Study on constitutive modeling and processing maps

Materials and Design 90 (2016) 804–814

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Materials and Design journal homepage: www.elsevier.com/locate/matdes

Study on constitutive modeling and processing maps for hot deformation of medium carbon Cr–Ni–Mo alloyed steel Chi Zhang a,b, Liwen Zhang a,⁎, Wenfei Shen a, Cuiru Liu a, Yingnan Xia a, Ruiqin Li a a b

School of Materials Science and Engineering, Dalian University of Technology, Dalian, 116024 Liaoning, China State Key Lab of Rolling Technologies and Automation, Northeastern University, Shenyang, 110819 Liaoning, China

a r t i c l e

i n f o

Article history: Received 1 August 2015 Received in revised form 23 October 2015 Accepted 9 November 2015 Available online 10 November 2015 Keywords: Medium carbon steel Hot deformation behavior Constitutive modeling Processing map

a b s t r a c t The hot deformation behavior of medium carbon Cr–Ni–Mo alloyed steel 34CrNiMo was studied in the wide temperature range of 900–1150 °C and the strain rate of 0.002–5 s−1. The thermo-mechanical analysis using Thermo-Calc software indicates austenite is the main phase during hot compression, which has a high occurrence tendency of DRX because of low stacking fault energy of such a phase. Constitutive equations for flow stress were developed by using the work hardening curve for the work hardening and dynamic recovery period and Avrami equation for the dynamic recrystallization period. All the factors in the equations can be illustrated in terms of Zener–Hollomon parameter. Besides, processing maps based on dynamic material model at the strains of 0.2, 0.4, 0.6, 0.8 and 1.0 were established. The concluded optimum hot working parameters are at 950–1100 °C & 0.01–0.2 s−1 and 1100–1150 °C & 0.01–5 s−1. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction 34CrNiMo, which is a medium carbon Cr–Ni–Mo alloyed steel, is a prime candidate material for manufacturing spindles. The spindle is one key component of mechanical machines to assure the safety and stability of operation. For example, the wind turbines, converting wind power to kinetic energy, are developed fast and the power can reach 5 MW or larger so far [1]. This demands the spindle processing excellent mechanical properties [2]. The spindles are mainly manufactured by forging or rolling. During these hot working processes, the microstructure and deformation resistance of metals or alloys undergo complex variations with different deformation conditions, including strain, strain rate, deformation temperature [3]. In order to control the forming process to ensure the good performance of spindles, it is of great importance to investigate the hot deformation behavior of the medium carbon Cr–Ni–Mo alloyed steel. The modeling of the hot flow stress and prediction of flow stress for unseen deformation conditions are quite important in metal forming process, since any feasible mathematical simulations needs accurate flow description. Lin and Chen have presented a critical review on some recent experimental results and constitutive equations for metals and alloys in hot working [4]. A number of constitutive equations were developed to describe the hot deformation behaviors of metals and alloys, including steel [5,6], magnesium alloy [7,8], aluminum alloy [9], titanium alloy [10], and superalloy [11]. The capability of these equations ⁎ Corresponding author at: School of Materials Science and Engineering, Dalian University of Technology, Dalian 116085, China. E-mail address: [email protected] (L. Zhang).

http://dx.doi.org/10.1016/j.matdes.2015.11.036 0264-1275/© 2015 Elsevier Ltd. All rights reserved.

to represent the flow behavior of material was also comparatively analyzed [12]. These equations exhibit a high accuracy comparing with the experimental data for different specific material. However, parts of the previous researches mainly focused on phenomenological modeling the flow stress using mathematical methods or computational programming. A relatively fewer attentions have been paid on modeling the flow stress based on the deformation mechanisms of metals, recently [13,14]. The start of hot deformation of metals is elastic deformation, followed with work hardening (WH) and dynamic recovery (DRV). After that when dynamic recrystallization (DRX) occurs, the flow stress will decreases with the further increasing of strain, resulting in a peak stress in the flow stress curves. Bergstrom and Aronsson have modeled the strain and temperature dependence of the strain hardening for mild steels using dislocation model [15]. And for the recrystallization kinetics, the Avrami relation has been well known to describe the static recrystallization since the late 1930s. It is found that the Avrami equation is also applicable for describing the DRX, recently [14]. Therefore, it is necessary to model the flow stress in terms of the deformation mechanisms of metals. In addition, the processing map, which is based on dynamic materials model (DMM), is an explicit representation of the response of a material, in terms of microstructure evolution mechanisms, to the imposed process parameters and consists of a superimposition of power dissipation and instability map [16,17]. The process condition with high efficiency of power dissipation and without flow instability is usually chosen as the optimum working parameters. Yang et al. [18], Lin et al. [19], Li et al. [20], and Sun et al. [10] have constructed the processing maps for low carbon bainitic steel, Ni-based superalloy, medium carbon steel and Ti alloy, respectively. The intrinsic workability for these

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materials has been discussed through the establishment of processing map. It is an effective supplementary technique to be used for optimizing the hot working parameters. In general, the constitutive equation and processing maps are the basements for optimizing hot working process to control the microstructure and properties. Some researches have been conducted to investigate the deformation behavior of medium carbon steel [5,21]. However, systematic investigation of the intrinsic workability of medium carbon Cr–Ni–Mo steel at high temperature is still very limited. This brings along challenges for controlling the processing of spindle manufacture. For the purpose of acquiring high performance of the spindles during processing, constitutive equation and processing map of the medium carbon Cr–Ni–Mo steel 34CrNiMo during hot deformation were developed in this work. 2. Experimental procedure Table 1 shows the chemical compositions of the tested steel, in which Cr, Ni and Mo were added. The occurrence tendency of DRX is different for the austenite and ferrite because of the stack fault energy preserving in the phases. Thus the equilibrium phase diagram at high temperatures for the steel was calculated using the software ThermoCalc for the chemical compositions. After casting, the ingot was forged to a bar with diameter of 200 mm. Cylindrical specimens with the diameter of 8 mm and the height of 12 mm were machined from the same area of the bar to acquire uniform initial microstructure. Isothermal compression tests were performed on a Gleeble 1500 thermal-mechanical simulator. The specimens were heated up to 1200 °C with a heating rate of 10 °C/s and held for 180 s to obtain equiaxed and homogeneous microstructure before compression. After that, the specimens were cooled down to 900, 950, 1000, 1050, 1100, and 1150 °C at a cooling rate of 10 °C/s and held for 30 s to eliminate the temperature gradient, respectively. Then the specimens were compressed to 1.0 strain with the strain rate of 0.002, 0.01, 0.1, 1 and 5 s−1, followed by fast nitrogen cooling. The compressed specimens were sectioned parallel to the compressing axis for microstructural observation. The specimens were mechanical polished and etched in an abluent solution of saturated picric acid. The region with a half radius from the compression axis at middle layer was examined using an optical microscope.

Fig. 1. Equilibrium phase constituents (mol%) predicted by Thermo-Calc software for the tested steel.

3.2. Flow stress curves and friction correction In the deformation tests, tantalum plates or solid lubricants are usually dedicated to decrease the friction in the interface between the sample and the die. Even using the tantalum plates, slight barreling of the edges of the specimen was still occurred after the deformation (Fig. 2a). This indicates that the lateral flow of material is suppressed by the friction. The existence of friction force would influence the compress pressure, resulting in that the detected flow stress curve is not the real one. This influence becomes more obvious at high deformation degrees [23]. Therefore, the measured flow stress should be corrected by considering the effects of friction. Besides, the temperature rising during deformation will also influence the results. In present study, the majority of deformation heat is thought to be dissipated by radiation and convection through the dies due to the small size of the sample [24]. And thermocouples were fixed at the surface of samples to trace and adjust the temperature. Thus, the influence of deformation heat was not considered during the stress correction. The effect of friction on the flow stress was considered using the following equations [24,25]: C2P 2½ expðC Þ−C−1

ð1Þ

2μr h

ð2Þ

3. Results and discussion

σ¼

3.1. Equilibrium phase diagram

with

The equilibrium phase diagram for the chemical compositions of tested steel was thermodynamic analyzed using Thermal-Calc software. The molar fractions (NPM) of phase constituent varying with the temperatures are shown in Fig. 1. It can be seen that the liquid is transformed to austenite with FCC structure at 1487–1417 °C. The fraction of austenite occupies about 99.9% in the temperature range of 1417– 750 °C. Ferrite with BCC structure is formed below 750 °C. If holding below 750 °C for a long time, M7C3-carbide and cementite will be formed. In general, the forging process is conducted at the temperature range of 850–1150 °C. So, the austenite is the main phase during forging. The austenite has low stacking fault energy (SFE) which has a high occurrence tendency of DRX during deformation [22]. And the DRX will have a big influence on both the flow stress and microstructural evolution.

C¼

where σ and P are the flow stress without and with friction, μ, r, and h are the friction coefficient, the radius and the height of the specimen. Ebrahimi and Najafizadeh have developed a function to calculate the friction coefficient by considering the geometry of the deformed sample [23]. The function is as follows: ðR=hÞb μ ¼  pffiffiffi  pffiffiffi 4= 3 − 2b=3 3 with b¼4

Table 1 Chemical compositions of the tested steel (wt.%). C

Si

Mn

P

S

Cr

Ni

Mo

Al

0.36

0.30

0.70

0.008

0.009

1.67

1.64

0.25

0.019

ð3Þ

ΔR h R Δh

ð4Þ

where R is the theoretical frictionless radius of the sample after deformation based on volume constancy, b is the barreling factor, ΔR is the difference between the maximum and minimum radiuses of the

806

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Fig. 2. Macro-photograph of the deformed samples (a) and dimension representations (b).

Fig. 3. True stress–strain curves of 34CrNiMo during hot compressive deformation. (a) 0.002 s−1; (b) 0.01 s−1; (c) 0.1 s−1; (d) 1 s−1; and (e) 5 s−1.

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deformed sample (ΔR= RM − RT), and Δh is the difference between the initial and final heights of the sample. It is difficult to detect the top radius of the sample (RT) directly. With approximation of the profile of the barreled specimens with an arc of a circle, RT can be determined by following equation [23]:

describes the relations between stress and strain during the deformation process that have not yet undergone DRX. In the DRX region, the evolution of dislocation density depends on the DRX kinetics. The Avrami equation has been demonstrated to characterize the kinetics of DRX behavior sufficiently as follows [14,30]:

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h0 RT ¼ 3 R0 2 −2RM 2 : h

X d ¼ 1− exp −K

" ð5Þ

The geometry parameters used in above equations are schematically shown in Fig. 2b. Thus, the experimental flow curves were corrected using above equations, which are shown in Fig. 3. After friction corrected, the effects of barreling on the flow stress curves weakened. It should be mentioned that due to the high deformation degree, this tendency has not been eliminated completely. The flow curves obtained at the lower strain rates and/or higher temperatures (0.002 s−1 & 900–1150 °C, 0.01 s−1 & 950–1150 °C, 0.1 s−1 & 1000–1150 °C, 1.0 s−1 & 1100–1150 °C) exhibit a typical form of DRX behavior. At initial stage, the flow stress increases with the increasing of strain, causing by the effects of WH. New DRX grains appear during deformation. These new grains produce softening, which decreasing the WH rate until eventually there is a clear stress peak. The flow stress then decreases with the increasing of strain until a steady state is attained. The steady state stress reflects the dynamic equilibrium between strain hardening and strain softening due to the formation of new grains and the associated grain boundary migration [22]. And for the other flow curves that deformed at relatively higher strain rates and/or lower temperatures, the softening caused by DRX is less obvious. However, DRX occurs in these conditions since the stress decreases slightly with the increasing strain in the range of 0.4–0.6. Because the effect of barrelling is not totally corrected, the barrelling causes the deviation of stress at high strains, especially at strain higher than 0.6. This makes the difference between peak stress and steady state stress small. Moreover, the occurrence of DRX was confirmed by optical observations which will be illustrated in Section 3.4. Therefore, DRX is likely to be occurred in the evaluated conditions for the tested steel. By comparison, it can be seen that the critical strain for peak stress decreases with the increasing of temperatures and decreasing of strain rates. This is because that the DRX is thermal activated, which would be facilitated with the increasing of temperature. And the rate of mobile dislocation generation would become slow with the increasing of strain rate, making the peak strain moves to right side in the flow stress curves.



ε−εc εp

N # ð8Þ

where N is the Avrami's power and K is a constant, and εp is the peak strain corresponding to the peak stress. The softening fraction caused by the DRX can be also estimated by the decrease in flow stress. Xd ¼

σ p −σ σ p −σ ss

ð9Þ

where σp and σss is the peak stress and steady state stress. Then the flow stress during DRX period can be expressed by combining Eqs. (8) & (9). ( "

#)   ε−εc N ðε ≥ε c Þ σ ¼ σ p − σ p −σ ss 1− exp −k εp

ð10Þ

3.3.1. Determination of mechanical parameters and Zener–Hollomon parameter The εc for the initiation of DRX is the prerequisite for the constitutive modeling. The critical conditions for the initiation of DRX can be determined by analyzing the relation between strain hardening rate (θ = dσ/ dε) and flow stress according to the method supplied by Poliak and Jonas [31]. The flow stress dependence of the strain hardening rate is illustrated in Fig. 4. The variation of θ reflects the deformation mechanism changes. The critical stress (σc) is identified as the point at which the second derivative of the θ with respect to stress, i.e. ∂2θ/∂2σ, is zero. Then the εc can be obtained by converting the σc back to the flow stress curves. And the peak stress σp is determined at the point with θ =0, indicating that a balance between strain hardening and softening reaches. After that, the peak strain εp can be determined. The saturation stress σsat is defined by the extrapolation of the θ − σ plot to θ = 0. In Fig. 4, it can be seen that the value of σsat is almost similar to the σp. So, the value of σsat is regarded to be equal to σp in this work. The yield stress σ0 was identified on the flow curves in terms of 2% offset in total strain [14]. The Zener–Hollomon parameter is an important value to characterize the combined effects of strain rate and temperature on the

3.3. Constitutive modeling the flow stress curves When no DRX occurring, the evolution of dislocation density (ρ) for metals during hot deformation is supposed to be the coupled effects of generation by WH and annihilation by DRV. The dependence of the dislocation density on strain can be written as follows [15,26,27]: dρ ¼ h−rρ dε

ð6Þ

where h is the athermal work-hardening rate or coefficient and r represents the rate of DRV at a fixed deformation condition. By integrating the above equation and doing some algebraic operations, the flow stress during WH and DRV period can be expressed as [28,29]:    0:5 σ ¼ σ 2sat − σ 2sat −σ 20 expð−rε Þ

ðεbεc Þ

ð7Þ

where σ0 and σsat are the yield stress and saturated stress, respectively, and εc is the critical strain for the occurrence of DRX. This equation

Fig. 4. Strain hardening rate varies with the stress at different temperatures at a strain rate of 0.1 s−1.

808

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sides of Eq. (12), it can be expressed by: ln ε_ ¼ ln A þ n ln ð sinhðασ ÞÞ−

Fig. 5. The variations of standard deviation of n with the values of α.

deformation process. Some parameters in Eqs. (6)–(10) are thought to be related to Zener–Hollomon parameter [14,29]. The well-known Zener–Hollomon parameter is introduced as follows [32]:

Q Z ¼ ε_ exp RT

ð11Þ

where ε_ the represents strain rate (s−1) and T is the absolute temperature (K), Q is the activation energy (J/mol), and R is the gases universal constant (8.314 J/mol·K). The Arrhenius equation is widely used to describe the relationship of flow stress, strain rate and temperature [3,33]:

Q ε_ ¼ AF ðσ Þ exp − RT

ð12Þ

in which

F ðσ Þ ¼

8