The Study and Application on Ductile Fracture Criterion of Dual Phase Steels During Forming

Abstract

High-strength steel exhibits complex fracture behavior due to the interplay between shear and necking mechanisms during stamping and forming processes, posing challenges to achieving the dimensional accuracy and reliability demanded for automotive body panels. Existing prediction methods often fail to simultaneously account for both tensile and shear fracture characteristics, thereby limiting their predictive accuracy under diverse stress conditions. To address this limitation, we propose a ductile fracture criterion that integrates both tensile and shear mechanisms, calibrated using a single tensile–shear test to facilitate practical engineering applications. This study investigates the fracture characteristics of DP780 dual-phase steel through numerical analysis and tensile–shear experiments. The findings establish a relationship between stress triaxiality and ultimate fracture strain across varying stress states, represented by the B–W curve. Simulations reveal distinct stress triaxiality behaviors under different loading conditions: under uniaxial tensile loading, triaxiality ranges from 0.33 to 0.6, with fracture strain decreasing monotonically as triaxiality increases. Under shear loading, triaxiality ranges from 0 to 0.33, with fracture strain increasing monotonically as triaxiality rises. Additional bending simulations validate that this criterion, along with the B–W curve, reliably predicts the fracture behavior of DP780, offering an effective tool for predicting fracture in dual-phase steels during stamping and forming processes.

1. Introduction

High-strength dual-phase steels, characterized by a relatively low yield-to-tensile strength ratio, exhibit fracture mechanisms that differ markedly from those in conventional alloy steels. Their failure modes are significantly influenced by the internal microstructure, resulting in distinct damage mechanisms. Recently, researchers have increasingly focused on factors influencing shear fracture in dual-phase steels and the associated damage mechanisms. Additionally, extensive theoretical and experimental research worldwide on ductile fracture phenomena in metals has led to the proposal of several fracture prediction criteria grounded in damage mechanics theory [1,2,3,4].

In the investigation of specific fracture behaviors, Shih et al. [5] analyzed the springback control process of advanced high-strength steel during bending, revealing that shear rupture in steel sheets is primarily influenced by three factors: the die radius of curvature, thickness-to-width ratio, and applied tensile force. Luo and Wierzbicki [6] further investigated the fracture behavior of DP780 steel sheets under bending conditions in 2010, employing a modified Mohr–Coulomb fracture criterion. Their findings indicated that shear fracture occurs with smaller die radii, while larger radii shift failure behavior toward tensile fracture. In a study on DP600 high-strength steel, Björklund et al. [7] linked shear fracture in dual-phase steel to stress triaxiality and loading parameters. Li Mei and Zhao Yixi et al. [8] thoroughly investigated the failure mode of dual-phase steel. The results demonstrated that the steel exhibits shear fracture when subjected to small-radius bending before reaching necking. In addition, in [9], a new failure specimen tested in a uniaxial tension machine has been presented where first local failure occurs at high stress triaxiality (tension-dominated mode) whereas the following second and final failure happens at low stress triaxiality (shear-dominated mode). Morin et al. [10] explored fracture behavior in butterfly specimens under combined tension and shear loading, while two-dimensional experiments and different cruciform specimen geometries have been used to analyze anisotropic plasticity.

Despite these advancements, research on shear rupture failure in metallic materials, particularly duplex steel body panels, remains limited. Few studies provide accurate predictions of shear rupture behavior in duplex steel under complex forming conditions, and no unified guideline exists to forecast the competing failure behaviors of duplex steel—namely, shear and necking rupture. Therefore, establishing a fracture prediction criterion for duplex steels is urgently needed, both from theoretical and practical engineering perspectives.

Numerous scholars have found that the toughness fracture criterion based on stress triaxiality can effectively predict the rupture failure of steel plates. In a study by Li S.L. et al. [11], by introducing the stress triaxiality parameter, the relationship between the model parameters and the stress triaxiality is established, so that the model can predict the fatigue life under different stress triaxiality states; however, the paper does not take into account the specific effect of the maximum principal stress on tensile fracture. Fu, Q. T. et al. [12] established the relationship between the B–W curve [13] and the performance parameters of advanced high-strength steel sheet materials through parameter fitting. This approach simplifies the calibration of the B–W curve and alleviates the challenges associated with parameter calibration. However, the multi-parameter solution based on the Modified Mohr–Coulomb (MMC) criterion is still difficult. At present, the B–W rupture failure criterion based on stress triaxiality and fracture strain for advanced high-strength duplex steel sheet material has been less studied, and a relatively uniform expression has not been established. Expanding on these findings, Nahshon, K. et al. [14] investigated the fracture behavior of low-stress triaxiality scenarios, such as pure shear, by introducing shear stress, while Kim Lau, N. et al. [15] explored the fracture behavior of shear stress in high-stress triaxiality by modifying the model. Yuanli, B. et al. [16] combined numerical simulation to obtain fracture stress–strain data within the range of stress triaxiality and derived a segmented functional curve governed by stress triaxiality, known as the B–W fracture criterion. With advancements in ductile fracture criteria, however, scholars have discovered that ductile fracture is influenced by more than just stress triaxiality. For instance, Yuanli, B. and Tomasz, W. [17] developed a Modified Mohr–Coulomb (MMC) model by introducing a hardening criterion based on the Mohr–Coulomb theory and successfully predicted the fracture of TRIP690. Lou, Yanshan et al. [18] constructed a shear-controlled ductile fracture criterion based on equivalent strain, stress triaxiality, and normalized maximum shear, which accurately predicted the fracture behavior of DP780, and then constructed a criterion with a variable threshold of stress triaxiality. Based on the shortcoming that DF2014 cannot describe the high-stress triaxial failure fracture well, Hu, Qi et al. [19] proposed the Hu criterion that can predict the fracture strain under different stress states more accurately. Quach Hung et al. [20] presents a new ductile fracture criterion that considers the effects of maximum tensile and shear stresses, as well as micro-mechanical influences. The results indicate that this model can more accurately predict fracture behavior in materials under low-stress triaxiality conditions. However, this model is primarily applicable to predicting toughness fracture in isotropic materials under monotonic loading. Further research is needed to assess its applicability for anisotropic materials or those under cyclic loading conditions Recent years have seen numerous studies by scholars utilizing various models to investigate failure mechanisms. Kang, L. et al. [21] used an improved ductile fracture model to predict the ductile fracture behavior of high-strength steels under high-stress triaxial conditions, which improves the prediction accuracy of the fracture behavior under complex stress states. Fuhui Shen et al. [22] present the improved Bai–Wierzbicki model, based on the Hill48 framework and incorporating anisotropic damage initiation and evolution parameters, which can more accurately predict the anisotropic damage and fracture behaviors of materials. However, while specific damage initiation criteria have been developed for different loading directions, a unified criterion has not yet been established, limiting the model’s applicability. Sung-Ju Park et al. [23] compared six common fracture criteria and found that the ones that take into account both triaxiality and Rhodes angle factors are better predictors of the fracture characteristics of AH36 and DH36 steel plates, and they suggested that further research is needed on the prediction of fracture in the range of low-stress triaxiality. It has been found that toughness fracture criteria based on damage accumulation and failure strain have limitations that restrict their ability to accurately and conveniently predict the unique rupture phenomena observed in advanced high-strength steel plates.

Stress triaxiality, a key field parameter describing the stress state, plays a critical role in influencing the plastic deformation of dual-phase steel materials. It effectively characterizes the material’s stress state under complex deformation conditions and is widely utilized in failure prediction models. Research findings indicate that stress triaxiality values continuously change during plastic deformation. This study aimed to conduct tensile experiments on DP780 steel specimens with varying notch radii to achieve high-stress triaxiality. Additionally, shear specimens with different angles were designed to attain low stress triaxiality.

In previous research, we investigated the toughness fracture behavior of advanced high-strength duplex steel body panels, evaluating various fracture criteria to identify a toughness criterion suitable for duplex steel panel forming that unifies necking and shear fracture mechanisms. We validated that the selected criterion accurately predicts toughness fracture under various deformation paths, demonstrating its broad applicability. These core findings, now published, provide an essential foundation for developing a criterion to predict fracture timing based on toughness parameters. In engineering practice, establishing a fracture-age prediction criterion for two-phase steel, based on the principles of ductile fracture, is essential. This thesis develops a ductile fracture criterion for forming dual-phase steel body panels, unifying necking and shear fracture mechanisms, with DP80 as the focus of study. We calibrate unknown parameters within this criterion through uniaxial tensile and shear testing, derive the associated constitutive equations, and conduct numerical simulations in Abaqus, incorporating a custom user-defined material subroutine to facilitate comparisons with empirical data. We conduct an in-depth analysis of the relationship between stress triaxiality, relevant eigenfield quantities, and ultimate strain. Based on our findings, we propose a set of criteria to predict the fracture behavior of advanced high-strength steel plates during stamping, providing theoretical support for the broad application of high-strength duplex steels in automotive body panels and enhancing quality control in duplex steel manufacturing.

2. Materials and Methods

2.1. Material Properties

In accordance with the standards for uniaxial tensile testing of thin sheets, plates conforming to the required specifications were chosen. DP780 steel, a widely utilized material, was selected for its suitability, with a specified thickness of 1 mm. The steel used in this study was sourced from Baosteel Metal Co., Ltd., Shanghai, China.

All the specimens in the plate material are unified plate and direction through the laser cutting, tensile specimen size according to national standards GB/T228.1-2010 [24], and in accordance with the provisions of the requirements of the range of design.

The chemical composition of the steel in question is detailed in [25], which indicates that the carbon (C), silicon (Si), manganese (Mn), phosphorus (P), sulfur (S), and aluminum (Al) contents of DP780 steel are 0.08%, 1.1%, 1.86%, 0.01%, 0.005%, and 0.05%, respectively.

The fundamental mechanical properties of the dual-phase steel were determined based on experimental data. The specific parameters are presented in

Table 1. Mechanical properties of DP780.

Hardness	Yield Strength (MPa)	Tensile Strength (MPa)	Elongation at Break (-)
HV110	780–810	913	0.1

.

As shown in Table 1, DP780 high-strength steel demonstrates exceptional strength, with a tensile strength of approximately 913 MPa. Additionally, it does not exhibit a distinct yield point, and the elongation at fracture is 0.1.

2.2. Material Model

Owing to the directional performance variations observed within the material plane, it has been demonstrated in practice that the Hill48 anisotropic yield criterion effectively represents the material’s anisotropy. The description of the theory above can be expressed in Equation (1) [26].

$$\overline{\sigma }=\sqrt{{F\left({\sigma }_y-{\sigma }_z\right)}^2+{G\left({\sigma }_z-{\sigma }_x\right)}^2+{H\left({\sigma }_x-{\sigma }_y\right)}^2+2L{\sigma }_{yz}^2+2M{\sigma }_{zx}^2+2N{\sigma }_{xy}^2}$$

(1)

where, $\overline{\sigma }$ is the equivalent stress, $F$, $G$, $H$, $L$, $M$, and $N$ indicate to as the anisotropy parameter.

The anisotropy parameters can be expressed in Equation (2) [27].

$$r={\displaystyle \frac{{\epsilon }_b}{{\epsilon }_t}}={\displaystyle \frac{ln\frac{b}{b_o}}{ln\frac{t}{t_o}}}$$

(2)

where $r$ is logarithmic strain, $r$ is expressed as the ratio of the strain ${\epsilon }_b$ in the width direction of the sheet to the strain ${\epsilon }_t$ in the thickness direction.

The meanings of the symbols used are as shown in

Table 2. The main parameters of anisotropic index.

Symbols	Meaning
${\epsilon }_b$	Strain rate perpendicular to the stretching direction
${\epsilon }_t$	Strain rate in the thickness direction
$b$	Width of the specimen after stretching
$b_0$	Original width of the specimen
$t_0$	Original thickness of the specimen
$t$	Thickness of the specimen after stretching

.

The relationship between the anisotropy parameter and the anisotropy index is as follows:

$$\left\{\begin{array}{c}F={\displaystyle \frac{r_0}{r_{90}(r_0+1)}}\\ G={\displaystyle \frac{1}{r_0+1}}\\ H={\displaystyle \frac{r_0}{r_0+1}}\\ N={\displaystyle \frac{(r_0+r_{90})(1+2r_{45})}{2r_{90}(r_0+1)}}\end{array}\right.$$

(3)

where $r_0$, $r_{45}$, $r_{90}$ represent the Lankford coefficients when the tensile axis of the plate is oriented at angles of $0°$, 45°, and 90°, respectively, relative to the rolling direction [28].

The anisotropy parameters of the studied duplex steels were provided by the company, as shown in

Table 3. Anisotropic parameters of dual-phase steel sheets.

Type of Steel	F	G	H	L	M	N
DP780	0.56	0.56	0.44	1.50	1.50	1.50

.

2.3. Research Methodology

Metal materials exhibit different damage mechanisms under various stress states. If the damage mechanism is not well understood, predicting material damage and fracture becomes challenging. There are generally two methods to achieve different triaxial stress states in specimen materials The first method involves directly applying different triaxial stresses to the specimen, but this requires specialized loading equipment, making it difficult to implement. The second method involves preparing specimens with different specifications and geometries, achieving different triaxial stress states through varied loading methods. This approach is used to examine certain microscopic characteristics within the specimen under different stress states during plastic deformation. This study adopts the latter method. Ductile fracture specimens under different triaxial stress states were obtained through tensile–shear tests, and the experimental process is shown in Figure 1.

3. Experiments

3.1. Static Tensile Testing of Advanced High-Strength Steel DP780

Uniaxial tensile testing is a fundamental experiment. To investigate the strength and plasticity characteristics of dual-phase steel, uniaxial tensile tests are conducted, and the experimental data are analyzed. This allows for the generation of displacement–load curves and stress–strain curves for DP780, thereby providing a comprehensive understanding of the mechanical properties of advanced high-strength dual-phase steel.

3.1.1. Experimental Plan

The steel plate material used in this experiment is DP780 thin steel plate. For the experiment, the provisions of the plate thickness of 1 mm the notched radius of the thin plates was selected as 0 mm, 5 mm, and 10 mm, with a length of 120 mm and a width of 20 mm. The specific dimensions are shown in Figure 2.

Tensile testing was conducted using a universal testing machine in the laboratory, with a loading rate of 0.2 mm/min. Due to significant errors associated with visual observation of crack formation, necking was assessed by observing the load–displacement curve. When material failure and crack initiation occur, the load–displacement curve exhibits a marked decrease.

3.1.2. Experimental Procedure

Prior to the experiment, the universal testing machine was accurately calibrated. The tensile and shear specimens were subjected to uniaxial tensile test at room temperature.by CMT5305 which supplied by Shenzhen Meters Industrial Systems Ltd, Shenzhen, China The test machine is an electronic tensile testing machine as the main body, which can automatically complete different specifications of the sheet specimens of fully automatic tensile testing, a maximum test force of 200 kN–300 kN, and a test force display value of the relative error of ±1%, as shown in Figure 3. The distance between the machine’s grips was adjusted, and the force and displacement were recoded via sensors. The load–displacement curve was displayed on the computer screen, and the stress–strain curve was simultaneously recorded. After the specimen fractured, the machine was shut down, and the experimental data were saved.

From the fracture images in Figure 4, it is evident that all specimens, regardless of notch size, fractured in the middle section. There was no significant necking at the fracture location, and the fracture surface was angled with respect to the tensile direction. When the two broken parts of the specimen were closely aligned, a crack was still visible in the middle, with the widest part of the crack located at the center of the fracture cross-section. This also indicates that the fracture initiation occurred at the center of the necking region.

3.1.3. Analysis of Experimental Results

Experiments were conducted on thin plates with different radii, with each configuration tested three times. The data from these three experiments were combined to generate accurate engineering stress–strain curves, as illustrated in the figure below.

By comparing the stress–strain curves in Figure 5, it is evident that DP780 steel exhibits no distinct yield plateau; rather, its stress–strain curve forms a consistently smooth arc. With Figure 5, we see that the three stress–strain curves obtained from the same direction are not in perfect agreement with each other. This phenomenon can be attributed to the anisotropic properties of advanced high-strength steel plates, which is a distinctive feature of DP780 steel. This finding indicates that DP780 steel demonstrates a high capacity to withstand tensile forces without breaking, reflecting its high tensile strength. However, the material shows relatively limited ability to undergo plastic deformation before fracture, indicating lower plasticity. This combination suggests that while DP780 can bear significant loads, it may be more prone to brittle fracture under high strain conditions, especially in applications requiring extensive deformation.

In practical tensile testing, once plastic deformation occurs, the material experiences permanent, irreversible deformation. As a result, the dimensions of the thin plate will deviate from their initial values. Therefore, the stress–strain curves used in this study are intended to represent the true stress–strain behavior.

The true stress–strain curve requires the use of the instantaneous cross-sectional area. The true stress is determined using this instantaneous area, and the strain is calculated using the instantaneous length, which can be expressed in Equation (4).

$$\left\{\begin{array}{c}\mathrm{e}=In(1+{\epsilon }_{nom})\\ \mathsf{\sigma }={\sigma }_{nom}(1+{\epsilon }_{nom})\end{array}\right.$$

(4)

where $\mathrm{e}$, ${\epsilon }_{nom}$ represent the real strain versus engineering strain $\mathsf{\sigma }$, ${\sigma }_{nom}$ represent the real versus engineered stresses.

Given the relatively slow stretching rate of the thin plates and the non-uniform deformation, the stress calculations incorporate the initial cross-sectional area. As a result, the calculated stress is significantly higher than the actual values. A comparison of one set of experimental data shows that after the elastic limit, the true stress becomes markedly greater than the engineering stress as the material deforms, and the disparity between the two continues to increase. After processing with the aforementioned Equation, the true stress–strain curve is obtained, as illustrated in Figure 6.

3.1.4. Numerical Simulation

In Abaqus, three thin plate models of different sizes were developed in Figure 7, ensuring that the loading speed, boundary conditions, and other parameters were as consistent as possible with the experimental conditions. By simulating the tensile process, stress field characterization metrics reflecting the mechanical performance of the thin plates, such as the range of stress triaxiality values shown in

Table 4. Experimental data.

Test Specimens	Triaxial Stress Degree
tensile test specimens R0	0.33~0.45
tensile test specimens R5	0.45~0.48
tensile test specimens R10	0.48~0.53

, were obtained. The resulting data will be used for further research.

3.2. Shear Test of Advanced High-Strength Steel DP780

The mechanisms of hole formation, development, and growth in advanced high-strength dual-phase (DP) steel are particularly distinctive. Research has shown that the fracture of DP steel is primarily characterized by shear fracture. To investigate the relationship between low-stress triaxiality and fracture strain, and to understand how stress triaxiality changes when transitioning from a pure shear stress state to a uniaxial tensile stress state, shear tests were conducted.

3.2.1. Experimental Scheme

To investigate the variation of the stress field when transitioning from a pure shear state to a uniaxial tensile state, several sets of specimens were carefully designed following the methodology outlined in the referenced study. The notch angles were set at 0°, 45°, and 90°, with the dimensions as shown in the Figure 8.

The figure above shows shear specimens with different notch angles. A total of three sets of specimens were prepared, each containing three specimens, with each labeled with a unique identifier. For example, S90001 denotes the first shear test of the 90° tensile specimen.

3.2.2. Analysis of Experimental Results

Figure 9 shows the fracture patterns of the shear specimens. It was observed that the fracture development followed the trajectory initiated at the designed notch, with progressive accumulation of changes leading to complete failure.

Since the shear tests were conducted on a tensile testing machine, some slip occurred during the tensile process. Additionally, the stress state at different notch angles was influenced by both shear and axial tensile stresses. Therefore, the stress–strain curves obtained from the testing machine do not accurately reflect the true stress–strain relationship. However, the displacement and load curves are correct. For this reason, in finite element simulations, the stress–strain curves are used for uniaxial tension, while the displacement and load curves are used as references for shear simulations.

By observing the shear fracture results in Figure 9, it is found that the occurrence of tensile shear fracture due to the notch angle of the specimen contains different proportions of shear fracture; 0° specimen shows pure shear fracture, with the notch angle increasing gradually to the transformation of necking fracture, and in 90° when the necking fracture occurred, the proportion of shear fracture is reduced to a minimum. Observation of Figure 10 found that in the shear specimen load displacement curve with the increasing shear angle, the resulting load also continues to rise, and in the shear angle of 90° when a sharp rise in the specimen fracture tensile force reached nearly 3500 N, it can be seen that 90° specimen has been approximated for the tensile state.

At the same time, it can be seen that the load–displacement curve of the shear specimen does not show a yield plateau. The trend of the curves obtained from different experimental groups is consistent. Although there are some differences in the load–displacement curves for the same group of materials, these differences are small. The displacement values at the fracture point are basically similar.

3.2.3. Numerical Simulation

The model was constructed based on the actual dimensions of the specimen, using shell elements to ensure computational efficiency and result accuracy. The mesh size was set to 0.2 mm, with finer mesh refinement applied in critical areas (such as near the notch) to effectively capture local stress and strain variations. Quasi-static loading tests were conducted in tension mode at a loading rate of 0.2 mm/min. Material properties in Abaqus were set to a Young’s modulus of 215,000 MPa and a Poisson’s ratio of 0.3. The part was defined as deformable, with the analysis step configured as dynamic explicit. The time duration was set to 0.1, with a friction coefficient of 0.01, and the boundary condition was fully fixed in the Z direction. The uniaxial tension and shear numerical simulations discussed in the text were both conducted using this model setup and properties.

Similar to the uniaxial tensile test, in Abaqus, thin plate models at 0°, 45°, and 90° orientations were developed as shown in Figure 11. Care was taken to ensure that parameters such as loading speed and boundary conditions were kept consistent with the experimental conditions as much as possible. Through simulation, stress field characterization quantities reflecting the mechanical properties of the thin plates, such as stress triaxiality, can be obtained. The value range of stress triaxiality is shown in

Table 5. Experiment data.

Specimen Type	Stress Triaxiality
shear specimen 0°	0~0.16
shear specimen 45°	0.16~0.33
shear specimen 90°	0.33~0.5

. The data obtained can be used in subsequent research.

3.3. Experimental Results

In this chapter, the specific mechanical properties of DP780 high-strength steel are further understood through simple specimen single tensile and shear experiments, and the following main conclusions are mainly obtained through the study in this chapter:

(1)

The results of shear and single tensile tests show that there is no obvious yield plateau of DP780 steel, which indicates that the tensile strength of DP780 is relatively large and the plasticity is low.

(2)

Through the finite element simulation, it can be found that the stress triaxiality of the tensile state is relatively high, and the stress triaxiality of the shear simulation is relatively low.

(3)

Ensure that the loading speed, boundary conditions, and other parameters with the actual experiment as far as possible to maintain consistency, through the simulation, to reflect the mechanical properties of the thin plate stress field characterization of the stress triaxial degree of change range

4. Development of Toughness Fracture Criteria and Forming Fracture Criteria for Automotive Sheet Metal

Stress triaxiality, as an important field characteristic for describing stress states, influences the plastic deformation of dual-phase steel materials. It effectively reflects the material’s stress state under complex deformation conditions and is applied in many failure prediction models for steel materials. Currently, many researchers have conducted in-depth studies on methods to accurately determine stress triaxiality. This study involves the design of tensile and shear specimens at different angles to obtain stress triaxiality under various stress states. By investigating the relationship between low- and high-stress triaxiality distributions and fracture limit strains in advanced high-strength dual-phase steel, this study aims to establish a failure criterion based on failure strain to predict the fracture of dual-phase steel.

4.1. Methods for Determining Stress Triaxiality

The expression for the stress triaxial is as follows:

$$\eta ={\displaystyle \frac{{\sigma }_m}{\overline{\sigma }}}={\displaystyle \frac{\left({\sigma }_1+{\sigma }_2{+\sigma }_3\right)/3}{\frac{1}{\sqrt{2}}\sqrt{{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2+{\left({\sigma }_1-{\sigma }_3\right)}^2}}}$$

(5)

In Equation (5) [29], $\eta$ represents the stress triaxial, ${\sigma }_1$ represents the maximum principal stress, ${\sigma }_2$ denotes the intermediate principal stress, ${\sigma }_3$ is the minimum principal stress, ${\sigma }_m$ refers to the hydrostatic stress (mean stress), and $\overline{\sigma }$ represents the equivalent stress.

Stress triaxiality is a dimensionless parameter. It is always positive in a tensile state, always negative in a compressive state, and equal to 0 in a pure shear state. It is 1/3 in uniaxial tension and −1/3 in uniaxial compression. This study, which combines simple uniaxial tensile and shear experiments, assumes that the material maintains a constant strain path during plastic deformation. Additionally, it assumes that the material deformation follows a thick anisotropic quadratic yield function. Under proportional loading conditions, the stress triaxiality and equivalent fracture strain expressions, which are expressed in terms of the strain ratio, are derived from the thick anisotropic yield criterion and its corresponding flow expressed in Equations (6) and (7).

$${\displaystyle \frac{{\sigma }_m}{\overline{\sigma }}}={\displaystyle \frac{\sqrt{1+2r}(1+\beta )}{3\sqrt{1+\left(\frac{2r}{1+2r}\right)\beta +{\beta }^2}}}$$

(6)

$$d\overline{\epsilon }={\displaystyle \frac{1+r}{\sqrt{1+2r}}}\sqrt{1+\left({\displaystyle \frac{2r}{1+2r}}\right)\beta +{\beta }^2}d{\epsilon }_1$$

(7)

The above expression involves the strain ratio $\beta$, $\beta ={\displaystyle \frac{{d\epsilon }_2}{{d\epsilon }_1}}$ and $r$ represents the thick anisotropic coefficient.

The thick anisotropic coefficient r is an important parameter for evaluating the compressive forming performance of sheet metal. According to experimental data, the average value of the thick anisotropic coefficients in all directions is determined to be 0.785.

4.2. Establishment of the Toughness Fracture Criterion

Current research indicates that the mechanisms of toughness fracture are classified into tensile-type toughness fracture under high stress triaxiality and shear-type toughness fracture under low stress triaxiality. Tensile-type toughness fracture is characterized by fracture surfaces covered with voids and a significant increase in void volume, while shear-type toughness fracture is characterized by little change in void volume but significant elongation of voids along the direction of the shear band. When considering both tensile-type fracture under high stress triaxiality and shear-type fracture under low stress triaxiality, the primary factor influencing tensile-type fracture is the maximum principal stress, while the main factor for shear-type fracture is the maximum shear stress. It can be determined that both types of ductile fracture can occur in a material only when the maximum tensile stress of plastic deformation σ1 is greater than 0. This paper proposes a toughness fracture criterion applicable to different deformation paths [30,31].

$$\mathrm{D}={\int }_0^{\overline{{\epsilon }_f}}({\displaystyle \frac{{\sigma }_1}{{\sigma }_1-{\sigma }_m}}+D_1{\displaystyle \frac{{\sigma }_1}{{\sigma }_1-{\sigma }_3}}{\displaystyle \frac{{\sigma }_3^{\prime }}{{\sigma }_1^{\prime }}}\left)d\overline{{\epsilon }_p}\right({\mathsf{\sigma }}_1>0)$$

(8)

In Equation (8), ${\epsilon }_f$ is the equivalent plastic strain at critical fracture. The above equation considers the maximum principal stress ${\sigma }_1$ and stress triaxiality, as well as the maximum shear stress ${\sigma }_1-{\sigma }_3$.

Since anisotropic stretching not only increases the void volume but also causes changes in the shape of the voids, a coefficient $D_1$ is introduced in the integral term of Equation (8) to account for this effect.

Since Equation (8) contains two unknown material constants, $D_1$ and D, two experimental procedures are required for numerical simulation. To accurately determine the parameter values, the displacement changes at the moment of fracture in tensile specimens and the displacement changes at the moment of fracture in shear specimens are selected as boundary conditions for the finite element simulation. Boisse et al. [32] pointed out that the failure accumulation value at the critical fracture moment is generally considered a material constant, and the critical value can be predicted by simple tensile and shear tests to estimate the critical fracture value of high-strength dual-phase steel during the forming process. Therefore, equations can be formed based on the results of these two experiments to determine $D_1$. Substituting the simulation results for tensile specimen $R_0$ and shear specimen $S_0$ into Equation (9) results in:

$$D_1={\displaystyle \frac{{\left[{\int }_0^{\overline{{\epsilon }_f}}({\sigma }_1/({\sigma }_1-{\sigma }_m\left)d\overline{{\epsilon }_p}\right)\right]}_{S_0}-{\left[{\int }_0^{\overline{{\epsilon }_f}}({\sigma }_1/({\sigma }_1-{\sigma }_m\left)d\overline{{\epsilon }_p}\right)\right]}_{R_0}}{{\left[{\int }_0^{\overline{{\epsilon }_f}}({\sigma }_1{\sigma }_3^{\prime }/({\sigma }_1^{\prime }\left({\sigma }_1-{\sigma }_3\right)\left)d\overline{{\epsilon }_p}\right)\right]}_{R_0}-{\left[{\int }_0^{\overline{{\epsilon }_f}}({\sigma }_1{\sigma }_3^{\prime }/({\sigma }_1^{\prime }\left({\sigma }_1-{\sigma }_3\right)\left)d\overline{{\epsilon }_p}\right)\right]}_{S_0}}}$$

(9)

In the above Equation, the subscript denotes the specimen name. By substituting the values of each integral term calculated from the finite element analysis into Equation (9), $D_1$ is determined to be 1.5. Substituting $D_1$ into Equation (8) yields a critical damage fracture value D of 0.164 for DP780 steel. The integral value on the left side of Equation (9), which represents the accumulated damage of material points D, is defined as the critical threshold for fracture. For DP780 dual-phase steel, when this value reaches 0.164, it is considered that the material will have reached the critical state of ductile fracture.

4.3. Study on the Fracture Characteristics of Advanced High-Strength Dual-Phase Steel Under Uniaxial Tension

4.3.1. Relationship Between Fracture Strain and Stress Triaxiality Under Uniaxial Tension

The previous Abaqus 6.14 software simulation of DP780 high-strength steel in tensile and shear tests has provided the range of stress triaxiality values. To obtain the stress triaxiality values for different states, three types of thin steel plates with varying notch radii were subjected to unidirectional tensile tests. Each type of steel plate was tested three times, and the average values of the stress triaxiality and corresponding ultimate fracture strains were calculated for each type of specimen. The experimental data are shown in

Table 6. Specimen size in test.

Notch Radius	Specimen Name	Original Width (mm)	Width After Deformation (mm)	Average Value (mm)	Original Thickness (mm)	Thickness After Deformation (mm)	Average Value (mm)
R0	D00002 D00003 D00004	12.5 12.5 12.5	8.9 8.8 9.0	8.9	1 1 1	0.7 0.66 0.6	0.65
R5	D00052 D00053 D00054	10 10 10	9.4 8.8 9	9	1 1 1	0.6 0.68 0.6	0.62
R10	D00102 D00103 D00104	10 10 10	9.2 8.9 9.2	9.1	1 1 1	0.66 0.68 0.7	0.68

The unit size of the above specimen is mm, because the first specimen is to verify whether the data of the tensile machine are standard or not, so it is omitted.

.

From the tensile tests of specimens with notches of different radii, we can conclude that the maximum tensile stretch of the specimen at the moment of fracture is 10 per cent.

Table 7. Specimen size at the moment of rupture /mm.

Notch Radius	Original Length (mm)	Length After Deformation (mm)	Elongation (-)
R0	120	132	0.1
R5	120	125.37	0.044
R10	120	126.54	0.055

shows the calculation results of triaxial stress and fracture strain. The experimental data are shown in

Table 8. The calculation results of triaxial stress and fracture strain.

Specimen Type	Fracture Strain	Stress Triaxiality
R0 test piece	0.65	0.33
R5 test piece	0.48	0.5
R10 test piece	0.55	0.48

.

Figure 12 presents a graph of the relationship between the fracture strain and stress triaxiality produced by unidirectional stretching. It is observed that as the stress triaxiality increases, the fracture strain follows a monotonically decreasing trend. In the stress triaxiality of 0.33~0.6, by analyzing the data from Table 8, it is found that it is more consistent with the Johnson–Cook criterion, to within the stress triaxiality range of 0.33 to 0.6, analysis of the data reveals a better consistency with the Johnson–Cook criterion, using MATLAB R2024a software to fit the curve and determine the parameter values of the J–C criterion, and finally fitted to the expression of the J–C curve for Equation (10).

$${\epsilon }_f=0.398+2.181\exp (-6.55{\sigma }^{\ast })$$

(10)

4.3.2. Numerical Modelling

According to the real size of the experimental specimen, a model was established using Abaqus 6.14 finite element software. The shell element was selected to ensure result accuracy while optimizing computational efficiency; therefore, the model in this study adopts shell elements, and the mesh near the gap was further refined.

The results indicate that when material necking occurs, the plastic strain and stress triaxiality in the necking region reach their maximum values, and fracture will occur first in the necking area. Taking the tensile displacement at the time of fracture of the actual experimental specimen as the boundary condition, the stress distribution corresponding to the different notch radii of the thin plate at the end of the simulation is shown in Figure 13, and it is found that the higher equivalent stresses are concentrated at the root of the notch of the specimen. Figure 14 shows the stress triaxiality distribution, revealing that the maximum value of stress triaxiality occurs at the center of the specimen, suggesting that this is the location where defects are likely to appear in the single tensile specimen, consistent with previous experimental results.

4.3.3. Comparison of Finite Element Simulation and Experimental Results

The stress triaxiality is not uniformly distributed throughout the necking cross-section, and most researchers typically select the center of the smallest diameter when fitting the failure parameter; therefore, in this paper, the stress triaxiality value at the center of the unit is selected. We extract the results of finite element calculation, Figure 15 gives the finite element calculation process of different specimens at the centre of the minimum diameter of the stress triaxiality of the change rule, Figure 15 can be seen in the whole experimental process of the stress triaxiality is not a stable value, the overall view of the upward trend. Figure 15 specific stress triaxiality is combined with the data after tensile deformation and combined with the empirical formula 6 assumes that the stress triaxiality remains unchanged throughout the deformation process, as can be seen from the results of the empirical formula used in the calculation of the results of the numerical simulation of a more consistent calculated average value.

Figure 15 shows that the stress triaxial simulation value of R0 specimen is in the range of 0.3–0.43, the average value is 0.36, the experimental calculated value is 0.33, and the error is 0.03. R5 specimen stress triaxial simulation value interval is 0.4–0.64, the average value is 0.52, the experimental calculated value is 0.49, and error 0.03. R5 specimen stress triaxial simulation value interval is 0.4–0.47, the average value is 0.435, the experimental calculated value is 0.47, and error 0.035.

This confirms that the J–C curves obtained through the experimental calculation of stress triaxiality in the range of 0.33 to 0.6 can effectively describe the material’s plastic deformation behavior. The curve represents the ultimate fracture strain corresponding to different stress triaxiality values, indicating that the material is considered to have ruptured if it exceeds this ultimate fracture strain.

4.4. Fracture Characteristics of Advanced High-Strength Duplex Steels Under Combined Tensile and Shear Deformation

4.4.1. Numerical Modelling

A review of the literature reveals that, to date, no relevant Equation can accurately characterize the ultimate fracture strain at the fracture of a shear specimen. Therefore, this paper proposes using finite element calculations to obtain the stress state at the fracture. By comparing the displacement–load curves calculated by the finite element method with those measured in actual experiments, the study ensures that the curves closely match and are consistent. By analyzing the stress triaxiality curve at the notch of the shear specimen, this analysis provides the relationship curve between stress triaxiality and ultimate fracture strain in the low-stress triaxiality state.

Figure 16 shows the experimentally measured displacement and load curves of specimens at different angles alongside the displacement and load curves at the nodes corrected by software simulation. The results are found to be relatively satisfactory, with a small error when compared to the actual values. This indicates that the established finite element model has a high degree of accuracy, as the simulation of stress triaxiality and fracture strain closely reflects the actual deformation of the material and changes in the stress field.

Figure 17 displays the deformation and stress distribution at the end of the simulation for different angles of shear specimens, and it can be observed that the maximum stress is mainly distributed in the notch. Figure 18 shows the stress triaxiality diagram, where it is observed that from the 0° specimen to the 90° specimen, the notch stress triaxiality steadily increases, indicating a transition from a pure shear state to a uniaxial tensile state. The areas with high stress are prone to fracture, indicating that the simulation results are consistent with the actual experimental fracture results shown in Figure 9.

4.4.2. Comparison of Finite Element Simulation and Experimental Results

Figure 19 shows that the stress triaxiality of the 0° specimen is 0.04, while, under ideal conditions, the stress triaxiality value should be 0 in a pure shear state. This discrepancy arises because the stress region rotates during the stretching process, making it difficult for the specimen to achieve a pure shear state. The sudden change in geometry at the notch edge often causes stress concentration, resulting in some discrepancy between the theoretical and simulated stress triaxiality values. To eliminate the bias in stress triaxiality results, the reference processing method is applied, prioritizing the theoretical value of stress triaxiality while using the simulated strain value.

4.4.3. Relationship Between Fracture Strain and Stress Triaxiality in the Shear Case

The results of shear experiment were conducted in

Table 9. The calculation results of triaxial stress and fracture strain.

Specimen Type	Fracture Strain	Stress Triaxiality	Theoretical Stress in Three Degrees
0° shear specimen	0.34	0.04	0
45° shear specimen 90° shear specimen	0.46 0.65	0.28 0.48	0.16 0.33

.

Figure 20 presents a graph of the relationship between the fracture strain and stress triaxiality produced by shearing specimens at different angles; as stress triaxiality increases, the fracture strain tends to increase as well, following a monotonically increasing function. Within the stress triaxiality range of 0 to 0.33, analysis of the measured data from Table 9 revealed that it aligns more closely with the Bao–Wierzbicki criterion. The parameter values of the B–W criterion were determined through curve fitting using MATLAB R2024a software, resulting in the following expression for the B–W curve.

$${\epsilon }_f=0.3447+1.34{\times \sigma }^2$$

(11)

4.5. Fracture Characterization of Advanced High-Strength Duplex Steels Under Biaxial Stretching

Through this research, we have derived the stress triaxiality range of 0 to 0.58 corresponding to the ultimate fracture strain, to further understand the relationship between high stress triaxiality and fracture strain. Curve fitting from single tensile and shear experiments has demonstrated that the simulation with the established criterion accurately reflects the stress field of the specimen during failure. However, due to experimental limitations, this section explores the relationship between stress triaxiality and corresponding fracture strain by establishing a biaxial tensile model for simulation.

Numerical Modelling

This study found that the literature [33] already provides a detailed exposition of the bidirectional tensile specimen. The specific dimensions of the biaxial tensile specimen in this paper refer to the model size provided in the literature [34]. The structure of the biaxial tensile specimen is shown in Figure 21a, with the center-thinning region of the thin plate having a thickness of 0.7 mm and the rest of the thickness being 2 mm. By introducing an open slit in the cross-shaped wall, the maximum deformation of the specimen is concentrated in the center region, effectively avoiding the occurrence of concentrated stress. The shear stress in the center region of the specimen is close to zero, and the axial principal stresses are uniformly distributed. Since it is a symmetric model, to save calculation time, a 1/4 model is used with axisymmetric constraints and local mesh refinement, as shown in Figure 21b.

Figure 22 shows the thickness variation cloud diagram of the biaxial model. It can be observed that thinning first occurs in the central region of the thin plate. Figure 23 shows the corresponding stress triaxiality cloud diagram of the thinning region, with a stress triaxiality value of 0.667.

Figure 24 illustrates the change in equivalent plastic strain over time at the moment of fracture. The specimen fails at 0.044 steps, at which point the equivalent plastic strain is observed to be 0.56.

Zengli Peng et al. [35] demonstrated that equivalent fracture strain is a monotonically increasing function of stress triaxiality. In this study, the stress triaxiality of the biaxial tensile specimens ranges from 0.577 to 0.667, aligning with the findings of Hoon Huh [36]. The stress triaxiality results obtained from the simulation are consistent with Hoon Huh’s findings. By combining the equivalent plastic strain at 0.577 and 0.667 with the trend of the curve in literature [18], an approximate curve fitting can be achieved, as shown in Figure 25.

An approximate curve expression is obtained by Gaussian curve fitting in MATLAB.

$$\mathrm{f}\left(\mathrm{x}\right)={1.25}^{-{\left({\displaystyle \frac{x-0.678}{0.0075}}\right)}^2}+{0.46}^{-{\left({\displaystyle \frac{x-0.667}{0.553}}\right)}^2}$$

(12)

4.6. Establishment of B–W Curves

The relationship between ultimate fracture strain and stress triaxiality for the range of 0 to 0.667 has been derived from the study in the above sections, and this relationship is plotted across the entire range of stress triaxiality, as shown in Figure 26.

As shown in the figure, the relationship between strain and stress triaxiality is not monotonic, as different ranges of fracture strain correspond to different stress triaxiality values, ranging from a minimum value of zero to a maximum value of 0.667 within the studied range. The curves obtained in this section consist of three segments that fall within different stress triaxiality ranges, where the stress triaxiality at the shear stage aligns with the Bao–Wierzbicki fracture criterion, and during the unidirectional stretching stage, it aligns with the Johnson–Cook fracture criterion. An empirical polynomial equation was derived from biaxial stretching, as shown in Equation (12).

Each point on the strain and stress triaxiality curve obtained in this thesis represents a combination of critical stress triaxiality and critical ultimate fracture strain at the moment of toughness initiation, and this curve is tentatively referred to as the toughness initiation fracture B–W curve. This curve shows that the plastic behavior of the material is affected by its own voids and indicates that there is a significant relationship between the strain state and stress triaxiality at the moment of material fracture. If the equivalent plastic strain and stress triaxiality of the material unit are above the curve, it predicts that the material is already in a failure or critical failure state. When this combination is above the curve, it indicates that the material is already in a state of failure or is at risk of critical failure, while if it is below the curve, the material is in a state of safety. Therefore, the B–W curve represents the threshold between safety and failure in the use of components made of these materials, and it can be used to predict the occurrence of rupture during the stamping process of advanced high-strength dual-phase steels. This curve can be employed for damage simulation and safety design of components, serving as a predictive tool for rupture during the stamping process of advanced high-strength dual-phase steels. The validity and applicability of the model can be verified by comparing the experimentally measured B–W curves with the predicted curves obtained from the relevant model simulations.

5. Fracture Test for Bending and Forming

In view of the high strength of DP780 steel, the YZ32-160S CNC four-column hydraulic press was selected from China Jiangsu Zhibo Hydraulic Machinery Co., Made in Nantong, China with a tonnage of 160 tons and a crimping force of 15 tons. The hydraulic press can automatically set and adjust the pressure, after positioning the sheet, increase the crimping force to press the sheet tightly, then set the speed to eject the punch upwards at a constant speed of 10 mm/s until the sheet is bent, as shown in Figure 27.

To demonstrate that the new toughness fracture criterion and the B–W fracture curve criterion can overcome the limitations of traditional FLD forming limit diagrams, which cannot accurately predict the defects in high-strength dual-phase steel fractures, and can accurately predict the forming limits of dual-phase steel, a series of tensile bending experiments were conducted to verify the fracture behavior of specimens, as shown in Figure 28, with fillet radii of 1 mm, 4 mm, 5 mm, and 15 mm.

The strokes at the moment of fracture for drawing specimens with convex die fillet radii of 15 mm, 5 mm, and 1 mm were selected, and the stroke results were used as the standard. The depth of draw at this time was compared with the stroke results to verify the applicability and accuracy of the criterion by simulating the depth of draw when the sheet material reaches the threshold value of fracture. The actual forming depth of the sheet obtained from the experiment is shown in

Table 10. Forming depth under different punch radius.

Radius of a Rounded Corner (mm)	R = 1	R = 4	R = 5	R = 15
Forming depth (mm)	6	8	10	32

.

5.1. Simulation Modelling

To verify whether the shear and tensile fracture failure criteria can accurately predict the stamping performance of dual-phase steel, corresponding simulations and experiments were conducted for comparative validation. A finite element model of a U-shaped stamping part was designed, as shown in Figure 29b. The schematic of the die required for the experiment is shown in Figure 29a, with key dimensions including the punch fillet radius $R_p$, which can be adjusted by changing the punch, and the die entry radius $R_{d1}$ = 5. Since the sheet thickness is 1 mm, the die gap $G$ was designed to be approximately 1.1 mm the sheet thickness. During stamping, the sheet dimensions were set to 250 mm 30 mm 1 mm. To improve computational efficiency, a 1/2 model was used for analysis. Due to the high rigidity of the punch, die, and blank holder, they were defined as analytical rigid bodies, while the sheet was modeled using S4R shell elements with a mesh size of $1\mathrm{m}\mathrm{m}\ast 1\mathrm{m}\mathrm{m}$. Boundary conditions were applied according to the experimental setup.

The drawing and bending model were established in Abaqus, as shown in Figure 28, with the sheet material being DP780, having dimensions of 250 mm, 30 mm, and 1 mm. The radius of the convex die fillet is variable, and the radius of the concave die fillet is 10 mm. The crimping force is 150KN, and the friction coefficient between the sheet and the mold is 0.05 [37].

5.2. Analysis of Simulation Results

Figure 30 shows that when the corner radius is 15 mm, the fracture accumulation value of the criterion reaches 0.164, which exceeds the predetermined fracture threshold, indicating that material rupture occurs at this point. Figure 31 shows that the thinning rate at this point is 3.4%, further confirming that the shear fracture does not exhibit obvious thinning and necking characteristics. It is observed that the location of the fracture aligns with the typical fracture location of dual-phase steel in the concave die with a larger corner radius. Figure 32 shows the relationship between the downward stroke of the convex die and the load force applied to the convex die when material rupture occurs. When the load force reaches its peak and begins to decline, indicating material failure, the corresponding stamping stroke is about 32 mm, which is consistent with the actual experimental data. This further demonstrates that the established toughness criterion can accurately predict the stamping and forming behavior of dual-phase steel.

5.3. Validation of B–W Curves

Several cells in region A of Figure 33 were selected, and the relationship between stress triaxiality and equivalent plastic strain of these cells was obtained through simulation. This relationship was then compared with the B–W judgment curve to evaluate the accuracy of the criterion.

Figure 33 shows the equivalent plastic strain obtained from the simulation of the specimen with a radius of 15 mm, indicating that the equivalent plastic strain at the side wall is the highest, approximately 0.55. Figure 34 illustrates the variation in stress triaxiality of the specimen, with the stress triaxiality at the side wall of the fracture being positive, approximately 0.42, which means that the fracture that occurred in this place is a tensile fracture. The unit around the fracture point in the fracture A region is selected to show the relationship between stress triaxiality and equivalent plastic strain. The points shown in Figure 35 show that when the corner radius is 15 mm, the fracture accumulation value of the criterion reaches 0.164, which exceeds the predetermined fracture threshold, indicating that material rupture occurs at this point. Figure 31 shows that the thinning rate at this point is 3.4%, further confirming that the shear fracture does not exhibit obvious thinning and necking characteristics. It is observed that the location of the fracture aligns with the typical fracture location of dual-phase steel in the concave die with a larger corner radius. Figure 32 shows the relationship between the downward stroke of the convex die and the load force applied to the convex die when material rupture occurs. When the load force reaches its peak and begins to decline, indicating material failure, the corresponding stamping stroke is about 32 mm, which is consistent with the actual experimental data. This further demonstrates that the established toughness criterion can accurately predict the stamping and forming behavior of dual-phase steel.

The combination of stress triaxiality and equivalent plastic strain at the moment of fracture is represented within this unit, with the dotted line illustrating the dynamic trajectory of stress triaxiality and equivalent plastic strain throughout the tensile bending process. These values change continuously during bending, following a complex path. The final analysis reveals that some units have already reached a failure state, aligning with the experimental fracture results in this region. This correlation confirms that the B-W curve established in this study serves as an effective criterion for predicting the fracture behavior of duplex steel.

A bending simulation example demonstrates that the newly established toughness fracture criterion is able to predict the fracture of DP780 high-strength steel, and the proposed B–W curve can be used as a criterion to accurately predict the stamping of dual-phase steels in conjunction with the established toughness fracture criterion.

6. Conclusions

In this paper, a ductile fracture criterion suitable for engineering applications is adopted, unifying the two phenomenological characteristics—tensile and shear—to predict the failure of high-strength steel plates in tensile–flexural operations. A series of tensile tests on specimens of various sizes and shear tests at different angles were conducted to comprehensively characterize the fracture behavior of DP780 steel plates. The plastic strain and stress triaxiality at fracture were measured, leading to the establishment of a criterion (B–W curve) for predicting the ultimate fracture of DP780 steel. A detailed finite element model was also developed based on the tensile bending test to validate the applicability and accuracy of the toughness fracture criterion in actual forming processes.

The main conclusions of this study are as follows:

(1)

Single tensile and shear tests were conducted using custom-designed tensile and shear specimens. Results indicated that DP780 steel exhibits no obvious yield plateau, with relatively high tensile strength and low plasticity.

(2)

Finite element simulations revealed that, in static tensile tests, the stress triaxiality of tensile specimens ranges from 0.33 to 0.53, with tensile stress playing a primary role. In shear specimens, the stress triaxiality ranges from 0 to 0.50, with shear force being the dominant factor.

(3)

The toughness fracture criterion that unifies tension and shear behaviors was selected, and the ultimate fracture threshold of DP780 was determined to be 0.164 based on unidirectional tensile and shear test data. The tensile bending test and finite element simulations showed that, upon reaching this threshold, the stamping stroke aligns with experimental observations, verifying that the established toughness criterion can accurately predict the stamping and forming behavior of dual-phase steel.

(4)

Combining experimental data and finite element simulations, it was observed that the relationship between stress triaxiality and strain at fracture in DP780 high-strength steel is a monotonically decreasing function during unidirectional stretching, closely aligning with the Johnson–Cook curve. In shear behavior, the relationship is a monotonically increasing function, resembling the Bao–Wierzbicki curve. The B–W curve, which combines ultimate fracture strain and stress triaxiality, is proposed as a predictive criterion. Through tensile bending experiments and finite element simulations, it was confirmed that, at the ultimate fracture threshold, the combination of ultimate fracture strain and stress triaxiality aligns with experimental fracture results, proving that the B–W curve established in this paper is an effective predictor of fracture in dual-phase steels.