Predictive Modeling of Spring-Back Behavior in V-Bending of SS400 Steel Sheets under Elevated Temperatures Using Combined Hardening Laws

Abstract

This research presents an innovative methodology for accurately predicting spring-back tendencies in V-bending of SS400 steel sheets under elevated temperatures. The study leverages extensive tensile test data to determine parameters for pure isotropic and kinematic hardening laws at varying temperatures, crucial inputs for Finite Element Method (FEM) simulations. While using pure isotropic or kinematic hardening laws alone has limitations, especially at elevated temperatures, a hybrid approach is recommended for robust predictive models in ABAQUS 6.13 software. To address this challenge, a novel method is introduced, utilizing flow stress curve ratios between elevated and room temperatures as a function of equivalent strain to derive combined hardening law parameters. Rigorous comparison of simulation and experimental results confirms the model’s effectiveness in predicting spring-back in the V-bending of SS400 steel sheets, particularly under elevated temperatures. This innovative approach enhances understanding of material behavior at high temperatures and improves predictive capabilities for designing and optimizing complex V-bending processes.

1. Introduction

Steel metal forming is widely used in various industries, including the aerospace, automotive, and shipbuilding industries [1]. V-shaped parts processed through bending technology make up a significant portion of products made from steel sheets. The spring-back phenomenon greatly affects the final accuracy of the product in the bending process and is influenced by factors such as bending radius, material, thickness, bending speed, and heating temperature [2,3,4,5,6,7]. Heating temperature is particularly influential in affecting the spring-back. Numerous studies [8,9,10,11,12] have shown that elevated temperatures reduce spring-back and have used both FEM simulation and experimental methods to evaluate this effect.

In the investigation regarding the impact of temperature on the mechanical properties of structural steel, K. W. Poh [13] examined the stress–temperature–strain relationship. A mathematical formulation was developed to describe the interrelation among these parameters. Moreover, the scientific team led by F. Stachowicz, T. Trzepiecin’ski, and T. Pieja conducted comprehensive research on AMS 5604 steel plates [8] across varying temperature conditions. They presented a comprehensive tabulation of mechanical properties for AMS 5604 panels at diverse temperature settings. Numerous experimental inquiries, supplemented by numerical simulations, have been conducted to delve into bending processes and elastic phenomena. Dongye Fei et al. [14] directed their attention towards the elastic rebound phenomenon ensuing from the forming deformation of cold rolled steel in V-shaped bending experiments. Their investigation encompassed finite element simulations employing Abaqus/Standard software and the USDFLD subroutine to account for elastic modulus changes due to plastic deformation, thus enhancing bending accuracy. M.L. Garcia-Romeu et al. [15] pursued a study to quantify the elastic deformation during sheet metal bending processes. Similarly, M. Zhan et al. [16] delved into the elastic mechanisms and principles of pipe bending through numerical analysis techniques. O. Tekaslan [17] conducted an experimental study to quantify post-forming elasticity in 0.5 mm thick steel plates bent in a V die. Likewise, Z. Tekiner [18] explored the elastic behavior of steel under varying bending angles. V. Hsu and Shien [19] introduced a computational approach grounded in bending theory, analyzing 29 samples to elucidate the symmetric sheet metal forming process. Their investigation emphasized the ratio of the punch (or die) radius to plate thickness as a critical factor impacting bending, particularly pronounced in planar strain conditions. Furthermore, Müderrisoğlu et al. [20] proposed an enhanced design methodology for car body panel contouring, focusing on input parameter influence on final contour quality. It was revealed that during product shaping, the desired geometric form deviates due to elasticity phenomena post-deformation. The bending angle corresponding to the mold design does not persist after removal from the mold due to elastic influences. F. Stachowicz et al. [8] performed an extensive study involving AMS 5604 stainless steel plate forming at temperatures ranging from 20 °C to 700 °C. The results indicated a correlation between temperature variation and elastic deformation following spring-back, with higher temperatures leading to reduced elastic angles.

Finite Element Method (FEM) simulation has proven to be a valuable tool in predicting spring-back in metal forming processes. Researchers [21,22,23,24,25,26,27,28] have proposed various methods to predict formability and reduce spring-back, including combining bending test data with FEM results, changing the shape of the tool, optimizing the sheet dimensions, and using flexible FEM simulations. Accurate numerical solutions require reliable descriptions of material properties, including elastic behavior and hardening models. S. Nishino et al. [21] developed a method to predict formability by combining bending test data with Finite Element Method (FEM) simulation results. I. N. Chou and C. Hung [22] investigated methods to reduce spring-back in U-bending processes using FEM analysis. A.P. Karafillis and M.C. Boyce [23] proposed a technique to compensate for spring-back in sheet metal forming by adjusting the shape of the tool using FEM. Ben Said and L. et al. [24] conducted numerical and experimental studies comparing conventional bending methods with rubber-pad cushion bending for AA1050-H14 aluminum thin sheets, using finite-element simulations with specific material models and hardness variations. Then, a comprehensive study involving numerical and experimental investigations assessed the use of an NC lathe machine in the SPIF process for producing dome parts from AA1060-H14 aluminum alloy sheets. The study revealed that the NC turning machine could efficiently manufacture axisymmetric components with improved accuracy compared with a traditional three-axis NC milling machines [25]. K.M. Zhao et al. [26] discovered that repetitive displacement cycles led to a stable bending process and verified the FEM simulation using various hardening laws, such as isotropic, kinematic, and combined hardening models. W. Gan and R.H. Wagoner [27] evaluated numerical integration errors and the impact of parameters such as the compressive stage, number of repeat tension/compression cycles, radius ratio, thickness, and material properties on spring-back prediction. D. Nguyen et al. [28] studied the simulation and prediction of spring-back after roll-bending a PCM sheet with various angular radii using a combined hardening law with a newly proposed method. Accurate numerical solutions require the use of reliable descriptions of materials, elastic behavior, and hardening models in mechanical models implemented in simulation algorithms [29,30,31,32,33,34,35,36].

The preceding discussion has critically examined recent domestic and international studies concerning the impact of thermal effects on workpieces in machining operations. These investigations have collectively underscored that within the realm of bending machining, the intricate phenomena of elastic recovery post-forming and forming forces are significantly influenced by temperature variations. This underscores their fundamental importance in the precise fabrication of intricate details from sheet metal materials. The application of machining in heated environments has garnered substantial attention from scholars across pressure processing and metal cutting domains. Notably, the concept of a heated workpiece has been explored through diverse thermal sources including electrical energy, laser beams, magnetic induction, furnaces, and even oxygen gas. In the pursuit of comprehensively exploring the ramifications of thermal-assisted bending in the manufacturing of mechanical components, emerging research directions gain prominence. These forthcoming inquiries necessitate a meticulous focus on specific aspects: a systematic analysis and optimal selection of an electromagnetic induction heating method that aligns with the experimental objectives of this research; a discerning evaluation and choice of material models to serve as the basis for deriving model parameters, crucial inputs for finite element analysis encompassing both tensile/compression and bending processes of SS400 steel plates, with consideration for variations in temperature; and the subsequent validation of these analyses through a combination of simulations and empirical testing of V-shaped components derived from SS400 steel plates, executed under varying temperature conditions. Furthermore, the endeavor involves the conceptualization of novel methodologies to ascertain material model parameters, evaluating their suitability for the intended analyses, or potentially formulating mathematical models encapsulating the intricate relationships between input parameters, such as workpiece heating temperature, punch radius, and punch displacement, with consequential output parameters including bending force, elastic angle upon retraction, part angle, and part radius.

In this study, we explore spring-back tendencies in V-bending parts made from SS400 steel sheets under elevated temperatures. We conducted tensile tests at various temperature thresholds (room temperature, 300 °C, and 600 °C). To model stress–strain behaviors under isotropic hardening laws, we utilized Voce’s hardening model with parameters determined via the least-squares method in Excel 2013’s Solver tool. We further conducted computational analyses of V-bending processes in ABAQUS 6.13 software, considering both room temperature and elevated scenarios, incorporating pure isotropic and kinematic hardening laws. Comparisons between simulations and experimental results revealed disparities, with isotropic hardening overestimating spring-back tendencies, while kinematic hardening underestimated them. In response, we developed a novel methodology to derive parameters for combined hardening laws, improving the prediction of spring-back behaviors, especially under elevated temperatures. Our research also introduces an innovative technique for determining hardening parameters in V-bending SS400 steel sheets at elevated temperatures, based on a back stress formulation that correlates flow stress ratios between elevated and room temperatures. The strong alignment between simulation outcomes and empirical data underscores the reliability and potential utility of our proposed method.

2. Materials and Hardening Model

2.1. Materials

In this study, SS400 steel sheet material, according to JISG 3101 [37], was used for tensile testing. The chemical composition of the material is presented in

Table 1. Chemical composition of SS400 steel sheets.

C	Si	Mn	P	S	Cr
0.19–0.21	0.05–0.17	0.4–0.6	0.04	0.05	≤0.3

. The tests were conducted at different temperature conditions on a Hung Ta H-200A tensile testing machine using heating generation (Figure 1a). The mechanical properties of the material were determined at room temperature (32 °C) and elevated temperatures of 300 °C and 600 °C. The tensile test specimens were cut from a 6 mm thick steel sheet using wire cutting in the rolling direction and conformed to the national standard TCVN 197-85 (197-2000) [38], as shown in Figure 1b–d, which show the wire cutting process and post-surface treatment with armor paper, respectively. The outcomes of the tensile tests conducted under various temperature conditions are depicted in Figure 1e. The values for Poisson’s ratio and Young’s modulus were deduced from the experimental tests conducted following the material test conditions outlined in

Table 2. Test conditions and mechanical properties of the material.

Parameters	Levels
Temperature (°C)	32; 300; 600
Bending speed (mm/s)	1
Thickness sheet (mm)	6
Poisson’s ratio	0.3
Young’s modulus (MPa)	213,000 (32 °C); 184,410 (300 °C); 107,640 (600 °C)

. The material parameters essential for the Finite Element Method (FEM) analysis are also listed in Table 2.

2.2. Hardening Models

This study used Voce’s material behavior model [39] to express the stress flow curves such as Equation (1) for isothermal environmental temperature conditions. To describe the V-bending process, both the pure isotropic and kinematic hardening laws were first used and then compared with corresponding experimental data. The Von-Mises yield surface, which represents the material’s yielding behavior, translates and expands with plastic strain and is defined as follows (Equation (1)):

$${\sigma }_{iso}=\sqrt{3J_2}$$

(1)

where:

${\sigma }_{iso}$ is the uni-axial equivalent yield stress;

$J_2$ is the second invariant of the deviatoric stress tensor.

The stress difference ($\xi$) is measured from the center of the yield surface and can be expressed as (Equation (2)):

$$\xi =\sqrt{3J_2}-{\sigma }_{iso}$$

(2)

In Equation (3), the deviatoric part of the current stress $\left(S_j\right)$ is defined as:

$$S_j={\sigma }_j-{\sigma }_{m}I$$

(3)

where:

${\sigma }_j$ is the current stress;

${\sigma }_m$ is the mean stress;

$I$ is the identity matrix.

Now, let us discuss the implications of pure isotropic and kinematic hardening:

Pure Isotropic Hardening: In the case of pure isotropic hardening, the yield locus only evolves in size, and there is no translation (Equation (4)):

$${\sigma }_{iso}=\sqrt{3J_2}+\xi$$

(4)

Kinematic Hardening: For the kinematic hardening model, the size of the yield surface remains constant (${\dot{\sigma }}_{iso}=0$), and the translation of the yield locus is determined by the back stress (α) (Equation (5)):

$${\sigma }_{iso}=\sqrt{3J_2}+\xi -\alpha$$

(5)

The evolution of kinematic hardening is described by the increment in back stress (α) as a function of equivalent plastic strain, as shown in Equation (6):

$$d{\alpha }_j=\frac{C}{{\sigma }_{iso}}\left({\sigma }_j-{\alpha }_j\right)d{\epsilon }_{eq}^{pl}-\gamma {\alpha }_{ij}d{\epsilon }_{eq}^{pl}$$

(6)

where $C$ and $\gamma$ are material coefficients related to kinematic behavior.

In Equation (7), the back stress ($\alpha$) curve is obtained by offsetting tensile stress–strain curve data about the yield stress value (${\sigma }_Y$) and fitting them to the back-stress evolution law:

$$\alpha =C\left(\frac{{\sigma }_j-{\sigma }_Y}{{\sigma }_{iso}}\right)d{\epsilon }_{eq}^{pl}-\gamma {\alpha }_{ij}d{\epsilon }_{eq}^{pl}$$

(7)

These equations illustrate how isotropic and kinematic hardening variables influence the total flow stress and the yield surface in the context of V-bending simulations.

To apply this to the FEM simulation, the hardening parameters of isotropic hardening law could be adopted according to Equation (8) and of the kinematic hardening law described by the increase in back stress ($\alpha )$ as a function of the equivalent plastic strain, as shown in Equation (9).

$$\overline{\sigma }={\sigma }_Y+A(1-exp(-B{\epsilon }_{eq}^{pl}))$$

(8)

where ${\sigma }_Y$ is the yield strength, $A$ and $B$ are hardening parameters, $\overline{\sigma }$ is the equivalent stress, and ${\epsilon }_{eq}^{pl}$ is the equivalent strain.

$$\alpha =\frac{C}{\gamma }(1-e^{-\gamma {\epsilon }_{eq}^{pl}})$$

(9)

where $C$ and $\gamma$ are material parameters that describe the kinematic behavior and can be determined using the least-squares method deviation with the SOLVER tool in Excel 2013 software. The implementation procedures and obtained parameters are shown in Figure 2 and

Table 3. Determination of hardening parameters using pure isotropic and kinematic hardening laws at elevated temperatures.

Temperature (°C)	${\mathit{\sigma }}_{\mathit{Y}}$ (MPa)	A (MPa)	B	C (MPa)	γ
32	348	188.86	28.3293	5350	28.3293
300	199.3	171.56	3.452	592.2251	3.452
600	72.43	36.89	6.0145	221.8749	6.0145

, respectively. These material parameters will be used as input data in the FEM simulations performed using Abaqus software. It is crucial to emphasize that the fitting of stress–strain curve data is primarily confined to very small equivalent strain regions. Consequently, this stress–strain curve fitting procedure is applied prior to the initiation of necking in the experimental tests. Consequently, the softening segments of the curves (as illustrated in Figure 1 and Figure 2) are deliberately excluded from the scope of the simulation. It is imperative to recognize that the principal aim of these simulations is to effectively capture and predict the spring-back phenomenon within the pre-necking phases.

3. Experiment and Simulation of V-Shaped Bending at Room Temperature

3.1. Experimental Procedure

In this study, the V-shaped bending process was performed at room temperature with changes in vertical displacement (H) levels of 10 mm, 16 mm, and 22 mm, resulting in corresponding bending angles of 136°, 112°, and 88°, respectively. For each displacement level, three repetitions of the bending process were performed, resulting in a total of nine V-bending experiments at room temperature. Data before and after spring-back were measured and are presented in Figure 3 and

Table 4. Comparison of measurements before and after spring-back in V-bending experiment.

H (mm)	Bending Angle (°)	Measure Angle θexp (°)				Spring-Back (°)
		Levels			Average	Levels			Average
		1	2	3		1	2	3
10	136	141.5	142.5	141.5	141.833	5.5	6.5	5.5	5.833
16	112	117.67	117.88	119.72	118.423	5.67	5.88	7.72	6.423
22	88	96	95	94.5	95.167	8.0	7.0	6.5	7.167

, respectively. These spring-back measurements will be compared with the simulation results in the next section to evaluate the accuracy of pure isotropic and kinematic hardening laws and to propose a new method for predicting spring-back at room temperature.

3.2. Simulation Procedure

To predict the spring-back after V-bending of an SS400 steel sheet, both pure isotropic and kinematic hardening laws were employed using ABAQUS software. The input parameters for the simulation are presented in Table 2 and Table 3. The simulation model is shown in Figure 4a, where the workpiece dimensions are 5 mm × 35 mm × 110 mm (thickness × width × length). The results of the V-bending simulation at room temperature are presented in Figure 4b–d. A comparison between the simulation results using pure isotropic and kinematic hardening laws and the corresponding experimental data (Table 4) are presented in

Table 5. Comparison of experimental and simulation results for spring-back angle.

H (mm)	Bending Angle (°)	Measure Angle (°)			Deviation (Δθ)(°)
		Experiment (${\mathsf{\theta }}_{\exp }$)	Simulation (${\mathsf{\theta }}_{\mathrm{sim}}$)
			Isotropic	Kinematic	Isotropic ( ${\mathsf{\Delta }\mathsf{\theta }}_{\mathrm{iso}}$)	Kinematic ( ${\mathsf{\Delta }\mathsf{\theta }}_{\mathrm{kine}}$)
10	136	141.83	142.50	139.05	0.667	−2.783
16	112	118.42	119.98	117.11	1.557	−1.313
22	88	95.17	96.47	93.94	1.303	−1.227

. The deviation of the measured angle (Δθ) after spring-back was calculated as Equation (10).

$$\Delta \mathsf{\theta }={\mathsf{\theta }}_{sim}-{\mathsf{\theta }}_{exp}$$

(10)

where ${\mathsf{\theta }}_{sim}$ and ${\mathsf{\theta }}_{exp}$ are the simulatied and experimental measure angles, respectively.

3.3. Combined Hardening Law

From the comparison data presented in Table 5, it was observed that the deviations between the simulation predictions using pure isotropic and kinematic hardening laws and the corresponding experiments were significant. The spring-back prediction using the pure isotropic hardening law was larger than the corresponding experiment by approximately 1.6°, while the prediction using the pure kinematic hardening law was smaller than the corresponding experiment by approximately 2.8°. Therefore, a new model was proposed by combining the isotropic and kinematic hardening laws as shown in Equation (11). The new parameters for this model were determined by using the data obtained from the V-bending simulations using both isotropic and kinematic hardening laws to calculate the values of back stress (α) for various equivalent strains $({\epsilon }_{eq}^{pl}$) as follows: at each equivalent strain position, the value of the new back stress $({\alpha }_{comb})$ was calculated based on the previous back stress (α) using Equation (12), in which ${\mathsf{\Delta }\mathsf{\theta }}_{iso}$ and ${\mathsf{\Delta }\mathsf{\theta }}_{kine}$ are the deviations of the measure angles after the simulation of V-bending compared with the corresponding experiments using pure isotropic and kinematic hardening laws, respectively (Table 5). The average value of the new back stress (${\alpha }_n$) was calculated using Equation (13).

$$\overline{\sigma }={\sigma }_Y+A_1(1-exp(-B_1{\epsilon }_{eq}^{pl}))+\frac{C_1}{{\gamma }_1}(1-e^{-{\gamma }_1{\epsilon }_{eq}^{pl}})$$

(11)

$${\alpha }_{comb}=(\frac{\mathsf{\Delta }{\alpha }_{iso}}{\mathsf{\Delta }{\alpha }_{iso}-{\alpha }_{kine}})\times \alpha =0.583\times \alpha$$

(12)

The relationship between the new back stress (${\alpha }_{comb}$) and equivalent strain $({\epsilon }_{eq}^{pl}$) was used to determine the parameters of the new proposed hardening law $C_1$and ${\gamma }_1$ using Excel 2013 calculation software based on the least-squares method. The values of $C_1$ and ${\gamma }_1$ were calculated as 3538.925 MPa and 33.526, respectively. The equivalent stress vs. equivalent strain value was then recalculated using Equation (14) and Figure 5a.

$${\overline{\sigma }}_{iso}({\epsilon }_{eq}^{pl})=\overline{\sigma }({\epsilon }_{eq}^{pl})-\alpha ({\epsilon }_{eq}^{pl})$$

(13)

The data obtained from Equation (13) are used to calculate the new parameters, $A_1$ and $B_1$, of Equation (14), using Microsoft Excel 2013. The resulting values are $A_1$= 55,004 MPa and $B_1$ = 33,527, respectively.

$${\overline{\sigma }}_{iso}={\sigma }_Y+A_1(1-exp(-B_1{\epsilon }_{eq}^{pl}))$$

(14)

The results of a V-bending simulation performed on SS400 steel sheet using the proposed combined hardening law are depicted in Figure 5b–d and

Table 6. Comparison of spring-back results between experiment and simulation with combined hardening law.

Bending Angle (°)	Measure Angle (°)		Deviation (Δθ) (°)
	Experiment (${\mathsf{\theta }}_{\mathit{e}\mathit{x}\mathit{p}}$)	Simulation (${\mathsf{\theta }}_{\mathit{s}\mathit{i}\mathit{m}}$)
88	95.170	95.48	−0.313
112	118.423	119.11	0.686
136	141.833	141.50	0.330

. Table 6 demonstrates that the proposed combined hardening law has higher accuracy in predicting spring-back in V-bending compared with the pure isotropic and kinetic hardening laws. The maximum deviation between the simulation and experimental results for the spring-back angle was found to be 0.668°. Therefore, it can be concluded that the proposed combined hardening law can be utilized to predict spring-back in other cases involving SS400 steel sheets.

4. Experiment and Simulation of V-Shaped Bending at Elevated Temperatures

4.1. Experimental Procedure

The experimental setup for the V-bending of SS400 steel sheets is depicted in Figure 6a. To conduct the heated V-bending test, a magnetic induction heating device is used to heat the blank to elevated temperatures of 300 °C and 600 °C. To minimize heat radiation during the V-bending process, the entire die, punch, and blank are placed inside an insulated box surrounded by insulating glass wool. The temperature controller mounted on the system ensures that the required stable temperature of the blank is maintained during the V-bending process.

Table 7. Comparison of V-bending results before and after spring-back at elevated temperatures of 300 °C and 600 °C.

H (mm)	Bending Angle (°)	Measure Angle (°)
		300 °C				600 °C
		Level			Avg.	Level			Avg.
		1	2	3		1	2	3
10	136	139.1	140.15	138.9	139.38	137	137.45	137.2	137.22
16	112	116.2	115.5	115.2	115.63	113.15	114	113.5	113.55
22	88	93.5	93.0	93.0	93.167	90.5	90	90	90.167

and Figure 6b,c show the experimental results of the V-bending process at 300 °C and 600 °C, respectively.

4.2. Simulation Procedure for V-Bending at Elevated Temperatures

To simulate spring-back at elevated temperatures, this study used a proposed hardening law at room temperature combined with the determination of the back stress function at elevated temperatures. The method to determine the back stress function (${\alpha }_{n\left(T\right)}$) at elevated temperatures is described in Equation (15).

$${\alpha }_{n\left(T\right)}=\left(\frac{{\sigma }_T}{{\sigma }_R}\right)\times {\alpha }_{\left(T\right)}=\left(\frac{{\sigma }_{yT}+A_T\times (1-exp(-B_T\times {\epsilon }_{eqT}^{pl}))}{{\sigma }_{yR}+A_R\times (1-exp(-B_R\times {\epsilon }_{eqR}^{pl}))}\right)\times {\alpha }_{\left(T\right)}$$

(15)

where R and T are indices for the flow stress equation at room- and elevated temperatures, respectively, and α(T) is the back stress function at elevated temperatures (see Equation (2) and Table 3).

The obtained data on the back stress value (${\alpha }_{n\left(T\right)}$) based on the equivalent strain (${\epsilon }_{eq}^{pl}$) at elevated temperatures were used to determine the parameters of the new combined hardening law for elevated temperatures, C1T and γ1T, using Equation (4) and shown in

Table 8. The hardening parameters of combined hardening laws at elevated temperatures.

T (°C)	${\mathit{\sigma }}_{\mathit{Y}}$ (MPa)	${\mathit{A}}_1$ (MPa)	${\mathit{B}}_1$	${\mathit{C}}_1$ (Mpa)	${\mathit{\gamma }}_1$
300	199.3	53.3422	6.9049	280.9023	2.0133
600	72.43	29.2536	6.3814	36.5690	4.7288

and Figure 7a. The new parameters, A1T and B1T, in Equation (8) of the combined hardening law were determined using Excel 2013 calculation and are shown in Table 8. Figure 7b shows the simulation results for spring-back prediction based on the proposed combined hardening law.

Table 9. Comparison of spring-back results between experiments and simulations using combined hardening law at elevated temperatures.

Bending Angle (°)	Measure Angle (°)
	300 °C			600 °C
	Experiment (${\mathsf{\theta }}_{\mathit{e}\mathit{x}\mathit{p}}$)	Simulation (${\mathsf{\theta }}_{\mathit{s}\mathit{i}\mathit{m}}$)	Δθ (°)	Experiment (${\mathsf{\theta }}_{\mathit{e}\mathit{x}\mathit{p}}$)	Simulation (${\mathsf{\theta }}_{\mathit{s}\mathit{i}\mathit{m}}$)	Δθ (°)
88	93.167	93.86	0.693	90.167	90.52	0.353
112	115.63	115.24	−0.393	113.55	113.15	−0.4
136	139.38	138.79	−0.593	137.22	136.77	−0.45

compares the results between the experiment and simulation at 300 °C and 600 °C. The deviation angle (Δθ) between the simulation and experiment is calculated using Equation (3). Table 9 shows that as the bending angle increases, the spring-back angle decreases, which can be determined by subtracting the bending angle from the simulation or experimentally measured angle. The comparison results indicate that the proposed method offers high predictive accuracy and good agreement with experimental values in describing the spring-back phenomenon after V-bending at elevated temperatures.

5. Conclusions

This study conducted an extensive experimental investigation, encompassing both tensile and V-bending tests, with the inclusion of thermal assistance. The results from the tensile tests, conducted at three distinct temperature levels (32 °C, 300 °C, and 600 °C), revealed a noteworthy trend: as the temperature increased, the mechanical properties of the material exhibited a discernible decline, while the formability of the sheet metal notably improved. A pioneering framework was introduced for determining parameters related to the combined hardening law within the context of V-bending, both under standard room conditions and at elevated temperatures. The validity of this approach was substantiated through a rigorous comparative analysis, which involved finite element analysis using ABAQUS software and experimental results obtained from V-bending trials. Impressively, the predictions for V-bending at room temperature closely matched the corresponding experimental findings. Furthermore, a novel model rooted in the back stress function, derived from the relationship between flow stress at elevated temperatures and room temperature, consistently demonstrated its predictive capabilities by aligning with the associated experimental dataset. Consequently, the proposed methodology stands as a powerful tool for simulating spring-back predictions across diverse bending processes involving SS400 steel sheets at elevated temperatures. The ingenious amalgamation of experimental exploration, computational analysis, and the conceptualization of novel methodologies in this research makes a significant contribution to the understanding of material behavior under thermal-assisted bending conditions. Moreover, the outcomes hold substantial promise for optimizing bending procedures in practical industrial applications, where accurate spring-back predictions can significantly influence process efficiency and product quality, marking a distinct and innovative contribution to the field.