Influence of Deformation Temperature and Strain Rate on Martensitic Transformation of Duplex Stainless Steel and Its Corresponding Kinetic Model

Abstract

For investigating the effect of temperature and strain rate on martensitic transformation and establishing the corresponding kinetic model for newly TRIP (transformation-induced plasticity) aided duplex stainless steel (DSS), the tensile tests are conducted at temperatures of 20–150 °C and strain rates of 0.0001–150 s⁻¹. The stepped cross-section tensile specimen is proposed and designed for obtaining microstructure at specific strain during dynamic tensile testing. The results demonstrate that the deformation mechanism of austenite in TRIP-aided DSS is highly sensitive to temperature and strain rate. As the deformation temperature increases, strain-induced martensitic transformation is inhibited, and the deformation mechanism transforms from martensitic transformation to the co-occurrence of martensitic transformation and twinning, and finally, twinning is the main deformation mechanism. This leads to reduced strength with an initial increase followed by a decrease in elongation. As the strain rate increases, martensitic transformation is inhibited, resulting in a reduction in strength and plasticity during quasi-static tensile testing, while during dynamic tensile testing, strength increases due to enhanced resistance to dislocation motion, and plasticity displays no significant variation because of the combination of adiabatic softening and martensitic transformation suppression. Moreover, during tensile deformation, a plastic temperature rise model is established for newly developed DSSs. Based on this model, the Ludwigson–Berger model for martensitic transformation was modified to couple the effect of temperature and strain rate by considering the non-uniform distribution of temperature rise within the material and its variation with strain rate, as well as the suppression of dynamic strain rate on martensitic transformation. This new model could accurately describe the characteristics of martensitic transformation in newly developed DSSs at different deformation temperatures and strain rates.

1. Introduction

Duplex stainless steels (DSSs) consist of ferrite and austenite phases, thus combining the advantages of ferritic stainless steels (FSSs) and austenitic stainless steels (ASSs) [1,2]. The application of DSS covers a broad range of fields, such as petrochemicals, nuclear power, ocean engineering, shipbuilding, food processing, biomedicine, and papermaking [3,4,5,6]. However, for traditional DSS, it is not possible to significantly improve its strength through precipitation strengthening, solution strengthening, and refinement strengthening [7,8,9], and its austenite has high stability and no phase transformation takes place during heat treatment and deformation. Thus, although DSS possesses higher strength compared to ASS and FSS, its strength is still relatively low compared to advanced high-strength steel.

Recently, DSS having significantly higher strength and higher plasticity has been developed by adding Mn and N elements instead of Ni element, managing austenitic stability to trigger martensitic transformation during deformation (the occurrence of transformation-induced plasticity (TRIP) effect) [10]. Furthermore, considering the poor deformation coordination of the two-phase structure during hot working, our previous study developed a new processing route of strip casting in the fabrication of TRIP-aided DSS [11,12]. For these newly developed DSSs, occurring TRIP effect occurs, and the mechanical properties are intricately linked to martensitic transformation behavior in austenite during deformation. Extensive research shows that optimizing alloy composition [13] and processing parameters (such as solution treatment temperature [14,15], solution treatment time [16], and aging time [14]) can improve the internal characteristics of austenite (such as nitrogen content, austenitic content, and size), leading to the optimized strain-induced martensitic transformation behavior and thus the enhanced plasticity and strength of DSSs.

Besides internal factors of austenite in DSS, external factors such as deformation temperature and strain rate also significantly influence the martensitic transformation behavior, thereby affecting the properties of DSS. Regarding the influence of deformation temperature on the transformation of austenite to martensite, a consensus has been reached. According to martensitic transformation kinetic theory, elevated temperatures reduce the transformation driving force, thereby suppressing the occurrence of martensitic transformation.

However, the influence of strain rate on martensitic transformation is extremely complex, as it is coupled with adiabatic heating that leads to a change in temperature during deformation. Moreover, the temperature increase occurs non-uniformly within the microstructure and varies continuously during deformation. Wang [17] suggested that for Mn-N bearing lean DSS, the adiabatic heating induced by increased strain rates during deformation elevates the deformation temperature, thereby enhancing austenite stability and suppressing martensitic transformation. A similar phenomenon was observed in the metastable austenite of ASS by Talonen et al. [18,19] and in the metastable austenite of low-carbon quenching and partitioning steel by Huang et al. [20]. However, Lee et al. [21,22] found that in 304 L stainless steel, the shear bands developed during deformation increased with the strain rate, which brought more sites for martensite nuclei and thus promoted the martensitic transformation. Cao et al. [23] found through non-destructive real-time magnetic permeability monitoring that in SUS304 metastable austenitic stainless steel, high-strain-rate deformation produces higher martensite content than quasi-static deformation at low strain levels, but when strain exceeds a critical value, the deformation heating increases austenite stability and suppresses martensitic transformation with increasing strain rates. Overall, although many scholars conducted extensive research on the impact of strain rate on martensitic transformation, the influence mechanism has not been completely elaborated, which has also been noted by Vázquez-Fernández et al. [24], Pun et al. [25], and Langi et al. [26]. This could be related to the following factors: (1) During deformation at different strain rates, especially strain rates above 1 s⁻¹, varying amounts of deformation heating are generated, leading to different temperature rises in the specimen. Moreover, these temperature rises are very complex phenomena and their accurate estimation is very difficult because these temperature rises were related to multiple factors, including strain, strain rate, temperature reached by the specimen, and martensitic transformation [27,28,29]; (2) When deforming at dynamic strain rates, the extremely brief deformation duration makes interrupted testing at different strain levels unfeasible, thereby preventing determination of martensite content evolution.

In addition to analyzing the factors influencing martensitic transformation, various kinetic models describing the martensitic transformation have been established, the empirical models such as Angel model [30], Ludwigson–Berger model [31], Sugimoto model [32], and Shin model [33], and the physical-based models based on the theory of shear band interaction such as Olson–Cohen model [34]. Subsequently, based on the above models, many newly developed martensitic transformation models were developed. According to the Olson–Cohen model, Stringfellow et al. [35] proposed a new model by introducing the effect of stress triaxiality on martensitic transformation. Based on this new model, Iwamoto et al. [36] introduced the effect of strain rate on martensitic transformation. Subsequently, building upon this foundation, Dan et al. [37] incorporated the influence of adiabatic heating during high-strain-rate deformation on the martensitic transformation. In addition, Beese and Mohr [38] developed a stress state-dependent kinetic model by introducing the Lode angle parameter and the effect of stress triaxiality. Kim et al. [39] proposed a newly modified martensitic transformation kinetic model considering the impact of stress conditions on martensitic transformation, by introducing the Lode angle parameter, temperature, and a non-linear function of the stress triaxiality. The above models consider the influence of stress conditions (temperature, strain rate, and strain) on martensitic transformation and can better describe the martensitic transformation behavior of Q&P steels, ASSs, and TRIP steels. However, the existing models have not taken into account the coupling effect of multiple factors, such as temperature and strain rate (containing high strain rate), and the prediction of martensitic transformation behavior during actual forming processes is difficult.

In this study, in combination with ABAQUS simulation, a stepped tensile specimen was designed to obtain the microstructure under different strains for the high-speed dynamic tensile and achieve the measurement of the change in strain-induced martensite content with strain. The microstructure evolution and deformation mechanism of DSS under various deformation temperatures (20–150 °C) and strain rates (0.0001–150 s⁻¹) were investigated in depth, and its effect on mechanical properties was analyzed. Moreover, a temperature rise model for the high-speed dynamic tensile was established by the measurement of temperature rise during tensile. Finally, the Ludwigson model for martensitic transformation was modified to couple the effect of temperature and strain rate for accurately describing the martensitic transformation behavior in newly developed DSSs at a wide range of deformation temperature and strain rate (including dynamic strain rate).

2. Experimental Procedure

The chemical composition of the investigated steel is detailed in

Table 1. Composition of duplex stainless steel used in this study (wt. %).

C	N	Si	Mn	Cr	Ni	Cu	Sn	Fe
0.14	0.25	0.35	5	20	0.4	0.5	0.2	Bal.

. The ingot was hot-rolled to 4.8 mm and held at 1050 °C for 8 min for solution treatment. Subsequently, it was cold-rolled to 1 mm and finally subjected to another solution treatment at 1050 °C for 15 min.

The mechanical properties of the investigated steel were measured through uniaxial tensile testing, including tensile tests at different temperatures (20, 30, 40, 55, 85, and 150 °C) and different strain rates (quasi-static: 0.0001, 0.001, 0.005, and 0.1 s⁻¹; high-speed dynamic: 4 and 150 s⁻¹). The strain rate for the tensile tests performed at various temperatures was set at 0.0001 s⁻¹. The deformation temperature for tensile tests performed at various strain rates was 20 °C (ambient temperature). Among these, the quasi-static tensile tests, including those conducted at various temperatures, were performed using a universal tensile testing machine (Model: AI 7000 LA10, Gotech Testing Machines Inc., Taichung, China), and the high-speed dynamic tensile tests were carried out on a high-speed tensile testing machine (Model: ZWICK HTM 5020, ZwickRoell, Ulm, Germany).

The high-speed dynamic tensile stepped specimens were designed through simulation with ABAQUS 6.14 software, leading to different strains experienced by different regions of the gauge section after fracture, and obtaining the microstructure under the given strain during high-speed dynamic tensile. Based on the conventional high-speed dynamic tensile specimen, a notch was designed at the mid-region of the parallel section, and it was symmetrical on both sides (Figure 1). The notch length controls the tensile time to ensure the strain rate of a specific area in the gauge length section (outside the notch), while the notch depth ensures the strain of this specific area after fracture. The notch length (L_x), depth (d), and circular radius (R) of the stepped specimens were determined through ABAQUS simulations. There are different parameter combinations corresponding to different given strains: for specimens designed to obtain the deformed microstructure with a strain of 0.15, L_x = 3.12 mm, d = 0.23 mm, and R = 1.73 mm; for specimens designed to obtain the deformed microstructure with a strain of 0.25, L_x = 4.95 mm, d = 0.11 mm, and R = 3.27 mm. Figure 2 shows the distribution of true stress and true strain by using ABAQUS simulations when the stepped specimen for obtaining a deformed microstructure with a strain of 0.15 began necking. Obviously, the true strain at a specific area in the gauge length section (outside the notch) is approximately 0.15, demonstrating the reliability of the design.

The real-time deformation temperature of the specimen was measured using a thermal imager (FLIR SC620, Wilsonville, Oregon, USA), and the plastic temperature rise was obtained by subtracting the ambient temperature. The microstructure observation was conducted by X-ray diffractometer (XRD, Model: D/Max-2500PC, Rigaku Corporation, Tokyo, Japan), electron backscatter diffraction (EBSD, Model: Symmetry, Zeiss, Oberkochen, Germany), and transmission electron microscope (TEM, Model: FEI Tecnai G² F20, FEI Company, Hillsboro, USA). The XRD specimens were obtained by cutting from the tensile specimens, then mechanical grinding using emery paper, and finally electrolytic polishing in a solution (the ratio of alcohol to perchloric acid is 7:1). During XRD measurement, the scanning diffraction angle range was from 40° to 100°, and the scanning speed was 1°/min. In this measurement, in order to avoid texture and residual stress effects, the (200)_γ, (220)_γ, and (311)_γ austenite peaks along with the (200)_α and (211)_α ferrite peaks were identified, followed by peak integration, then the volume fraction of retained austenite content was calculated using the formula established in our previous studies [11,12]. It should be noted that while this method does not employ full profile Rietveld refinement and may have some accuracy limitations, it does not compromise the analysis of austenite content trends and has been widely adopted in research [40,41,42]. The EBSD specimens were obtained using a method similar to XRD specimen preparation, which was tested on an Oxford Symmetry detector (Oxford Instruments, Oxford, UK) installed on a Zeiss Crossbeam 550 scanning electron microscope (Zeiss, Oberkochen, Germany). During EBSD measurement, the acceleration voltage was 20 kV, the working distance was 14.1 mm, and the step size was 0.05 μm. The EBSD data analysis was performed on the AztecCrystal 2.1 software. The thin film for TEM observation was prepared by first extracting from the ND plane of the specimens, then mechanically thinning to 50 µm, and subsequently performing twin-jet electro-polishing in the solution of alcohol and perchloric acid with a ratio of 7:1 at 32 V and −25 °C.

3. Results and Discussion

3.1. Microstructure and Mechanical Properties

Figure 3 presents the engineering stress–strain curves and corresponding work hardening rate curves, and the temperature-dependent and strain rate-dependent evolution of mechanical properties. With the deformation temperature increasing, the ultimate tensile strength declines gradually, and its maximum value obtained at ambient temperature is 948 MPa, corresponding to the uniform elongation of 64%, the uniform elongation exhibits a trend of first increasing and then decreasing, reaching the higher value of 74%~80% at 30~40 °C and the corresponding ultimate tensile strength of 817~891 MPa (Figure 3a,b).

With the strain rate increasing, the ultimate tensile strength initially decreases in the quasi-static tensile stage and then increases in the high-speed tensile stage, and the elongation first decreases in the quasi-static tensile stage and then displays no relatively significant variation in the high-speed tensile stage (Figure 3d,e). It is quite obvious that the mechanical properties of the investigated steel exhibit pronounced dependence on both deformation temperature and applied strain rate.

The work hardening rate curves generally display a trend of decreasing, then increasing, and finally decreasing again, and three variation stages are observed (Figure 3c,f). The enhanced strain hardening during plastic deformation likely results from the widespread formation of deformation-induced martensite. However, during deformation when the temperatures are above 40 °C or strain rates are above 4 s⁻¹, the work hardening rate does not increase (Figure 3c,f). This may be attributed to insufficient strain-induced martensitic transformation.

In order to figure out the deformation mechanism of the investigated steel, the microstructures at different strains during deformation at ambient temperature (0.0001 s⁻¹) are first analyzed by TEM, XRD, and EBSD, as shown in Figure 4, Figure 5a, and Figure 6a. When the true strain reaches 0.05, many deformation bands predominantly composed of ε-martensite form within the austenite grains, which exhibit an S-N (Shoji-Nishiyama) relationship with austenite (Figure 4(a1–a3)). This indicates that strain-induced martensitic transformation occurred during deformation. As the true strain further increases to 0.2, the martensite fraction enhances and the austenite fraction reduces, Figure 4(b1–b6) and Figure 5a. The ε-martensite within each austenite grain exhibits a banded distribution along specific orientations, which is indicated by the black dashed line in some austenite grains (Figure 6a). In addition to banded ε-martensite, blocky α’-martensite also developed, predominantly distributed along deformation bands, Figure 4(b1–b5). The α’-martensite primarily nucleates at ε-martensite band junctions and surrounding areas (Figure 6a), which is also observed by our previous study on TRIP steel [43], and exhibits a K-S (Kurdjumov-Sachs) or N-W (Nishiyama-Wassermann) relationship with austenite, Figure 4(b3,b6). When the true strain reaches 0.47, the majority of austenite has converted to α’-martensite, and the K-S or N-W relationship is maintained between the α’-martensite and austenite. Moreover, during deformation, no deformation twins are observed. Overall, the deformation mechanism of austenite in the investigated steel at ambient temperature is dominated by martensitic transformation.

To elucidate the relationship between deformation mechanisms and temperature and strain rate, and further clarify the correlation between mechanical properties and temperature as well as strain rate, the microstructures deformed under different temperatures and strain rates were observed by XRD (Figure 5) and EBSD (Figure 6). The XRD profiles are shown in Figures S1 and S2. Figure 5 clearly demonstrates that the austenite fraction progressively decreases with increasing true strain due to strain-induced martensitic transformation, and this reduction rate gradually diminishes with both rising deformation temperature and increasing strain rate. This indicates that higher deformation temperatures and strain rates inhibited the martensitic transformation.

The suppression of strain-induced martensitic transformation because of rising deformation temperature is harmful to enhancing the work hardening rate, postponing the start of necking, and thus the increase in elongation [44]. This contradicts the observed phenomenon where elongation first increases and then decreases with temperature, possibly due to other deformation mechanisms induced in austenite, in addition to martensitic transformation, as temperature changes. The microstructures of the deformed investigated steel at various temperatures with a strain of 0.2 were analyzed by EBSD (Figure 6). As stated above, based on the EBSD analysis (Figure 6a), in combination with TEM observation (Figure 4), martensitic transformation predominantly governs the deformation mechanism of austenite at ambient temperature. As deformation temperature increases from ambient temperature to 40 °C, the content of ε-martensite and α’-martensite significantly reduces, consistent with the XRD measured results (Figure 5a), and substantial deformation twins appear at the initial stage of deformation, with deformation twin boundary length per unit area of 0.42 µm (Figure 6b). This indicates that martensitic transformation and twinning compete to become the dominant deformation mechanisms. With deformation temperature continuing to increase to 55 °C, only a very small amount of α’-martensite and ε-martensite forms, but the number of deformation twins increases, with deformation twin boundary length per unit area of 0.61 µm (Figure 6c). This demonstrates that twinning has become the dominant deformation mechanism of austenite. At 150 °C, only deformation twins develop, indicating that twinning still remains the dominant deformation mechanism in austenite, and the number of twins exhibits a reduction compared to that at 55 °C, with deformation twin boundary length per unit area of 0.48 µm (Figure 6d). Overall, with increasing deformation temperature from ambient temperature to 150 °C, the deformation mechanism transforms from martensitic transformation to the coexistence of martensitic transformation and twinning, and finally, twinning is the main deformation mechanism. This phenomenon can be connected to the fact that as temperatures rise, stacking fault energy increases.

During deformation, different deformation mechanisms of austenite in steel, such as martensitic transformation, mechanical twinning, and dislocation slip, intrinsically determine different strain hardening behavior and eventually exhibit different strength and ductility characteristics. At ambient temperature, strain-induced martensitic transformation is the dominant deformation mechanism of austenite. The martensite developed during deformation acts as a barrier to dislocation glide, thereby elevating localized strength and restricting further plastic deformation within transformed regions, promoting strain redistribution to adjacent untransformed regions. This significantly enhances the work hardening capacity (Figure 3c,f), effectively delays necking formation, and ultimately results in higher elongation and tensile strength.

With increasing deformation temperature, the two mechanisms of strain-induced martensitic transformation and twinning compete with each other to become the dominant deformation mechanism of austenite. Similar to strain-induced martensite, the deformation twins developed during deformation can also impede dislocation motion, thereby enhancing the work hardening rate and contributing to improved elongation [45]. This synergistic interaction between the concurrent operation of martensitic transformation and twinning ultimately achieves a further improvement in elongation. However, the contribution of twinning to tensile strength is weaker than that of martensite transformation [46]. Consequently, the investigated steel exhibits lower tensile strength compared to that deformed at ambient temperature. Moreover, with the deformation temperature further rising, when the deformation mechanism is governed by twinning, the tensile strength and elongation of the investigated steel undoubtedly decrease. With the further increase in the deformation temperature, the number of deformation twins exhibits a reduction, which leads the tensile strength and elongation of the investigated steel to decrease further. Ultimately, with the increase in deformation temperature, the change in deformation mechanism led to the tensile strength decreasing, and an initial rise followed by a decline in the elongation of the investigated steel.

According to the XRD analysis of the evolution of strain-dependent austenite content, during the deformation process at different strain rates, higher strain rates significantly inhibit the strain-induced martensitic transformation. As stated above in this section, martensitic transformation is the dominant deformation mechanism of austenite in the investigated steel at ambient temperature. During deformation at ambient temperature, the plastic temperature increases with strain rate, as analyzed in the following section. Moreover, Choi et al. [47] found that strain concentrates preferentially in ferrite compared to austenite during deformation, and this strain partitioning exhibits significant enhancement under elevated strain rates. That is to say, as the strain rate increased, austenite accommodated progressively less strain. More importantly, our previous studies found that compared with quasi-static tensile deformation, strain-induced martensite nucleates exclusively on a single habit plane of the primary slip systems within parent austenite grains during high-speed dynamic tensile, enhancing martensite variant selectivity, heightening strain energy, and finally inhibiting the occurrence of martensitic transformation [48]. Therefore, strain-induced martensitic transformation behavior is progressively suppressed with increasing strain rate from 0.0001 to 150 s⁻¹.

Analogous to deformation temperature, variations in strain rate inevitably lead to corresponding alterations in strength and ductility characteristics. During the quasi-static tensile stage, an increase in strain rate inhibits martensitic transformation, resulting in a decline in both uniform elongation and tensile strength. However, during the high-speed dynamic tensile, elevating the strain rate necessitates higher mobile dislocation velocities, which intensifies the impact of short-range obstacles (such as lattice resistance, atomic thermal vibration resistance, and electron cloud resistance) on dislocation glide [49,50]. Moreover, elevating the strain rate caused a more rapid dislocation multiplication and entanglement within the ferrite matrix in a short time, thereby elevating the resistance to dislocation glide and substantially hindering further slip [51]. As a result, the enhanced strain rate substantially elevates resistance to dislocation motion, thereby resulting in improvements in ultimate tensile strength. By contrast, as the strain rate increases, the ductility reduction caused by martensitic transformation suppression is countervailed by the adiabatic softening-induced ductility improvement, consequently resulting in no significant variation in uniform elongation. Ultimately, increasing strain rate (from 0.0001 to 150 s⁻¹) triggers an initial reduction followed by a subsequent enhancement in the ultimate tensile strength, while the uniform elongation exhibits an initial decrease with no significant variation in the later stage.

3.2. Measurement of Plastic Temperature Rise

Under high-strain-rate conditions, the deformation-induced heat generated during plastic deformation within the metallic material causes the actual deformation temperature to exceed the nominal deformation temperature, because the extremely short duration of deformation prevents significant heat transfer from occurring. Cao et al. [52] found that for the same material, the difference in plastic heating caused by geometric variations affects martensitic transformation, although the maximum temperature difference is only about 13 °C, indicating that plastic heating significantly influences martensitic transformation. Given the pronounced temperature dependence of the austenitic stability of the investigated steel, it is essential to measure the actual deformation temperature at different strain rates to modify the relationship between strain rate and martensitic transformation. To establish the relationship between strain rate and actual deformation temperature, the plastic temperature rise in different strain rates during tensile testing was measured, as shown in Figure 7 and Figure 8. Due to the limited acquisition frequency of the thermal imager, the temperature rise was only measured at strain rates of 0.1 s⁻¹ or below. The plastic temperature rise increased with strain rate (Figure 8) because higher strain rates reduced both deformation time and heat exchange time, resulting in greater deformation heat accumulation. When deformed at a strain rate of 0.0001 s⁻¹, the 80 min tensile test duration enabled near-complete thermal equilibrium between the specimen and environment, and the maximum temperature rise remained below 3.5 °C, so the deformation can be treated as isothermal. When deformed at strain rates of 0.001, 0.005, and 0.1 s⁻¹, the temperature rise reached 13.8, 40.2, and 50.4 °C, respectively, and these deformation processes could not be regarded as isothermal.

3.3. Plastic Temperature Rise Model

For TRIP-aided DSS, the plastic temperature rise during deformation stems from the synergistic effects of plastic work, the latent heat of martensitic transformation, heat exchange between the material and the environment, as well as heat radiation. Therefore, the principle of energy conservation in uniaxial tensile tests can be described as follows:

$$C\rho \dot{T}=\beta \sigma \dot{\epsilon }-\rho l_m{\dot{f}}_m+{\displaystyle \frac{hA(T_{\infty }-T_s)}{V}}+\dot{\phi }$$

(1)

where C represents the specific heat capacity (J/kg/K) and is 432 J/kg/K for the investigated steel, ρ represents the material density (Kg/m³) and is 0.766 × 10⁴ kg/m³ for the investigated steel; $\dot{T}$ represents the rate of temperature rise (K/s); σ represents the true stress (MPa); $\dot{\epsilon }$ represents the strain rate (s⁻¹); β represents the Quinney-Taylor coefficient, which is generally recognized as a constant and usually takes the value of 0.9 for DSS; ${\dot{f}}_m$ represents the martensitic transformation rate; l_m represents the latent heat of martensitic transformation and is −1.50 × 10⁴ J/kg for investigated steel; h represents the convective heat transfer coefficient (W/m²/K); A represents the superficial area of the gauge region of the tensile specimen and is 1.08 × 10⁻³ m² for investigated steel; T_s and T_∞ represent the temperature of the tensile specimen and ambient temperature, respectively; V represents the volume of the gauge region of the tensile specimen and is 5 × 10⁻⁷ m³ for the investigated steel; $\dot{\phi }$ represents the heat radiation during tensile deformation.

In this study, the heat radiation ($\dot{\phi }$) in Equation (1) is negligible due to the relatively low-temperature rise, so the contribution of heat radiation to the temperature variation in the specimen during deformation can be neglected. The latent heat of martensitic transformation can also induce the temperature variation in the specimen during deformation, which can be calculated by the following:

$$\Delta T_m=-{\displaystyle \frac{l_m\Delta f_m}{C}}$$

(2)

where Δf_m is the fraction of martensitic transformation during deformation, and it is measurable by X-ray diffraction. By combining Equations (1) and (2), the temperature rise model can be expressed as follows:

$$C\rho \dot{T}=\beta \sigma \dot{\epsilon }+{\displaystyle \frac{C\rho \Delta T_m}{\Delta t}}-{\displaystyle \frac{hA\Delta T}{V}}$$

(3)

By multiplying Equation (3) by V and then integrating with respect to dt, we can obtain the following:

$$VC\rho \Delta T=V\beta {\displaystyle {\int }_0^{\epsilon }\sigma }d\epsilon +VC\rho \Delta T_m-hA\Delta T\Delta t$$

(4)

where Δt is the duration of the tensile test, which can be calculated by $\dot{\epsilon }=\Delta \epsilon /\Delta t$. According to Equation (4), the temperature rise can be computed as follows:

$$\Delta T={\displaystyle \frac{\beta {\displaystyle {\int }_0^{\epsilon }\sigma }d\epsilon +C\rho \Delta T_m}{C\rho +\frac{hA\Delta t}{V}}}$$

(5)

Clearly, the value of h can be obtained through optimal fitting based on experimental data from quasi-static strain rates (Figure 8) using Equation (5), and it is found to be 10.5 W/m²K.

Figure 9 shows the comparison between the values of the plastic temperature rise model calculated by Equation (5) and the experimental values, and the model demonstrated excellent predictive ability.

During high-speed tensile deformation, the deformation time is extremely short, resulting in almost no heat exchange between the specimen and the environment. Consequently, this process can be treated as adiabatic, and the corresponding plastic temperature rise can be calculated using Equation (6).

$$\Delta T={\displaystyle \frac{\beta {\displaystyle {\int }_0^{\epsilon }\sigma }d\epsilon +C\rho \Delta T_m}{C\rho }}$$

(6)

The calculated results based on Equations (5) and (6) showed significant differences when the strain rate is lower than 0.1 s⁻¹, but almost no difference when the strain rate is higher than 0.1 s⁻¹ (Figure 10). Therefore, a strain rate of 0.1 s⁻¹ can be regarded as the critical strain rate to distinguish between the adiabatic and non-isothermal states. A strain rate of 0.0001 s⁻¹ can be regarded as the critical strain rate to distinguish between the isothermal and non-isothermal states because the plastic temperature rise at this strain rate can be negligible. Finally, the expression of the plastic temperature rise model was obtained, as shown in Equation (7), where ${\dot{\epsilon }}_{it}$ (0.0001 s⁻¹) is the isothermal critical strain rate, and ${\dot{\epsilon }}_{ad}$ (0.1 s⁻¹) is the adiabatic critical strain rate.

$$\Delta T=\left\{\begin{array}{cc}0& for \dot{\epsilon }\le {\dot{\epsilon }}_{it}\\ {\displaystyle \frac{\beta {\displaystyle {\int }_0^{\epsilon }\sigma }d\epsilon +C\rho \Delta T_m}{C\rho +\frac{hA\Delta t}{V}}}& for {\dot{\epsilon }}_{it}\le \dot{\epsilon }\le {\dot{\epsilon }}_{ad}\\ {\displaystyle \frac{\beta {\displaystyle {\int }_0^{\epsilon }\sigma }d\epsilon +C\rho \Delta T_m}{C\rho }}& for \dot{\epsilon }\ge {\dot{\epsilon }}_{ad}\end{array}\right.$$

(7)

3.4. Kinetic Model of Martensitic Transformation

As stated in Section 1 (Introduction), various kinetic models describing the martensitic transformation have been proposed, such as the Angel model [30], the Ludwigson–Berger model [31], the Sugimoto model [32], the Shin model [33], and the Olson–Cohen model [34]. Using various models mentioned above, our previous study [48] established different martensitic transformation kinetic models based on the variation in metastable austenite content with strain in DSS at specific deformation temperature and strain rate and found that the Ludwigson–Berger model demonstrated superior predictive capability based on the comparative analysis. Therefore, in this study, a temperature–strain rate coupled kinetic model for martensitic transformation is established based on the Ludwigson–Berger model.

Ludwigson and Berger proposed the Ludwigson–Berger model [31] to characterize the association of volume fraction of austenite and strain in metastable austenitic stainless steels, and this model can be expressed as follows:

$${\displaystyle \frac{1}{V_{\gamma }}}-{\displaystyle \frac{1}{V_{\gamma 0}}}={\displaystyle \frac{k_p}{p}}{\epsilon }^p$$

(8)

where V_γ corresponds to the real-time austenite volume fraction during the tensile process; while V_γ0 represents the austenite volume fraction before deformation (60.86%); k_p is a constant, lower value of k_p corresponds to higher austenite stability; p is the autocatalytic factor, and lower value of p corresponds to fewer autocatalytic propagation phenomena [53].

3.4.1. Temperature-Dependent Ludwigson–Berger Model

During deformation at strain rates above 0.0001 s⁻¹, temperature rise became significant, influencing the martensitic transformation. Additionally, as the strain rate varies, the impact of temperature rise correspondingly changes. Therefore, in this study, it is necessary to first precisely describe the effect of temperature on martensitic transformation. To introduce the effect of temperature on martensitic transformation, the constants k_p and p in the conventional Ludwigson–Berger model are defined as functions of the deformation temperature, as shown in Equations (9) and (10), respectively.

$$k_p=k_p(T)=\left({\displaystyle \frac{m_1}{T}}+m_2\right)/\exp \left({\displaystyle \frac{m_3}{T}}\right)$$

(9)

$$p=p(T)=n_1+n_2\exp (n_{3}T)$$

(10)

where m₁ to m₃ and n₁ to n₃ are constants, T is the deformation temperature. During deformation at a strain rate of 0.0001 s⁻¹, the deformation temperature T is the ambient temperature T_0, and the corresponding expression for the martensitic transformation rate as a function of deformation temperature is expressed as:

$${\displaystyle \frac{1}{V_{\gamma }}}-{\displaystyle \frac{1}{V_{\gamma 0}}}={\displaystyle \frac{k_p(T_0){\epsilon }^{p(T_0)}}{p(T_0)}}$$

(11)

Obviously, the values of m₁ to m₃ and n₁ to n₃ can be determined through optimal fitting of changes in austenite content at various deformation temperatures using Equation (11) (Figure 11). These values are listed in

Table 2. Parameters of temperature–strain rate coupled Ludwigson–Berger model.

Parameter	Value
m1	−0.6659
m2	0.0378
m3	−214.7257
n1	0.6804
n2	6.6707
n3	−0.0534
λ1	1.9843
λ2	0.1599
η1	0.1105
η2	0.0139

.

3.4.2. Temperature–Strain Rate Coupled Ludwigson–Berger Model

During deformation at strain rates above 0.0001 s⁻¹, the temperature rise at quasi-static conditions (0.0001–0.1 s⁻¹) is obviously different from that at high-speed dynamic conditions (above 0.1 s⁻¹), which is an adiabatic process. Within the quasi-static strain rate range, as the strain rate increases, the plastic temperature rise becomes more pronounced, leading to a progressively strengthened inhibiting effect on martensitic transformation. Under these conditions, the plastic temperature rise contributes significantly to the effect of strain rate on martensitic transformation. Zou et al. [54] found that plastic temperature rise predominantly governed the impact of strain rate on martensitic transformation under quasi-static deformation conditions. In contrast, when the specimen is within the dynamic strain rate stage, the variation in plastic temperature rise with increasing strain rate shows a relatively weak change (Figure 9), indicating that the martensitic transformation inhibition induced by plastic temperature rise remains essentially constant during strain rate variations. However, according to the evolution of austenite content with strain under different strain rates (Figure 5b), the inhibitory effect on martensitic transformation significantly intensifies with increasing strain rate under high-speed dynamic conditions. This demonstrates that under these conditions, the enhanced suppression of martensitic transformation with increasing strain rate is not governed by the plastic temperature rise. According to our previous study, this enhanced suppression with increasing strain rate could be governed by the selectivity of martensite variants during martensitic transformation [48].

First, within the quasi-static strain rate range, the effect of strain rate on martensitic transformation is introduced in the kinetic model for martensitic transformation by considering plastic temperature rise, because plastic temperature rise dominated the influence of strain rate on martensitic transformation. Due to the occurrence of plastic temperature rise, the real deformation temperature T is the sum of the plastic temperature rise ∆T and ambient temperature T₀:

$$T=\Delta T+T_0$$

(12)

The corresponding kinetic model of martensitic transformation considering temperature rise is expressed as follows:

$${\displaystyle \frac{1}{V_{\gamma }}}-{\displaystyle \frac{1}{V_{\gamma 0}}}={\displaystyle \frac{k_p(\Delta T+T_0){\epsilon }^{p(\Delta T+T_0)}}{p(\Delta T+T_0)}}$$

(13)

From Equation (13), the change in austenite content with strain at strain rates of 0.001, 0.005, and 0.1 s⁻¹ can be calculated (Figure 12). Obviously, a pronounced difference exists between the predicted and measured values, and compared with experimental measurements, the predicted results display a more pronounced inhibitory effect of strain rate on the martensitic transformation behavior. Here, the introduction of ΔT only takes into account the temperature rise in the macroscopic region, which refers to the directly measured temperature rise. In DSSs, since ferrite is softer than austenite, it experiences more strain than austenite, as well as more than the strain undertaken by the specimen. Thus, the plastic temperature rise in ferrite is higher than that of austenite, and the measured value. When the temperature rise in the macroscopic region of the material is incorporated into the martensitic transformation model, the temperature of austenite during deformation is overestimated. This overestimation will lead to an increase in the calculated suppression of martensitic transformation.

To solve the above problems, the correction parameter is introduced into the temperature-dependent functions k_p(T) and p(T) to consider the temperature inhomogeneity within the material caused by non-uniform strain distribution, namely, considering the temperature rise in the microscopic region. Considering this temperature inhomogeneity is associated with strain rate, the expression of the correction parameters $h(\dot{\epsilon })$ is formulated as a strain rate-dependent function, as shown in Equations (14) and (15).

$$h(\dot{\epsilon })=1+\lambda \ln ({\dot{\epsilon }}^{*})$$

(14)

$${\dot{\epsilon }}^{*}={\displaystyle \frac{\dot{\epsilon }}{{\dot{\epsilon }}_{\mathrm{it}}}}$$

(15)

where λ is a constant, and since the influence of temperature on parameters k_p and p is different, the value of λ in Equation (14) is also different for parameters k_p and p. The expressions for parameters k_p and p are given in Equations (16) and (17), respectively.

$$k_p=k_p\left(T\right)h_1\left(\dot{\epsilon }\right)=k_p\left(T\right)\times \left[1+{\lambda }_1\ln ({\dot{\epsilon }}^{*})\right]$$

(16)

$$p=p(T)h_2(\dot{\epsilon })=p(T)\times [1+{\lambda }_2\ln ({\dot{\epsilon }}^{*})]$$

(17)

The corresponding kinetic model of martensitic transformation considering the temperature rise in the microscopic region is expressed as follows:

$${\displaystyle \frac{1}{V_{\gamma }}}-{\displaystyle \frac{1}{V_{\gamma 0}}}={\displaystyle \frac{k_p(\Delta T+T_0)h_1(\dot{\epsilon }){\epsilon }^{p(\Delta T+T_0)h_2(\dot{\epsilon })}}{p(\Delta T+T_0)h_2(\dot{\epsilon })}}$$

(18)

Obviously, the values of λ₁ and λ₂ can be determined through optimal fitting of changes in austenite content at strain rates of 0.001, 0.005, and 0.1 s⁻¹ using Equation (18) (Figure 13). These values are listed in Table 2. Additionally, this model is also employed to predict the variation in austenite content during dynamic strain rate deformation, with the predicted values being markedly higher than the experimental data (Figure 13), indicating an overestimation of the plastic temperature rise induced suppression of martensitic transformation. This aligns with the earlier analysis in this section, that is, under quasi-static and high-speed dynamic tensile conditions, the key factors governing the influence of strain rate on martensitic transformation differ significantly.

Next, at the dynamic strain rates, deformation operates within the adiabatic regime. As the strain rate increased, the plastic temperature rise remained largely constant, resulting in no significant variation in temperature inhomogeneity within the material during deformation. That is to say, the correction function $h(\dot{\epsilon })$ is a constant, which is equivalent to the value at a strain rate of 0.1 s⁻¹. Therefore, the correction function $h(\dot{\epsilon })$ loses validity during deformation under dynamic conditions and should be modified by introducing a function $H_1(\dot{\epsilon })$ to maintain the phenomenon that, with increasing strain rate, the inhibition of martensitic transformation induced by plastic temperature rise remains constant under dynamic tensile conditions. This new correction function is referred to as $h^{\ast }(\dot{\epsilon })$, as shown in Equations (19) and (20) (Note that, except for the newly introduced function $H_1(\dot{\epsilon })$, the constants λ in these two functions $h(\dot{\epsilon })$ and $h^{\ast }(\dot{\epsilon })$ are identical.).

$$h^{\ast }(\dot{\epsilon })=1+\lambda H_1(\dot{\epsilon })\ln ({\dot{\epsilon }}^{\ast })$$

(19)

$$H_1(\dot{\epsilon })=\left\{\begin{array}{ll}0& for \dot{\epsilon } \le {\dot{\epsilon }}_{it}\\ 1& for {\dot{\epsilon }}_{it}<\dot{\epsilon } \le {\dot{\epsilon }}_{ad}\\ {\displaystyle \frac{\ln ({\dot{\epsilon }}_{ad}/{\dot{\epsilon }}_{it})}{\ln (\dot{\epsilon }/{\dot{\epsilon }}_{it})}}& for \dot{\epsilon }>{\dot{\epsilon }}_{ad}\end{array}\right.$$

(20)

where the function $H_1(\dot{\epsilon })$ is a piecewise function with the critical strain rate ${\dot{\epsilon }}_{it}$ (0.0001 s⁻¹) and ${\dot{\epsilon }}_{ad}$ (0.1 s⁻¹) as the boundary. When the strain rate is ${\dot{\epsilon }}_{it}$ or below, the deformation process operates within the isothermal regime and no obvious plastic temperature rise takes place. When the strain rate is between ${\dot{\epsilon }}_{it}$ and ${\dot{\epsilon }}_{ad}$, the plastic temperature rise is directly proportional to the strain rate. When the strain rate is above ${\dot{\epsilon }}_{ad}$, the deformation process operates within the adiabatic regime, and plastic temperature rise remains basically constant. After introducing the new correction function, the expressions for parameters k_p and p are given in Equations (21) and (22), respectively.

$$k_p=k_p(T)h_1^{\ast }(\dot{\epsilon })=k_p(T)\times [1+{\lambda }_1{H}_1(\dot{\epsilon })\ln ({\dot{\epsilon }}^{*})]$$

(21)

$$p=p(T)h_2^{\ast }(\dot{\epsilon })=p(T)\times [1+{\lambda }_2{H}_1(\dot{\epsilon })\ln ({\dot{\epsilon }}^{*})]$$

(22)

Moreover, at dynamic strain rates, the enhanced suppression of martensitic transformation with increasing strain rate is related to the selectivity of martensite variants during martensitic transformation, according to the discussion above. Under quasi-static strain rates, martensitic nuclei form on multiple habit planes of the primary and secondary slip systems in the parent austenite phase, whereas under dynamic strain rates, martensitic nuclei form only on a single habit plane of the primary slip system in the parent austenite phase, significantly reducing the variety of martensite variants [24,48]. Therefore, in order to accurately describe the characteristics of martensitic transformation at strain rates above 0.1 s⁻¹, the suppression of martensitic transformation induced by the enhanced selectivity of the martensite variant with increasing strain rate needs to be taken into account. For considering this inhibitory effect in this study, a strain rate-dependent function $s(\dot{\epsilon })$ is introduced into k_p and p, as shown in Equations (23)–(25).

$$s\left(\dot{\epsilon }\right)={\displaystyle \frac{1}{1+\eta H_2(\dot{\epsilon })\ln ({\dot{\epsilon }}^{\circ })}}$$

(23)

$${\dot{\epsilon }}^{\circ }={\displaystyle \frac{\dot{\epsilon }}{{\dot{\epsilon }}_{ad}}}$$

(24)

$$H_2(\dot{\epsilon })=\left\{\begin{array}{ll}0& for \dot{\epsilon }\le {\dot{\epsilon }}_{ad}\\ 1& for \dot{\epsilon }>{\dot{\epsilon }}_{ad}\end{array}\right.$$

(25)

where $H_2(\dot{\epsilon })$ is a piecewise function with the critical strain rate ${\dot{\epsilon }}_{ad}$ (0.1 s⁻¹) as the boundary, indicating that the selectivity of the martensite variant becomes enhanced, thereby affecting the martensitic transformation behavior for the strain rate above 0.1 s⁻¹. η is constant and its value is different for temperature-dependent functions k_p(T) and p(T) since the influence of strain rate on parameters k_p and p is different. The expressions for parameters k_p and p are given in Equations (26) and (27), respectively. Obviously, the values of η₁ and η₂ can be determined through optimal fitting of changes in austenite content using Equations (8), (26), and (27) (Figure 14). These values are listed in Table 2.

$$k_p=k_p(T)\times h_1^{\ast }(\dot{\epsilon })\times s(\dot{\epsilon })={\displaystyle \frac{k_p(T)\times h_1^{\ast }(\dot{\epsilon })}{[1+{\eta }_1{H}_2(\dot{\epsilon })\ln ({\dot{\epsilon }}^{\circ })]}}$$

(26)

$$p=p(T)\times h_2^{\ast }(\dot{\epsilon })\times s(\dot{\epsilon })={\displaystyle \frac{p(T)\times h_2^{\ast }(\dot{\epsilon })}{[1+{\eta }_2{H}_2(\dot{\epsilon })\ln ({\dot{\epsilon }}^{\circ })]}}$$

(27)

Finally, the newly proposed kinetic model of martensitic transformation (Equations (8), (26) and (27)) considers the effect of coupling deformation temperature and strain rate on martensitic transformation, and regarding the effect of strain rate on transformation, the variations in martensite variant selectivity and plastic temperature rise during the change in strain rate are incorporated, inducing the changes in martensitic transformation behavior. This newly improved model can very well describe the martensitic transformation of the investigated steel under a wide range of deformation temperatures and strain rates (such as high-speed dynamic deformation), as shown in Figure 11 and Figure 14. In future research, we will integrate stress-state coupling into the established martensitic phase transformation kinetics model and implement the coupled model as a user subroutine in the finite element model to predict martensitic phase transformation under complex stress states during the hot stamping process.

4. Conclusions

The microstructure evolution and deformation mechanism of DSS under various deformation temperatures (20–150 °C) and strain rates (0.0001–150 s⁻¹) were investigated in depth, and its effect on mechanical properties was analyzed. Moreover, a temperature rise model for the high-speed dynamic tensile was established by the measurement of temperature rise during tensile. Finally, the Ludwigson model for martensitic transformation was modified to couple the effect of temperature and strain rate for accurately describing the martensitic transformation behavior in newly developed DSSs at a wide range of deformation temperature and strain rate (including dynamic strain rate). We have reached the following conclusions:

• As the deformation temperature increases from 20 to 150 °C, the strain-induced martensitic transformation is inhibited, and the deformation mechanism transforms from martensitic transformation (20 °C) to the co-occurrence of martensitic transformation and twinning (40–55 °C), and finally twinning was the main deformation mechanism (150 °C). This results in a monotonic decrease in tensile strength from 948 MPa to 652 MPa with increasing deformation temperature, while the elongation first increased from 64% to a range of 68–80% (30–55 °C), then decreased to 43% (150 °C);
• As the strain rate increases from 0.0001 to 150 s⁻¹, martensitic transformation is inhibited. This leads to a monotonic decrease in tensile strength from 948 to 760 MPa and a reduction in elongation from 64% to 40% as the strain rate increases from 0.0001 to 0.1 s⁻¹. However, when the strain rate rises from 4 to 150 s⁻¹, the tensile strength increases monotonically from 708 to 871 MPa due to enhanced resistance to dislocation motion. Meanwhile, the plasticity remains largely unchanged owing to the combined effects of adiabatic softening and suppressed martensitic transformation;
• A plastic temperature rise model is established for newly developed DSSs during tensile deformation. Based on this model, the Ludwigson–Berger model for martensitic transformation was modified to couple the effect of temperature and strain rate by considering the non-uniform distribution of temperature rise within the material and its variation with strain rate, as well as the suppression of dynamic strain rate on martensitic transformation. This new model could accurately describe the characteristics of martensitic transformation in newly developed DSSs over a wide range of deformation temperatures and strain rates.