Fig. 10. In the surface layer with the depth range between 0.3 and 0.6 mm, the hardness drop becomes smaller, which is attributable to more rich martensite formation in this range than that in the neighboring layers, as shown in Fig. 11. As for the general tendency of hardness decrease from surface to center, it may be also related to the carbon and nitrogen gradients besides with the effects of microstructure morphologies. Fig. 11 represents the profile of residual stresses on the middle section of the specimen. The data on the surface agree qualitatively with the measured values by X-ray diffraction method on the surface. And, the value of axial stress almost equals to that of tangential stress. Fig. 12 displays the calcu-
Fig. 12. Comparison of distortion.
lated results of final distortion, which also agree well to the measured displacements in the radial direction. The small discrepancy between the measured and calculated values may be caused by the following factors. Constitutive equations should be the functions of carbon and nitrogen contents, and transformation plasticity should be added while both effects neglected at present in the first approximation. But due to the lack of related experimental results about the effects of carbon and nitrogen contents on the related parameters used in the constitutive equations, they are neglected at present. These factors will be considered in hereafter research and calculations.
5. Conclusions
Fig. 10. Hardness distribution.
A numerical modeling is developed to predict the profiles of carbon and nitrogen as well as residual stresses and distortions after carburized and nitrided quenching. The predicted results agree well with the experimental results. This proves that the program can be used effectively to calculate the metallo-thermo-mechanical behavior during heat treatment and chemical treatment. Of course, there are some aspects to be improved, which can be realized by considering the effects of carbon and nitrogen contents on constitutive equations. The models about the interactions
D.-Y. Ju et al. / Journal of Materials Processing Technology 143–144 (2003) 880–885
between diffusion of nitrogen and carbon also need to be established. Acknowledgements The authors wish to express their gratitude for the financial support by High Technology Research Center, Saitama Institute of Technology, Japan. Mr. H. Kamisugi and Mr. H. Hirano (Dowa Mining Co. Ltd., Japan) are also acknowledged for the cooperation during the experiments involved in this project.
References [1] M. Kitamura, I. Tsukatani, T. Inoue, ISIJ Int. 34 (1994) 115. [2] E.J. Mittemeijer, M.A.J. Somers, Surf. Eng. 13 (1997) 483. [3] L. Torchane, P. Blilger, J. Dulcy, M. Gantois, Metall. Mater. Trans. A 27 (1996) 1823. [4] F. Xinmin, S. Fei, H. Fei, Ordance Mater. Sci. Eng. 22 (1999) 15 (in Chinese). [5] S. Tsuji, I. Ishigami, K. Yamanaka, Trans. Jpn. Inst. Met. 28 (1987) 48. [6] T. Inoue, D.-Y. Ju, K. Arimoto, in: Proceedings of the First International Conference on Quenching and Control of Distortion, ASM International, Chicago, IL, 1992, p. 205. [7] K.T. Raic, Surf. Coat. Technol. 92 (1997) 22.
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[8] G.H. Geiger, A.P. Poirier, Transport Phenomena in Metallurgy, Addison-Wesley, Reading, MA, 1973. [9] K.T. Raic, Scand. J. Metall. 22 (1993) 50. [10] R. Collin, M. Brachaczek, D. Thulin, J. Iron Steel Inst. 207 (1969) 1122. [11] R. Collin, S. Gunnarson, D. Thulin, J. Iron Steel Inst. 210 (1972) 785. [12] L. Sproge, J. Agren, J. Heat Treat. 6 (1988) 9. [13] G. Xinshu, in: Proceedings of the Seventh International Conference on Heat Treatment of Materials, vol. I, Moscow, 1990, p. 262. [14] K. Bongartz, et al., Metall. Trans. A 11 (1980) 1883. [15] K.T. Raic, et al., J. Metall. 1 (1995) 105. [16] T. Inoue, K. Arimoto, J. Mater. Eng. Perform. ASM Int. 6 (1997) 51. [17] C. Mgbokwere, M. Callabresi, J. Eng. Mater. Technol. 122 (2000) 305. [18] D. Pont, in: Proceedings of the Third World Congress on Computational Mechanics, vol. 2, Chiba, August 1994, p. 1723. [19] K. Arimoto, G. Li, A. Arvind, W.T. Wu, in: Proceedings of the 18th ASM Heat Treating Process Conference and Exposition, October 1998, p. 121. [20] H.C.F. Rozendaal, E.J. Mittemeijer, P.F. Colijn, P.J. Vander Schaaf, Metall. Trans. A 9 (1983) 395. [21] I. Ishigami, Y. Yokoyama, K. Miura, F. Uratani, H. Hoshino, J. Soc. Mater. Sci. Jpn. 49 (2000) 1235 (in Japanese). [22] W.A. Johnson, R.F. Mehl, Trans. AIME 135 (1939) 416. [23] D.Y. Ju, M. Narazaki, H. Kamisugi, H. Hirano, J. Shanghai Jaotong Univ. E 5 (1) (2000) 165. [24] C.L. Magee, Nucleation of Martensite, ASM, New York, 1968. [25] L. Cheng, A. Bottger, T.H. Keijser, E.J. Mittemeijer, Scripta Metall. Mater. 24 (1990) 509.
Numerical modeling and simulation of carburized and nitrided quenching process Dong-Ying Ju a,∗ , Chuncheng Liu b , Tatsuo Inoue c a
Department of Mechanical Engineering, Saitama Institute of Technology, Saitama, Japan b High Technology Research Center, Saitama Institute of Technology, Saitama, Japan c Department of Energy Conversion Science, Kyoto University, Kyoto 606-8501, Japan
Abstract In order to study metallo-thermo-mechanical behavior during carburized and nitrided quenching, a numerical model when considering quantitative effects of diffused carbon and nitrogen gradients and kinetics of phase transformation is proposed. Coupled calculations of diffusion, phase transformation and stress/strain give the final distributions of carbon and nitrogen contents as well as residual stress and distortion. Within the numerical simulation method, the effects of both transformation and lattice expansion induced from carbon and nitrogen absorption are considered when calculating the evolution of the internal stress and strain. In order to verify the method of numerical simulation, the simulated distributions of carbon and nitrogen content and residual stress/strain of a cylinder during carburized and nitrided quenching are compared with the measured results. © 2003 Published by Elsevier Science B.V. Keywords: Carburizing; Nitriding; Quenching; Metallo-thermo-mechanical behavior; Numerical simulation
1. Introduction Nitriding and carburizing are thermochemical surface treatments of large importance in engineering practice [1–4]. They are widely used in machine parts such as automotive gears or shafts due to their beneficial effects on fatigue resistance, tribological and anti-corrosion properties. The reasons for the property improvement are ascribed to the compound layers composed of fine nitrides or carbides of iron or alloy elements, and residual surface stresses are also attributed to this process. As a known fact, residual stresses are always accompanied by the changes in shape and size of workpiece, which often results in significant distortion of machine parts. This also leads to undesired effects such as increasing noise from transmission gears. In order to improve such kind of defects, manufacturing engineers need to have a thorough understanding how various process parameters influence the final profiles of carbon/nitrogen, stress/distortion and microstructures of heat-treated components. To do this, in recent years, there have been a great number of experimental and theoretical interests in mathematical description of carburizing and/or nitriding processes [5–17]. Numerical simulation by the finite element method (FEM), due to its benefit in reducing
∗
Corresponding author.
0924-0136/$ – see front matter © 2003 Published by Elsevier Science B.V. doi:10.1016/S0924-0136(03)00378-9
time-consuming and trial-and-error experiments that plague engineers, is looking forward to being used widely in engineering problems. Some FEM softwares have realized the simulation of carburizing, like HEARTS [6], ABAQUS [17], SYSWELD [18] and DEFORM-HT [19]. However, in these commercialized programs, some problems such as effect of diffusion of nitrogen or simultaneous diffusion of carbon and nitrogen on the following thermo-mechanical simulation still remain imperfect. With respect to this, the numerical modeling for simulating the metallo-thermo-mechanical behavior during carburized and nitrided quenching is developed in this paper. The purpose of the modeling aims to make it capable to calculate not only diffusion of multi-element during chemical treatment but also the evolution of microstructure, stress and distortion during carburized and nitrided quenching by considering the effect of C/N diffusion on distortion and phase transformation kinetics. This paper motivates to propose a numerical modeling of coupled diffusion of carbon/nitrogen, heat conduction, phase transformation and stress/strain. Within the numerical simulation method, the effects of both transformation and lattice expansion relevant to describing carbon and nitrogen absorption are considered when calculating the evolution of the internal stress and strain. In order to verify the numerical method, the simulated distributions of carbon and nitrogen contents, temperature and variation of microstructure as well as
D.-Y. Ju et al. / Journal of Materials Processing Technology 143–144 (2003) 880–885
residual stress/strain of a cylinder during carburized and nitrided quenching are compared with the measured results.
2. Numerical simulation models The modeling developed by the authors can be used to predict the evolution of temperature, microstructure, stress/distortion and profiles of chemical component such as carbon and nitrogen and so on during some processes of thermochemical treatment and heat treatment. The models involved in the FEM procedure can be divided into four categories of analysis: heat conduction, diffusion of C/N, transformation kinetics and stress/strain analysis by use of elasto-plastic constitutive equations. The following depictions are based on 3D coordinate, though it is also capable to axisymmetric problems in the program. When the boundary conditions are involved, the prescribed flux boundary conditions are employed. 2.1. Heat conduction The heat conduction equation governing temperature field T is ∂ ∂T ˙ ρcT − k − rv = 0 (1) ∂xi ∂xi ∂T ni = h(T − Tw ), ∂xi
of each constituent DNI (N), which is designed to be calculated by the following formula: QN ˆ DNI (N) = DNI,0 f(C, N)h(grad C, grad N) exp − RT (6) ˆ with the functions f(C, N) and h(grad C, grad N) of carbon and nitrogen concentrations and the gradients, respectively. And QN in Eq. (6) is the activation energy for diffusing nitrogen, and DNI,0 and R the material constants. The coefficient of surface reaction rate βN in Eq. (4) of nitriding process under partial pressure pH2 of hydrogen in the mixed gas is given by QN βN = β0 pH2 exp − , (7) RT where β0 is a constant [20]. The equation governing carbon diffusion kinetics can be simply obtainable by exchanging N and C in Eqs. (3)–(6). When the coefficient of surface reaction rate βC for carbon is decided by using experimental method [21], the carbon diffusion in steel can be expressed by the following boundary condition: ∂C −DC ni = βC (C − Cw ). (8) ∂xi 2.3. Phase transformation kinetics
with heat transfer boundary condition −k
881
(2)
where ρ, c and k denote density, specific heat and heat conductivity, respectively; rv the internal heat source, h and Tw the mean heat transfer coefficient and environmental temperature, respectively, on the boundary and ni the unit normal. The value of ρ, c and k should be calculated according to the mixture rule X = XI ξI , where X is a material property and ξ I is volume fraction of the Ith phase.
The microstructures involved in the present simulation are assumed to be composed of three constituents: austenite, pearlite or bainite and martensite. The diffusion type transformation from austenite to pearlite is calculated based on the modified Johnson–Mehl relation [22]. In this case of carbon steel S45C, the volume fraction of pearlite ξ p is expressed as [23]: t ξp = 1 − exp − f1 (T)f2 (C)f3 (σ)(t − τ)3 dτ , (9)
0
T − a1 a2
a3
a4 − T a5
a6
2.2. Diffusion of nitrogen and carbon
f1 (T) = a0
The following equations are the related diffusion formulas used for nitriding: ∂ ∂N ˙ − N DN − rN = 0 (3) ∂xi ∂xi
f2 (C) = exp{−a7 (C − C0 )},
(11)
f3 (σ) = exp{a8 σm },
(12)
with boundary condition −DN
∂N ni = βN (N − Nw ). ∂xi
(4)
Here, N, Nw and rN , respectively, denote diffused mass percent, environment potential and internal source of nitrogen. Global diffusivity DN is also assumed to be given by the mixture law DN = DNI (N)ξI (5)
,
(10)
where σ m is mean stress, C and C0 the current and initial carbon contents, respectively, and ai (i = 0, 1, 2, . . . , 8) are some transformation kinetic parameters. Diffusionless transformation is controlled by ξM = 1 − exp{φ1 T + φ2 (C − C0 ) + φ3 (N − N0 ) + φ4 σm +φ5 σe + φ6 },
(13)
by modified Magee’s rule [24], where ξ M is the volume fraction of martensite, φi (i = 1, 2, . . . , 6) are material parameters. N and N0 mean the current and initial nitrogen contents (wt.%) in the nodes, respectively.
882
D.-Y. Ju et al. / Journal of Materials Processing Technology 143–144 (2003) 880–885
In the above equations, the effects of C/N profile on transformation kinetics are considered. Due to the small penetrating depth of N atoms and high contents of C in the nitrogen diffusion layer, little pearlite transformation occurs in this layer. It should not matter greatly even though no effect of nitrogen is taken into account in diffusion type transformation kinetics. 2.4. Constitutive equations Total strain rate ε˙ ij is assumed to be sum of rates of elastic strain ε˙ eij , plastic strain ε˙ pij , thermal strain ε˙ th , transformation ij tr df strain ε˙ ij and diffusion induced strain ε˙ ij of carbon and nitrogen which is newly introduced in this paper. ε˙ ij =
ε˙ eij
p + ε˙ ij
+ ε˙ th ij
+ ε˙ trij
+ ε˙ df ij .
(14)
Here the four terms are given by 1+ν ν σij − σkk δij , E E K ∂F ∂F ∂F ∂F p ˆ ξ˙I ε˙ ij = G σkl + T˙ + ∂σkl ∂T ∂ξI ∂σij εeij =
(15)
(16)
I=1
with yield function p
F = F(σij , εij , κ, T, ξI ), ε˙ th ij =
K
(17)
αI T˙ δij ,
(18)
βJ→I ξ˙I δij .
(19)
I=1
ε˙ trij
=
K
If no transformation occurs during diffusion of carbon and nitrogen, the microstructure is only composed of austenite. Thus, the strain rate ε˙ df ij simply depends on the diffusion rate of carbon and nitrogen in the form: 1 a3 (Ct1 , Nt1 ) ε˙ df = − 1 ij 3 a3 (Ct0 , Nt0 ) 1 1 ˙ δij , × (21) C˙ + N Ct1 − Ct0 Nt1 − Nt0 where a is the lattice parameter of austenite, which is a function of current carbon and nitrogen contents. 3. Materials and condition of experiments and simulation The steel used in this paper is the carbon steel (JIS-S45C). The heat pattern is composed of five steps, as shown in Fig. 1. In the first step, the specimen is heated up to 860 ◦ C from room temperature and then mixed C/N gas is provided with certain carbon and nitrogen potentials, which are both 0.9 wt.% in the present case. The second step is for carburizing and nitriding for 270 min. After that, the specimen is cooled to 850 ◦ C for 10 min and then is held at this temperature for 30 min as diffusion period to decrease the gradients of carbon and nitrogen. The final step is for quenching of the specimen into oil quenchant of 60 ◦ C. The sizes of the cylindrical specimen are shown in Fig. 2, in which the holes (nos. 1 and 2) are drilled to insert thermocouples for temperature measurement during quenching. The mesh division for FEM calculation is also depicted in Fig. 2. Due to the symmetry, the one-fourth of the cross-section is
I=1
Here, E and ν are the Young’s modulus and Poisson’s ratio, respectively, T the temperature, and βJ→I the dilatation due to the microstructure change from the Jth to Ith phase. αI is thermal expansion coefficient of the Ith phase. Compared with the previous constitutive equations [16,17] of the authors, a new term ε˙ df ij in (14) due to carbon or nitrogen diffusion is introduced in this paper. When carbon or nitrogen is absorbed as interstitial atoms, the lattice parameters change to induce the volumetric dilatation. Some researchers proposed formulas of the lattice parameters depending on carbon and nitrogen contents [25]. If transformation (α → β) occurs between time interval t0 and t due to the diffusion of carbon and nitrogen, the strain ε˙ df ij is
T 1 Vβ (Ct1 , Nt1 )/nβ,Fe ε˙ df − 1 ξ˙α→β δij , (20) ij = 3 VαT (Ct0 , Nt0 )/nα,Fe where nI ,Fe means the number of Fe atoms in single unit cell of certain microstructure. VαT and VβT denote the aαT and aβT lattice parameter before and after α → β transformation, while Ct and Nt are the carbon and nitrogen contents at time t, respectively.
Fig. 1. Heat treatment process.
Fig. 2. Specimen and FE mesh.
D.-Y. Ju et al. / Journal of Materials Processing Technology 143–144 (2003) 880–885
Fig. 3. Carbon profiles at different times.
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Fig. 5. Carbon contents vs. time.
used for meshing by 160 nodes and 133 elements by use of quadrilateral iso-parameter elements with four nodes. The thermal and diffusion boundary conditions are both the flux type, and hardness and residual stresses are measured after experiments, the results of which are used to compare with the calculated ones.
4. Results and discussions Figs. 3 and 4 show the carbon and nitrogen profiles in the surface layer on the middle section after carburizing and nitriding, respectively. Because of the high surface reaction rate from nitrogen gas to surface and low diffusion rate of nitrogen in austenite, the depth of diffused nitrogen is only 500 m or so after 6 h. From the time 270–310 min, while gas is not provided and the environment is kept at high temperature, carbon and nitrogen atoms still diffuse into the inner side from surface layer, such that the carbon content at the surface drops after 270 min, and that the same situations occur in the diffused nitrogen. The variations of carbon and nitrogen contents versus time for certain special positions (corner and certain positions on the middle section with 0, 0.5 and 1.0 mm beneath the middle surface) are also represented in Figs. 5 and 6. Temperatures calculated by the program are compared with measured ones, which are shown in Fig. 7. The cal-
Fig. 4. Nitrogen profiles at different times.
Fig. 6. Nitrogen contents vs. time.
culated temperatures agree well with the measured ones. A small discrepancy between both data could be decreased by improving the heat exchange coefficient on the surface. Fig. 8 shows the calculated microstructure evolution during quenching with temperature change, which follows that the volume fraction of martensite is higher at certain depth (0.5 mm) than that on the surface in spite of the decrease in cooling rate from surface to center. This is due to the different carbon and nitrogen contents at the corresponding positions. As we know, diffused carbon and nitrogen prohibit martensite transformation, so positions with high carbon and nitrogen contents form smaller volume fraction of martensite. Thus, the distributions of microstructures in the outer layer have the profiles as shown in Fig. 9.
Fig. 7. Comparison of cooling curves.
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Fig. 8. Microstructures at 0.5 mm from the surface.
Fig. 11. Stress profiles on the middle section.
Fig. 9. Microstructure in the surface layer.
The measured hardness distribution on the middle section of the cylinder is depicted in Fig. 10. In the surface layer with the depth range between 0.3 and 0.6 mm, the hardness drop becomes smaller, which is attributable to more rich martensite formation in this range than that in the neighboring layers, as shown in Fig. 11. As for the general tendency of hardness decrease from surface to center, it may be also related to the carbon and nitrogen gradients besides with the effects of microstructure morphologies. Fig. 11 represents the profile of residual stresses on the middle section of the specimen. The data on the surface agree qualitatively with the measured values by X-ray diffraction method on the surface. And, the value of axial stress almost equals to that of tangential stress. Fig. 12 displays the calcu-
Fig. 12. Comparison of distortion.
lated results of final distortion, which also agree well to the measured displacements in the radial direction. The small discrepancy between the measured and calculated values may be caused by the following factors. Constitutive equations should be the functions of carbon and nitrogen contents, and transformation plasticity should be added while both effects neglected at present in the first approximation. But due to the lack of related experimental results about the effects of carbon and nitrogen contents on the related parameters used in the constitutive equations, they are neglected at present. These factors will be considered in hereafter research and calculations.
5. Conclusions
Fig. 10. Hardness distribution.
A numerical modeling is developed to predict the profiles of carbon and nitrogen as well as residual stresses and distortions after carburized and nitrided quenching. The predicted results agree well with the experimental results. This proves that the program can be used effectively to calculate the metallo-thermo-mechanical behavior during heat treatment and chemical treatment. Of course, there are some aspects to be improved, which can be realized by considering the effects of carbon and nitrogen contents on constitutive equations. The models about the interactions
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between diffusion of nitrogen and carbon also need to be established. Acknowledgements The authors wish to express their gratitude for the financial support by High Technology Research Center, Saitama Institute of Technology, Japan. Mr. H. Kamisugi and Mr. H. Hirano (Dowa Mining Co. Ltd., Japan) are also acknowledged for the cooperation during the experiments involved in this project.
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