Fracture Behavior of the Hot-Stamped PHS2000 Steel Based on GISSMO Failure Model

Abstract

Hot-stamped steel is currently the most widely used lightweight material in automobiles, and accurately predicting its failure risk during the simulation is a bottleneck problem in the automobile industry. In this study, the fracture failure behavior of the hot-stamped PHS2000 steel manufactured by Ben Gang Group (Benxi, China) is investigated by experiments and simulation. Static tension and high-speed tension tests are conducted to obtain the elastic-plastic stress-strain relations, and a Swift + Hockett–Sherby model is proposed to describe the hardening behavior under static and high-speed loads. Tests under five kinds of stress states, namely static shear, static tensile shear, notched static tension, center-hole static tension, and static punching, are conducted to obtain the ultimate fracture strains under different stress states for establishing a failure model. The finite element method (FEM) is used to inversely achieve the fracture parameters of the material, and the GISSMO model in LS-Dyna is adopted to describe the fracture characteristics of the material. A fracture card is further established for simulation analysis by combining fracture characteristics with high-speed tension curves and simultaneously loading size effect curves of meshes. Finally, the card is applied in the simulation of the three-point bending test. High-precision results of fracture simulation matching the experimental results are obtained. This research proves that the proposed fracture card is accurate and can be widely used in the simulation of fracture behaviors of the hot-stamped PHS2000 steel.

1. Introduction

Nowadays, with the rapid development of the automobile industry, a sharp increase occurs in the number of cars in China [1]. The fuel consumption and exhaust emissions generated by automobiles have become increasingly serious social problems. Therefore, the policies and regulations on fuel economy are becoming more strict [2]. The light weight of automobiles is an important way to reduce fuel consumption. For every 1% reduction in automobile weight, fuel consumption can be reduced by 0.7% [3]. However, with the increasing demand from consumers for functions such as video entertainment and intelligent driving, more and more parts have been installed on the automobile, and therefore the overall weight of the automobile has to be reduced. Thus, while ensuring strength, stiffness, and safety, the lightweight of the white body is a key way [4,5,6,7,8,9,10,11].

Hot-stamped steel is a new material with higher strength than traditional cold-rolled steel. Its tensile strength can reach 1200~2000 MPa. Compared with other lightweight materials such as aluminum alloys, hot-stamped steel materials have the advantages of controllable manufacture-use-maintenance cost, good processability, and good lightweight potential, and thereby the proportion of applications is increasing in the automobile field [12,13,14,15,16,17,18,19]. The application of hot-stamped steels as a structural part of the body in white (BIW) can reduce the weight and ensure safety performance. Therefore, hot-stamped steels have been widely used in the structural parts, such as A-pillars and B-pillars [20,21]. However, the ductility of the steel after hot stamping sharply decreases, and consequently the possibility of fracture increases [22]. It is very necessary to accurately simulate the fracture process in order to understand and predict the fracture failure behavior of the hot-stamped steel [23,24,25,26].

To obtain an accurate simulation result of the fracture behavior, two types of mechanical behaviors, namely elastic-plastic behavior and fracture failure behavior, should be considered and determined. Generally, in the previous reports, the accurate description of elastic-plastic mechanical behavior could be focused on. However, only conventional uniaxial tensile tests were used to obtain the maximum failure plastic strain as the criterion for failure behavior [27]. In fact, the failure strain of a material is closely related to its stress state (stress triaxiality) and strain rate. The failure strain obtained by only the uniaxial tensile test will misrepresent the maximum strain during metal forming and collision processes, leading to incorrect failure prediction results [28]. Similarly, directly using fracture elongation as a failure criterion can lead to overly conservative designs [29]. Therefore, it is necessary to use hardening models to extrapolate and predict the hardening characteristics of the material.

In this paper, the widely used 2000 MPa grade hot-stamped PHS2000 steel is selected as an object. Key characteristics of its hardening and failure behaviors during the hot stamping application are investigated. Based on the GISSMO model, fracture behavior is explored through a combination of experiments and simulation. A fracture card is established in LS-Dyna for the simulation analysis to improve simulation accuracy.

2. Material and Experiments

2.1. Material and Equipment

PHS2000 with a thickness of 1.4 mm was selected as the test steel. The steel plate was heated to 930 °C, held for 5 min, stamped at 680 °C, and cooled at a cooling rate of 15 °C/s. The tested specimens were manufactured from the hot-formed steel plate for tensile and fracture failure tests. The tests were conducted on the Zwick-type universal testing machine, Zwick-type high-speed tensile testing machine, low-speed/high-speed camera, and non-contact analysis system. The finite element analysis (FEA) was finished in the LS-Dyna R11 software platform. HyperMesh and LS-Present were used as the pre-processing software and HyperView was used as the post-processing software.

2.2. Uniaxial Tensile Tests under Static and Dynamic Loads

A total of 2 types of uniaxial tension tests, namely the static and dynamic tests, were performed in this paper, as shown in

Table 1. Methods of static and high-speed tensile test for the tested steel.

Test Type	Strain Rate (/s)	Load Measurement	Deformation Measurement
Static test	0.001	Universal testing machine	CCD camera
Dynamic test	0.1, 1, 10	High-speed tensile testing machine	High-speed camera
	100, 500	Strain gauge	High-speed camera

. The static tension tests were conducted at the strain rates of 0.001/s, according to the Chinese National Standard GB/T228.1-2010 “Metallic materials—Tensile testing—Part 1: Method of test at room temperature” [30]. The dynamic tension tests were conducted at 5 strain rates, namely 0.1, 1, 10, 100, and 500/s, according to the International Standard ISO 26203-2-2011 “Metallic materials—Tensile testing at high strain rates—Part 2: Servo-hydraulic and other test systems” [31]. The dimensions of the specimens for static and dynamic tests are shown in Figure 1a and Figure 1b, respectively.

Wire-cut electrical discharge machining (WEDM) was adopted to manufacture the specimens. When cutting, the temperature was controlled as much as possible to prevent affecting the material performance. The static and dynamic tensile tests were conducted on the Zwick-type universal testing machine and Zwick-type high-speed tensile testing machine, respectively. For the high-speed test, 1 end of the specimen was fixed and the other end was stretched at 5 different speeds, increasing the strain rate from 0.1/s to 500/s. It is difficult for conventional extensometers to measure the deformation process due to the fast loading speed. Therefore, a strain measurement system was combined with the digital image correlation (DIC) technology to analyze deformation. The force sensor on the testing machine was responsible for collecting load information. Before the test, speckles were sprayed on the surface of the specimen for the measurement of the strain. When the strain rate was lower than 100/s, the collection frequency was consistent with the frequency taken by DIC photos to ensure synchronization of displacement and load data. The load force was provided by the testing machine. The view field of the camera was located in the center of the tensile specimen, as shown in Figure 2. The images were collected at a sampling frequency based on the loading speed. For example, when the strain rate was 500/s, the loading speed was 5 m/s. In this situation, the sampling frequency was 500 kHz. Thus, the images and loads over time were recorded and saved. After that, the images were imported into GOM analysis software (Figure 2). A scale bar was first virtually installed to determine the proportional relationship between pixel size and actual size, and then a 10 mm virtual extensometer was established, which changed with the deformation length of the specimen. Therefore, the strain of the specimen could be calculated through the length change of the virtual extensometer.

When the strain rate exceeded 100/s, the strain gauges were used to measure the force in order to avoid the occurrence of unstable test results at a high strain rate. The strain gauge was a half-bridge type, that is, two strain gauges were pasted on the upper and lower surfaces of the lower clamping end (the longer end) of the specimen, respectively. The pasted position was located between the clamping line and the parallel section, 5 mm away from the clamping line, as shown in Figure 1b. The strain gauges were connected with the collector. In the test, the voltage signal was recorded by the collector and used to calculate the load by Formula (1). Here, E_m is the elastic modulus, S₁ is the cross-sectional area of the clamping end, U is the output voltage of the strain gauge, and K is the sensitivity coefficient of the strain gauge. The load curve obtained by the strain gauges was calibrated through the force sensor of the used tensile testing machine. Each test was repeated three times to ensure the reliability. Finally, the engineering stress-strain curve at each strain rate was obtained.

$$F=\frac{E_m\times S_1\times 4U}{k}$$

(1)

2.3. Static Tests under Different Stress States

In order to calibrate the fracture limit curve and verify the failure model, static tests under different stress states need to be performed to cover the entire plane stress triaxiality range. In this paper, five stress states were considered and thereby five types of specimens including R-notched static tension, center-hole tension, static tensile shear, static shear, and static punching, were designed and adopted to measure the fracture properties. Among them, R-notched static tension was divided into R5-notched static tension and R20-notched static tension according to different fillet radii. Thus, six static tests were conducted in total. Their corresponding specimens are shown in Figure 3. The failure positions of these tests corresponded to different stress triaxiality values and the testing results could be used to characterize the mechanical properties of the material under different stress states. In the test, the Zwick-type universal testing machine was used and the loading speed was set as 2 mm/min. For the static tests under different stress states, the size of the deformation area was very small, which made it difficult to accurately test the strain. Therefore, the DIC system was used to measure the force-displacement curve in the paper. Each test was repeated three times to ensure the reliability.

3. Static and High-Speed Tensile Properties

3.1. Tensile Mechanical Properties

Car collision is a dynamic process closely related to the dynamic deformation of materials. Therefore, it is necessary to consider the effect of the strain rate on the mechanical properties of materials. In this study, the strain rate was set to 0.001, 0.1, 1, 10, 100, and 500/s, respectively. The test at each strain rate was conducted three times to ensure reliability. The middle one of the three curves under the same strain rate conditions was further selected as the curve at that corresponding strain rate. Thus, the stress-strain curves of the PHS2000 steel under different strain rates are shown in Figure 4. It can be seen that the yield strength and tensile strength of PHS2000 after hot stamping do not significantly change with the increase in strain rate, but its total elongation significantly increases.

In the static tensile state, the total elongation of PHS2000 was about 7.22%, as shown in Figure 4. With the increase in the strain rate, the total elongation changed. At a strain rate of 0.1/s, the elongation was 8.27%. However, when the strain rate increased to 10/s, the elongation increased sharply to 11.39%, with an increased ratio of 57.76%. After that, with the further increase in the strain rate, the change in the total elongation was not very obvious. When the strain rate reached the tested maximum value of 500/s, the elongation was about 11.52%, which was only 39.26% higher than that of 0.1/s. This indicates that the total elongation of the PHS2000 steel underwent a rapid change at a strain rate of 0.1/s. The hot-formed PHS2000 steel contained full martensite microstructures with a lot of dislocations, and thereby it was hard and brittle, which contributed to a high static tensile strength and resulted in a low total elongation. When the strain rate increased, the adiabatic temperature difference phenomenon occurred in the gauge length of the PHS2000 steel. Generally, at high temperatures, due to the thermal softening effect, the strength decreases and the total elongation increases. Therefore, during the high-speed tensile deformation process, the total elongation of the PHS2000 steel suddenly increased when the tensile strain rate exceeded 0.1/s. However, the strength of the steel did not have a significant change due to the combined effect of high strain rate and temperature rise.

3.2. Extrapolation of Stress-Strain Curves

The data after necking should be deleted from the stress-strain curve because of the reduction in the cross-sectional area of the specimen after necking. In this paper, the data before necking in the true stress-strain curve were fitted together and extrapolated to the strain of 1 using a combined model of Swift and Hockett–Sherby. Swift, Hockett–Sherby and their combined model can be expressed as follows.

Swift constitutive function:

$$\sigma =C\cdot {\left({\epsilon }_{pl}+{\epsilon }_D\right)}^m$$

(2)

where, σ is the effective stress, ε_pl is the plastic strain, and ε_D is the yield strain. m means the work-hardening coefficient and c is a material constant.

Hockett–Sherby constitutive function:

$$\sigma ={\sigma }_{Sat}-\left({\sigma }_{Sat}-{\sigma }_i\right)e^{-a{\epsilon }_{pl}^p}$$

(3)

where, σ is the effective stress, σ_sat is the saturation stress, and σ_i is the initial yield stress. ε_pl is the plastic strain. a, p are material constants.

Swift + Hockett–Sherby constitutive function:

$$\sigma =\left(1-\alpha \right)\left\{C\cdot {\left({\epsilon }_{pl}+{\epsilon }_D\right)}^m\right\}+\alpha \left[{\sigma }_{Sat}-\left({\sigma }_{Sat}-{\sigma }_i\right)e^{-a{\epsilon }_{pl}^p}\right]$$

(4)

where, the definitions of the variables are the same to the Formulas (2) and (3).

The fitted and extrapolated curves using the Swift + Hockett–Sherby constitutive function are shown in Figure 5. It can be clearly seen in Figure 5 that with the increase in strain rate, the tensile strength of the PHS2000 steel shows a slowly increasing trend, which means that the material has a low strain sensitivity. Moreover, the curves at different strain rates do not intersect with each other, meeting the requirements for the engineering analysis.

3.3. Numerical Simulation and Benchmark Analysis of High-Speed Tensile Tests

The obtained high-speed tensile curves were applied in the simulation. In this study, the # Define Curve function was used in LS-Dyna to input the curve data at each dynamic strain rate as a table into the K file of the material card. # MAT_PIECEWISE LINEAR_PLASTICITY material card was selected to refer to the stress-strain table for the high-speed tension.

In order to verify the validity of the true stress-strain curve obtained by fitting, the same loading conditions were used for modeling and simulation. * BOUNDARY_PRESCRIBED_MOTION keyword was used to input the speed curve for loading. * DATABASE_CROSS_ SECTION_PLANE keyword was used to define a cross section for outputting the tensile load (black solid line in Figure 6). Two nodes at the parallel segment of the specimen were selected to define * DATABASE_HISTORY_NODE keyword (white nodes in Figure 6). The relative displacement between these two nodes was calculated to obtain the tensile displacement. After that, the simulated and experimental force-displacement curves were compared to implement benchmarking between experiments and simulations.

The material card was imported into the mathematical model of the high-speed tensile test for simulation calculation. The strain rate was set as 0.1, 10, 100, and 500/s. The comparison of the experimental and simulated engineering stress-strain curves is shown in Figure 7. It can be seen that the simulated curves coincide well with the experimental curves before the failure of the specimen, with a high simulation accuracy at each strain rate. In particular, when the strain rate is 0.1/s, as shown in Figure 7a, the average tensile strength of three experimental results is 2052.03 MPa. The simulated tensile strength is 2052 MPa, basically the same as the experimental value. The material card can meet the accuracy requirements for use. As the failure model was not defined in the card, the simulated curves deviates from the experimental curves after the fracture of the tensile specimen.

4. Material Card Establishment and Fracture Simulation

4.1. Establishment of the Fracture Card of the PHS2000 Steel

Lemaitre [32] proposed the concept of stress triaxiality to describe the stress state that affects the mechanical properties of materials. It can be expressed by the Formula (5).

$$TS=\frac{{\sigma }_1+{\sigma }_2+{\sigma }_3}{3\overline{\sigma }}$$

(5)

where $\overline{\sigma }$ is the von Mises equivalent stress. σ1, σ2, and σ3 are three principal stresses, respectively. The mechanical significance of stress triaxiality is the proportion of volume stress of the characterized section in the total stress. The larger the value of stress triaxiality, the greater the proportion of volume stress of the characterized section.

The PHS2000 material card without considering fracture was used in the simulations of the R5-notched static tension, R20-notched static tension, central-hole tension, static tensile shear, static shear, and static punching tests, respectively. The stress triaxiality, Lode angle, and effective plastic strain of the failure section at the failure time point were calculated through the FEM calculation. The fracture characteristics were inversely calibrated by combining the experimental and simulated results. Thus, the fracture characteristic curve was obtained, as shown in Figure 8. It can be seen that when the stress triaxiality of the failure section increases, the failure strain shows an increasing-decreasing-increasing trend, indicating that the PHS2000 steel exhibits different mechanical properties under different stress states.

The obtained curve was input into the K file of the material card and # MAT_ADD_EROSION material card was enabled to describe the material fracture characteristic curves. Thus, a material card considering the fracture failure for collision simulation analysis was produced.

4.2. Verification and Optimization of the Fracture Card

The PHS2000 material card with fracture characteristics was adopted in the simulations of the above tests at different stress states. The simulated stress-strain curves with failure could be obtained. Considering the error of the FEM calculation, it is necessary to optimize the fracture characteristic curve. Through a closed-loop procedure of comparing the calculated curve with the actual curve, modifying the characteristic curve, and re-importing it into the simulation, the optimized fracture characteristic curve was achieved when the calculated and experimental curves were in good agreement.

A total of 6 static tests under different stress states, namely R5-notched static tension, R20-notched static tension, center-hole tension, static tensile shear, static shear, and static punching, were simulated. The experimental and simulated results of the PHS2000 steel are shown in Figure 9a–f. In Figure 9, the red curves represent the simulated results and the green curves represent the experimental results. The simulation curve matches the experimental curve well in each test. The simulated and experimental results, such as failure time and failure position, are also close, which proves that the results are reliable.

4.3. Optimization of the Mesh Coefficient

To improve the simulation accuracy of fracture characteristics of the material, a mesh size of 0.5 mm was adopted in the deformed zone for numerical simulation. The main application scenario of the material failure card is collision simulation. In the mainstream car manufacturer as the target user, a mesh size of 5~10 mm is used for collision simulation. Therefore, errors may be caused due to differences in mesh size. In order to avoid these unnecessary errors and improve the simulation accuracy, it is necessary to introduce the fracture mesh coefficient (LCREGD).

Firstly, the nominal mesh sizes were set as 0.5 mm, 1.0 mm, 2.5 mm, and 5 mm, respectively, for the static tensile simulation. The effective strains at fracture were extracted under the conditions of these four mesh sizes, as shown in Figure 10 and then imported as the mesh coefficient LCREGD into the material card. After that, the material card was used in the static tensile simulation for benchmarking analysis. Considering the mesh coefficient, the overlap degree between the simulated and experimental curves could be observed. By the optimization of the mesh coefficient, the accurate simulated results were achieved, which eliminated the errors caused by mesh size differences, and improved the matching degree with the mesh size used in collision analysis by the mainstream car manufacturer. Therefore, the material card in this paper has a high practical engineering value.

4.4. Application of the Fracture Card of the PHS2000 Steel

To further verify the accuracy of the fracture card of the PHS2000 material on the hardening and failure behaviors, the 3-point bending test of an hot-stamped PHS2000 B-pillar part was designed. The B-pillar part was in good condition before the test, and there were no macroscopic defects on the surface, meeting the test requirements. The test temperature was an ambient temperature of 22.9 °C. The humidity was 49% R.H. In the 3-point bending test, the B-pillar part was connected with the mounting plates through bolts and fixed on the testing platform, with full restraint at the upper and lower ends, as shown in Figure 11a. A pressure head with a bend diameter of 17 mm was used to perform quasi-static loading on Position 1, as shown in Figure 11b. The loading direction was the vertical compression direction from the outside of the vehicle body to the inside of the passenger compartment. The loading speed was 10 mm/min and the final bending stroke was 130 mm. The bent B-pillar part is shown in Figure 11c. It can be seen that the curved degree of the entire part is greatly reduced, and the bending effect is very obvious. The surrounding area of Position 1 is enlarged, as shown in Figure 11d. The large strain and the bending wrinkle are mainly distributed near Position 1. Local cracking can be found at the bending areas of the rounded corners of the B-pillar part in Figure 11d. The crack extended from the rounded corners to the top surface of the B-pillar part, finally penetrating into a connected crack on the top surface and sidewall of the B-pillar part. After the completion of the 3-point bending test, the peak loading force was 25.54 kN, and the loading displacement at the time of failure was 52.57 mm.

The PHS2000 material card with fracture characteristics was adopted in the simulations of the 3-point bending test of the PHS2000 steel. Based on the experimental condition, a three-point bending simulation model for the B-pillar was established. The translational degrees of freedom of the clamping positions of the fixture was constrained in the x, y, z directions. A cylindrical rigid wall was used to simulate the pressure head. In the simulation, the fracture card with the GISSMO failure criterion was loaded to calculate the three-point bending failure diagram of the B-pillar. The simulated results are shown in Figure 12. Figure 12a shows the displacement distribution at the time of failure. It is obvious that the maximum displacement is concentrated near the pressure head and it is 54.09 mm. The displacement value gradually decreases towards both ends of the B-pillar part. The distribution of effective plastic strain corresponding to the peak loading force is shown in Figure 12b. The nucleation of cracks in the simulation also existed near the rounded corners and then extended to become a connected crack on the top surface and sidewall of the B-pillar part. The simulated peak loading force was 25.02 kN, and the simulated loading displacement at the time of failure was 54.09 mm. The maximum effective plastic strain was about 18%. Comparing Figure 12b and Figure 11d, it can be observed that the simulated result matches the experimental result very well. The relative error (RE) of the loading displacement at the time of failure is only 2.89%. The RE of peak loading force is only about 2.03%. It proves that the PHS2000 fracture card can accurately predict the failure status of the material.

The experimental and simulated force-displacement curves of the 3-point bending test are shown in Figure 13. In Figure 13, the consistency between the experiment and simulation is excellent. The bending test starts when the pressure head contacts the B-pillar part. With the increase in displacement, the force increases. The fracture occurs when the loading force reaches the maximum value. After that, the loading forces decreases until the designed loading stroke is reached. The experimental and simulated force-displacement curves have the same changing trend. Especially before the fracture, the experimental and simulated curves almost coincide. After the fracture, the force becomes unstable and the the simulation curve deviates slightly from the experimental curve, which is caused by the generated fracture. However, overall, the changing trend of simulated force is consistent with the experimental one.

In general, the proposed fracture failure model based on various static and dynamic mechanical tests in this paper can better describe the fracture failure behavior of the hot-stamped PHS2000 steel, which improves the accuracy of the simulation calculation. The fracture failure of the PHS2000 steel can be accurately predicted. The research method established in this paper for predicting the fracture failure behavior is meaningful and transferable. This method can be promoted and used in other metal materials. It is of great significance for the design, development, and optimization of hot-stamped parts.

5. Conclusions

In this paper, the experiments and the FEM simulation were used to investigate the fracture behavior of the PHS2000 steel. The high-speed tensile and fracture properties were obtained. The material card for the simulation analysis was established and verified in LS-Dyna. The high-precision simulation results were obtained. The conclusions are presented as follows.

(i) The sensitivity of strength to strain rate of the PHS2000 steel was extremely low. The static tensile strength of the PHS2000 steel could reach 2044 MPa, and it was 2057 MPa, hardly changed at a strain rate of 500/s. By contrast, the total elongation of the PHS2000 steel was highly sensitive to the strain rate. The total elongation had a sudden rise at the strain rates of 0.1 and 100/s.

(ii) The tensile curves of the PHS2000 steel at different strain rates were obtained by static and dynamic tensile tests. A combined Swift + Hockett–Sherby model was used to extrapolate the fitted curve to a strain of 1 to obtain the high-speed stress-strain relations for simulation analysis.

(iii) The mechanical properties of the PHS2000 steel under different stress triaxiality loads were measured through different tests. A material fracture card based on the GISSMO failure model was further established.

(iv) The proposed PHS2000 fracture card was applied in the simulation of the 3-point bending tests. The simulated results were in good agreement with the experimental results, proving that the accuracy of the fracture card was high and it could meet the requirements of the FEM collision analysis.