Hot Deformation Behavior and Optimization of Processing Parameters for 4Cr16MoCu Martensitic Stainless Steel

Abstract

The hot deformation behavior of 4Cr16MoCu martensitic stainless steel alloyed with 1% Cu was investigated through hot compression tests at temperatures ranging from 900 to 1150 °C and strain rates of 0.001 to 1 s⁻¹. The addition of Cu is strategically employed to synergistically enhance precipitation hardening and corrosion resistance, yet its complex interplay with hot deformation mechanisms remains poorly understood, demanding systematic investigation. The results revealed a narrow forging temperature range and significant strain rate sensitivity, with deformation resistance increasing markedly at higher strain rates. An Arrhenius constitutive model incorporating a seventh-degree polynomial for strain compensation was developed to describe the flow stress dependence on deformation temperature and strain rate. The model demonstrated high accuracy, with a correlation coefficient (R²) of 0.9917 and an average absolute relative error (AARE) of 3.8%, providing a reliable theoretical foundation for practical production applications. Furthermore, a hot processing map was constructed based on the dynamic material model (DMM), and the optimal hot working parameters were determined through microstructural analysis: an initial forging temperature of 1125 °C, a final forging temperature of 980 °C, and a strain rate of 0.1 s⁻¹. These conditions resulted in a fine and uniform grain structure, while strain rates above 0.18 s⁻¹ were identified as unfavorable due to the risk of uneven deformation.

1. Introduction

4Cr16Mo martensitic stainless steel (MSS) has been extensively utilized in high-quality plastic mold steels due to its exceptional mechanical properties and corrosion resistance [1,2,3]. These steels typically contain alloying elements such as chromium (Cr) and molybdenum (Mo) which are integral to their performance. However, the high content of these elements, particularly in the case of 4Cr16MoCu steel, increases the deformation resistance during forging, posing significant challenges during manufacturing [4,5]. To further enhance the material’s performance, the incorporation of 1wt.% Cu in this steel has been proven effective in improving its machinability and corrosion resistance. However, the introduction of copper, owing to its unique characteristics, can degrade the hot workability of the steel, necessitating the prevention of “copper embrittlement” during hot forging operations. Additionally, a certain amount of primary carbides were found in 4Cr16MoCu experimental specimens, which need to be removed through forging processes during subsequent production. Yet, there remains a dearth of research on the forging and machinability aspects of this material. Therefore, while the addition of copper to the steel enhances the application performance of the material, it also poses challenges related to its influence on the alloy’s hot workability and microstructure evolution. In particular, the low impact toughness observed in large ingots of plastic mold steel represents a critical quality issue, necessitating precise control over the microstructure during free-forging to ensure final product quality [6].

One of the most critical microstructural evolution phenomena during hot deformation is discontinuous dynamic recrystallization (DRX), which is typical in high stacking-fault-energy metals such as MSS [7,8]. DRX plays a pivotal role in the microstructural evolution of alloys with low, medium, and high stacking fault energy (SFE) [9,10,11]. By optimizing key forging parameters, the challenges associated with coarse and uneven grain structures can be effectively addressed, thereby improving impact toughness and enhancing the overall quality of plastic mold steel blooms [12,13,14].

In 1969, Sellars and Jonas et al. introduced a widely recognized phenomenological model based on the Arrhenius-type equation to study material behavior during hot deformation [15,16]. This model, which establishes the relationship among stress, temperature, and strain rate, has been successfully applied to investigate the hot deformation behavior of various alloys [17,18]. The processing map (PM) technology based on the dynamic materials model (DDM) has been accepted as a powerful approach to evaluate the deformation mechanisms and optimize the process parameters of materials due to its convenience and accuracy [19,20]. It has been used in a wide range of materials including aluminum alloy, titanium, magnesium, aluminum, Ni-based alloy, as well as steels [21,22,23].

Despite extensive research on the hot deformation characteristics of various martensitic stainless steels [24,25], the specific hot deformation behavior of 4Cr16MoCu steel remains unexplored. To address this gap, we conducted hot compression tests on a Gleeble-3500 thermomechanical simulator under varying temperatures and strain rates. A strain-compensated constitutive equation was developed by incorporating modified flow stress values, and a hot processing map was constructed. By integrating the hot processing map with microstructural evolution analysis, the optimal hot working region was identified, providing a scientific basis for determining reasonable processing parameters in the production and manufacturing of 4Cr16MoCu MSS.

2. Experimental

Material and Experimental Procedures

The material was formulated by modifying the chemical composition of 4Cr16Mo MSS to include an additional 1 wt.% Cu, resulting in a composition of Fe-0.32C-15.94Cr-1.08Mo-0.46Si-0.60Mn-1.07Cu-0.88Ni. The incorporation of Ni, a pivotal constituent in the majority of Cu-bearing steels, is crucial due to its efficacy in mitigating hot shortness during the forging operation [26,27].

In accordance with the Gleeble hot compression specimen standards, 4Cr16MoCu steel was machined into cylindrical samples with dimensions of Φ8 mm × 12 mm using wire electrical discharge machining technology. The experiments were conducted on a Gleeble-3500 testing machine with varying parameters; specifically, the hot compression deformation temperatures were set at 900 °C, 950 °C, 1000 °C, 1050 °C, 1100 °C, and 1150 °C, and the constant strain rates were 0.001 s⁻¹, 0.01 s⁻¹, 0.1 s⁻¹, and 1 s⁻¹. The detailed process flow for the thermal compression of the specimens is illustrated in Figure 1. Initially, the specimens were heated to 1200 and homogenized for 300 s. Subsequently, they were cooled to the respective deformation temperatures, followed by a 60 s soak time prior to compression to eliminate internal temperature gradients, thereby achieving uniform temperature and microstructure distribution. Upon deforming the specimens to a strain of 0.6, water quenching was applied to rapidly cool the specimens, preserving the deformed microstructure. To obtain the microstructure of the material after hot compression, the deformed and water-quenched samples were sectioned along the axial direction. The cut surfaces were ground, polished, and etched using an aqueous solution of HCl and HNO₃. Subsequently, these etched surfaces were observed under a metallographic optical microscope.

3. Results and Discussion

3.1. Hot Deformation Flow Curve

The variation in flow stress during hot compression is governed by the interplay between work hardening and softening mechanisms. Figure 2 illustrates the representative true stress–strain curves of 4Cr16MoCu MSS under different deformation conditions. As shown in the figure, the flow stress increases with decreasing temperature and increasing strain rate. During the initial stage of hot compression, a substantial generation of dislocations occurs within the alloy, leading to work hardening and a rapid increase in stress with strain [28]. Subsequently, a balance is achieved between dynamic softening and work hardening, resulting in a steady-state flow stress [29]. At strain rates of 0.001 s⁻¹ and 0.01 s⁻¹, the flow stress exhibits a relatively flat trend during the steady-state deformation stage. This phenomenon is attributed to the enhanced thermal activation energy at lower strain rates, which promotes recovery softening and counteracts the work hardening effect [30]. In contrast, at higher strain rates of 0.1 s⁻¹ and 1 s⁻¹, the flow stress continues to increase gradually even after reaching the steady-state deformation stage. This behavior is caused by the accelerated generation of dislocations at faster compression rates, which intensifies the work hardening effect [31,32].

3.2. Constitutive Equation and Hot Deformation Activation Energy

3.2.1. Establishment of Constitutive Equation

During thermal deformation, the primary influencing factors on the true stress of metallic materials are the deformation rate and deformation temperature [33]. In this study, the Arrhenius constitutive model is employed to establish the following equation:

$$\dot{\epsilon }=AF\left(\sigma \right)\mathit{exp}\left(-{\displaystyle \frac{Q}{RT}}\right)$$

(1)

wherein, F(σ) is a function of stress, and this expression can be further divided into three forms: the form for low stress levels (ασ < 0.8), the form for high stress levels (ασ > 1.2), and the form for all stress levels after the coefficients of the Arrhenius equation are modified and optimized by Sellar based on the hyperbolic sine function.

(1)

Form for low stress levels (ασ < 0.8):

$$\dot{\epsilon }=A_1{\sigma }^{n_1}exp\left(-{\displaystyle \frac{Q}{RT}}\right)$$

(2)

(2)

Form for high stress levels (ασ > 1.2):

$$\dot{\epsilon }=\mathit{exp}\left(\beta \sigma \right)\mathit{exp}\left(-{\displaystyle \frac{Q}{RT}}\right)$$

(3)

(3)

The fully stress-level form after Sellars’ modification and optimization of the coefficients in the Arrhenius equation based on the hyperbolic sine function:

$$\dot{\epsilon }=A{\left[\mathit{sinh}\left(\alpha \sigma \right)\right]}^n\mathit{exp}\left(-{\displaystyle \frac{Q}{RT}}\right)$$

(4)

In the equation, $\dot{\epsilon }$ represents the strain rate, with units of s⁻¹; σ denotes the flow stress corresponding to a specific strain, with units of MPa; n is the stress exponent; α is the stress level parameter, obtained through the formula α = β/n₁; Q represents the deformation activation energy, with units of KJ/mol; R is the ideal gas constant, typically taken as 8.314 J/(mol·K); T is the absolute temperature, with units of K; A, A₁, A₂, β, and n₁ are all material constants, which can be obtained through fitting.

Taking the characteristic point with a true strain of 0.3 as an example, the constitutive equation at this true strain value (0.3) is constructed. By taking the logarithm of both sides of Equations (2) and (3) and rearranging, we obtain

$$\mathrm{l}\mathrm{n}\dot{\mathsf{\epsilon }}=\mathrm{l}\mathrm{n}{\mathrm{A}}_1-{\displaystyle \frac{\mathrm{Q}}{\mathrm{R}\mathrm{T}}}+\mathrm{l}\mathrm{n}\mathsf{\sigma }$$

(5)

$$\mathrm{l}\mathrm{n}\dot{\mathsf{\epsilon }}=\mathrm{l}\mathrm{n}{\mathrm{A}}_2-{\displaystyle \frac{\mathrm{Q}}{\mathrm{R}\mathrm{T}}}+\mathsf{\beta }\mathsf{\sigma }$$

(6)

Substitute the strain rate $\dot{\epsilon }$ and stress σ corresponding to the strain of 0.3 into Equations (5) and (6), and establish the relationship plots of $\mathrm{l}\mathrm{n}\dot{\epsilon }-$ lnσ and $\mathrm{l}\mathrm{n}\dot{\epsilon }-$σ as shown in Figure 3a and Figure 3b, respectively, with linear fitting applied. The average values of the slopes calculated are designated as n₁ and β, yielding n₁ = 6.93758 and β = 0.055773333. Subsequently, the stress level parameter α for 4Cr16MoCu steel is obtained as 0.008039. By taking the logarithm of both sides of Equation (4) simultaneously, we derive

$$\mathit{ln}\left[\mathit{sinh}\left(\alpha \sigma \right)\right]=-{\displaystyle \frac{1}{n}}lnA+{\displaystyle \frac{1}{n}}{\displaystyle \frac{Q}{RT}}+ln\dot{\epsilon }$$

(7)

Next, substitute the $\dot{\epsilon }$ and σ values corresponding to the true strain of 0.3 into Equation (7). Scatter plots for $\ln \left[\sinh \left(\mathsf{\alpha }\mathsf{\sigma }\right)\right]-\ln \dot{\mathsf{\epsilon }}$ and $\ln \left[\sinh \left(\mathsf{\alpha }\mathsf{\sigma }\right)\right]-1/\mathrm{T}$ were then constructed separately and subjected to linear fitting, as illustrated in Figure 3c,d. Subsequently, the slopes of the fitted lines were calculated and averaged to obtain the stress exponent n = 5.0255 and the thermal deformation activation energy Q = 469.7088 KJ/mol for 4Cr16MoCu steel.

Zener and Hollomon introduced the deformation rate Z factor for temperature compensation [10], which quantifies the influence of deformation temperature and strain rate on the hot deformation behavior of metallic materials:

$$\mathrm{Z}=\dot{\mathsf{\epsilon }}\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{\mathrm{Q}}{\mathrm{R}\mathrm{T}}}\right)={\mathrm{A}\left[\mathrm{Sinh}\left(\mathsf{\alpha }\mathsf{\sigma }\right)\right]}^{\mathrm{n}}$$

(8)

Taking the logarithm of Equation (8) yields

$$\ln \left[\sinh \left(\mathsf{\alpha }\mathsf{\sigma }\right)\right]=-{\displaystyle \frac{1}{\mathrm{n}}}\mathrm{l}\mathrm{n}\mathrm{A}+{\displaystyle \frac{1}{\mathrm{n}}}{\displaystyle \frac{\mathrm{Q}}{\mathrm{R}\mathrm{T}}}+{\displaystyle \frac{1}{\mathrm{n}}}\mathrm{l}\mathrm{n}\dot{\mathsf{\epsilon }}$$

(9)

Substituting the experimental results into Equation (9), a relationship graph of $\ln Z-\mathrm{l}\mathrm{n}[\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\alpha \sigma \left)\right]$ is established as shown in Figure 4. Through linear fitting, the intercept lnA = 38.95096, which determines A = 8.2449 × 10¹⁶. From Figure 4, the linear correlation coefficient R² for lnZ − ln[sinh(ασ)] is 0.98844, indicating a high degree of linear fit between the two variables and thus a high level of accuracy.

In summary, substituting the calculated parameters α, n, Q, and A into Equation (4) yields the constitutive equation for 4Cr16MoCu steel at a true strain of 0.3 as follows:

$$\dot{\mathsf{\epsilon }}=8.2449 \times {10}^{16}{\left[\sinh \left(0.0081\mathsf{\sigma }\right)\right]}^{5.0255}\exp \left(-{\displaystyle \frac{469.7088}{\mathrm{RT}}}\right)$$

(10)

3.2.2. Establishment of Strain-Compensated Model

The flow stress under a single strain can be predicted through the classical Arrhenius model. However, this model neglects the influence of strain on the flow stress. Therefore, a strain-compensation modification is applied to the classical Arrhenius model to establish a relationship between strain (ε) and flow stress (σ). Based on the true stress-true strain data of 4Cr16MoCu steel under various conditions during hot compression tests, the parameter-solving process described earlier is repeated with a true strain interval of 0.05. The parameters (α, n, Q, A) of 4Cr16MoCu steel under strain conditions ranging from 0.05 to 0.9 are determined separately. The calculation results are shown in

Table 1. Material parameters of the constitutive equations with different strain variables.

True Strain	α	n	Q (KJ/mol)	lnA
0.05	0.0107	6.4472	576.2463	48.6392
0.1	0.0091	5.8427	532.3549	44.8246
0.15	0.0085	5.5857	510.5521	42.8791
0.2	0.0085	5.2924	489.6513	40.6982
0.25	0.0082	5.1712	474.6575	39.3945
0.3	0.0081	5.0255	469.7088	38.9510
0.35	0.0080	4.8268	453.8576	38.5051
0.4	0.0078	4.8616	457.3048	37.9121
0.45	0.0077	4.8040	453.0186	37.5417
0.5	0.0077	4.8075	451.5751	37.4570
0.55	0.0076	4.7907	448.8429	37.2464
0.6	0.0075	4.8067	446.1386	37.0091
0.65	0.0075	4.8188	445.0589	36.9167
0.7	0.0075	4.7665	441.1404	36.5856
0.75	0.0074	4.7937	440.0870	36.5127
0.8	0.0073	4.8315	442.5761	36.1681
0.85	0.0073	4.9002	439.2116	36.5026
0.9	0.0073	4.9644	441.8771	36.7634

.

These four parameters all exhibit a nonlinear correlation with the strain variable ε. A seventh-degree polynomial, as shown in Equation (11), is employed to fit the relationship between strain and the model parameters. The results are presented in Figure 5, where the correlation coefficients R for each parameter against the strain are very close to 1. The specific values of these coefficients are detailed in

Table 2. Seventh-degree polynomial coefficients for α, n, Q, lnA parameter.

Stress Level Parameter (α)	Stress Exponent (n)	Strain Activation Energy (Q)	lnA
B0 = 0.0146	C0 = 7.6244	D0 = 644.7489	E0 = 53.6681
B1 = −0.1091	C1 = −33.1578	D1 = −1790.6077	E1 = −119.4691
B2 = 0.7925	C2 = 235.8938	D2 = 9861.2017	E2 = 399.5431
B3 = −3.0455	C3 = −1028.2501	D3 = −37,700	E3 = −779.8986
B4 = 6.5608	C4 = 2576.0709	D4 = 91,000	E4 = 933.1345
B5 = −7.9708	C5 = −3594.5561	D5 = −127000	E5 = −648.6062
B6 = 5.0996	C6 = 2593.1982	D6 = 93,344.5432	E6 = 190.1419
B7 = −1.3358	C7 = −752.7588	D7 = −27,634.6160	E7 = 11.0290

.

$$\left\{\begin{array}{c} \mathsf{\alpha }\left(\mathsf{\epsilon }\right)={\mathrm{B}}_0+{\mathrm{B}}_1\epsilon +{\mathrm{B}}_2{\mathsf{\epsilon }}^2+{\mathrm{B}}_3{\mathsf{\epsilon }}^3+{\mathrm{B}}_4{\mathsf{\epsilon }}^4+{\mathrm{B}}_5{\mathsf{\epsilon }}^5+{\mathrm{B}}_6{\mathsf{\epsilon }}^6+{\mathrm{B}}_7{\mathsf{\epsilon }}^7\hfill \\ \mathsf{n}\left(\mathsf{\epsilon }\right)={\mathrm{C}}_0+{\mathrm{C}}_1\epsilon +{\mathrm{C}}_2{\mathsf{\epsilon }}^2+{\mathrm{C}}_3{\mathsf{\epsilon }}^3+{\mathrm{C}}_4{\mathsf{\epsilon }}^4+{\mathrm{C}}_5{\mathsf{\epsilon }}^5+{\mathrm{C}}_6{\mathsf{\epsilon }}^6+{\mathrm{C}}_7{\mathsf{\epsilon }}^7\hfill \\ \mathsf{Q}\left(\mathsf{\epsilon }\right)={\mathrm{D}}_0+{\mathrm{D}}_1\epsilon +{\mathrm{D}}_2{\mathsf{\epsilon }}^2+{\mathrm{D}}_3{\mathsf{\epsilon }}^3+{\mathrm{D}}_4{\mathsf{\epsilon }}^4+{\mathrm{D}}_5{\mathsf{\epsilon }}^5+{\mathrm{D}}_6{\mathsf{\epsilon }}^6+{\mathrm{D}}_7{\mathsf{\epsilon }}^7\hfill \\ \mathrm{lnA}\left(\mathsf{\epsilon }\right)={\mathrm{E}}_0+{\mathrm{E}}_1\epsilon +{\mathrm{E}}_2{\mathsf{\epsilon }}^2+{\mathrm{E}}_3{\mathsf{\epsilon }}^3+{\mathrm{E}}_4{\mathsf{\epsilon }}^4+{\mathrm{E}}_5{\mathsf{\epsilon }}^5+{\mathrm{E}}_6{\mathsf{\epsilon }}^6+{\mathrm{E}}_7{\mathsf{\epsilon }}^7\hfill \end{array}\right.$$

(11)

Deriving the stress σ from Equation (4) and rearranging, the flow stress of the material can be expressed by Equation (12):

$$\mathsf{\sigma }={\displaystyle \frac{1}{\mathsf{\alpha }}}{\sinh }^{-1}\left[{\left({\displaystyle \frac{\mathrm{Z}}{\mathrm{A}}}\right)}^{{\displaystyle \frac{1}{\mathrm{n}}}}\right]$$

(12)

Furthermore, based on the definition of the hyperbolic sine function, Equation (12) can also be represented as

$$\mathsf{\sigma }={\displaystyle \frac{1}{\mathsf{\alpha }}}\mathrm{l}\mathrm{n}\left\{{\left({\displaystyle \frac{\mathrm{Z}}{\mathrm{A}}}\right)}^{{\displaystyle \frac{1}{\mathrm{n}}}}+{\left[{\left({\displaystyle \frac{\mathrm{Z}}{\mathrm{A}}}\right)}^{{\displaystyle \frac{2}{\mathrm{n}}}}+1\right]}^{{\displaystyle \frac{1}{2}}}\right\}$$

(13)

By substituting the strain-dependent flow stress parameter relationship (Equation (11)) obtained from the above calculations into Equation (13), the high-temperature flow stress model for 4Cr16MoCu steel under different deformation temperatures and train rates is obtained, as shown in Equation (14):

$$\left\{\begin{array}{c}\mathsf{\sigma }={\displaystyle \frac{1}{\mathsf{\alpha }\left(\mathsf{\epsilon }\right)}}\ln \left\{{\left({\displaystyle \frac{\mathrm{Z}\left(\mathsf{\epsilon }\right)}{\mathrm{A}\left(\mathsf{\epsilon }\right)}}\right)}^{{\displaystyle \frac{1}{\mathrm{n}\left(\mathsf{\epsilon }\right)}}}+{\left[{\left({\displaystyle \frac{\mathrm{Z}\left(\mathsf{\epsilon }\right)}{\mathrm{A}\left(\mathsf{\epsilon }\right)}}\right)}^{{\displaystyle \frac{2}{\mathrm{n}\left(\mathsf{\epsilon }\right)}}}+1\right]}^{{\displaystyle \frac{1}{2}}}\right\}\\ \mathrm{Z}(\epsilon )=\dot{\mathsf{\epsilon }}\exp \left({\displaystyle \frac{\mathrm{Q}\left(\mathsf{\epsilon }\right)}{\mathrm{R}\mathrm{T}}}\right)\end{array}\right.$$

(14)

3.2.3. Verification of the Developed Constitutive Model

Using Equation (14), the stress values corresponding to different true strains, deformation temperatures, and strain rates for 4Cr16MoCu steel were calculated. The actual experimental values were compared with the model-calculated values to validate the model; the results are shown in Figure 6, in the figure, the solid line corresponds to the experimental results, whereas the colored dots represent the values predicted by the model. From Figure 6b, it can be observed that there is a slight discrepancy between the flow stress data and the actual values for the condition of 900 °C + 0.01 s⁻¹. However, the errors between the calculated values and the experimental values under the remaining deformation conditions are relatively small. This indicates that the predicted values calculated by the equation are generally in good agreement with the actual stress–strain curves.

To quantitatively assess the accuracy of the present equations, as illustrated in Equations (15) and (16), the correlation coefficient (R) and the Average Relative Error (AARE) are introduced for quantitative analysis of the model [34,35].

$$\mathrm{R}={\displaystyle \frac{{\sum }_{\mathrm{i}=1}^{\mathrm{N}}({\mathrm{E}}_{\mathrm{i}}-\overline{\mathrm{E}})({\mathrm{P}}_{\mathrm{i}}-\overline{\mathrm{P}})}{\sqrt{{\sum }_{\mathrm{i}=1}^{\mathrm{N}}{({\mathrm{E}}_{\mathrm{i}}-\overline{\mathrm{E}})}^2-{\sum }_{\mathrm{i}=1}^{\mathrm{N}}{({\mathrm{P}}_{\mathrm{i}}-\overline{\mathrm{P}})}^2}}}$$

(15)

$$\mathrm{A}\mathrm{A}\mathrm{R}\mathrm{E}={\displaystyle \frac{1}{\mathrm{N}}}{\sum }_{\mathrm{i}=1}^{\mathrm{N}}\left|{\displaystyle \frac{{\mathrm{E}}_{\mathrm{i}}-{\mathrm{P}}_{\mathrm{i}}}{{\mathrm{E}}_{\mathrm{i}}}}\right|\times 100\%$$

(16)

In these equations, $E_i, \overline{E}$ represent the experimental stress values and their average, respectively; $P_i, \overline{P}$ denote the stress values predicted by the model and their average, respectively; N stands for the number of experimental data points; and i indexes the experimental sequences.

As shown in Figure 7, the solid line represents the best-fit curve between the experimental values and the model-predicted values. The results indicate that the correlation coefficient and the average relative error are 0.9917 and 3.8%, indicating a relatively small overall error. Therefore, this strain-compensated constitutive equation can effectively predict the flow stress during the hot deformation process and provide a theoretical basis for formulating a reasonable forming process for large-scale forgings.

3.3. Effects of Strain on Processing Maps and Microstructural Evolution

The thermomechanical processing map is established based on the dynamic materials model (DMM) proposed by Prasad et al. [36], which allows for the determination of safe and unstable regions during the hot working of materials, ultimately to determine the optimum hot deformation condition for materials [37,38].

3.3.1. Establishment of Power Dissipation Maps

This model posits that the total energy P absorbed by a hot-worked workpiece instantaneously during deformation consists of a dissipative component G and a co-dissipative component J according to Equation (17).

$$\mathrm{P}=\mathsf{\sigma }\dot{\mathsf{\epsilon }}=\mathrm{G}+\mathrm{J}={\int }_0^{\dot{\mathsf{\epsilon }}}\mathsf{\sigma }\mathrm{d}\dot{\mathsf{\epsilon }}+{\int }_0^{\mathsf{\sigma }}\dot{\mathsf{\epsilon }}\mathrm{d}\mathsf{\sigma }$$

(17)

Once the temperature and strain during the hot working of the workpiece are determined, the relationship between the stress σ experienced by the workpiece and the strain rate can be expressed by Equation (18):

$$\mathsf{\sigma }=\mathrm{K}{\dot{\mathsf{\epsilon }}}^{\mathrm{m}}$$

(18)

In this equation, K represents the flow stress at a strain rate of 1, m is the strain rate sensitivity exponent, which governs the power dissipation G and the dissipation covariant J during the thermal deformation process of the material. The value of m can be obtained through curve fitting of the lnσ-lnε relationship, with its specific definition given by

$$\mathrm{m}={\left[{\displaystyle \frac{\partial \mathrm{J}}{\partial \mathrm{G}}}\right]}_{\mathsf{\epsilon },\mathrm{T}}={\left.{\displaystyle \frac{\partial \mathrm{l}\mathrm{n}\mathsf{\sigma }}{\partial \dot{\mathsf{\epsilon }}}}\right|}_{\mathsf{\epsilon },\mathrm{T}}$$

(19)

H. Ziegler [39] pointed out that for the steady-state rheological behavior of viscoplastic solids, the value of m ranges between 0 and 1, and a higher value of m indicates a greater energy requirement for microstructural evolution. When m is constant, the dissipation covariant J can be simplified from Equations (3)–(17) as follows:

$$\mathrm{J}={\int }_0^{\mathsf{\sigma }}\dot{\mathsf{\epsilon }}\mathrm{d}\mathsf{\sigma }={{\int }_0^{\mathsf{\sigma }}\dot{\left({\displaystyle \frac{\mathsf{\sigma }}{\mathrm{K}}}\right)}}^{{\displaystyle \frac{1}{\mathrm{m}}}}\mathrm{d}\mathsf{\sigma }={\displaystyle \frac{\mathrm{m}\mathsf{\sigma }\dot{\mathsf{\epsilon }}}{1+\mathrm{m}}}$$

(20)

For nonlinear dissipative systems, due to the continuous nonlinear variation of J, for the sake of computational convenience, it is compared with the ideal linear dissipative factor J_max to generate a dimensionless parameter, namely η, which represents the energy dissipation efficiency arising from microstructural evolution:

$$\mathsf{\eta }={\displaystyle \frac{\mathrm{J}}{{\mathrm{J}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}}={\displaystyle \frac{2\mathrm{m}}{\mathrm{m}+1}}$$

(21)

In the equation, η is termed the energy dissipation efficiency factor; J_max represents the maximum dissipation covariant when m = 1, at which point the material is in an ideal linear dissipative state. The distribution plot drawn by the η values obtained from different temperatures T and strain rates $\dot{\epsilon }$ forms the power dissipation map. The microstructure evolution patterns of the workpiece during hot working deformation can be inferred from the distribution of η values in the power dissipation map.

Next, taking the strain of 0.2 as an example, the power dissipation map and instability map for 4Cr16MoCu are plotted. Initially, based on the stress–strain curves depicted in Figure 2, the stress value σ corresponding to the strain of 0.2 is extracted. A σ − $\mathrm{l}\mathrm{n}\dot{\epsilon }$ scatter plot is constructed and subjected to linear fitting, with the results presented in Figure 8a. It is evident from the figure that there is a good linear correlation between the two variables. Subsequently, a lnσ − ln$\dot{\epsilon }$ curve is plotted and fitted using a cubic polynomial, as shown in Figure 8b. The cubic spline function, as expressed in Equations (3)–(22), is then solved:

$$\mathrm{l}\mathrm{n}\mathsf{\sigma }=\mathrm{k}+{\mathrm{b}}_1\mathrm{l}\mathrm{n}\dot{\mathsf{\epsilon }}+{\mathrm{b}}_2{\left[\left.\mathrm{l}\mathrm{n}\dot{\mathsf{\epsilon }}\right]\right.}^2+{\mathrm{b}}_3{\left[\left.\mathrm{l}\mathrm{n}\dot{\mathsf{\epsilon }}\right]\right.}^3$$

(22)

The coefficients k, b₁, b₂, b₃ of the cubic function in the figure are used to calculate the values under various conditions. A $\left(\mathrm{l}\mathrm{n}\sigma \right)\prime -\mathrm{l}\mathrm{n}\dot{\epsilon }$ scatter plot is constructed and fitted, with the results displayed in Figure 8c. Combining Equation (19), by taking the partial derivative of Equation (22) again, the value of m can be obtained. Further, η can be derived using Equation (21).

$$\mathrm{m}={\displaystyle \frac{\partial \mathrm{l}\mathrm{n}\mathsf{\sigma }}{\partial \mathrm{l}\mathrm{n}\dot{\mathsf{\epsilon }}}}={\mathrm{b}}_1+2{\mathrm{b}}_2\ln \dot{\mathsf{\epsilon }}+3{\mathrm{b}}_3{\left(\ln \dot{\mathsf{\epsilon }}\right)}^2$$

(23)

For subsequent comparison of hot working maps under different true strain values, similarly, the η values at strains of 0.4, 0.6, and 0.8 are calculated using the aforementioned method. The specific values are presented in

Table 3. Energy dissipation efficiency factor values (η) under different strain conditions.

Strain	Strain Rate (s−1)	Energy Dissipation Efficiency Factor Values (η)
		900 °C	950 °C	1000 °C	1050 °C	1100 °C	1150 °C
0.2	0.001	0.2827	0.1981	0.1692	0.2929	0.2283	0.1888
	0.01	0.2112	0.3005	0.2933	0.2867	0.3165	0.3221
	0.1	0.1359	0.2411	0.2287	0.2415	0.2759	0.3066
	1	0.0565	0.0183	0.0691	0.1508	0.0858	0.1335
0.4	0.001	0.2763	0.1837	0.1196	0.2931	0.2433	0.2080
	0.01	0.2242	0.3114	0.2955	0.3291	0.4097	0.4263
	0.1	0.1364	0.2299	0.2505	0.2723	0.3467	0.3982
	1	0.0022	0.1174	0.0526	0.1019	0.0104	0.0936
0.6	0.001	0.2698	0.1642	0.1234	0.3127	0.2614	0.2341
	0.01	0.2448	0.3263	0.3080	0.3372	0.4043	0.4009
	0.1	0.1543	0.2369	0.2609	0.2695	0.3398	0.4156
	1	0.0233	0.1689	0.0586	0.0844	0.0085	0.0269
0.8	0.001	0.2922	0.2022	0.1478	0.2979	0.2916	0.1014
	0.01	0.2415	0.3288	0.3023	0.3240	0.3736	0.3955
	0.1	0.1497	0.2355	0.2504	0.2635	0.3073	0.3815
	1	0.0039	0.1438	0.0491	0.0960	0.0509	0.0412

.

At a given strain, plotting the values of η at various temperatures and strain rates onto a contour map constitutes the power dissipation map. The power dissipation efficiency map on the T − lg($\dot{\epsilon }$) plane is plotted using Origin software (OriginPro 2018C), with the results shown in Figure 9. Combining the results from Table 3, it can be observed that the variation trend of η values for 4Cr16MoCu steel under different strains tends to be consistent: the maximum η values at various strains are concentrated in the low strain rate region near 0.01~0.1 s⁻¹, with temperature ranges mainly focused on approximately 950~1000 °C and 1100~1150 °C. The combination of these two temperature ranges can be preliminarily considered as the optimal hot working region for 4Cr16MoCu steel. Comparing the η values under the same parameters among the four strains, the results indicate that the η value of 4Cr16MoCu steel also increases as the strain increases. This is due to the increasing number of dislocations within the steel as strain progresses, which subsequently increases the driving force required for dynamic recrystallization to occur within the microstructure. According to the dynamic materials model theory, an increase in the η value also represents an increase in the energy utilized for microstructural evolution.

3.3.2. Construction of Instability Maps

Generally, a higher power dissipation efficiency factor (η) indicates that the material consumes more energy during microstructural evolution, making the material more prone to recrystallization and suggesting that the material will exhibit better processability [40]. However, high η values can also be observed in the region of processing instability. Therefore, based on the principle of extreme values in irreversible thermodynamics of large plastic deformation, Prasad and Murthy proposed a processing instability criterion:

$$\mathsf{\xi }\left(\dot{\mathsf{\epsilon }}\right)={\displaystyle \frac{\partial \mathrm{l}\mathrm{n}\left(\frac{\mathrm{m}}{\mathrm{m}+1}\right)}{\partial \mathrm{l}\mathrm{n}\dot{\mathsf{\epsilon }}}}+\mathrm{m}<0$$

(24)

The physical significance of Equation (24) is that when the entropy production rate of a system is less than the strain rate applied to the system, i.e., $\mathsf{\xi }$ < 0, plastic deformation will localize, leading to flow instability.

Similarly, by calculating the ξ values and constructing contour plots of ξ as a function of temperature and strain rate, the instability parameter region for 4Cr16MoCu steel can be determined. For a strain of 0.2, the value of ln(m/(m + 1)) is first calculated using the m values obtained in Equation (23) Then, a cubic spline function is used to fit the functional relationship between ln(m/(m + 1)) and ln$\dot{\epsilon }$, and the coefficients e, f, g, h are obtained through regression analysis.

$$\ln \left({\displaystyle \frac{\mathrm{m}}{\mathrm{m}+1}}\right)=\mathrm{e}+\mathrm{f}\ln \dot{\mathsf{\epsilon }}+\mathrm{g}{\left(\ln \dot{\mathsf{\epsilon }}\right)}^2+\mathrm{h}{\left(\ln \dot{\mathsf{\epsilon }}\right)}^3$$

(25)

$$\mathsf{\xi }\left(\dot{\mathsf{\epsilon }}\right)={\displaystyle \frac{\partial \mathrm{l}\mathrm{n}\left(\frac{\mathrm{m}}{\mathrm{m}+1}\right)}{\partial \ln \dot{\mathsf{\epsilon }}}}+\mathrm{m}=\left(\mathrm{l}+2\mathrm{m}\left(\ln \dot{\mathsf{\epsilon }}\right)+3\mathrm{n}{\left(\ln \dot{\mathsf{\epsilon }}\right)}^2\right)+\mathrm{m}<0$$

(26)

By substituting the coefficients e, f, g, h and their corresponding m values into Equation (26), the ξ values can be obtained, as shown in

Table 4. The corresponding ξ value under different strain conditions.

Strain	Strain Rate (s−1)	ξ Value
		900 °C	950 °C	1000 °C	1050 °C	1100 °C	1150 °C
0.2	0.001	0.0291	0.1804	0.4833	0.1730	0.2901	0.4347
	0.01	−0.0202	0.3442	0.2482	0.1361	0.2703	0.3118
	0.1	−0.1928	−0.3459	−0.1740	0.0086	−0.0823	0.0042
	1	−0.4888	−1.8713	−0.7117	−0.2097	−0.7677	−0.4883
0.4	0.001	−0.2678	0.5979	0.5095	0.2009	0.2071	0.8331
	0.01	0.2020	0.1821	0.3315	0.2041	0.3250	0.5915
	0.1	−0.6080	−0.0939	−0.2128	−0.0752	−0.1673	−0.1980
	1	−2.6978	−0.0754	−1.0734	−0.6370	−1.2697	−1.5352
0.6	0.001	0.0389	0.7488	0.6622	0.1904	0.0514	0.1461
	0.01	0.0951	0.2028	0.3627	0.2163	0.5210	0.5398
	0.1	−0.3503	−0.0802	−0.1923	−0.0990	−0.4243	−0.1603
	1	−1.2738	0.0845	−0.9424	−0.7557	−2.7846	−2.1076
0.8	0.001	−0.3490	0.5778	0.6415	0.2171	0.1301	0.5128
	0.01	0.2139	0.1997	0.3574	0.2161	0.5265	0.4551
	0.1	−0.6865	−0.1153	−0.2081	−0.0617	−0.3963	−0.0376
	1	−3.0502	−0.2427	−1.0012	−0.6163	−2.6278	−0.9651

.

Similarly, using Origin software, different ξ values are substituted onto the T − lg($\dot{\epsilon }$) plane, and contour maps are plotted to represent the instability maps of the steel under various strains, as shown in Figure 10. The negative ξ value regions in Figure 10 indicate the hot forging parameters where instability occurs during the hot working of 4Cr16MoCu steel. By comparing the instability maps at different strains, it can be seen that the instability phenomenon in 4Cr16MoCu steel is mainly controlled by the strain rate. When the strain rate exceeds 0.1 s⁻¹, the material is highly susceptible to instability. For steel grades with poor plasticity, such as this one, forging at a high deformation rate during hot forging can easily lead to instability phenomena, such as cracking.

Subsequently, contour plots are formulated utilizing the ξ values to delineate the zones within the workpiece where deformation instability manifests, termed the instability map. Through the superposition of the power dissipation map onto the instability map, the hot working map pertinent to the specified deformation conditions is derived. Figure 11 illustrates the establishment of hot working maps corresponding to distinct strain conditions, where gray-shaded regions denote instability zones, whereas the unshaded areas signify the suitable processing domains.

By analyzing the hot working maps across four strain levels, it is evident that for 4Cr16MoCu steel, the flow instability zone predominantly concentrates in the high strain rate region of approximately 0.01 s⁻¹ and higher. The maximum permissible thermal deformation rate of the material, as deduced from Figure 11a, is approximately 0.018 s⁻¹. Throughout the entire hot forging process, minimal variations are observed in the instability zone, suggesting inadequate plastic deformation characteristics for this steel grade. Considering the inconsistency in strain across various parts of the steel component during hot forging, along with the η values under different strain parameters depicted in the figure, the peak η value occurs in two regions: one within the range of 950–1000 °C at a strain rate of 0.01 s⁻¹ with an η value of approximately 0.32, and another within the range of 1100–1150 °C at a strain rate of 0.01 s⁻¹ with an η value reaching approximately 0.4. Given the practical requirements of hot forging to determine the initial and final forging temperatures, these two sets of parameters can be adopted as the suitable hot forging parameters for 4Cr16MoCu steel, namely an initial forging temperature of 1125 °C, a final forging temperature of 980 °C, and a strain rate set at 0.01 s⁻¹.

Figure 12 illustrates the microstructures of 4Cr16MoCu specimens subsequent to selecting representative hot compression parameters. Specifically, Figure 12a and b depict the microstructures of the instability zones at 900 °C + 1 s⁻¹ and 1000 °C + 1 s⁻¹, respectively. Both specimens exhibit localized deformation bands internally, with grains predominantly undergoing deformation and elongation along the direction of hot compression. As temperature increases, grain size augments. The microstructure exhibits a mixed grain phenomenon attributed to incomplete recrystallization, which degrades the workpiece properties and predisposes it to instability [41]. Figure 12c,d display the microstructures of the regions at 1100 °C + 0.001 s⁻¹ and 1100 °C + 0.01 s⁻¹, respectively. The results indicate the occurrence of dynamic recrystallization in both samples. However, a lower deformation rate leads to the formation of larger grains, adversely affecting the material properties. Consequently, maintaining a strain rate of 0.01 s⁻¹ results in a more uniform microstructure of the workpiece.

4. Conclusions

• The forging temperature range of 4Cr16MoCu stainless steel is relatively narrow. During the forging process, its deformation resistance increases significantly with the rise in deformation rate, exhibiting a pronounced sensitivity to strain rate.
• The Arrhenius model constitutive equation was established using corrected flow stress. The dependence of flow stress on strain rate and temperature during hot difformation can be expressed as

  $$\dot{\mathsf{\epsilon }}=8.2449 \times {10}^{16}{\left[\sinh \left(0.0081\mathsf{\sigma }\right)\right]}^{5.0255} \exp \left(-{\displaystyle \frac{469.7088}{RT}}\right)$$

  (27)
  This demonstrates the reliability of the model for practical applications. The findings provide valuable insights into the hot deformation behavior of 4Cr16MoCu steel, and the established equation can serve as a theoretical foundation for optimizing processing parameters in industrial applications.
• Based on the dynamic materials model, a hot processing map for 4Cr16MoCu steel was established, and the optimal hot working process window for 4Cr16MoCu steel was determined in conjunction with microstructural evolution. The optimal conditions include an initial forging temperature of 1125 °C, a final forging temperature of 980 °C, and a strain rate of 0.1 s⁻¹, which result in a fine and uniform grain structure. To avoid uneven deformation, high strain rate areas of 0.18 s⁻¹ and above should be avoided during hot processing.