Numerical modeling and anvil design of high-speed forging process for railway axles

Abstract

Railway axles, which are an important component of railway vehicles, are generally manufactured using high-speed forging processes. To investigate the microstructure evolution and flow behavior during forging processes, a series of isothermal hot compression and heating tests were conducted in the temperature range of 900–1200 °C and strain rate range of 0.01–20 s⁻¹. A strain-compensated Arrhenius constitutive relationship was identified for 25CrMo4 steel, and microstructure evolution kinetics, comprising dynamic recrystallization, metadynamic recrystallization, static recrystallization, and grain growth, were determined. A coupled thermomechanical–metallurgical numerical model was established for the high-speed forging of 25CrMo4 steel axles by using the TRANSVALOR Forge software package. The grain size evolution during the multipass high-speed forging process was predicted, and a full-scale axle was fabricated through high-speed forging to verify the predicted results. The predicted and experimentally observed grain sizes had good agreement. Finally, a series of geometric constraints are proposed for the design of round anvils. The reliability and applicability of the proposed constraints were validated by considering the surface quality and microstructure requirements as well as the forming force during chamfering. The results indicated that the proposed constraints can suitably guide the design of anvils for high-speed forging processes.

Introduction

25CrMo4 steel (EA4T) has excellent strength and ductility at room temperature, and it has been widely used to produce railway axles [1]. Railway axles, which are an important component of railway vehicles, usually comprise two parts: the body section and journal parts (Fig. 1). To obtain well-performing railway axles, the axles are primarily manufactured through high-speed forging technology, which involves chamfering, rounding, and final forming, during which the axles are rotated by manipulators and forged incrementally between anvils. Thus, the final quality of the forged axles is largely dependent on the geometric structure of anvils and process parameters of high-speed forging. Due to the multipass hot forming, temperature, and deformation degree are extremely inhomogeneous in different parts of an axle, steels experience a complex thermomechanical history that includes work hardening (WH), dynamic recovery (DRV), dynamic recrystallization (DRX), static recrystallization (SRX), metadynamic recrystallization (MDRX), and grain growth (GG). WH originates from the accumulation of dislocation during deformation. Under an increase in deformation, thermally activated processes tend to restore the microstructure to its original state through the annihilation and rearrangement of dislocations, which is known as DRV [2]. When the strain reaches a critical value, new grains nucleate through the formation and migration of high angle grain boundaries driven by the stored energy. This process is known as DRX [3]. SRX and MDRX—which occur during the interpass time of hot forging or after hot forging—considerably influence microstructure evolution. SRX occurs when the strain is less than the critical value of DRX during the hot forming process. MDRX is characterized by the continual growth of DRX nuclei formed during the hot deformation process [4]. After recrystallization (recrystallization fraction volume > 0.95), a fine new structure is formed and further GG occurs to reduce the energy of the system [5]. Because of the aforementioned connected metallurgical phenomena, fine and coarse grains usually coexist along the forged axles, which results in the axles having anisotropic properties and low fatigue resistance [6]. Thus, to attain well-performing railway axles, a reliable microstructure-based finite-element (FE) simulation platform must be developed for investigating and controlling the grain size evolution of forged axles and for guiding the structural design of anvils.

Fig. 1

Schematic of a railway axle

Numerous studies have examined the thermal and mechanical properties of 25CrMo4 during hot forming. Xu et al. [7] investigated the hot deformation behavior of 25CrMo4 in a temperature range of 950–1150 °C and strain rate range of 0.1–20 s⁻¹ by performing hot compression tests and established an Arrhenius-type constitutive equation. The peak stress and peak strain were considered to determine the parameters of the constitutive equation. Huo et al. [8] studied the constitutive relationship of 25CrMo4 steel using the internal-state-variable method. They derived a constitutive equation based on physical laws and used evolutionary programming to determine the constants in the equation. Some researchers have attempted to investigate the microstructure evolution during hot forming. Hibbe et al. [9] investigated the void closure and healing behavior in open-die forging through an experimental examination of the bond strength. Jang et al. [10] developed an analysis program that considers the DRX, SRX, and GG physics of C-Mn steel to determine the microstructure evolution during hot forging. Sherstnev et al. [11] proposed a physical model based on the dislocation density, subgrain size, and misorientation to calculate the grain size evolution after hot rolling. Huo et al. [12] established a microstructure model that covers the DRX and GG of 25CrMo4. They implemented the model into DRFORM-3D to predict the grain size distribution during hot-cross wedge rolling. Wang et al. [6] integrated microstructure evolution kinetics into an existing model of hot forging and investigated the austenite grain size distribution along a hot-forged AP1000 primary coolant pipe. Regarding the anvil design for hot forging, Buteler et al. [13] compared the stretching of cylinders with a plane and with 135° anvils by analyzing their stress distributions. Dyja et al. [14] proposed two types of anvils, namely radial-trapezoid anvils and shape of forcing-through radial anvils. They investigated the free hot forging of long products with the aforementioned two anvils through numerical modeling and laboratory experiments. Du et al. [15] studied the influence of the design of round anvils on high-speed forging processes by analyzing the stress–strain distribution and forming precision by using a 3D FE model. However, most previous studies have only focused on the DRX or SRX of the microstructure during hot forming. Comprehensive studies on the material flow behavior and grain size evolution kinetics, including the DRX, SRX, MDRX, and GG, have been lacking. Previous studies on anvil design have merely focused on the dimensions and stress–strain results of hot-forged parts. Few studies have examined the microstructure evolution during the fabrication of round anvils. Thus, difficulties are encountered in developing a comprehensive macro-micromodel to investigate the microstructure evolution and to guide anvil design in high-speed forging.

The aim of this study was to investigate the microstructure evolution and design constraints of round anvils during the hot high-speed forging of railway axles. For this purpose, a constitutive model and the microstructure evolution kinetics were investigated through hot compression tests under various deformation conditions by using a Gleeble-2000 thermal–mechanical simulator. The determined equations were implemented in TRANSVALOR Forge NXT 1.1. The microstructure evolution throughout the heating and high-speed forging process chain of a railway axle was analyzed. A full-scale high-speed forging experiment was performed to fabricate a 25CrMo4 axle for verifying the validity and effectiveness of the simulation results. On the basis of the developed model, the effects of the anvil structure on the forging process were investigated by considering the surface qualities, microstructure, and forming force.

Experimental methods

Material and experimental equipment

25CrMo4 steel, supplied by Masteel Rolling Stock Co., Ltd., was used in this study. The chemical composition (wt.%) of 25CrMo4 steel is summarized in Table 1. The chemical composition was determined using a Thermo iCAP7600 inductively coupled plasma optical spectrometer. The initial microstructure of 25CrMo4 was determined using Nital etchant (4% solution of alcohol and nitric acid). The original microstructure of 25CrMo4 steel consisted of ferrite, pearlite, and bainite, as displayed in Fig. 2.

Table 1 Chemical composition of the studied 25CrMo4 steel (wt.%)

Fig. 2

Initial microstructure of 25CrMo4 steel

Compression specimens (12 mm in length and 8 mm in diameter) were wire-cut from a billet. The machined cylinder samples were compressed according to a specific heating and deformation schedule by using a Gleeble-2000 thermal–mechanical simulator. Tantalum plates were used with lubricant at elevated temperatures to minimize the friction between the anvils of the Gleeble simulator and the samples. Hot compressed samples were quenched in water after each test.

The quenched samples were sliced along the axial section and then etched with a picric acid solution to identify grain boundaries. The microstructure was evaluated using a Zeiss Axio Imager M2m optical microscope, and the grain size was determined using the line intercept method in accordance with ASTM E112.

Three types of tests, namely hot single compression tests, hot double compression tests, and static GG tests, were performed under isothermal conditions. Hot single compression tests were used to obtain strain–stress curves under different temperatures and strain rates. The DRX behaviors were also investigated using the aforementioned tests. Hot double compression tests were used to study the MDRX and SRX kinetics, and static GG tests were used to analyze the austenite grain growth behavior of 25CrMo4 steel.

Hot single compression tests

Two series of hot single compression tests were performed. In the first series of tests (Fig. 3), the specimens were initially heated to 1200 °C with a heating rate of 10 °C/s and then soaked for 180 s for full austenization. Subsequently, the samples were cooled to different test temperatures with a cooling rate of 20 °C/s and then held for 120 s. The specimens were then compressed to a true strain of 0.9 using different strain rates. To investigate the effect of the initial grain size on the DRX kinetics, a second series of hot single compression tests was conducted. The specimens were initially heated to 1200 °C; maintained for 180, 5400, and 14,400 s to achieve different initial austenite grain sizes; and cooled to 1050 °C with a cooling rate of 20 °C/s. After a soaking time of 120 s, the specimens were compressed to a true strain of 0.9 with a strain rate of 0.01 s⁻¹. Subsequently, water quenching was conducted to retain the microstructure.

Fig. 3

Experimental plan for the hot single compression tests

Hot double compression tests

Figure 4 displays the experimental plan for two series of hot double compression tests. In the first series of tests, the specimens were heated to 1200 °C with a heating rate of 10 °C/s and held for 300 s. The specimens were then cooled to the test temperature with a cooling rate of 20 °C/s, soaked for 60 s, and then compressed under various prestrain values. Three temperatures and four strain rates were used in the predeformation process. After the specimens were subjected to prestrain, they were unloaded and held for 1–300 s to allow SRX or MDRX to occur. The specimens were then reloaded again under the aforementioned process parameters. The level of softening could be determined from the stress difference between the flow curves measured for the first and second loadings. Finally and immediately thereafter, water quenching was conducted. The effects of different initial austenite grain sizes on SRX or MDRX kinetics were investigated using a second series of hot double compression tests (Fig. 4). The prestrain values for the investigation of SRX and MDRX were selected according to the critical strain of DRX, which is determined in section 3. Only the SRX or MDRX mechanism was assumed to contribute to the microstructure evolution when the prestrain values were less than or greater than the critical strain of DRX, respectively [2].

Fig. 4

Experimental plan for the hot double compression tests

Static GG tests

Static GG tests were conducted as illustrated in Fig. 5. Samples were heated to 700 °C and then held for 1800 s to achieve a uniform temperature distribution. Subsequently, the samples were heated to a specific temperature in the range of 900–1200 °C with a heating rate of 0.25 °C/s and then held at that temperature for 0–18,000 s. After different holding times, the samples were quenched in water immediately for microstructure investigation.

Fig. 5

Schematic of the static GG tests

Constitutive model and microstructure evolution kinetics

Formulation of the constitutive model

Typical true stress–strain curves for the hot single compression tests were plotted from the load–displacement data in Gleeble-2000 postprocessing software (Fig. 6). Most of the curves indicated typical DRX behavior, that is, the stress reached a peak value and then declined to a steady state. The stress was significantly sensitive to the deformation temperature and strain rate. At the same strain, a higher temperature and lower strain rate led to a lower stress value. This result is mainly attributed to the competition between the dislocation storage caused by WH and the dynamic softening caused by DRV and DRX. When steels or alloys are subjected to deformation under high temperatures and low strain rates, dislocations can easily climb and undergo cross slip. This phenomenon enhances the grain boundary mobility and thus promotes the occurrence of DRV and DRX and decreases flow stress [16].

Fig. 6

Typical true stress–strain curves of 25CrMo4 steel under different deformation conditions: (a) effect of the temperature on the flow stress when $\dot{\varepsilon}$= 0.1 s⁻¹ and (b) effect of the strain rate on the flow stress when T = 1200 °C

For hot forming processes, Arrhenius-type equations are usually used to describe the constitutive relationship between the deformation temperature and the strain rate. The aforementioned relationship can be expressed as follows [17,18,19,20]:

$$ \Big\{{\displaystyle \begin{array}{c}\dot{\varepsilon}=A\cdotp F\left(\sigma \right)\cdotp \exp \left(-\frac{Q}{R_UT}\right)\\ {}F\left(\sigma \right)=\Big\{\begin{array}{c}{\sigma}^{{\mathrm{n}}_1},\alpha \sigma <0.8\\ {}\exp \left(\beta \sigma \right),\alpha \sigma >1.2\\ {}{\left[\sinh \left(\alpha \sigma \right)\right]}^{\mathrm{n}},\mathrm{for}\mathrm{all}\sigma \end{array}\end{array}} $$

(1)

where $\dot{\varepsilon}$ is the strain rate (s⁻¹), σ is the true stress (MPa), Q is the activation energy of deformation (J·mol⁻¹), R_U is the universal gas constant (8.314 J·mol⁻¹·K⁻¹), and T is the absolute temperature (K). Moreover, n₁, α, β, n, and A (s⁻¹·MPa⁻¹) are material constants. The values of n₁ and β can be obtained from the slopes of the plots of lnσ versus ln$\dot{\varepsilon}$ and σ versus ln$\dot{\varepsilon}$, respectively. The value of α can then be determined from the calculated values of n₁ and β as α = n₁/β. By substituting the value of α into Eq. (1), the values of n and Q can be calculated from the slopes of the plots of ln[sinh(ασ)] versus ln$\dot{\varepsilon}$ and (1000/T) versus ln $\dot{\varepsilon}$, respectively. In addition, A can be calculated from the intercept of the plot of ln[sinh(ασ)] versus ln$\dot{\varepsilon}$. Detailed derivations of the aforementioned formulae are available in a previous study [19]. The influence of the initial austenite grain size on flow behavior was not considered in this study.

In general, the combined effect of the temperature and strain rate on the hot deformation behavior of metals and alloys can be characterized by the Zener–Hollomon parameter as follows [21]:

$$ Z=\dot{\varepsilon}\exp \left(\frac{Q}{R_UT}\right) $$

(2)

By combining Eqs. (1) and (2), the flow stress under hot deformation can be derived as follows (the detailed derivations are presented in Appendix):

$$ \sigma =\frac{1}{\alpha}\ln \left\{{\left(\frac{\dot{\varepsilon}\cdotp \exp \left(Q/{R}_UT\right)}{A}\right)}^{\frac{1}{n}}+{\left[{\left(\frac{\dot{\varepsilon}\cdotp \exp \left(Q/{R}_UT\right)}{A}\right)}^{\frac{2}{n}}+1\right]}^{0.5}\right\}=\frac{1}{\alpha}\ln \left\{{\left(\frac{Z}{A}\right)}^{\frac{1}{n}}+{\left[{\left(\frac{Z}{A}\right)}^{\frac{2}{n}}+1\right]}^{0.5}\right\} $$

(3)

To enhance the predictive accuracy of Eq. (3), the effect of strain on flow stress should be considered. Thus, a fifth-order polynomial equation was used to capture the relationship between the material constants (α, n, Q, and A) and the strain. Strain values of 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.4, 0.5, 0.6, 0.7, and 0.8 and the corresponding stresses were substituted in the aforementioned equation under different temperatures and strain rates. The relationships between material constants and true strain are plotted in Fig. 7. All the material parameters were obtained as functions of the deformation degree. The flow stress can be predicted using Eqs. (3) and (4).

$$ {\displaystyle \begin{array}{c}Q=2.622\times {10}^5+9.272\times {10}^5\varepsilon -4.593\times {10}^6{\varepsilon}^2+1.017\times {10}^7{\varepsilon}^3-1.077\times {10}^7{\varepsilon}^4+4.427\times {10}^6{\varepsilon}^5\\ {}\alpha =0.02137-0.1297\varepsilon +0.5969{\varepsilon}^2-1.283{\varepsilon}^3+1.330{\varepsilon}^4-0.5338{\varepsilon}^5\\ {}\begin{array}{c}n=6.153-1.215\varepsilon -22.27{\varepsilon}^2+54.51{\varepsilon}^3-48.10{\varepsilon}^4+15.58{\varepsilon}^5\\ {}\ln A=21.30+88.39\varepsilon -444.1{\varepsilon}^2+992.9{\varepsilon}^3-1055{\varepsilon}^4+433.5{\varepsilon}^5\end{array}\end{array}} $$

(4)

Fig. 7

Relationship between material constants of (a) Q, (b) α, (c) n and (d) lnA and strain

Microstructure evolution kinetics

DRX kinetics

For steels or alloys subjected to a high deformation temperature and large deformation degree, DRX is initiated easily when the critical condition is reached. The effect of DRX on the microstructure evolution is characterized by its volume fraction. The aforementioned effect can be represented by a modified Avrami equation as follows [22, 23]:

$$ {X}_{drx}=1-\exp \left[{X}_{\mathrm{d}1}{\left(\frac{\varepsilon -{\varepsilon}_{\mathrm{c}}}{\varepsilon_{0.5}-{\varepsilon}_{\mathrm{c}}}\right)}^{{\mathrm{X}}_{\mathrm{d}2}}\right]\left(\varepsilon >{\varepsilon}_{\mathrm{c}}\right) $$

(5)

where X_drx is the volume fraction of DRX; X_d1 and X_d2 are DRX constants; and ε, ε_c, and ε_0.5 represent the true strain, critical strain of DRX, and the strain when X_drx is 50%, respectively. Moreover, X_drx can be determined using the following Eq. [24]:

$$ {X}_{drx}=\frac{\sigma_{WH}-\sigma }{\sigma_{sat}-{\sigma}_{ss}}\left(\varepsilon >{\varepsilon}_c\right) $$

(6)

where σ_sat is the saturation stress (MPa), σ_ss is the steady-state stress (MPa), and σ_WH is the flow stress (MPa) once the dynamic recovery is the sole softening mechanism. σ_WH can be expressed as follows [23]:

$$ {\sigma}_{WH}={\left[{\sigma}_{sat}^2+\left({\sigma}_0^2-{\sigma}_{sat}^2\right){e}^{-\varOmega \varepsilon}\right]}^{0.5}\left(\varepsilon <{\varepsilon}_c\right) $$

(7)

where σ₀ is the yield stress (MPa) and Ω is the DRV coefficient. On the basis of Eq. (7), the following equation can be derived:

$$ \ln \left(\frac{{\sigma_{WH}}^2-{\sigma}_{sat}^2}{\sigma_0^2-{\sigma}_{sat}^2}\right)=-\varOmega \varepsilon $$

(8)

The data prior to the critical strain of DRX were used to calculate the value of Ω through the method described in [16]. Thereafter, X_drx was calculated using Eq. (6).

The values of ε_0.5 and ε_c can be calculated using the following empirical equations [17, 18, 25, 26]:

$$ \left\{\begin{array}{c}{\varepsilon}_{\mathrm{c}}={A}_c\cdotp {d}_0^{n_{2_{\mathrm{c}}}}{\dot{\varepsilon}}^{n_{3_{\mathrm{c}}}}\exp \left(\frac{Q_{\mathrm{c}}}{R_UT}\right)\\ {}{\varepsilon}_{0.5}={A}_{0.5}\cdotp {d}_0^{n_{2_{0.5}}}{\dot{\varepsilon}}^{n_{3_{0.5}}}\exp \left(\frac{Q_{0.5}}{R_UT}\right)\end{array}\right. $$

(9)

where d₀ is the initial austenite grain size; A_c, A_0.5, n_2c, ${n}_{2_{0.5}}$, n_3c, and ${n}_{3_{0.5}}$ are the material parameters for the critical strain or 50% volume fraction of DRX; and Q_c and Q_0.5 are the activation energy with respect to the critical strain of DRX and 50% volume fraction of DRX, respectively.

By taking the logarithm of Eq. (9), the following equation is obtained:

$$ \left\{\begin{array}{c}\ln {\varepsilon}_{\mathrm{c}}=\ln {A}_c+{n}_{2_{\mathrm{c}}}\cdotp \ln {d}_0+{n}_{3_{\mathrm{c}}}\cdotp \ln \dot{\varepsilon}+\frac{Q_{\mathrm{c}}}{R_UT}\\ {}\ln {\varepsilon}_{0.5}=\ln {A}_{0.5}+{n}_{2_{0.5}}\cdotp \ln {d}_0+{n}_{3_{0.5}}\cdotp \ln \dot{\varepsilon}+\frac{Q_{0.5}}{R_UT}\end{array}\right. $$

(10)

The critical strain was calculated on the basis of Kocks–Mecking plots by using the method proposed by Poliak and Jonas [27]. In addition, ε_0.5 was obtained from the flow curves when X_drx reached 50%. Initial austenite grain sizes of 113, 163, and 202 μm were achieved when the specimens were heated to 1200 °C for 180, 5400, and 14,400 s, respectively. The values of n_2c (or ${n}_{2_{0.5}}$)_, n_3c (or ${n}_{3_{0.5}}$), and Q_c (or ${Q}_{2_{0.5}}$) were calculated from the linear fitting results for the plots of lnε_c (or lnε_0.5) versus lnd₀, ln$\dot{\varepsilon}$, and (−1/R_UT), respectively, according to the methods in [28, 29]. The values of A_c and A_0.5 were calculated by averaging the intercepts of the fitted lines of the aforementioned plots.

On the basis of the preceding discussion, X_drx is expressed as follows:

$$ \left\{\begin{array}{c}\begin{array}{c}\begin{array}{c}{X}_{\mathrm{drx}}=0,\left(\varepsilon <{\varepsilon}_{\mathrm{c}}\right)\\ {}{X}_{\mathrm{drx}}=1-\exp \left[-0.693{\left(\frac{\varepsilon -{\varepsilon}_{\mathrm{c}}}{\varepsilon_{0.5}-{\varepsilon}_{\mathrm{c}}}\right)}^{1.274}\right],\left(\varepsilon >{\varepsilon}_{\mathrm{c}}\right)\end{array}\\ {}{\varepsilon}_{\mathrm{c}}=4.4503\times {10}^{-4}{d}_0^{0.3322}{\dot{\varepsilon}}^{0.1887}\exp \left(\frac{52339}{R_UT}\right)\end{array}\\ {}{\varepsilon}_{0.5}=0.2068{d}_0^{-6613}{\dot{\varepsilon}}^{0.1973}\exp \left(\frac{46806}{R_UT}\right)\end{array}\right. $$

(11)

The size of the dynamic recrystallized grains (D_drx) strongly depends on the deformation conditions. The parameter D_drx is expressed using the deformation temperature, strain rate, and initial grain size as follows [30, 31]:

$$ {D}_{\mathrm{drx}}={A}_{\mathrm{D}1}\cdotp {d}_0^{{\mathrm{D}}_2}{\dot{\varepsilon}}^{{\mathrm{D}}_3}\exp \left(-\frac{Q_{\mathrm{D}}}{R_UT}\right) $$

(12)

where A_D1, D₂, and D₃ are material constants and Q_D is the activation energy of the dynamic recrystallized grains. According to the method proposed in [28], the values of D₂, D₃, and Q_D were determined by linear fitting for the plots of lnD_drx against lnd₀, ln$\dot{\varepsilon}$, and (−1/R_UT), respectively. The values of D₂, D₃, and Q_D were determined to be 0.4526, −0.2122, and 99,066 J/mol, respectively, according to the measured size of the dynamic recrystallized grains in the samples. In addition, A_D1 was calculated to be 1.881 × 10⁴ from the intercepts of the fitting lines of the aforementioned plots.

SRX and MDRX kinetics

In this study, the 0.2% offset strain method was used to calculate the softening fraction caused by SRX and MDRX. The relative softening fraction (F) can be measured using the following Eq. [32]:

$$ F=\frac{\sigma_{\mathrm{m}}-{\sigma}_2}{\sigma_{\mathrm{m}}-{\sigma}_1} $$

(13)

where σ_m is the stress at the end of the prestrain and σ₁ and σ₂ are the yield stresses determined at an offset strain of 0.2% for the first and second deformations, respectively.

According to Sun and Millitzer [33], recrystallization plays a dominant role in material softening only after the softening fraction reaches a critical value. Typically, the critical value of the softening fraction is considered as 0.2. Therefore, the MDRX or SRX volume fraction X_mdrx/srx can be obtained using the following Eq. [33]:

$$ {X}_{\mathrm{m} drx/ srx}=\frac{F-0.2}{1-0.2}=\frac{F-0.2}{0.8} $$

(14)

In general, the kinetics of SRX and MDRX can be described by the Avrami equation as follows [2223, 33, 34]:

$$ \Big\{{\displaystyle \begin{array}{c}{X}_{srx}=1-\exp \left[{X}_{\mathrm{s}1}{\left(\frac{t}{t_{0.5s}}\right)}^{X_{\mathrm{s}2}}\right]\\ {}{X}_{mdrx}=1-\exp \left[{X}_{\mathrm{m}1}{\left(\frac{t}{t_{0.5m}}\right)}^{X_{\mathrm{m}2}}\right]\end{array}} $$

(15)

where X_srx/mdrx is the volume fraction of SRX/MDRX, X_s1/m1 and X_s2/m2 are material parameters, and t_0.5s/m is the time required for the occurrence of 50% volume fraction of SRX/MDRX. The following empirical equation is widely used to determine t_0.5 [5, 1731]:

$$ \Big\{{\displaystyle \begin{array}{c}{t}_{0.5_{\mathrm{s}\mathrm{rx}}}={T}_{\mathrm{s}1}{\varepsilon}^{T_{\mathrm{s}2}}{d}_0^{T_{\mathrm{s}3}}{\dot{\varepsilon}}^{T_{\mathrm{s}4}}\exp \left(\frac{Q_{\mathrm{Ts}}}{R_UT}\right)\\ {}{t}_{0.5_{\mathrm{m}\mathrm{drx}}}={T}_{\mathrm{m}1}{\varepsilon}^{T_{\mathrm{m}2}}{d}_0^{T_{\mathrm{m}3}}{\dot{\varepsilon}}^{T_{\mathrm{m}4}}\exp \left(\frac{Q_{\mathrm{Tm}}}{R_UT}\right)\end{array}} $$

(16)

where T_s1, T_s2, T_s3, T_s4 T_m1, T_m2, T_m3, and T_m4 are material parameters; ε is the prestrain; and Q_Ts and Q_Tm represent the activation energy for SRX and MDRX, respectively.

By taking the logarithm of Eq. (15), the following equation is obtained:

$$ \Big\{{\displaystyle \begin{array}{c}\ln \left[\ln \left(\frac{1}{1-{X}_{srx}}\right)\right]=\ln \left(-{X}_{s1}\right)+{X}_{s2}\ln \left(\frac{t}{t_{0.5s}}\right)\\ {}\ln \left[\ln \left(\frac{1}{1-{X}_{mdrx}}\right)\right]=\ln \left(-{X}_{m1}\right)+{X}_{m2}\ln \left(\frac{t}{t_{0.5m}}\right)\end{array}} $$

(17)

When inputting the X_srx (or X_mdrx) value and interval time corresponding to different deformation conditions into Eq. (17), the relationship between ln[ln(1/(1 − X_srx))] (or ln[ln(1/(1 − X_mdrx))]) and ln(t/t_0.5s) (or ln(t/t_0.5m)) can be linear-fitted, as displayed in Fig. 8. Consequently, X_s1 (or X_m1) and X_s2 (or X_m2) can be determined.

Fig. 8

Relationship between ln[ln(1/(1 − X_srx))] (or ln[ln(1/(1 − X_mdrx))]) and ln(t/t_0.5s) (or ln(t/t_0.5m)) for (a) SRX and (b) MDRX under different deformation conditions

The following equation is obtained by taking the logarithm of Eq. (16):

$$ \Big\{{\displaystyle \begin{array}{c}\ln {t}_{0.5s}=\ln {T}_{s1}+{T}_{s2}\ln \varepsilon +{T}_{s3}\ln {d}_0+{T}_{s4}\ln \dot{\varepsilon}+\left(\frac{Q_{Ts}}{R_UT}\right)\\ {}\ln {t}_{0.5m}=\ln {T}_{m1}+{T}_{m2}\ln \varepsilon +{T}_{m3}\ln {d}_0+{T}_{m4}\ln \dot{\varepsilon}+\left(\frac{Q_{Tm}}{R_UT}\right)\end{array}} $$

(18)

The values of t_0.5s (or t_0.5m) under different deformation conditions can be derived from the relationship between X_srx (or X_mdrx) and t. After differentiating Eq. (18), the values of T_s1, T_s2, T_s3, T_s4, T_m1, T_m2, T_m3, T_m4, Q_Ts, and Q_Tm can be obtained by performing linear fitting according to the methods used in [35]. Thus, the kinetics of SRX and MDRX under the unloading condition can be described as follows:

$$ {\displaystyle \begin{array}{c}\Big\{\begin{array}{c}{X}_{srx}=1-\exp \left[-0.6927{\left(\frac{t}{t_{0.5}}\right)}^{0.5413}\right]\\ {}{t}_{0.5_{\mathrm{srx}}}=8.528\times {10}^{-11}{\varepsilon}^{-1.444}{d}_0^{0.2163}{\dot{\varepsilon}}^{-0.4538}\exp \left(\frac{227278}{R_UT}\right)\end{array}\\ {}\Big\{\begin{array}{c}{X}_{\mathrm{mdrx}}=1-\exp \left[-0.5903{\left(\frac{t}{t_{0.5}}\right)}^{0.7611}\right]\\ {}{t}_{0.5_{\mathrm{mdrx}}}=1.724\times {10}^{-9}{\varepsilon}^{-0.7841}{d}_0^{0.3627}{\dot{\varepsilon}}^{-0.6127}\exp \left(\frac{189338}{R_UT}\right)\end{array}\end{array}} $$

(19)

The SRX grain size (D_srx) is represented using the deformation temperature, strain rate, prestrain, and initial grain size as follows [31, 36, 37]:

$$ {D}_{\mathrm{s}\mathrm{rx}}={D}_{\mathrm{s}1}{\varepsilon}^{D_{\mathrm{s}2}}{\dot{\varepsilon}}^{D_{\mathrm{s}3}}{d}_0^{D_{\mathrm{s}4}}\exp \left[-\frac{Q_{\mathrm{Ds}}}{R_UT}\right]{X}_{\mathrm{s}\mathrm{rx}}^{0.5} $$

(20)

where D_s1, D_s2, D_s3, and D_s4 are material constants and Q_Ds is the activation energy.

The average size of the dynamic recrystallized grains (D_srx) was measured. By taking the logarithm of and differentiating Eq. (14), D_s2, D_s3, D_s4, and Q_Ds were determined to be 23.65, −0.04326, 0.4676, and 21,981 J/mol, respectively, from the linear fitting results for the plots of lnD_srx versus lnε, ln$\dot{\varepsilon}$, lnd₀, and (−1/R_UT), respectively. The parameter D_s1 was calculated to be 0.3715 [33].

The MDRX grain size is less affected by the prestrain and initial grain size; thus, the MDRX grain size (D_mdrx) can be expressed as a function of the forming temperature, strain rate, and volume fraction of MDRX as follows [31, 36]:

$$ {D}_{\mathrm{m}\mathrm{drx}}={D}_{\mathrm{m}1}{\dot{\varepsilon}}^{D_{\mathrm{m}2}}\exp \left[-\frac{Q_{\mathrm{Dm}}}{R_UT}\right]{X}_{\mathrm{m}\mathrm{drx}}^{0.5} $$

(21)

where D_m1 and D_m2 are material constants and Q_Dm is the activation energy. The values of D_m1, D_m2, and Q_Dm were calculated to be 2.9173 × 10³, −0.1129, and 46,797 J/mol, respectively.

Static GG kinetics

Figure 9 presents the average austenite grain size of 25CrMo4 under different holding times and heating temperatures. The average austenite grain size increased with an increase in holding time and heating temperature. For a heating temperature of 900–950 °C, austenite exhibited a slow GG under different holding times. The GG rate was almost flat. However, the average grain size increased significantly with an increase in holding time in the heating temperature range of 1000–1200 °C.

Fig. 9

Average grain size at different heating temperatures and holding times

In this study, the following Arrhenius-type equation was adopted to predict the GG of austenite [5]:

$$ {d}_{\mathrm{g}\mathrm{g}}^{Gg1}-{d}_0^{Gg1}={G}_{\mathrm{g}2}\exp \left(\frac{Q_{\mathrm{Ggr}}}{R_UT}\right)t $$

(22)

where d_gg is the average grain size after holding time t, d_o is the initial grain size, Gg1 and Gg2 are the growing constants of 25CrMo4 steel, and Q_Ggr is the activation energy for GG. The austenite GG kinetics for 25CrMo4 steel was mathematically modeled with multiple nonlinear fitting methods by using the following piecewise functions:

$$ {d}_{\mathrm{gg}}^{1.89}={d}_0^{1.89}+4.12\times {10}^5t\exp \left(-\frac{164327}{R_UT}\right)\left(900{}^{\mathrm{o}}\mathrm{C}<T\le 950{}^{\mathrm{o}}\mathrm{C}\right) $$

$$ {d}_{\mathrm{gg}}^{2.7}={d}_0^{2.7}+6.31\times {10}^7t\exp \left(-\frac{150672}{R_UT}\right)\left(950{}^{\mathrm{o}}\mathrm{C}<T\le 1200{}^{\mathrm{o}}\mathrm{C}\right) $$

(23)

Model verification

To verify the reliability of the developed Arrhenius equations with strain compensation for 25CrMo4 steel, the established model was input into TRANSALOR Forge NXT 1.1. The hot compression test was numerically simulated for the following two deformation conditions: 1) a temperature of 950 °C with a strain rate of 0.1 s⁻¹ and 2) a temperature of 1050 °C with a strain rate of 0.01 s⁻¹. The predicted forming forces were compared with the experimental forming forces, as displayed in Fig. 10. The predicted forces were in good agreement with the corresponding experimental results. The aforementioned results indicate that the developed Arrhenius model can effectively predict the flow behavior of 25CrMo4 steel at elevated temperatures.

Fig. 10

Comparison of the predicted and experimental forming loads at a temperature of 950 °C with a strain rate of 0.1 s⁻¹ and a temperature of 1050 °C with a strain rate of 0.01 s⁻¹

The microstructure evolution kinetics involving DRX, SRX, MDRX, and GG were also input into TRANSALOR Forge NXT 1.1. To simulate DRX, MDRX, and SRX behaviors and further validate the reliability of the established microstructure kinetics, three compression tests with different anvil motion profiles (motions A, B, and C) were numerically and experimentally performed, as displayed in Fig. 11. Motions A, B, and C were specially designed for the validation of the grain size evolution under DRX, MDRX, and SRX, respectively. A total reduction of 7 mm (ε = 0.875) without interval unloading was selected for motion A. According to the critical strain calculated from Eq. (9), a prereduction of 3 mm (ε = 0.295) and holding time of 60 s were selected for motion B to simulate the occurrence of MDRX. Similarly, a prereduction of 1.65 mm (ε = 0.148) and holding time of 30 s were selected for motion C to simulate the occurrence of SRX. Fig. 12(b)–(d) illustrates the predicted average grain size distribution and metallography of the core for the three hot compression tests with different motions. Fig. 12(a) presents a comparison of the average grain size in the core of the samples obtained from numerical simulation and experiments under the same conditions. The average grain size simulated through FE modeling was in good agreement with the experimental results. However, the experimental average grain sizes were marginally larger than the simulated average grain sizes. This discrepancy is attributable to the errors introduced by factors such as the simplification in the FE compression model and the measurement operation in the experiment. Overall, the established microstructure kinetics suitably describe the recrystallization and GG behavior of 25CrMo4 steel during hot forming processes.

Fig. 11

Profiles of the anvil motions: (a) motion A, (b) motion B, and (c) motion C

Fig. 12

a Comparison of the experimental data with the simulation results and numerical grain size distributions for b motion A, c motion B, and d motion C

Numerical simulation and experimental verification of the high-speed forging process

To understand the microstructure evolution of 25CrMo4 during high-speed forging, an FE model of high-speed forging was established. The derived constitutive equations and microstructure evolution kinetics (DRX, MDRX, SRX, and GG) were implemented in TRANSALOR Forge NXT 1.1 through a user-defined subroutine.

FE simulation procedure

A numerical 3D model was developed to replicate the high-speed forging process. The aforementioned model, which comprised upper and lower round anvils as well as a heated billet, was developed using FORGE, as displayed in Fig. 13. Some hypotheses were adopted in the condition of permissive scope of the engineering calculation accuracy in the simulation process. Two round anvils were considered as the rigid material. Because of the multipass forming and large strain at elevated deformation temperatures during high-speed forging, the elastic deformation of forged axles was assumed to be negligible, which adopted a rigid-plastic material model in the FE simulation [38]. The adopted billet had a square surface area and the dimensions of 250 × 250 × 1380 mm³. The schematic of the anvils used in the aforementioned model is presented in Fig. 14.

Fig. 13

FE model of the high-speed forging process

Fig. 14

Schematic of the anvils

A case related to the axle illustrated in Fig. 1 was analyzed. The axle was hot-formed through multipass forging, during which the billet was first chamfered from the left side and then rounded to form the left body section. Subsequently, rough rounding and final forging were performed to manufacture the left journal part. The aforementioned processes were also conducted for the right side of the workpiece to forge the right body section and journal part. The process parameters of the high-speed forging test are detailed in Table 2.

Table 2 Process parameters used in the high-speed forging simulation

The thermal parameters used in the FE model are presented in Table 3 [39,40,41]. The thermal conductivity and specific heat capacity of 25CrMo4 at different temperatures are listed in Table 4. To obtain the precise microstructures and temperature distribution of the billets before high-speed forging, the occurrence of step heating [Fig. 15(a)] and the transfer of the billet from the furnace to the anvil in 100 s were considered in the simulation. The initial temperature of the billet in the heating process was 900 °C, and the initial average austenite grain size was 15 μm, which was obtained using the GG test at 900 °C. To avoid excess interfacial slip between the tool and the workpiece, the friction factor was set as 1.0 in the FE model [12].

Table 3 Primary thermal parameters of the FE model

Table 4 Thermal conductivity and specific heat capacity of 25CrMo4 at different temperatures

Fig. 15

Schedule of step heating (a) and (b) simulation results after transferring process

Figure 15(b) illustrates the distributions of the temperature and average grain size before the forging process. The temperature was marginally lower on the surface of the billet than inside it, with a difference around 150 °C. The average grain sizes on the surface of and inside the billet were 220 and 211 μm, respectively. The results were input into the simulation of the high-speed forging process as the initial state.

FE simulation results and discussion

Figure 16 displays the profiles of the temperature, effective strain, and grain size after multipass forging. As depicted in Fig. 16(a), a railway axle with satisfactory surface quality and without folds or dents was obtained. Moreover, the maximum temperatures mainly occurred in the inner section of the axle body (approximately 1170 °C) due to deformation-induced adiabatic heating [12]. The minimum temperature was approximately 890 °C. This temperature was observed on the journal surfaces. The temperature range of the surface of the axle body was 960–980 °C [Fig. 16(b)]. The temperature distribution was neither uniform nor symmetrical due to the partial and stepwise forming process in high-speed forging and the severe heat loss caused by the high contact pressure on the axle periphery. In addition, Fig. 16(c) indicates that the maximum effective strain (approximately 6.5) occurred in the core region of the left journal part; thus, the accumulated strain in this region was sufficiently large to guarantee complete DRX according to the developed kinetic model. Consequently, the average grain size in the aforementioned section was smaller than that in the body section. The left journal was the first part to be forged. In the subsequent forging process of the right sections, the recrystallized grains grew rapidly due to the elevated temperature in the core region of the left sections. Consequently, the average grain size in the center of the left journal was marginally larger than that in the right journal. Similarly, the fine-grained area in the right body was larger than that in the left body, as illustrated in Fig. 16(d). Although the average grain sizes of the grains grown in the left journal and body were approximately 55–65 μm, a relatively homogeneous microstructure was achieved along the entire axle, with the average grain size in the axle being 25–55 μm.

Fig. 16

Schematic of the (a) railway axle and the (b) temperature, (c) effective strain, and (d) grain size profiles of the axle after multipass forging

To better illustrate the grain size evolution during the complex multipass forging process, six points in two areas along the radial direction of the forged axle were tracked, as displayed in Fig. 17. Given the symmetry of the axle and forging process, area 1 in the left journal and area 2 in the left body were considered. Points 3 and 6 were approximately 30 and 22 mm from the surface, respectively. Furthermore, the points were equally distributed in the radial direction.

Fig. 17

Points tracked for investigating grain size evolution in the cross section of the axle

Figure 18 illustrates the grain size, temperature, and total strain evolution for the six tracked points of the axle. For areas 1 and 2, the grains were refined during the chamfering process from a coarse size of approximately 220 μm to a fine size of approximately 50 μm. In area 1, the grains at points 1–3 are further refined to approximately 40 μm because of the DRX in the subsequent left rounding and left journal forging processes [Fig. 18(c)]. During the aforementioned processes, the left journal part had an elevated temperature of approximately 1150–1240 °C at its center [Fig. 18(a)]. Moreover, this left journal part was subject to severe deformation, as depicted in Fig. 18(c). Due to the severe deformation and low heating loss of the aforementioned part, the temperatures at points 1 and 2 increased to 1240 and 1220 °C, respectively. The processes used in the fabrication of the left part of the axle were also used in the fabrication of the right part. The recrystallized grain growth in area 1 was stimulated at elevated temperatures. Marginal winding of the axle occurred in the left journal part during the forming of the right axle body part, which increased the total strain in area 1 [Fig. 18(c)]. Thus, the accumulated strain in area 1 promoted recrystallization and marginally decreased the grain size. The grain size at point 3 was smaller than those at points 1 and 2 throughout the forming process. As illustrated in Fig. 18(c), the heating loss was severe for point 3, which was near the surface. Therefore, a high temperature drop occurred at point 3. According to the microstructure kinetic model developed in section 3, a relatively low temperature (in the range of 900–1200 °C) is beneficial for obtaining fine recrystallized grains.

Fig. 18

Evolution in the grain size, temperature, and total strain at (a and c) area 1 and (b and d) area 2 during high-speed forging

Figure 18(b) depicts the grain size evolution at points 4–6. The grain size in area 2 mainly decreased during the chamfering process. Moreover, the rounding process resulted in a further decrease in the grain size by approximately 30 μm. However, almost no deformation occurred in area 2 during the left journal forming and right chamfering processes, as illustrated in Fig. 18(d). GG was the dominant mechanism in area 2 and marginally increased the grain size. Furthermore, during axle body forging, the grain sizes at points 4 and 6 increased marginally, whereas that of point 5 decreased. Because the temperature at point 4 was approximately 1200 °C, the recrystallized grains produced during the forging of area 2 grew marginally. By contrast, at point 6, which was located near the surface, a considerable temperature drop inhibited the recrystallization [Fig. 18(d)]. The GG at points 4 and 6 led to a marginal increase in the grain size. At point 5, sufficient deformation and low heat transmission were guaranteed for the recrystallization; therefore, the grains were refined again at point 5 during axle body forging.

Experimental validation

A full-scale axle was forged on a 12.5-MN high-speed forging hydraulic press system to validate the simulation results of high-speed forging. As displayed in Fig. 19, the hydraulic system comprised a high-speed forging press and a pair of manipulators. Both these parts were driven by the hydraulic system.

Fig. 19

A 12.5-MN high-speed forging hydraulic press system

The billet was first heated in the furnace according to the heating plan displayed in Fig. 15(a) and then transferred to the press in approximately 100 s. All the process parameters of the high-speed forging experiment were similar to those of the FE model described in section 4.1. To retain the microstructures at elevated temperatures, the forged axle was placed into a water pool for quenching. Due to the length and limited hardenability of the axle, the left part of the forged axle was quenched, and the average grain sizes at seven positions along the axial direction near the surface layer were investigated.

Figure 20 displays the microstructure morphology at seven locations on the experimentally forged axle as well as the grain size distribution obtained from the FE simulation. The initial austenite grain boundaries were revealed in metallographic images, which were used to measure the average grain sizes. A comparison of the average experimental and simulated grain sizes is presented in Fig. 21.

Fig. 20

Average grain size at different areas in the forged axle

Fig. 21

Comparisons of the average simulated and experimental grain sizes at different areas in the forged axle

Application of the models for designing round forging anvils

Reasonable anvil design is crucial for guaranteeing forging precision and quality. Chamfering is a key forging process for refining coarse grains. Section 4.2 describes an instance in which this process was performed. To determine the appropriate rules of anvil design for the high-speed forging of railway axles, this study used the FE model established in section 4.1 to investigate the geometric constraints for the design of round anvils in terms of surface quality, microstructure, and forming force during chamfering.

Geometric constraints

The working zone radius R and transition radius r₁ are the two most important parameters that control the forming quality of railway axles. Fig. 22(a) illustrates the geometry of the square billet. In this figure, a refers to the side length and r refers to the radius. A schematic of the chamfering process is presented in Fig. 22(b). The arc BFC of the working zone, with its center at point P, is tangential to the transition arc CK of radius r₁. During the chamfering process, the working zone compresses the billet from point A to point F. Consequently, the arc is tangential to the billet at points B and C. Because of the dramatic increase in the forming force with the further vertical movement after point F of the working zone, the reduction with a value of 2|AF| is considered as the maximum one (h_max) for the anvil. The geometric correlations for the chamfering of the billet are described as follows.

Fig. 22

Schematics of (a) the initial geometry of the billet and (b) the chamfering process in the transverse section

The arc BFC, with its center at (0, y₀), can be described using the following equation:

$$ {x}^2+{\left(y-{y}_0\right)}^2={R}^2 $$

(24)

The tangent point C is on the line CQ, which can be expressed as follows:

$$ x+y=\frac{\sqrt{2}}{2}a $$

(25)

By combining Eqs. (24) and (25), the following equation can be obtained:

$$ 2{y}^2-\left(\sqrt{2}a+2{y}_0\right)y+\frac{1}{2}{a}^2+{y}_0^2-{R}^2=0 $$

(26)

According to the single-root condition of Eq. (26), y₀ can be solved as follows: ${y}_0=\frac{\sqrt{2}a-2\sqrt{2}R}{2}$.

Consequently, h_max can be expressed as follows:

$$ {h}_{\mathrm{max}}=2\mid AF\mid =2\left( AO- FO\right)=\sqrt{2}\left(a-2r\right)+2r-2R+2\mid {y}_0\mid $$

(27)

Thus, the maximum reduction of chamfering (h_max) is influenced by the dimensions of the initial billet and the working zone radius R. Moreover, the coordinates of C can be calculated as follows:

$$ \Big\{{\displaystyle \begin{array}{c}{x}_C=\frac{\sqrt{2}a-2{y}_0}{4}\\ {}{y}_C=\frac{\sqrt{2}a+2{y}_0}{4}\end{array}} $$

(28)

According to the geometric correlations, the coordinates of H can be expressed as follows:

$$ \Big\{{\displaystyle \begin{array}{c}{x}_H=\left(1+\frac{r_1}{R}\right){x}_C\\ {}{y}_H={r}_1+R+{y}_0-\frac{1}{2}h\end{array}} $$

(29)

Point C is on the transition arc with the center H and radius r₁. Consequently the following equation is obtained:

$$ {\left({x}_C-{x}_H\right)}^2+{\left({y}_C-{y}_H\right)}^2={r}_1^2 $$

(30)

The minimum value of the transition radius r₁ can be obtained from the aforementioned equation.

Effects of round anvil geometry on the chamfering process

In general, the design of anvils is adapted according to the diameter of the axle. In addition, the length of the anvil should be sufficiently long for meeting the clamping requirements of the press. In this study, the die height h was set as 150 mm, and the anvil length l was set as 500 mm. According to an empirical equation (L ≤ 0.5l) [42], the working zone length L was set as 240 mm.

Effects of R on the chamfering process

To investigate the effects of R on the chamfering process, three R values were considered. The corresponding h_max and minimum r₁ values were calculated according to the geometric constraints in Section 5.1 (Table 5). The reduction of the chamfering was set as 86 mm, and the thermal parameters specified in Section 4.1 were adopted in the computation.

Table 5 Maximum reduction and transition radius for different working zone radii

Figure 23 illustrates the surface quality and average grain size distribution for the deformed zone with different R values. A clear dent [Fig. 23(a)] appeared on the surface with an R value of 115 mm due to excessive reduction. The material flow along the transverse direction increased dramatically once the reduction exceeded h_max. The deformed billet began to contact the transition arc, and a dent was formed near point C [Fig. 22(b)]. The dent was improved by increasing the value of R under the same r₁ and reduction values. As displayed in Fig. 23, the coarse grains of the heated billet were refined to a size of 40–70 μm after chamfering. The average grain size in the core was approximately 40–60 μm for the R values of 115 and 125 mm. The average grain size was 50–60 μm when R was 135 mm. Fig. 24(a) presents the variation in the forming force of the upper die with different R values. The forming force of the first stroke increased with a decrease in R, whereas the forming forces of the other strokes exhibited no obvious deviation with different R value. As depicted in Fig. 25(a), the reduction (86 mm) exceeded h_max when R was 115 mm, which resulted in severe material flow and spreading of the billet in the horizontal direction. The friction force in the transition arc area increased, which led to an increase in the forming force. However, the reduction was less than h_max when the R value was 135 mm. In this case, almost no material flow contacted the transition arc, as displayed in Fig. 25(b), which reduced the deformation resistance. Thus, the forming force was comparatively low when R was 135 mm [Fig. 24(a)]. Although the forming force with R of 135 mm was the most economical option, given the comprehensive considerations of the surface quality, microstructure, and forming force, 125 mm was determined to be the optimal R value for achieving a reduction of 86 mm. The results indicate that the developed geometric constraints can meet the requirements for surface quality, microstructure, and forming force.

Fig. 23

Surface quality and average grain size distribution for different R values: (a) 115, (b) 125, and (c) 135 mm

Fig. 24

Variation in the forming forces of the upper die with the (a) R and (b) r₁ values

Fig. 25

Profiles of the contact between the billet and anvils in the first stroke for R values of (a) 115 and (b) 135 mm

Effects of r ₁ on the chamfering process

The effect of the transition radius r₁ on the chamfering was also investigated, and the results are displayed in Fig. 26. No obvious dent was observed on the surface for three r₁ values under an R value of 125 mm and h_max value of 86 mm. Thus, r₁ has no significant influence on the surface quality when the R and corresponding h_max values are set. Fig. 26 also illustrates the average grain size distribution with different r₁ values when the reduction was set as 86 mm and R was set as 125 mm. After chamfering, the austenite grains were refined from a size of 215 μm to a size of approximately 40–70 μm. Moreover, the area occupied by the fine grains (green and blue) in the cross section decreased when r₁ increased from 130 to 170 mm. The effect of r₁ on the forging force during chamfering is presented in Fig. 24(b). The forging force increased with an increase in r₁ for the first two strokes. The r₁ value had almost no effect on the forging force for the last two strokes. An increase in r₁ caused a decrease in the arc length of the working zone, which led to material flow in the horizontal direction and enlarged the contact area between the deformed billet and the anvils. Consequently, the large friction force caused by the anvils increased the forging force. According to comprehensive considerations of the surface quality, microstructure, and forming force, 150 mm was calculated to be the optimal r₁ value for achieving a reduction of 86 mm and R value of 125 mm.

Fig. 26

Surface quality and average grain size distribution for different r₁ values: (a) 130, (b) 150, and (c) 170 mm

The simulation results and the proposed geometric constraints exhibited high agreement with respect to the relationship between R, h_max, and r₁. Therefore, the geometric constraints considered in this study can achieve suitable R and r₁ values for round anvils. The results and findings presented in this section indicate the reliability and applicability of the developed geometric constraints for the design of round anvils through high-speed forging processes.

Summary

In this study, an FE model for high-speed forging and thermal–mechanical–metallurgical equations for 25CrMo4 steel were established. The microstructure evolution was analyzed throughout the heating and high-speed forging process chain for a railway axle. The effects of the anvil structure on the forging process were investigated on the basis of the developed FE model by considering the surface qualities, microstructure, and forming force. The main conclusions of this study are as follows:

• (1) A set of strain-compensated Arrhenius constitutive equations and a model of microstructure evolution kinetics that involves DRX, MDRX, SRX, and GG were developed on the basis of hot compression tests. Comparisons between the experimental and simulation results indicated that the flow behavior and microstructure evolution of 25CrMo4 steel can be well predicted using the established thermomechanical–metallurgical FE model.

• (2) The coarse grains were mainly refined during the chamfering and rounding processes from a size of approximately 220 μm to a size of approximately 40 μm. The refined grains grew marginally to a size of approximately 50 μm till the end of the high-speed forging process.

• (3) A set of suitable geometric constraints are proposed for the design of round anvils. By considering the surface quality, microstructure, and forming force during the chamfering, the reliability and applicability of proposed constraints were demonstrated.

Appendix

According to Eq. (1), the following equation can be obtained

$$ \dot{\varepsilon}=A\cdotp {\left[\sinh \left(\alpha \sigma \right)\right]}^n\cdotp \exp \left(-\frac{Q}{R_UT}\right) $$

(31)

From Eq. (31), the following equation can be obtained:

$$ {\left[\frac{\dot{\varepsilon}\cdotp \exp \left(Q/{R}_UT\right)}{A}\right]}^{\frac{1}{n}}=\sinh \left(\alpha \sigma \right) $$

(32)

Moreover, sinh(ασ) can be expressed as follows:

$$ \sinh \left(\alpha \sigma \right)=\frac{\exp \left(\alpha \sigma \right)-\exp \left(-\alpha \sigma \right)}{2} $$

(33)

From Eq. (33) and the inverse function of Eq. (33), the following equation is obtained:

$$ \sigma =\frac{1}{\alpha}\ln \left\{\sinh \left(\alpha \sigma \right)+{\left[{\left(\sinh \left(\alpha \sigma \right)\right)}^2+1\right]}^{\frac{1}{2}}\right\} $$

(34)

By combining Eqs. (32), (34) and (2), the following equation is obtained:

$$ \sigma =\frac{1}{\alpha}\ln \left\{{\left(\frac{\dot{\varepsilon}\cdotp \exp \left(Q/{R}_UT\right)}{A}\right)}^{\frac{1}{n}}+{\left[{\left(\frac{\dot{\varepsilon}\cdotp \exp \left(Q/{R}_UT\right)}{A}\right)}^{\frac{2}{n}}+1\right]}^{0.5}\right\}=\frac{1}{\alpha}\ln \left\{{\left(\frac{Z}{A}\right)}^{\frac{1}{n}}+{\left[{\left(\frac{Z}{A}\right)}^{\frac{2}{n}}+1\right]}^{0.5}\right\} $$

(35)