Comparison of Flow Behaviors at High Temperature of Two Press Hardening Boron Steels with Different Hardenability

Abstract

The deformation behavior and the constitutive description of materials are important in the design and numerical simulation of manufacturing processes. In this work, the rheological behavior of two quenchable boron steels was derived in the range of interest for the Press Hardening (PH) process. For this study, the following steps were performed: (i) Design of the specimen geometry adopted during the hot tensile tests in the Gleeble^®3180 physical simulator. (ii) Performance of hot tensile tests at temperatures between 750 and 850 °C and strain rates of 0.01–1 s⁻¹. (iii) The description of the material constitutive model by means of the Arrhenius-type equation and the comparison between the experimental and predicted data. Firstly, the method was validated on USIBOR^®1500 boron steel by means of a comparison with existing literature data, and good agreement with other scientific works was found. Finally, this method was also adopted on USIBOR^®2000, a steel that belongs to the new generation of Ultra-High-Strength Steels (UHSS). A comparison between the rheological behavior of these two steels was carried out. The results show that USIBOR^®2000 steel exhibits greater strength compared to USIBOR^®1500 steel. The Arrhenius constitutive model well predicts the flow behavior of both steels.

1. Introduction

The current requirements for automotive component design include the development of materials with a high strength-to-weight ratio to lighten vehicles. To meet this demand, several Ultra-High-Strength Steels (UHSS) have been developed [1,2]. Compared with other lightweight materials, UHSS have lower cost and higher ultimate tensile strength. The high mechanical properties make such steels suitable for manufacturing automotive structural components (e.g., bumpers, doors, pillar). In addition, the high values of ultimate tensile strength make it possible to adopt thinner blank, reducing the global weight of automobiles [3]. Generally, this class of steels is adopted in Press-Hardening (PH) technology due to its low formability at ambient temperature [3,4,5,6]. The PH process is a thermo-mechanical sheet-metal-forming process in which the blank is first heated to the complete austenitization temperature and then formed and quenched in the tools [3,4,5].

One of the most widely adopted UHSS in the automotive field is 22MnB5 steel, also known by the commercial name USIBOR^®1500, produced by ArcelorMittal steel industry. This steel exhibits, after the quenching phase, an ultimate tensile strength of about 1500 MPa [7]. By modifying some chemical characteristics of the USIBOR^®1500 steel, ArcelorMittal has created a new ultra-high-strength steel, named USIBOR^®2000 [8]. Compared with the USIBOR^®1500, this new steel makes it possible to manufacture products with higher mechanical properties (ultimate tensile strength of about 2000 MPa after the quenching phase) and to save weight by up to 10–15 percent more than components manufactured in USIBOR^®1500 steel [9].

Knowledge of the thermo-mechanical properties of such new materials is essential for accurately designing and simulating PH process by means of Finite Element (FE) codes [10]. Many studies have investigated the high-temperature flow behavior of the USIBOR^®1500 steel [11,12,13,14,15,16].

Some authors [11,15,16] have adopted the Arrhenius-type constitutive equation to predict the flow behavior of this ultra-high-strength boron steel. Naderi et al. [17] compared the Molinari–Ravichandran model with the Voce–Kocks one, Hochholdinger et al. [18] adopted Norton–Hoff, Nemat–Nasser and Tong–Wahlen models and Dastgiri, M. S., et al. [19] used the Johnson–Cook and Khan–Huang–Liang modified models.

In many works, the Gleeble thermo-mechanical physical simulator has been exploited to derive the flow curves of materials [11,12,13,14,15,16,20,21,22]. This system is able to reproduce high heating rates (up to 10³ K/s) thanks to the Joule effect [23]. However, to reproduce rapid cooling rates, careful specimen design can be required if the water-cooled jaws surrounding specimen and the external cooling are not sufficient to achieve the required cooling rate [24,25]. Usually, the external cooling system is avoided due to oxidation problems caused by the absence of vacuum in the test chamber.

In this work, the flow behavior of both USIBOR^®1500 and USIBOR^®2000 steels was investigated in the range of the interest of the PH process. In particular, the explored temperature and strain rate range are 750–850 °C and 0.01–1 s⁻¹, respectively. To this end, Gleeble^®3180 system was adopted. With the aim to characterize the two UHSS in the austenitic field, a cooling rate of at least 27 K/s is required [13].

To reproduce a cooling rate greater than or equal to 27 K/s with the Gleeble system without the support of the external cooling system, accurate specimen design was performed by means of a transient thermo-electrical Finite Element (FE) model developed in COMSOL Multiphysics.

Experimental stress–strain curves were used to develop the Arrhenius-type constitutive equation for both UHSS. Since, to the authors knowledge, no comparison of the constitutive models of the two steels has previously been undertaken, this work presents useful and innovative results.

2. Materials and Methods

The aim of this work was the comparison of the flow behavior of the USIBOR^®1500 and USIBOR^®2000 steels in the austenitic field. To this end, the method outlined in Figure 1 was developed.

The method was initially assessed on USIBOR^®1500 steel. After its validation, this method was applied to USIBOR^®2000 steel. Finally, a comparison between the two steels was carried out in terms of the flow curves and materials constants of the Arrhenius model.

For the experimental tests, samples in USIBOR^®1500 and in USIBOR^®2000 were obtained by laser cutting from, respectively, 1.3-mm- and 2-mm-thick sheets.

In the first step of the methodology, the specimen geometry to be used during thermo-mechanical tests was obtained. The specimen geometry was chosen to guarantee a cooling rate of at least 27 K/s from 930 °C to the test temperature, avoiding high thermal gradients in the gauge length. The limit imposed on the cooling rate allows to characterize the material in the austenitic field. Moreover, the uniform thermal gradient during the thermo-mechanical physical simulation tests is necessary in order to obtain accurate stress–strain data [26].

To find the specimen geometry that met these requirements, the 3D transient FE thermo-electric model, already described by Palmieri et al. [24,25], was exploited.

Once the suitable geometry had been found, material flow behavior at high temperatures was derived thanks to the thermo-mechanical tests. The material flow behavior was investigated for temperatures of 750 °C, 800 °C and 850 °C and for strain rates of 0.01 s⁻¹, 0.1 s⁻¹ and 1 s⁻¹.

Finally, the experimental data obtained from hot tensile tests were adopted to derive the Arrhenius-type constitutive equation model. The accuracy of the analytical model was verified by comparing experimental data and the predicted ones.

To efficiently process such experimental data, a self-programmed code in MATLAB (version R2019b, MathWorks, Natick, MA, USA) was developed.

2.1. Specimen Design

A 3D FE model developed in COMSOL Multiphysics (version 5.6, Comsol Inc., Burlington, MA, USA) [24,25] was exploited to identify the specimen geometry that ensures during the thermo-mechanical tests low thermal gradient and a cooling rate able to characterize the material in the austenitic field. This model numerically simulates the thermo-mechanical tests in the Gleeble system, by coupling the thermal and the electrical physics.

The thermal physics was governed by the following equation:

$$\rho \times c_p\times \frac{\partial T}{\partial t}=div\left(k\times grad\left(T\right)\right)+Q_{joule}$$

(1)

where $\rho$ is the density of the material, $c_p$ is the specific heat, and the $k$ represents the thermal conductivity of the material. The latter two thermo-physical material properties were modeled as a function of the temperature, as highlighted in Figure 2. The density of material was considered to be constant and equal to 7860 kg/(m³).

$Q_{joule}$, in Equation (1), represents the heat generated by the Joule effect and it can be expressed as follows:

$$Q_{joule}={\sigma }_e\times \left[{\left(\frac{\partial V}{\partial x}\right)}^2+{\left(\frac{\partial V}{\partial y}\right)}^2+{\left(\frac{\partial V}{\partial z}\right)}^2\right]$$

(2)

where ${\sigma }_e$ and $V$ are, respectively, the electrical conductivity and the electrical potential. The electrical conductivity was defined as a function of the temperature (

Table 1. Electrical conductivity function.

Temperature (°C)	${\mathit{\sigma }}_{\mathit{e}}$ (S/m)
20–800 °C	1/(−2.599347 × 10−17 × T3 + 9.171830 × 10−13 × T2 − 2.524916 × 10−11 × T + 9.981349 × 10−8)
800–1300 °C	1/(−3.321429 × 10−13 × T2 + 1.168564 × 10−9 × T + 2.250362 × 10−7)

).

The electrical physics was governed by the following equation:

$$div\left({\sigma }_e\times \overrightarrow{grad}\left(V\right)\right)=0$$

(3)

A Proportional–Integrative–Derivative (PID) controller was implemented in the FE model in order to guarantee the required thermal cycle during the heating phase and the keeping phase at the complete austenitization temperature at the control point. Figure 3 shows the FE model of the specimen characterized by a total length of 150 mm and a grip region width of 30 mm.

The electrical potential was varied according to Equation (4) in order to ensure that the real temperature was equal to the imposed one ($T_{set}$) at the control point.

$$V=K_p\times \left(T-T_{set}\right)+K_I\times \int \left(T-T_{set}\right)\times dt+K_D\times \frac{\partial }{\partial t}\left(T-T_{set}\right)$$

(4)

where $K_p$, $K_I$, and $K_D$ are, respectively, the proportional, integral and derivative parameters of the PID controller. These parameters were defined after the calibration phase of the FE model.

Electric boundary conditions were imposed. Specifically, on the left grip, an electric potential equal to $V$ was applied, while on the right grip, an electric potential equal to zero (ground) was imposed; the other surfaces were insulated.

A convective heat exchange was modeled on the specimen surfaces to simulate the cooling phase from the complete austenitization temperature to the test temperature. The values of heat transfer coefficients in each specimen region (tapered, no-tapered and grip regions) were assigned after the calibration of the FE model. Three probes (control point, Probe 1 and Probe 2 in Figure 3) were positioned on the sample surface along its longitudinal axis, to assist during the calibration phase of the convective heat transfer coefficients. The influence of the length (L) and the width (w) of the sample’s tapered region on the cooling rate and on the thermal gradient at the end of the cooling phase was evaluated.

The length (L) of the tapered region was investigated in the range of 15–30 mm; instead, the width (w) of the tapered region was investigated in the range of 6–12 mm. The effect of these geometric parameters was explored with kriging techniques using the DACE (Design and Analysis of Computer Experiments) MATLAB toolbox.

The FE model mesh was realized with a brick-type element. A mapped square mesh was adopted in the tapered region of the specimen, with 2 mm as the maximum element size in the XY plane and one element in the sample thickness (Z axis). Unstructured mesh was adopted in the grip regions as well as in the other sample regions.

2.2. Thermo-Mechanical Tests

Hot tensile tests were conducted on the Gleeble^®3180 thermo-mechanical testing system to derive the flow behavior of the investigated materials. This system uses direct resistance Joule heating and the hydraulic ram attached to one of the two copper jaws to heat and to deform the sample, respectively.

During the experimental tests described here, K-type NiCr–NiSi thermocouples were adopted. The control thermocouple was spot welded at the center of the sample. This thermocouple provides feedback on the actual temperature which is instantly compared to the set value. The physical simulator appropriately adjusts the electric current for obtaining the desired temperature.

The second and the third thermocouple were spot welded at 5 mm and 10 mm, respectively, away from the center in order to check the temperature along the longitudinal direction of the samples. During the tests, in fact, the samples are mounted between two water-cooled jaws. Due to this configuration, a thermal gradient in the longitudinal direction of the sample is generated.

Hot tensile tests were assisted by an HZT071 extensometer in contact with samples. The extensometer characteristics are detailed in

Table 2. Characteristics of HZT071 extensometer.

Gauge length (mm)	10
Travel during tensile tests (mm)	5
Travel during compression test (mm)	2
Temperature range (°C)	Ambient to 1200 °C
Precision (µm)	±2

.

In Figure 4, the scheme of imposed thermo-mechanical cycles is shown. Specifically: (i) Specimens are heated to the complete austenitization temperature at a heating rate of 10 K/s up to a temperature of 705 °C, and then with a heating rate of 5 K/s up to a temperature of 930 °C. (ii) Samples are soaked for 4 min at the complete austenitization temperature. (iii) After the austenitization phase, specimens are cooled to the test temperature at a cooling rate optimized during the specimen design phase. (iv) Then, hot tensile tests are performed. Before the deformation phase, samples are maintained at the test temperature for 2 s in order to achieve a uniform temperature distribution. (v) Finally, specimens are cooled to room temperature.

Force control mode was adopted during the physical simulation of the (i), (ii), (iii) and (vi) phases; instead, strain control mode was adopted during the (iv) phase.

The choice of the values for the heating rates, the complete austenitization temperature and the soaking time at such temperature was driven by an analysis of the existing literature on the USIBOR^®1500 steel [27].

2.3. Arrhenius Constitutive Model

In this work, the Arrhenius-type constitutive equation is chosen to predict the materials flow stress. According to this model, the Zener–Hollomon parameter (Equation (5)) can be used to represent the effect of the strain rate and the temperature on the hot deformation for both investigated steels.

$$Z=\dot{\epsilon }\times \exp (\frac{Q}{ R\times T} )$$

(5)

where $\dot{\epsilon }$ is the strain rate (s⁻¹), $R$ is the gas constant (J/(mol∙K)), $T$ is the absolute temperature (K) and $Q$ is the activation energy (J/mol).

The Arrhenius constitutive equation that describes the relationship between strain rate, flow stress and temperature is:

$$\dot{\epsilon }=A\times F\left(\sigma \right)\times \exp (-\frac{Q}{ R\times T} )$$

(6)

where:

$$F\left(\sigma \right)=\left\{\begin{array}{c}{\sigma }^{n^{\prime }} \alpha \sigma <0.8 \\ \exp \left(\beta \times \sigma \right) \alpha \sigma >1.2 \\ \left(\mathit{sin}h(\alpha \times \sigma \right){)}^{n } \mathrm{for} \mathrm{all} \mathrm{stress} \mathrm{levels} \end{array}\right.$$

(7)

$n^{\prime }$, $\beta$, $\alpha$, $n$ are constant of material and $\alpha$ can be expressed as $\frac{\beta }{n\prime }$. Moreover, $\sigma$ is the stress.

Replacing Equation (7) in Equation (6), the following expressions are obtained:

$$\dot{\epsilon }=A_1\times {\sigma }^{n\prime }\times \exp \left(-\frac{Q}{ R\times T} \right) \alpha \sigma <0.8$$

(8)

$$\dot{\epsilon }=A_2\times \exp \left(\beta \times \sigma \right)\times \exp \left(-\frac{Q}{ R\times T} \right) \alpha \sigma >1.2$$

(9)

$$\dot{\epsilon }=A\times {(\mathit{sinh}\left(\alpha ·\sigma \right))}^n\times \exp \left(-\frac{Q}{ R\times T} \right) \mathrm{for} \mathrm{all} \mathrm{stress} \mathrm{level}$$

(10)

Turning to logarithmic expressions for Equations (8)–(10), the following expressions are obtained:

$$ln\left(\dot{\epsilon }\right)=\mathit{ln}(A_1)+n^{\prime }\times \mathit{ln}\left(\sigma \right)-\frac{Q}{ R\times T} \alpha \sigma <0.8$$

(11)

$$ln\left(\dot{\epsilon }\right)=\mathit{ln}(A_2)+\beta \times \sigma -\frac{Q}{ R\times T} \alpha \sigma >1.2$$

(12)

$$ln\left(\dot{\epsilon }\right)=\mathit{ln}\left(A\right)+n\times ln(\mathit{sinh}\left(\alpha \times \sigma \right))-\frac{Q}{ R\times T} \mathrm{for} \mathrm{all} \mathrm{stress} \mathrm{level}$$

(13)

Finally, to derive the activation energy $Q$, Equation (13) is differentiated and, considering the constant strain rate, the following expression is achieved:

$$Q=Rn\times \left[\frac{\partial \mathit{ln}\left(\mathit{sinh}\left(\alpha \sigma \right)\right)}{\partial \frac{1}{T}}\right]$$

(14)

3. Results and Discussion

3.1. Geometry of Specimens for Thermo-Mechanical Tests

The geometry of specimens to be adopted during the thermo-mechanical tests was chosen after studying, by means of both experimental test and FE analysis, the effect of geometric parameters, L and w, on the cooling rate from 930 °C to the test temperature and on the thermal gradient at the end of the cooling phase (before the deformation phase). First, this analysis was applied to the 1.3-mm-thick USIBOR^®1500 specimens and then to the 2-mm-thick USIBOR^®2000 specimens. To derive FE results relating to the USIBOR^®2000 steel, the FE model described in Section 2.1 was adopted with the hypothesis that the thermophysical properties of the two steels are the same. A preliminary calibration phase of the FE model was carried out. To this end, the heat transfer coefficients were defined for both the USIBOR^®1500 and USIBOR^®2000 specimens. The calibration phase was performed by comparing numerical and experimental thermal cycles on samples with L ranges between 20 and 25 mm and w ranges between 6 and 10 mm. Some experimental results are shown in Figure 5.

The results derived from the calibrated FE model were collected to obtain the metamodels in Figure 6 for the 1.3-mm-thick USIBOR^®1500 specimens and in Figure 7 for the 2-mm-thick USIBOR^®2000 specimens. These metamodels make it possible to understand how the geometric parameters L and w affect the cooling rate in the control point and the thermal gradient in the extensometer gauge length. The test temperature of 750 °C was chosen for the evaluation of the thermal gradient because, among the investigated test temperatures, is the lower one and consequently the most critical.

The results in Figure 6 and in Figure 7 confirm that a reduction in w and L of the tapered zone leads to an increase in the cooling rate and in the thermal gradient.

The rheological behavior of the investigated UHSS was derived in this work by adopting samples with L = 20 mm and w = 10 mm. Such a geometry guarantees for USIBOR^®1500 specimens 1.3 mm thick a cooling rate of 35.2 K/s and a thermal gradient of 5.36 K/mm. In this geometric configuration, the cooling rate value guarantees the mechanical characterization in the austenitic field and the thermal gradient is relatively low. The same geometry on USIBOR^®2000 specimens 2 mm thick exhibit a cooling rate of 30.77 K/s, 13% lower compared with that reached in USIBOR^®1500 specimens. Moreover, for USIBOR^®2000 samples, the thermal gradient at the end of the cooling phase is about 7 K/mm.

The difference in terms of cooling rate and thermal gradient between the two steels can be due to both the thermophysical properties and the different thickness of the specimens. Assuming the thermophysical properties of the two steels to be equal, from the experimental results reported in Figure 5, it can be observed that an increase in specimen thickness leads to a significant reduction in the cooling rate and a significant increase in the thermal gradient.

The results show that the chosen geometry always guarantees a cooling rate higher than the critical one (27 K/s). Specifically, the geometry with L = 20 mm and w = 10 mm can achieve a cooling rate of 30 K/s without using an external cooling system, both on 1.3-mm-thick USIBOR^®1500 specimens and 2-mm-thick USIBOR^®2000 specimens. Therefore, mechanical characterization tests were carried out by means of the physical simulator by imposing a cooling rate equal to 30 K/s.

As an example, Figure 8a shows numerical–experimental comparison of thermal cycles after the calibration phase for a USIBOR^®1500 sample. Experimental thermal cycles, highlighted with circular markers, were acquired by thermocouples welded on specimens; FE thermal cycles (continues lines) were derived by the FE probes. Figure 8b shows the thermal gradient at the end of the complete austenitization phase and at the end of the cooling phase of 750 °C. Experimental results are highlighted with circular markers.

A good agreement between numerical results and experimental ones was observed. The FE model well predicts the experimental tests.

3.2. Thermo-Mechanical Tests

Once specimen geometry was chosen, the stress–strain curves at different temperatures and strain rates were obtained by means of thermo-mechanical tests using the Gleeble^®3180 system.

As an example, Figure 9 shows the stress–strain curves obtained for a strain rate equal to 0.1 s⁻¹ for each investigated test temperature. In this figure, the dashed curves refer to the USIBOR^®2000 steel, instead, the curves with continuous lines refer to the USIBOR^®1500 steel.

The preliminary results shown in Figure 9 exhibit a significant effect of temperature on steel strength and elongation at break. Specifically, a reduction in temperature leads to an increase in the mechanical strength and a reduction in the elongation at break.

By comparing the two boron steels, however, a greater mechanical strength and a lower elongation at break was observed for USIBOR^®2000 compared to USIBOR^®1500.

The stress–strain curves, similar to those shown in Figure 9, were processed to obtain the flow curves at different strain rate and different test temperature to study the flow behavior of the two UHSS.

3.3. Flow Behavior of the USIBOR^®1500 Steel

True stress vs. true strain curves at different strain rates for a fixed temperature are given in Figure 10. The curves are obtained by extracting twenty data points from each experimental stress–strain curve, in the plastic section up to the necking, at regular intervals within the true strain range of 0.01–0.15.

From the curves shown in Figure 10, it is possible to observe that the temperature and the strain rate have significant effects on mechanical properties of the steel. Specifically, the increase in the temperature and the decrease in the strain rate led to the decreasing in the material resistance.

Under all investigated conditions, as the strain increases from 0.01 to 0.15, the flow stress increases. However, the slope of the stress–strain curves is higher in the initial stage of deformation due to the work hardening effect [16], which is in turn caused by the increase in dislocation density. In the second part of the stress–strain curve, the increase in the flow stress decreases due to the annihilation of dislocations, causing a softening effect [16]. Then, the flow stress remains constant as a result of the dynamic equilibrium between the work hardening and dynamic recovery effect [16,28,29,30].

The results of flow curves for the USIBOR^®1500 are in good agreement with the literature data [11,15,16]. As an example, in Figure 11, for a test temperature of 800 °C and strain rate of 0.1 s⁻¹, experimental results obtained in this work (curve with green square markers), are compared with those detailed in earlier scientific works. The experimental results are closer to those found by Zhou et al. [15] and Li et al. [16], who adopted specimens with a thickness value similar to our specimens. Specifically, Zhou et al. [15] adopted samples of 1.4 mm thick and Li et al. [16] adopted samples 1.2 mm thick.

3.4. Zener–Hollomon Factor for the USIBOR^®1500 Steel

In the Arrhenius-type constitutive equation, the Z parameter and material coefficients make it possible to estimate the flow stress. In this section, material coefficients and the relationship between such coefficients and the strain are determined. The parameters $n^{\prime }$ and $\beta$ can be obtained, respectively, from the slope of $ln\left(\dot{\epsilon }\right)$–$ln\left(\sigma \right)$ and ln$\left(\dot{\epsilon }\right)$–$\sigma$. In Figure 12a,b, the relationship between $ln\left(\dot{\epsilon }\right)$ and $ln\left({\sigma }_{max}\right)$ and that between $ln\left(\dot{\epsilon }\right)$ and ${\sigma }_{max}$ at different temperatures are shown for a strain equal to 0.15, corresponding at the maximum stress (${\sigma }_{max}$).

A linear relationship between $ln\left(\dot{\epsilon }\right)$ and $ln\left({\sigma }_{max}\right),$ as well as between $ln\left(\dot{\epsilon }\right)$ and ${\sigma }_{max},$ is observed; in fact, the experimental points are well interpolated by straight lines (dashed lines in Figure 12). By averaging the slopes of the lines, the mean values and the standard deviation of $n^{\prime }$ and $\beta$ are found to be, respectively, 9.739 ± 2.232 and 0.0544 ± 0.0041. By considering that $\alpha =\frac{\beta }{n\prime }$, it is possible to calculate the corresponding value of $\alpha$ coefficient equal to 0.00558. The material coefficient $n$ can be calculated from the slope of $ln\left(\dot{\epsilon }\right)$–$\mathit{ln}(\mathit{sinh}\left(\alpha ·\sigma \right))$ by linear fitting, and the material coefficients $Q$ and $A$ can be calculated from the slope and intercept of $\mathit{ln}(\mathit{sinh}\left(\alpha ·\sigma \right))$–$\frac{1}{T}$, respectively, by linear fitting. Figure 13a and Figure 13b show, respectively, the relationship between $ln\left(\dot{\epsilon }\right)$ and $\mathit{ln}(\mathit{sinh}\left(\alpha ·{\sigma }_{max}\right))$ and that between $\mathit{ln}(\mathit{sinh}\left(\alpha ·{\sigma }_{max}\right))$ and $\frac{1}{T}$ at different temperatures calculated at strain equal to 0.15.

The mean value and the standard deviation of $n$, $Q$ and $A$ are, respectively, 7.329 ± 1.07; −3.540 ± 0.926; 294.64 ± 49.9 kJ/mol.

To predict the flow stress at different strain rates, the relationship between the strain and the material coefficient were calculated. To this end, with the same procedure adopted for the true strain of 0.15, material coefficients were determined for all true strain values ranging from 0.01 and 0.15. A fifth-order polynomial fit was used to describe the relationship between the true strain and $\alpha$, $n$, $Q$, $\mathit{ln}\left(A\right)$. See Equations (15)–(18).

$$\alpha =A_5\times {\epsilon }^5+A_4\times {\epsilon }^4+A_3\times {\epsilon }^3+A_2\times {\epsilon }^2+A_1\times \epsilon +A_0$$

(15)

$$n=B_5\times {\epsilon }^5+B_4\times {\epsilon }^4+B_3\times {\epsilon }^3+B_2\times {\epsilon }^2+B_1\times \epsilon +B_0$$

(16)

$$Q=C_5\times {\epsilon }^5+C_4\times {\epsilon }^4+C_3\times {\epsilon }^3+C_2\times {\epsilon }^2+C_1\times \epsilon +C$$

(17)

$$lnA=D_5\times {\epsilon }^5+D_4\times {\epsilon }^4+D_3\times {\epsilon }^3+D_2\times {\epsilon }^2+D_1\times \epsilon +D_0$$

(18)

The coefficients of polynomial fitting are listed in

Table 3. Polynomial coefficient of α, n, Q, ln(A) for USIBOR^®1500.

Coefficients	$\mathit{\alpha }$	Coefficients	n	Coefficients	Q	Coefficients	lnA
$A_5$	−16.632	$B_5$	720,137	$C_5$	7 × 1010		8 × 106
$A_4$	17.215	$B_4$	−312,786	$C_4$	−3 × 1010	$D_4$	−4 × 106
$A_3$	−5.5086	$B_3$	50,615	$C_3$	6 × 109	$D_3$	625,764
$A_2$	0.8492	$B_2$	−3752.7	$C_2$	−5 × 108	$D_2$	−50,748
$A_1$	−0.071	$B_1$	120.58	$C_1$	2 × 107	$D_1$	2000.9
$A_0$	0.0083	$B_0$	6.4988	$C_0$	17,624	$D_0$	−0.7429

.

In Figure 14, the material coefficients are presented as a function of true strains (circular markers). Fitting curves are also shown as dashed lines.

To validate the adopted methodology, the reliability of the results was verified. To this end, flow curves of USIBOR^®1500 found in the literature [11,15,16] were adopted to derive, using self-programmed MATLAB code, the trend of the material coefficients as a function of the true strain. These curves were compared with those shown in Figure 14. This comparison is shown in Figure 15. The results of USIBOR^®1500 steel investigated in this work are indicated with filled circular markers. The material coefficients are compared starting from a deformation equal to 0.06, in a deformation range of 0.06–0.15, since a great variability on the flow behavior is observed at lower deformation values.

The results in Figure 15 show that the values of the material constants for the USIBOR^®1500 steel investigated in this work fall within the range defined by the literature curves. The adopted method is thus validated.

3.5. Comparison between Predicted and Experimental Data for the USIBOR^®1500 Steel

In this section, a comparison between experimental data and the computed stress–strain curves is shown for the USIBOR^®1500 steel.

Considering Equations (1) and (2), the relationship between the Zener–Hollomon parameter Z and the flow stress can be found. This relationship can be expressed as follows:

$$\sigma =\frac{1}{\alpha }\times ln\left\{{\left(\frac{Z}{A}\right)}^{1/n}+{\left[{\left(\frac{Z}{A}\right)}^{2/n}+1\right]}^{\frac{1}{2}}\right\}$$

(19)

Equation (19) makes it possible to determine the flows stress at different temperatures, strain rates and true strain. Figure 16 shows the experimental stress–strain curves and the predicted ones for the USIBOR^®1500 steel. The analytic flow curves are represented with solid lines, while on the contrary, the experimental ones are represented with circular markers. Figure 16 shows the flow curves for a constant value of the strain rate.

From the graphs in Figure 16 it can be deduced that the flow stress–strain curves predicted using the Arrhenius-type constitutive equation are similar to the experimentally determined curves. The Arrhenius model well captures both the hardening and the dynamic recovery phases.

The quantitative accuracy of the analytical model can be defined using the correlation coefficient $R$, the average absolute relative error ($AARE$), and the root mean square error ($RMSE$). These parameters are defined as follows:

$$R=\frac{{{\displaystyle \sum }}_{i=1}^n\left(E_i-\overline{E}\right)\times \left(P_i-\overline{P}\right)}{{{\displaystyle \sum }}_{i=1}^n{\left(E_i-\overline{E}\right)}^2\times {{\displaystyle \sum }}_{i=1}^n{\left(P_i-\overline{P}\right)}^2}$$

(20)

$$\mathit{AARE} \left(\%\right)=\frac{1}{N}{\displaystyle \sum }_{i=1}^n\left|\frac{E_i-P_i}{E_i}\right|$$

(21)

$$RMSE=\sqrt{\frac{1}{n}\times {\displaystyle \sum }_{i=1}^n{\left(E_i-P_i\right)}^2}$$

(22)

where $E_i$ represents the experimental data, $P_i$ the predicted data using the Arrhenius model, and $\overline{E}$ and $\overline{P}$ the mean values of the experimental and predicted data, respectively. Moreover, n is the total number of samples.

The values of the statistical parameters $R$, $AARE$ and $RMSE$ are equal to 0.999, 0.5906% and 1.0954 MPa.

A good agreement between experimental data and analytical ones is observed, since the correlation coefficients almost reach a value of 1; moreover, the $AARE$ and $RMSE$ coefficients are low.

3.6. Flow Behavior of the USIBOR^®2000 Steel

Once the method proposed in Section 2 had been validated, mechanical characterization tests were carried out on the USIBOR^®2000 specimens. The results are given in terms of the true stress vs. true strain curves at different strain rates for a fixed temperature in Figure 17.

As shown for the USIBOR^®1500 steel, for the USIBOR^®2000 steel, too, increasing temperature and decreasing strain rate led to a reduction in the material resistance. Moreover, the slope of the flow curves confirms the work hardening effect in the first part and a softening effect due to the dynamic recovery in the second part of the curves.

The flow curves of the new advanced high-strength steel USIBOR^®2000 were compared with those of the USIBOR^®1500 steel. This comparison is shown in Figure 18, Figure 19 and Figure 20 for strain rates equal to 1 s⁻¹, 0.1 s⁻¹ and 1 s⁻¹, respectively. Greater flow stress is recorded at each test temperature and strain rate for the USIBOR^®2000 steel.

3.7. Zener–Hollomon Factor for the USIBOR^®2000 Steel

The same procedure described in Section 3.4 for the USIBOR^®1500 steel was adopted to derive the Arrhenius-type constitutive equation for the USIBOR^®2000 steel.

The material constants of the Arrhenius model constitutive equation, i.e., $n\prime$, β, $n$, $Q$ and $A$ were, respectively, obtained from the plots of (i) $ln\left(\dot{\epsilon }\right)$ versus $ln\left(\sigma \right)$, (ii) $ln\left(\dot{\epsilon }\right)$ versus $\sigma$, (iii) $ln\left(\dot{\epsilon }\right)$ versus $\mathit{ln}\left(\mathit{sinh}\left(\alpha ·\sigma \right)\right)$, and (iv) $\mathit{ln}\left(\mathit{sinh}\left(\alpha ·\sigma \right)\right)$ versus $\frac{1}{T}$ by means of the linear-fitting method. These plots, evaluated at a deformation of 0.15, are represented in Figure 21.

The coefficients $n\prime$ and $\beta$ can be obtained from the slopes of $ln\left(\dot{\epsilon }\right)–ln\left({\sigma }_{max}\right)$ and $ln\left(\dot{\epsilon }\right)–{\sigma }_{max}$, respectively. The average values and the standard deviation of the coefficients $n\prime$ and β are, respectively, equal to 13.374 ± 3.818 and 0.0663 ± 0.009. Considering that $\alpha =\frac{\beta }{n\prime }$, $\alpha$ is equal to 0.00495.

The coefficient $n$ can be calculated from the slope of $ln\left(\dot{\epsilon }\right)–\mathit{ln}\left(\mathit{sinh}\left(\alpha ·{\sigma }_{max}\right)\right)$. The average value and the standard deviation of $n$ are 10.106 ± 2.22. Rather, the material coefficients $Q$ and $A$ can be derived from the slope and the intercept of $\mathit{ln}\left(\mathit{sinh}\left(\alpha ·{\sigma }_{max}\right)\right)–\frac{1}{T}$, respectively. The average values and the standard deviation of the coefficients $Q$ and $A$ are, respectively, equal to 499.7 ± 130.27 kJ/mol and 4.495 ± 1.44.

The material coefficient values have been determined so far for a true strain of 0.15; in order to calculate the material coefficients for strains ranging from 0.01 to 0.15 at intervals of 0.01, the same procedure as that shown above for the USIBOR^®1500 steel was adopted. A fifth-order polynomial fit was used to correlate the strain and the material coefficients, as expressed in Equations (15)–(18). The fitting curves and the material coefficients ($\alpha$, $n$, $Q$ and $\mathit{ln}\left(A\right)$) calculated for the specified strains are shown in Figure 22. In Figure 22, the fitting curves of the USIBOR^®1500 are also shown, with the aim of comparing the two UHSS.

The polynomial fitting results are listed in

Table 4. Polynomial coefficient of α, n, Q, lnA for the USIBOR^®2000.

Coefficients	$\mathit{\alpha }$	Coefficients	n	Coefficients	Q	Coefficients	lnA
$A_5$	−23.404	$B_5$	53,639	$C_5$	1 × 1010	$D_5$	1 × 106
$A_4$	16.638	$B_4$	−24,918	$C_4$	−6 × 109	$D_4$	−643,243
$A_3$	−4.9209	$B_3$	2214	$C_3$	1 × 109	$D_3$	139,528
$A_2$	0.7962	$B_2$	483.44	$C_2$	−1 × 108	$D_2$	−13,178
$A_1$	−0.0736	$B_1$	−83.194	$C_1$	5 × 106	$D_1$	541.37
$A_0$	0.0081	$B_0$	12.403	$C_0$	299,581	$D_0$	29.752

.

From Figure 22, it can be observed that the USIBOR^®2000 steel exhibits a lower value of the α coefficient and greater values of the other material coefficients ($n$, $Q$ and $\mathit{ln}\left(A\right)$). In the following, a physical interpretation of these results is given.

The constant α was obtained as the ratio between $\beta$ and $n\prime$, which are representative of the creep rate [31,32]. Since the behavior of the material during hot working is analogous to its behavior during creep, it is possible to state that low values of $\beta$ and $n\prime$ lead to a low value of α, indicating high resistance to deformation. Therefore, the lower the value of α, the higher the resistance to deformation. On the basis of Figure 22a, it can concluded that the new ultra-high-strength steel (USIBOR^®2000) is characterized by a greater resistance to deformation.

Zhang et al. [33] stated that the constant $n$ represents the workability of the material, meant as deformation resistance. From the comparison of $n$ between the two investigated steels in Figure 22b, a better workability can be observed for USIBOR^®1500, since a lower value of $n$ was calculated.

Considering a value of true strain in the range of 0.0625–0.15, it can be observed that for the two investigated steels, the value of the coefficient $n$ remains almost constant as the strain varies. For lower values of true strain (range between 0.01 and 0.05), the trend of $n$ is increasing for USIBOR^®1500 and decreasing for USIBOR^®2000. This different trend could be explained by the experimental variability of stress at low strain values.

The activation energy $Q$ stands for the energy required for plastic deformation. The greater value of $Q$ for USIBOR^®2000 (Figure 22c) confirms the greater deformation resistance of this steel. This behavior can be justified by the higher silicon content. In fact, Serajzadeh and Taheri [34] demonstrated that silicon has a more dominant effect than carbon on the flow behavior of steels. In particular, an increase in silicon percentage causes an increase in the activation energy.

Finally, the greater value of $\mathit{ln}\left(A\right)$ for the USIBOR^®2000 compared with the USIBOR^®1500 (Figure 22d) also proves the greater deformation resistance of the new USIBOR^®2000 steel.

3.8. Comparison between Predicted and Experimental Data for the USIBOR^®2000 Steel

Once the Zener–Hollomon parameter and the constants $A$, $n$ and α had been determined, the flow stress from Equation (19) could be derived. In Figure 23, with solid lines, the flow curves predicted by the Arrhenius-type constitutive equation are shown. These curves are presented at different test temperatures by setting a constant strain rate, which is equal to 1 s⁻¹ in Figure 23a, 0.1 s⁻¹ in Figure 23b, and 1 s⁻¹ in Figure 23c.

To verify the accuracy of the analytical model, in Figure 23, the experimental flow curves are superimposed on the predicted ones with circular markers.

A good agreement between the experimental and analytical results can be observed in Figure 23. This agreement is confirmed by the values of statistical parameters $R$, $AARE$, $RMSE$ (Equations (22)–(24)). The correlation coefficient $R$ is close to 1 (the exact value is 0.9992), and the average absolute relative error and the root mean square error are low (the exact values are 0.2512% and 1.2586 MPa, respectively, for $AARE$ and $RMSE$).

4. Conclusions

In this study, the flow behavior of two Ultra-High-Strength Steels, the USIBOR^®1500 and the USIBOR^®2000 steels, was investigated in the temperature and strain rate ranges of the press-hardening process. To this end, hot tensile tests in the Gleeble^®3180 system were carried out to perform experimental mechanical characterization of the investigated materials, and the Arrhenius-type constitutive equation was adopted to establish the constitutive model of both steels.

The following conclusions can be drawn:

• The experimental flow curves of the USIBOR^®1500 steel are in agreement with those found in the literature, especially for comparable sample thickness. This result makes it possible to validate the proposed methodology.
• In the range of temperature and strain rate explored, the Arrhenius-type constitutive equation fit the experimental data with good agreement for both the USIBOR^®1500 and USIBOR^®2000 steels. Furthermore, in the range of true strain employed in the experiments for the USIBOR^®1500 steel, the material coefficients of the Arrhenius-type model were in good agreement with those derived using the literature results. This confirms the validity of the adopted methodology.
• The two ultra-high-strength boron steels exhibit the same flow behavior; specifically, an increase in the test temperature and a reduction in the strain rate leads to a reduction in the flow stress. Both steels show the work hardening effect at low deformations and the softening recovery effect at greater deformations.
• From the comparison between the new USIBOR^®2000 steel and the USIBOR^®1500 steel, it was observed that the former exhibits greater strength, lower workability and requires grater activation energy to carry out plastic deformation.

Compared to the existing literature, this study establishes a constitutive model, not yet investigated, for the recently developed USIBOR^®2000 steel. Moreover, a comparison between the two UHSS most adopted in the press hardening process was presented.