Int. J. Microstructure and Materials Properties, Vol. 8, Nos. 1/2, 2013
Mathematical modelling and computer simulation of mechanical properties of quenched and tempered steel Božo Smoljan*, Dario Iljkić and Loreta Pomenić Faculty of Engineering, Department of Materials Science and Engineering, University of Rijeka, Vukovarska 58, HR-51000 Rijeka, Croatia Fax: 00385 51 651490 E-mail:
[email protected] E-mail:
[email protected] E-mail:
[email protected] *Corresponding author Abstract: A mathematical model and method of computer simulation for the prediction of mechanical properties of quenched and tempered steel were developed. Numerical modelling of hardness distribution in as-quenched steel components was performed based on experimental results of Jominy test. Modified Jominy test was designed for hardenability prediction of high-hardenability steels. Hardness of quenched and tempered steel was expressed as a function of maximal hardness of actual steel and representatives of chemical diffusivity of steel according to the time and temperature of tempering. Yield strength and fracture toughness distributions were estimated using the Hahn–Rosenfield approach. Fatigue resistance was estimated based on predicted microstructure and hardness. Distribution of other relevant mechanical properties was found out based on predicted as-quenched and tempered hardness of steel. Experimental investigation was performed on high-hardenability steel for tools and dies. The established procedure was applied in computer simulation of mechanical properties of a quenched and tempered steel workpiece. Keywords: high-hardenability steels; modified Jominy test; quenching; tempering; mathematical modelling; computer simulation; microstructure; mechanical properties; hardness; yield strength; fracture toughness; fatigue resistance. Reference to this paper should be made as follows: Smoljan, B., Iljkić, D. and Pomenić, L. (2013) ‘Mathematical modelling and computer simulation of mechanical properties of quenched and tempered steel’, Int. J. Microstructure and Materials Properties, Vol. 8, Nos. 1/2, pp.97–112. Biographical notes: Božo Smoljan received his BS and MS from the Faculty of Engineering of the University of Rijeka, Croatia, and his PhD from the Faculty of Mechanical Engineering of the Belgrade University, Serbia. Currently, he is Full Professor and Head of Department of Materials Science and Engineering at the Faculty of Engineering of the University of Rijeka, Croatia. He is President of the Croatian Society for Heat Treatment and Surface Engineering. He is author of three books and more than 100 scientific article Copyright © 2013 Inderscience Enterprises Ltd.
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B. Smoljan et al. publications. His fields of interest are mathematical modelling of materials, computer simulation of thermal processes, heat treatment of metals, material testing and material characterisation. Dario Iljkić received his BS and PhD from the Faculty of Engineering of the University of Rijeka, Croatia. Currently, he is Lecturer at Department for Materials Science and Engineering at the Faculty of Engineering of the University of Rijeka, Croatia. He is a Member of Croatian Society for Heat Treatment and Surface Engineering, a Member of Technical Committee for Casting on Croatian Standards Institute, a Member of the Editorial Board of Journal for Theory and Application in Foundry, and a Member of several organising committees of the scientific conferences, and an author or a co-author of ca 40 papers. His areas of research include material engineering and casting technology. Loreta Pomenić received her BS from the Faculty of Technology of the University of Zagreb, Croatia, MS from the Faculty of Chemistry and Chemical Technology of the University of Ljubljana, Slovenia, and her PhD from the Faculty of Engineering of the University of Rijeka, Croatia. Currently, she is Full Professor at Department of Materials Science and Engineering at the Faculty of Engineering of the University of Rijeka, Croatia. She is a Member of Croatian Society for Materials and Tribology, a Member of Administrative Committee of Croatian Society for Materials Protection, a Member of Croatian Society for Engines and Vehicles. Her fields of interest are materials chemistry, corrosion and metals protection. This paper is a revised and expanded version of a paper entitled ‘Mathematical modelling and computer simulation of mechanical properties of quenched and tempered steel’, presented at the 8th International Conference on Industrial Tools and Material Processing Technologies at Ljubljana, 2–5 October, 2011.
1
Introduction
Computer simulation of quenching includes several different analyses. Computer simulations of hardness distribution in quenched steel specimens could be consisted of computer simulations of specimen cooling, specimen hardening and, respectively, of prediction of mechanical properties (Smoljan, 2000). The objective of design of quenching is to estimate the results of quenching, concerning mostly with the estimation of mechanical properties. For the simulation of specimen cooling which is a thermodynamical problem, it is necessary to establish the appropriate algorithm which describes the cooling process, and to accept appropriate input data. The accuracy of numerical simulation of quenching directly depends on the applied input variables. Calibrated data are not as precise as experimentally acquired data, but they are useful for large spectra of specimen dimensions (Smoljan, 1995). Structure composition can be defined by kinetic equations of prior structure transformation or can be estimated by using Continuous Cooling Transformation (CCT) diagrams (Liščić and Totten, 1994). Structure transformation and hardness distribution can also be estimated based on time relevant to structure transformation (Rose and Wever, 1954).
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One of the most common methods of computer prediction of quenching results is based on the chemical composition of steel and on the specimen dimensions (Dobrzanski and Trzaska, 2004). Moreover, prediction of microstructure composition usually is based on semi-empirical methods derived from kinetic equations of microstructure transformation (Serajzadeh, 2004; Reti et al., 1997). The predicted microstructure composition can be used to predict mechanical properties, mostly focused on hardness. The structure transformations and hardness distribution can be estimated based on time, relevant for structure transformation. Cooling time from 800°C to 500°C, t8/5, usually is accepted, as relevant time for quenching. If the cooling time t8/5 is equal for two different specimens, i.e., quenched workpiece and Jominy specimen, the hardness of these two specimens could be equal to each other (Rose and Wever, 1954; Smoljan, 1999; Mayner et al., 1978). To accept the assumption that the equal cooling time t8/5 of several samples indicates their equal hardness, the history of cooling of these samples must be the same or similar. By involving the cooling time t8/5 in the mathematical model of steel hardening, the Jominy test results could be involved in the model. Since the critical cooling rate of martensite transformation of high-hardenability steels is less than minimal cooling rate of Jominy specimen, the application of the original Jominy test is not acceptable. Numerical modelling of the quenched hardness of high-hardenability steel could be performed based on relevant time of cooling by involving the results of modified Jominy test (Smoljan et al., 2009b, 2011).
2
Prediction of cooling time t8/5
The temperature field change in an isotropic rigid body without heat sources can be described by Fourier’s law of heat conduction:
δ (c ρ T ) = div λ grad T , δt
(1)
where
λ: ρ: c:
Coefficient of heat conductivity [W/(mK)] Density [kg/m3] specific heat capacity [J/(kgK)].
Characteristic initial condition is
−λ
δT = α (Ts − T f ), δn s
where Ts: Surface temperature [K] Tf: Quenchant temperature [K] Α: Heat transfer coefficient [W/(m2K)].
(2)
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Solution of equation (1) can be found out using the finite volume method (Smoljan, 1998; Patankar, 1980). Transient temperature field in an isotropic rigid body can be defined by 2-D finite volume formulation (Figure 1): 2 ⎛ 2 ⎞ 2 Tij1 ⎜ ∑ bI(i + n ) j + ∑ bJ i ( j + n ) + bij ⎟ = ∑ ( bI( i + n ) j T(1i + k ) j + bJ i ( j + n )Ti1( j + k ) ) + bij Tij0 m =1 ⎝ m =1 ⎠ m =1 i = 1, 2,..., imax , j = 1, 2,..., jmax , n = 2 − m, k = 3 − 2m,
(3)
where
Tij0 :
Temperature in the beginning of time step Δt [K]
Tij1 :
Temperature in the end of time step Δt [K]
Δt:
Time step [s].
bij = (ρijcijΔVij)/Δt, ΔVij [m3] is the volume of the control volume, bI(i+n)j = WI(i+n)j–1 and bJi(j+n) = WJi(j+n)–1, where variables WI(i+n)j and WJi(j+n) are the thermal resistances between ij and (i + k)j volume and between ij and i(j + k) volume. WI (i + n ) j =
WJi ( j + n ) =
1 ΔFI (i + n ) j 1 ΔFJi ( j + n )
lI ( i + k ) j (3− m ) ⎛ lIijm + ⎜⎜ λ(i + k ) j ⎝ λij ⎛ lJijm lJi ( j + k )(3− m ) + ⎜⎜ λi ( j + k ) ⎝ λij
⎞ ⎟⎟ ⎠
(4)
⎞ ⎟⎟ . ⎠
(5)
Thermal resistances for boundary volume are WI ( i + n ) j =
WJi ( j + n ) =
1 ΔFI ( i + n ) j
1 ΔFJi ( j + n )
⎛ lIij 1 ⎜⎜ + λ α ϕ I (i + n ) j cos TsI ( i + n ) j ⎝ ij
⎞ ⎟⎟ ⎠
(6)
⎛ lJij 1 ⎜⎜ + ⎝ λij α TsJ i ( j + n ) cos ϕ Ji ( j + n )
⎞ ⎟⎟ , ⎠
(7)
where Heat transfer coefficient at the boundary temperature Ts [W/(m2K)] cos φ: Direction cosines of heat flux.
αTs:
Discretisation system has N linear algebraic equations with N unknown temperatures of control volumes, where N is total number of control volumes. Time of cooling from Ta to specific temperature in a particular point is determined as sum of time steps, and in this way, the diagram of cooling curve in every grid-point of a specimen is possible to find out. M
t M = ∑ Δt m . m =1
(8)
Mathematical modelling and computer simulation Figure 1
3
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Control volume
Simulation of quenched workpiece hardness based on the cooling time t8/5
One of the most important factors for efficient simulation of hardening is the proper selection and use of representative cooling phenomena that is relevant for microstructure transformation. In the developed computer simulation of hardenability of quenched workpiece, the hardness at different workpiece points is estimated by the conversion of the cooling time t8/5 to the hardness (Figure 2) (Rose and Wever, 1954; Smoljan et al., 2007). This conversion is provided by the relation between the cooling time t8/5 and distance from the quenched end of the Jominy specimen. Figure 2
4
Conversion of the cooling time t8/5 to the hardness
Simulation of quenched workpiece hardness of high-hardenability steel
For prediction of hardness of the quenched workpiece, it is necessary that the cooling times t8/5 for austenite decomposition of investigated workpiece and the cooling times t8/5
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of Jominy specimen are in the same range. Because of high hardenability, the cooling times t8/5 for austenite decomposition of most steels for tools and dies are not comparable with the cooling times t8/5 of Jominy specimen, and there are limits in the application of the original Jominy test in computer simulation of quenching of this kind of steels (Smoljan et al., 2006). It is known that for tools and die steels, the cooling times t8/5 of austenite decomposition range from 200 s to 1000 s (Smoljan et al., 2009b). The cooling time t8/5 for bainite transformation of steels X38CrMoV51 and X45NiCrMo4 is greater than 1400 s. The cooling time t8/5 for pearlite transformation of steel X45NiCrMo4 is greater than 45,000 s. Original Jominy test gives the cooling times t8/5 up to a maximum of 200 s (Figure 2), and it is obvious that the original Jominy test is not suitable for prediction of hardenability of steels for tools and dies. To achieve cooling times t8/5 longer than 200 s in end quench test, Jominy-modified cylindrical specimen (JMC®) was designed for high-hardenability steels, i.e., steels for tools and dies (Figure 3). Other pieces of assembly are slowly cooled. Figure 4 shows the cooling times t8/5 at a depth of 0.8 mm from the surface of the JMC® specimen at different distances from the quenched end of the JMC® specimen. Maximal cooling time t8/5 of the JMC® specimen is nearly ten times longer than the maximal time given by the Jominy test. The JMC® specimen is ground flat along its length to a depth of 0.8 mm. The Vickers hardness is measured at intervals of 2 mm along its length beginning at the quenched end. The Vickers hardness values are plotted vs. distance from the quenched end. In Figure 5 are shown hardenabilty curves of steels X40Cr13, X210Cr12, 60WCrV7 and 60WCrV7 tested by both JMC® test and Jominy test. Figure 3
JMC® specimen and JMC® specimen holder
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Figure 4
Distance from the quenched end of JMC® specimen vs. cooling time t8/5
Figure 5
Hardenability curves of different steels: (a) JMC® test and (b) Jominy test
(a)
5
(b)
Comparison of cooling rates of cylindrical specimens and JMC® specimen
In order to compare the kinetics and history of cooling in the JMC® specimen and actual steel workpieces, the cooling curves of steel workpieces have been compared with the cooling curves in different locations of the JMC® specimen. Cooling curves at points at a depth of 0.8 mm from the surface and at different distances from the quenched end of the JMC® specimen have been given by computer simulation. Computer simulation of cooling curves of cylindrical specimens (workpieces), in which heights are equal to four diameters (4D), has been done at different points of the cross section at their half height. In Figure 6, the cooling curves of cylindrical specimens quenched in oil are compared with the cooling curves at different locations of the JMC® specimen. The cooling curves of cylindrical specimens quenched in oil and the cooling curves at different locations of the JMC® specimen are similar enough to encourage the usage of the JMC® specimen in the estimation of hardness of quenched steel specimens. Satisfying results of mechanical properties of cylindrical specimens of high-hardenability steels with small dimensions can be obtained by cooling in air (Smoljan et al., 2009b).
104 Figure 6
B. Smoljan et al. Cooling curves of cylindrical specimen (D = 500 mm)
Source: Smoljan et al. (2009b)
It is possible to establish a relation between critical diameters (which gives the microstructure of at least 50% martensite in the entire cross section) of cylindrical specimens and the JMC® distance (Figure 7). The relation given in Figure 7 can be used in the same way as the relation between distance on standard end quench test and diameter of round (Hoyt, 1952). Figure 7
Evaluation of the critical diameters of cylindrical specimens for steel X210Cr12 (austenitised at 970°C)
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Figure 7 shows the evaluation of the critical diameters of cylindrical specimens made of steel X210Cr12 (austenitised at 970°C), for different quenchants. From Figure 7 it is visible that quenching in water of cylindrical specimens made of steel X210Cr12, with diameters up to 327 mm, gives the microstructure of at least 50% martensite in the entire cross section. For quenching in oil, these diameters of cylindrical specimens are up to 268 mm, and for cooling in air these diameters are up to 71 mm (Smoljan et al., 2009b).
6
Prediction of quenched steel microstructure
The austenite decomposition results can be estimated based on time, relevant for structure transformation. Usually the characteristic cooling time relevant for structure transformation for most steels is the time t8/5. It can be accepted that if other heat treatment parameters are constant, the austenite decomposition results in some location of a cooled specimen depending only on time t8/5. Microstructure composition of as-quenched steel depends on the chemical composition, severity of cooling, austenitising temperature and steel history. Actual steel hardness HV is function of microstructure composition: HV = ⎡⎣ (%ferrite + %pearlite)HV( F + P ) + (%bainite)HV( B ) + (%martensite)HV( M ) ⎤⎦ 100 ,
(9) where HV(F+P): Steel hardness HV of ferrite + pearlite HV(B): Steel hardness HV of bainite HV(M): Steel hardness HV of martensite. Sum of all phase’s portions is equal to unity: [(%ferrite + %pearlite) + %bainite + %martensite] 100 = 1.
(10)
If the total hardness in some location is known and hardness of microstructure constituents separately is known, and if the phase fraction of one of the microstructure constituents is known, it is not difficult to predict fractions of other phases by equations (9) and (10). Both hardness (HV) of cooling microstructure with 90% or 50% of martensite and of 10%, 50% or 90% of (ferrite + pearlite), and hardness of microstructure constituents separately have to be known (Smoljan, 2002). It could be written for Jominy specimen that phase hardness depends on chemical composition and cooling rate parameter that corresponds to actual distance d of the Jominy specimen quenched end. It was adopted that the cooling rate parameter is equal to log(t8/5) (Smoljan, 2002). M d
HV
M max
= HV
− K M log
t8M5 d t8M5 max
,
(11)
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B. Smoljan et al. B HVdB = HVmax − K B log
t8B5 d t8B5 max
(12)
.
and HVdP + F = HVNP + F + K P + F log
where N: B max
HV
t8P5+NF t8P5+dF
(13)
,
Normalising :
Hardness of lower bainite.
Characteristic value of HV, KM, KB, KP+F and t8M5 d , t8M5 max , t8B5 d , t8B5 max , t8P5+NF in equations (11)–(13) has to be evaluated for investigated steel by regression analysis and quantitative microstructure analysis. Hardness of quenched structures with characteristic percentage of martensite can be predicted by using the diagram of hardness at different percentages of martensite vs. carbon content after Hodge and Orehoski (Smoljan and Butković, 1998) and Jominy curve.
7
Prediction of quenched and tempered steel mechanical properties
The reference value of hardness of quenched and tempered state, HRCtempered, can be estimated based on as-quenched hardness, HRCquenched, by (Smoljan et al., 2009a; Reti et al., 2009):
HRC tempered =
HRCquenched − HRC min K
+ HRCmin ,
(14)
where HRCmin is the material constant. K is the factor which can be expressed as
⎡ ⎛ Ttr ⎞n1 ⎤ K = exp ⎢ AB ⎜ ⎟ ⎥ , ⎢⎣ ⎝ a ⎠ ⎥⎦
(15)
where Ttr is the reference value of tempering temperature [K] and A, B, a and n1 are the material constants, which are established by regression analysis of hardness of quenched and tempered steel (Iljkić, 2010). The algorithm for prediction of hardness of tempered and quenched steel given by equations (14) and (15) was established by regression analysis. Mechanical properties of quenched steel or quenched and tempered steel directly depend on the degree of quenched steel hardening (Smoljan, 2006). One most tested relation in material science is the relation between hardness and ultimate tensile stress. Relation between hardness HV and ultimate tensile stress, Rm [MPa], is equal to Rm = 3.3 HV.
(16)
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By experimental work it was found out that the relation given by equation (16) is valid for tensile strength ranging between 400 MPa and 2500 MPa (Pavlina and Van Tyne, 2008). The relation between ultimate tensile stress, Rm [MPa], and yield strength, Rp0,2 [MPa], is given by (Just, 1976) R p 0,2 = ( 0.8 + 0.1S ) Rm + 170 S − 200.
(17)
Coefficient S which is the ratio between the actual hardness and hardness of martensite in Rockwell C hardness should be taken into account, since as-quenched and quenched and tempered steel properties depend on the degree of quenched steel hardening (Smoljan, 2006). Fracture toughness, KIc [MPam1/2], can be estimated from the mechanical properties obtained by tensile test. The Hahn–Rosenfield correlation can be successfully used for that purpose (Smoljan and Butković, 1998): K Ic =
ε f n 2 ERe 60
,
(18)
where
εf: n: E:
True fracture strain Strain-hardening exponent Modulus of elasticity [MPa].
True fracture strain can be expressed by percentage reduction of area, Z [%]: −1
Z ⎞ ⎛ ε f = ln ⎜1 − ⎟ . ⎝ 100 ⎠
(19)
For many steels, percentage reduction of area, Z [%], can be estimated from the tensile strength by (Just, 1976): Z = 96 − ( 0.062 − 0.029 S ) Rm .
(20)
The strain-hardening exponent, n, can be defined by n
Rm ⎛ n ⎞ , −⎜ ⎟ ≈0 R p 0,2 ⎝ 0.002e ⎠
(21)
where e is the base of the natural logarithm. In comparison with yield strength, the fatigue resistance properties additionally depend on microstructure, especially they depend on ferrite phase composition. The fatigue crack initiation threshold, ∆Kth, below which fatigue cracks would not initiate at workpiece points in the quenched and tempered state can be established as a function of the yield strength, microstructure and grain size diameter (Figure 8) (Smoljan et al., 2011). Results in Figure 8 are compiled from results listed in references (Ritchie, 1979; Vosikovsky, 1979; Lindley and Richard, 1978; Wilson, 1996).
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Figure 8
8
Variation of fatigue crack initiation threshold of steel with yield strength and grain size
Application
The established method for prediction of yield strength, fracture toughness and fatigue resistance is applied in the design of heat treating process of the blanking and forming punch of the die made of steel EN 60WCrV7 (Smoljan et al., 2011). The chemical composition of the investigated steel, EN 60WCrV7, was 0.55% C, 0.94% Si, 0.34% Mn, 0.015% P, 0.012% S, 1.27% Cr, 0.05% Mo, 0.12% Ni, 0.18% V and 2.10% W. Results of the JMC® test of the investigated steel are shown in Table 1. Table 1
JMC® test results of the investigated steel EN 60WCrV7
JMC® distance [mm]
2
4
6
8
10
15
20
25
30
35
40
Hardness HRC
63
63
63
63
62
62
61
60
58
56
55
JMC® distance [mm]
45
50
55
60
65
70
75
80
85
90
–
Hardness HRC
53
52
51
51
50
50
50
49
49
49
–
Geometry of the blanking and forming punch prepared for the heat treating processes is shown in Figure 9. After heating to 950°C for 2 h, punch was quenched in agitated oil with the severity of quenching H = 0.3. The tempering temperature in the process was 500°C. Based on the JMC® test results, the diagram of microstructure composition in dependency of cooling times, t8/5 is established. The calculated microstructure compositions vs. cooling time t8/5 of the investigated steel are shown in Figure 10. Hardness distribution in the quenched and tempered steel punch is shown in Figure 11. Critical locations for crack growth are locations 1–3 (Figures 9 and 11). The predicted values of microstructure and mechanical properties in critical locations of the as-quenched workpiece are given in Table 2. The predicted values of mechanical properties in critical locations of the quenched and tempered workpiece are given in Table 3.
Mathematical modelling and computer simulation Figure 9
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Geometry of blanking and forming punch
Figure 10 Microstructure compositions vs. cooling time t8/5
P: Pearlite; B: Bainite; M: Martensite. Microstructure and mechanical properties in critical locations of the as-quenched workpiece
Table 2
Critical location in Figure 11 Properties Phase fractions [%]
P
2
3
0
0
B
5.6
0
3.4
M
94.4
100
96.6
Hardness HRC Table 3
1 0
54
60
55
Mechanical properties in critical locations of the quenched and tempered workpiece Critical location in Figure 11
Properties
1
2
3
Hardness HV
628
775
646
Yield strength, Rp0,2 [MPa]
1784
2250
1842
Fracture toughness, KIc [MPam1/2]
44
27
42
Fatigue threshold, ∆Kth [MPam1/2]
3.8
3.8
3.8
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Figure 11 Hardness distribution in quenched and tempered steel die (see online version for colours)
9
Conclusion
A mathematical model for prediction of mechanical properties in quenched and tempered steel die was developed. The application of modified Jominy specimen, JMC®, in numerical simulation of high-hardenability steel quenching was investigated. By comparison of times of cooling of the JMC® specimen with times of cooling required to achieve martensite microstructure in quenching of high-hardenability steels, i.e., steels for tools and dies, it was found that the JMC® specimen is adequate for estimation of hardness of quenched workpieces made of high-hardenability steels. Based on the both, equal time of cooling and similarity of cooling curves in characteristic points of the investigated workpieces and JMC® specimen, it can be concluded that the JMC® test can be accepted as a very useful test for estimation of the hardness of the quenched workpieces made of high-hardenability steels. A developed mathematical model has been applied in computer simulation of a quenched and tempered steel die component. The results of the as-quenched hardness distribution and austenite decomposition in the quenched steel die component can be estimated based on calculated time of cooling from 800°C to 500°C, t8/5 in workpiece point and on results of the modified Jominy specimen (JMC® specimen). The prediction of distribution of hardness, yield strength, fracture toughness and fatigue crack initiation threshold can be calculated based on steel as-quenched hardness distribution and temperature of tempering. It can be concluded that mechanical properties of quenched and tempered die steel can be successfully calculated by the proposed method.
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