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The article delves into the critical challenge of improving the formability of advanced high-strength steels (AHSS) in automotive manufacturing, focusing on the hole expansion (HE) test. It highlights the limitations of traditional formability measurements and the significance of the HE ratio in evaluating the stretch-flangeability of AHSS. The study investigates the strain rate effects on two grades of AHSS, 980DP and 980CH, using both experimental and finite element (FE) simulation methods. Key findings include the influence of strain rate on the HE ratio, the importance of material anisotropy, and the role of fracture strain in determining edge formability. The research provides a comprehensive analysis of the deformation states at the hole edge, offering valuable insights into the behavior of complex-phased steels under varying deformation rates. The article also discusses the microstructural factors that contribute to the observed strain rate dependency, making it a crucial read for those interested in the advanced materials used in modern automotive engineering.
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Abstract
This study aims to investigate the influence and mechanism of strain rate variation on the hole expansion ratio of complex phase advanced high-strength steels. Two different speeds of hole expansion tests were conducted within the quasi-static range, accompanied by a uniaxial tension test with a deformation mode similar to hole expansion for supplementary analysis. Finite element simulation was utilized to analyze the detailed deformation behavior within the material, particularly at the hole edge where cracks occur during hole expansion. The mechanical and fracture properties obtained from the uniaxial tension test were incorporated into the simulation, taking into account the anisotropy of the material to predict the precise location of crack initiation within the hole edge. To account for the strain rate effect on the experimentally determined hole expansion ratio, a ductile fracture model was introduced and its necessity was validated by considering the occurrence of material fracture before and after crack initiation. By utilizing 3D solid elements, considering material anisotropy, and applying the ductile fracture model, the simulation provided reasonable predictions for the hole expansion ratio, which exhibited variation with strain rate.
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1 Introduction
The automotive industry is currently facing significant pressure to reduce exhaust emissions from vehicles due to the global movement toward carbon neutrality. To address this challenge, automotive manufacturers are focusing on developing lightweight structures using advanced high-strength steel (AHSS) sheets. The main objective of this development is to achieve weight reduction in vehicles, which not only addresses environmental concerns but also offers additional benefits such as improved fuel efficiency, enhanced strength and safety, and increased durability.
However, compared to conventional low-strength grade sheet steels, the formability of high-strength steels is generally insufficient. This poses a major challenge in the application of AHSS, particularly in the manufacturing of complex-shaped automotive body components. Among various formability indices, the stretch-flangeability of sheet materials, which measures their ability to resist edge fracture during complex shape forming, is notably poor in AHSS. Traditional formability measurements based on uniaxial tensile testing and forming limit diagrams do not accurately predict edge fracture (Ref 1). To evaluate the formability of sheared edges, the hole expansion (HE) test according to ISO 16630 standard using a conical punch is widely accepted. The HE ratio, which is the ratio of the final hole diameter to the initial diameter during metal forming, is gaining significant attention as a crucial parameter in the context of formability of AHSS, particularly for automotive components that require high strength and complex shapes, such as burring or flanging forming.
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In general, the deformation state at the edge of a hole is almost identical to the stress state of uniaxial tension (UT) (Ref 2). Therefore, previous research results on the HE have extensively discussed the correlation between the HE and mechanical properties of UT. Paul et al. (Ref 3) conducted to investigate the correlation between the HE ratio and tensile properties. They proposed a methodology to predict the HE ratio using the tensile properties of different steel grades. The HE ratio is correlated with the tensile strength of the steel, with a linear decrease in the HE ratio as tensile strength increases below 700 MPa (Ref 4). Paul (Ref 5) discovered a nonlinear relationship between the HE ratio and tensile properties such as yield stress, ultimate tensile stress, coefficient of normal anisotropy, total elongation, and post-uniform elongation. Yoon et al. (Ref 6) investigated the factors influencing the HE ratio in steel sheets and found that the intrinsic HE ratio is proportional to the strain rate sensitivity exponent and post-uniform elongation. Kim et al. (Ref 7) suggested a new approach to predict the HE ratio using total elongation and average normal anisotropy.
In the HE test, deformation occurs throughout the entire hole edge in the specimen, and cracks tend to occur in the direction with the maximum thinning of the thickness. Due to the anisotropic nature of steel sheets, which have different material properties in different directions, HE results in varying degrees of thinning at the circular edges depending on the direction. To accurately simulate this phenomenon using the finite element (FE) simulation, it is crucial to use an anisotropic yield function that provides a proper description of plastic anisotropy. Kuwabara et al. (Ref 8) conducted HE tests on 780 MPa grade dual-phase (DP) steel and used the Yld2000-2d non-quadratic yield function (Ref 9) in their FE analysis, resulting in predictions that closely matched experimental results. Furthermore, various previous research studies conducted the FE simulations using the Yld2000-2d model to incorporate the anisotropy of materials and simulate HE (Ref 10-14).
The HE experiments also aim to observe the occurrence of cracks that penetrate the entire thickness of the hole edge; therefore, it is considered important to analyze the strain distribution within the thickness direction of the material. As a result, many studies conducted using 3D solid elements instead of 2D shell elements in the FE simulation of HE to account for the through-thickness stress state around the hole (Ref 15-20). In addition, Kuwabara et al. (Ref 21) mentioned that the use of 3D elements could also be effective in reproducing the nonuniform contact between the tool and blank resulting from the material anisotropy in the HE experiments. Introducing 3D elements inevitably increases the computational time of the FE simulation, and to save the cost, the use of the relatively simple Hill '48 quadratic yield function (Ref 22) can be an alternative. In the studies by Chung et al. (Ref 23) and Yoon et al. (Ref 18), the anisotropic yield function Hill ‘48 incorporated into the FE analysis was used to successfully predict the HE ratio values in experiments for various steel grades.
In previous research studies, the HE test has typically been conducted at a fixed quasi-static rate of punch velocity. Paul (Ref 24) suggested the need to consider the influence of deformation rate in the HE tests, as it has not been sufficiently addressed, indicating open areas for future research in this field. A recent publication by Fietek et al. (Ref 25) observed that the HE ratio is rate-dependent for stainless steel and aluminum alloy. The punch force–displacement (F–D) results exhibited rate dependence, with increase in slope and decrease in maximum force and displacement to failure as punch velocity increased.
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In this study, the effect of strain rate on the edge formability during HE deformation for two different grades of AHSS, namely 980DP and 980CH (complex phase with higher formability), was investigated. HE tests were conducted under two different punch velocity conditions within the quasi-static speed range commonly employed for material certification in the automotive industry to compare and explore the detailed deformation states of the edge where cracks occur. Experiments and FE simulations were performed to validate their consistency. The material properties, including the necessary material anisotropy for FE analysis, were obtained from UT test, considering the close relationship between HE and UT. To simulate the detailed deformation state within the hole edge, 3D solid elements were introduced, and the computational time of the FE simulation was reduced by implementing the Hill '48 quadratic yield function to represent material anisotropy. A ductile fracture model was introduced to simulate the occurrence of cracks in HE, and the results comparing the presence and absence of the model were compared to demonstrate its necessity.
2 Experiments
2.1 Materials
In this study, two steel materials, 980CH and 980DP, produced by POSCO, were used. The thickness of each material is 1.5 and 1.2 mm, respectively. 980CH consists of ferrite and martensite as the main phases, with small fractions of bainite and retained austenite included to improve formability. 980DP also has ferrite and martensite as the main phases, with a small amount of bainite included.
2.2 Uniaxial Tension
The mechanical properties of 980CH and 980DP obtained through UT according to the ASTM E8 standard are listed in Table 1. Each experiment was performed five times, and the average values are given. The three different directions are also referred to as RD, DD, and TD, corresponding to the angular values of 0°, 45°, and 90° with respect to the rolling direction, respectively. The r value is defined as \(r=d{\varepsilon }_{w}^{p}/d{\varepsilon }_{t}^{p}\), where ‘w’ and ‘t’ denote the width and thickness direction and it was measured at 5% engineering strain. Two different testing speeds were conducted, with a slow speed of 36 mm/min and a fast speed of 180 mm/min, which correspond to strain rates of 0.01/s and 0.05/s, respectively. These two speeds were chosen to correlate the strain rate of the HE experiments, which will be discussed in detail in the following Section 2.3. The engineering stress–strain curves for all cases measured in the experiments are plotted in Fig. 1. Both materials exhibit anisotropic behavior, and although the differences in mechanical properties are not significant, they do show variations with the deformation rate. It can be observed that as the testing speed increases, there is a slight increase in stress level, and the total elongation varies, either increasing or decreasing depending on the case.
Table 1
Summary of mechanical properties in uniaxial tension
Material
Tensile speed, mm/min
Direction, ˚
Young’s modulus, GPa
Ultimate tensile strength, MPa
0.2 yield stress, MPa
Uniform elongation, %
Total elongation, %
r value
n value
980CH
1.5t
36
0 (RD)
200
1107
937
8.46
13.42
0.702
0.10
45 (DD)
199
1093
925
8.55
13.11
0.905
0.11
90 (TD)
210
1123
951
8.01
11.80
0.756
0.10
180
0 (RD)
207
1114
929
8.56
13.26
0.697
0.11
45 (DD)
203
1102
923
8.67
12.62
0.875
0.11
90 (TD)
213
1126
948
7.47
10.35
0.752
0.10
980DP
1.2t
36
0 (RD)
198
1052
735
7.18
12.71
0.610
0.07
45 (DD)
188
1043
709
7.20
13.28
0.923
0.07
90 (TD)
201
1082
723
6.90
12.64
0.793
0.06
180
0 (RD)
198
1059
741
7.01
11.60
0.598
0.07
45 (DD)
196
1052
720
7.08
13.57
0.919
0.07
90 (TD)
202
1091
731
6.66
12.09
0.789
0.06
Fig. 1
Uniaxial engineering stress–strain curves of (a) 980CH and (b) 980DP in three directions at different tested velocities
After conducting UT test, the cross-sectional area of the fractured surface was measured using an optical microscope to determine the local fracture strain of the specimen. There are several methods to measure the fracture strain using the UT fractured surface area, but in this study, the outer outline of the fractured surface was traced using software to create a polygon shape, and its area was measured to obtain the closest approximation to the actual cross-sectional area (Ref 26). The true fracture strain (TFS) is defined by the following equation:
where \({A}_{o}\) and \({A}_{f}\) are the specimen cross-sectional area before testing and after fracture, respectively (Ref 27). Optical measurements of the cross-sectional area were performed for all five fractured surfaces from repeated UT tests in each case and the selected representative images for each case are shown in Fig. 2. The results indicate that as the tensile speed increases for both materials, the cross-sectional area of the fractured surface decreases, indicating an increase in fracture strain. The calculated values of the TFS for all cases are summarized in Table 2.
Fig. 2
Area of cross section of tensile fractured surface at different tested velocities
Changes in cross-sectional area before and after failure and true fracture strain in uniaxial tension
Material
Tensile speed, mm/min
\({A}_{o}\), \({\text{mm}}^{2}\)
\({A}_{f}\), \({\text{mm}}^{2}\)
True fracture strain
980CH
1.5t
36
18.75
9.10
0.72
180
8.66
0.77
980DP
1.2t
36
15.00
5.72
0.96
180
5.60
0.99
2.3 Hole Expansion
The experiments were conducted in accordance with ISO 16630 standards, using a conical punch with a 60° angle. The specimens were square blanks measuring 90 mm, and the initial hole size at the center of the specimen was milled to 10 mm to eliminate any work hardening at the hole edge and to create a consistent initial edge condition for all cases. This is because considering the additional variable of edge quality, which varies depending on the material, during the hole punching process would go beyond the scope of this study. A blank holding force of 200 kN was applied to minimize the inflow of material at the flange during the HE. As shown in Fig. 3, an optical measurement system was set up on top of the testing machine, allowing real-time analysis of the deformation of the hole. This system enables accurate measurement of the crack initiation point and the corresponding HE ratio. HE testing is known for its inherent uncertainty and variation, with results that can be heavily influenced by human subjectivity (Ref 28). The traditional testing method involves manually stopping the experiment and extracting the specimen to measure the hole size and calculate the HE ratio. Therefore, it leaves room for variation in results depending on the operator. To address this issue, an optical measurement method has been introduced, where software is used to determine the crack initiation point, always ensuring consistent experimental results.
Fig. 3
Equipment used for hole expansion testing with optical measurement system
Similar to the UT test, the HE experiments were repeated five times for each condition, and also conducted with two different deformation rates: a slow speed of 0.2 mm/s and a fast speed of 1.0 mm/s. This choice was based on the evaluation methods of the HE ratio used by several steel manufacturers and automotive OEMs, which typically consider a range of 0.2 mm/s to 1.0 mm/s as the benchmark. Through a pre-study using FE simulation as shown in Fig. 7, it was determined that these two HE punch speeds correspond roughly to equivalent plastic strain rates of 0.01/s and 0.05/s, respectively, based on the highest peak during deformation. The equivalent plastic strain was measured based on the element that experienced the most deformation across the entire hole edge. This will be further discussed in Section 4.1 where the results of the FE simulation are presented. In other words, in order to ensure a fair and meaningful comparison, the UT test speed was carefully adjusted to match the strain rate level of the HE experiments. Figure 4 illustrates the HE ratio results for two different deformation speeds of 980CH and 980DP. HE ratio can define as follows:
where \({d}_{f}\) and \({d}_{o}\) are the final and initial diameter of the central hole. The experiments were repeated five times for each case, and the average values and standard deviations were plotted in the figure. It can be observed that for both materials, the HE ratio increases with increase in deformation speed.
Fig. 4
Experimental results diagram of hole expansion at different tested velocities
To accurately analyze the UT and HE conducted in the experiments, the FE simulations were performed. The commercial software ABAQUS/Explicit 2022 was utilized, and 8-node linear hexahedral solid elements (C3D8) were employed. In the FE analysis, the results are generally dependent on the mesh size. To reduce this dependency and ensure consistency between the results of UT and HE, the size of the FE elements of both cases in the region of concentrated deformation was set to 0.5 × 0.5 × thickness/5 (mm3). While the UT simulation was modeled using a full model, the HE simulation utilized axisymmetric model, employing a 1/4 quarter model to reduce computational time. The schematic diagram of the HE test tool and the blank mesh is illustrated in Fig. 5. The shape of the punch and die, and other setup details from the actual HE experiments were modeled accordingly. The flow stress of the material was determined by fitting the stress–strain curve from UT tests using the Swift–Voce hardening law. Instead of using a rate-dependent constitutive equation, the strain rate for both UT and HE tests was matched, allowing for the application of UT properties corresponding to each respective rate. The optimal parameters of the hardening model are summarized in Table 3, and the results of flow curve fitting are plotted in Fig. 6(a). It is noted that the flow curves of 980CH in Fig. 1(a) show obvious strain rate dependency, and the large post-uniform elongation in 980DP also indicates the importance of strain rate-dependent hardening model. However, for the simplicity of modeling, the strain rate-independent hardening law was used, but the hardening parameters were calibrated for different strain rates, which correspond to the HE tests.
Fig. 5
Schematic view of the finite element model of hole expansion
The deformation mode away from the hole edge is not perfectly uniaxial and it becomes biaxial stress states with superimposed bending as material position is far from the edge. Therefore, the hardening behavior referenced by the uniaxial tensile test may not represent the deformation mode near hole edge. In this study, to efficiently model the plastic deformation in the HE test an anisotropic yield function, Hill’ 48 model was employed. The yield function is identified by the uniaxial tensile stress as a reference state, while other deformation modes away from uniaxial tensile state can be estimated by the r values along different sheet orientations. More precise deformation under multiaxial stress state can be better captured by advanced yield functions such as Yld2000-2d etc.
The quadratic yield function proposed by Hill (Ref 22) was utilized to capture the anisotropy of the material.
where \({\sigma }_{ij}\) is the stress components in the sheet plane and \(\bar{\sigma }\) is the equivalent stress. Six anisotropic coefficients F, G, H, L, M, and N are defined as follows:
where \({R}_{ij}\) is specified material parameters for using Hill '48 yield function in ABAQUS software, which can be calculated from r value obtained from UT test (Ref 29).
where \({r}_{0}\), \({r}_{45}\), and \({r}_{90}\) are the r values under UT test along RD, DD, and TD, respectively. According to Eq 4 and 5, the parameter values for the Hill '48 yield function are listed in Table 4, and the corresponding yield surfaces are plotted in Fig. 6(b). Note that the stress coordinates on the yield surfaces were normalized by the uniaxial tension along the RD.
Table 4
Anisotropic coefficients for Hill '48 yield function
Material
Tensile speed, mm/min
F
G
H
L
M
N
980CH
1.5t
36
0.546
0.588
0.412
1.500
1.500
1.591
180
0.547
0.590
0.410
1.562
980DP
1.2t
36
0.478
0.621
0.379
1.565
180
0.473
0.625
0.375
1.559
When solving the FE problems of UT and HE, a ductile damage model composed of the damage initiation criterion and damage evolution law provided by ABAQUS was introduced to simulate the ultimate failure of the material. Damage initiation occurs when the plastic strain reaches a threshold, and the corresponding condition is given by Eq 6, in which \({\overline{\varepsilon }}^{pl}\) is the equivalent plastic strain, \({\overline{\varepsilon }}_{D}^{pl}\) is the equivalent plastic strain at the onset of damage and \({\omega }_{D}\) is the damage indicator, which is a state variable that increases monotonically with plastic deformation.
Once the damage initiation criterion is met, the material undergoes softening and experiences a degradation in stiffness. To accurately model this behavior, a damage evolution law is necessary. The degradation of material stiffness is represented by a scalar damage variable, denoted as D. The stress in the material is then calculated using the following Eq 7:
$$\sigma ^{*} = (1 - D)\,\bar{\sigma }$$
(7)
where \({\sigma }^{*}\) is the damaged true stress and \(\bar{\sigma }\) is the undamaged true stress. As the damage variable D increases, the stress decreases due to the effect of material softening. Eventually, when the D value of the element reaches 1, the element is deleted, effectively simulating material fracture. Assuming a linear degradation behavior for stiffness, the damage variable is defined by the following Eq 8:
where \({L}^{e}\) is the characteristic length of the element and \({\overline{u} }_{f}^{pl}\) is the effective plastic displacement.
Equations 6 to 8 can be summarized as follows: in order to utilize the ductile damage model, two material constants, namely \({\overline{\varepsilon }}_{D}^{pl}\) and \({\overline{u} }_{f}^{pl}\), need to be determined. \({\overline{\varepsilon }}_{D}^{pl}\) is typically assumed to be equal to the true fracture strain measured in experimental tests (Ref 30); therefore, for each case, the experimental data for TFS obtained from a Table 2 were employed. On the other hand, \({\overline{u} }_{f}^{pl}\) serves as a parameter that regulates the rate of stiffness degradation according to Eq 8. A higher value of \({\overline{u} }_{f}^{pl}\) can effectively delay the occurrence of fracture. To determine the appropriate value of \({\overline{u} }_{f}^{pl}\), an inverse method was used to ensure a good match between the predicted fracture point in the UT simulation and the corresponding experimental data. To reduce complexity, \({\overline{u} }_{f}^{pl}\) is considered to be an intrinsic material constant that is independent of the deformation rate. For the materials 980CH and 980DP, 0.1 and 10-5 are, respectively, applied to the constant \({\overline{u} }_{f}^{pl}\) in this study.
4 Results
In this section, the analysis will compare experimental and FE simulation results to examine the material behavior and formability at the hole edge, specifically focusing on the influence of strain rate. The content will be divided into two parts. Section 4.1 will analyze the changes in material behavior due to strain rate before crack initiation in the HE, without considering material fracture. This will involve discussing the differences in stress–strain curves, r values, and friction coefficients before material fracture and their potential impact on edge formability in the HE. Section 4.2 will consider the effect of strain rate variations on the fracture mechanism, taking into account material fracture. It will describe how these variations ultimately affect the HE ratio.
4.1 Without Fracture Consideration
The strain rate changes of the material were examined according to the HE test speed described in Section 2.3. As shown in Fig. 7, the plastic deformation and its rate were plotted against the punch stroke for 980CH. At punch speeds of 0.2 and 1.0 mm/s, the maximum strain rates are approximately 0.01/s and 0.05/s, respectively. Therefore, the UT and HE speeds were set to have the same strain rate in this study.
Fig. 7
Predicted equivalent plastic strain and its strain rate evolution for the element experiencing maximum deformation at the hole edge during hole expansion of 980CH
In Fig. 8, the F–D behavior of the HE punch is compared between experimental results and FE simulations. The results matched well with the experimental data, except for the post-crack force drop behavior. Additionally, there was minimal difference in the F–D behavior based on the deformation speed. Different friction coefficients were applied in the FE simulation based on the punch velocity. Increasing the friction coefficient from 0.08 to 0.12 and 0.15 for 980CH at 1.0 mm/s resulted in a higher F–D curve due to increased frictional forces, as shown in Fig. 9. The experimental results matched best when the friction coefficient was set to 0.08. Typically, as the deformation speed increases, the friction coefficient decreases (Ref 31). Therefore, a friction coefficient of 0.1 was applied at 0.2 mm/s, and 0.08 at 1.0 mm/s for all cases. To investigate the impact of friction coefficient variations on the material deformation behavior in the FE simulations, Fig. 10 shows the changes in equivalent plastic strain and stress triaxiality at the hole edge for 980CH at 1.0 mm/s. The results of elements at the hole edge in RD were analyzed, and the deformation distribution in the thickness direction was examined by dividing it into three layers (BOT/MID/TOP). Based on Fig. 10(b), it can be observed that, except for the early stage before a punch displacement of 10 mm, the entire region of the hole edge exhibits a deformation mode similar to UT (stress triaxiality = 0.33). This finding provides evidence that using material properties obtained from UT to simulate the HE is valid. The results also note that there was minimal variation in the deformation amount or state at the hole edge due to changes in the friction coefficient. As the deformation rate increases, the friction coefficient decreases, resulting in an increase in the frictional force applied to the material. However, since this effect has little impact on the hole edge in the HE test, it is determined that there will be no change in the HE ratio due to variations in the friction coefficient. A similar finding was also observed in a study by Lee et al. (Ref 12), where the friction coefficient between the punch and blank during the HE did not significantly affect the overall trend of thinning prediction. This can be attributed to the localized contact between the punch and blank occurring only around narrow regions, which had minimal influence on the deformation around the hole edge.
Fig. 8
Comparison of predicted and experimental data for punch force–displacement without consideration of fracture model
Comparison of changes in punch force–displacement behavior and experimental values with variations in friction coefficient in the case of 980CH at 1.0 mm/s
Prediction of changes in (a) plastic strain and (b) stress triaxiality behavior in the hole edge with variations in friction coefficient in the case of 980CH at 1.0 mm/s
Figure 11 shows the changes in plastic deformation and stress state for all cases based on punch displacement, considering the previously discussed variation in friction coefficient. Despite reflecting different material properties depending on the deformation rate, no significant differences were observed. As seen in Fig. 1, even with a fivefold difference in strain rate, there was no dramatic difference in material properties from the UT. This explains why there was no significant difference in the F–D behavior based on the deformation speed in the HE test.
Fig. 11
Prediction of changes in plastic strain of (a) 980CH and (b) 980DP, in stress triaxiality of (c) 980CH and (d) 980DP in the hole edge with variations in hole expansion punch velocity
To investigate the difference in hole edge thinning according to strain rate, the thickness of the hole edge at the 21-mm punch stroke just before crack initiation was analyzed, as shown in Fig. 12. In the experiment, three locations (0°, 45°, and 90°) were cut from the specimen, and the edge thickness was measured using an optical microscope. In the FE simulation, thickness variation data at the same locations were continuously extracted from 0° to 90°. Figure 13 shows that both experimental and simulation results exhibit high levels of thinning at 0° and 90°, while thinning is lower at 45°. This can be attributed to the inherent anisotropy of the material. As shown in Table 1, the r value is higher in the 45° direction compared to the 0° and 90° directions in all cases. According to the definition of the r value, a material with a higher r value has greater resistance to thinning, so it is reasonable to conclude that edge thinning is lowest in the 45° direction. The consistent correlation between r value and edge thickness in HE has been observed in the study by Narayanan et al. (Ref 20) and confirmed for 1180DP and 800CP steel sheets. The difference in thinning values between experimental and FE results indicates the limitations of the Hill ‘48 model. The objective of this study was not to accurately predict thinning at the hole edge, but to observe the differences in thinning based on strain rate. Therefore, confirming that the thinning difference due to strain rate variation is not evident in all directions within the same material is a significant achievement.
Fig. 12
Analysis of hole edge thickness in the hole expansion specimen at the point just prior to fracture (punch stroke = 21 mm)
Comparison of predicted and experimental data of hole edge thickness in the hole expansion specimen at the point just prior to fracture (punch stroke = 21 mm) for different orientations
Ishiwatari et al. (Ref 1) published research showing that the circumferential major strain gradient varies in the radial direction depending on the initial hole size at the same HE speed. They found that as the hole size decreases, the strain gradient increases, suppressing strain localization and delaying necking, which increases the HE ratio. If the change in strain rate due to HE testing speed causes a similar change in the strain gradient in the radial direction, it could explain the differences in the HE ratio. To verify this, we analyzed the strain gradient according to the HE deformation rate, as shown in Fig. 14. In RD, data for the distance from the hole center and the equivalent plastic strain were extracted from the TOP layer of the specimen just before fracture. The extraction points were selected as 20 nodes in the radial direction, starting from the hole edge and moving outward in sequence. However, in the FE simulation results in Fig. 14, no significant difference in the strain gradient according to the deformation rate was observed, although there may be slight differences depending on the material.
Fig. 14
Prediction of the behavior in plastic strain with distance from the hole center and distribution of plastic strain within the specimen at the point just prior to fracture (punch stroke = 21 mm)
Due to the absence of fracture behavior analysis in the previous section, it was not possible to identify any significant strain rate dependency that could cause variations in the HE ratio. Therefore, it is crucial to consider this aspect for further analysis. According to Hance (Ref 28), the fracture strain in UT is proportional to the HE ratio. Additionally, several studies have reported that the fracture strain in UT increases with increase in strain rate within the target strain rate range of this study (Ref 32, 33). These findings suggest that as the strain rate increases, the fracture strain in UT also increases, leading to an increase in the HE ratio. Therefore, analyzing fracture strain is key to establishing a connection between strain rate and the HE ratio.
The results of simulating the UT test using a ductile fracture model in FE analysis were compared with experimental results and are presented in Fig. 15. The tensile direction is RD, and ∆L represents the measured material deformation based on a gauge length of 50 mm for both the experiments and FE simulation results. In all cases, the results are in good agreement, particularly in terms of the fracture point. Additionally, when comparing the post-fracture specimen shapes in Fig. 16, although there were some differences in the fracture location, the shear phenomenon at a certain angle in the diagonal direction showed good agreement. These results confirm the reliability of the experimentally obtained mechanical and fracture properties of the material used in the FE simulation.
Fig. 15
Comparison of predicted and experimental data for force–displacement in uniaxial tension in RD
Using the same procedure, the results of the HE tests for two materials considering material fracture were compared in terms of punch velocity using experimental results and FE simulation, as shown in Fig. 17. The F–D behavior of the punch shows that the two sets of results match well overall, demonstrating high agreement in the fracture point. However, the FE simulation results for 980DP at 0.2 mm/s showed a delayed fracture occurrence compared to the experimental data. This is also evident in Fig. 18, where the FE prediction of the HE ratio for 980DP at 0.2 mm/s was higher than the experimental values, while the remaining cases generally showed good agreement between the experimental results and simulation. Note that in the FE simulation, fracture was determined to occur only when all five element layers (see Fig. 5) comprising the thickness were completely removed, as in the experiments. In Fig. 19, the shapes of the specimens at the fracture point in the HE test were summarized and compared between the experiment and FE results. By examining the circular regions marked by solid red lines, it can be observed that the fracture locations are consistent in all cases between the experiment and simulation. Similar to the results obtained in the UT test, these findings collectively indicate that the mechanical properties, anisotropy, and fracture properties of the material used in the FE analysis are appropriately applied, demonstrating high agreement with the experimental results.
Fig. 17
Comparison of predicted and experimental data for punch force–displacement with consideration of fracture model
In Section 4.1, the F–D behavior of the HE test showed good agreement between experimental and simulation results up to crack initiation. Different input material properties were applied based on the strain rate, resulting in slight variations in the flow curve, r value, and friction coefficient. Despite these differences, no significant differences in edge formability during HE were observed with respect to deformation speed. This suggests that the slight changes in material properties caused by a fivefold difference in strain rate did not significantly impact the HE ratio.
In Section 4.2, the focus was on the variation of fracture strain with respect to strain rate. A ductile fracture model was introduced in the FE analysis to consider material fracture. Experimentally, it was confirmed that the fracture strain in UT increases with tensile speed, and this information was incorporated into the simulation setup. As a result, the FE analysis accurately predicted the fracture point of the experimental results. Specifically, in UT, it predicted the shape of the inclined fracture surface, while in HE, it accurately predicted the fracture location at the circular hole. By incorporating the increased fracture strain in UT with a higher strain rate into the FE simulation, it was possible to achieve an improvement in the HE ratio with increased deformation speed, even in HE where the deformation mode is almost identical to UT.
The increase in fracture strain with increase in strain rate has been consistently observed in various studies. Roth and Mohr (Ref 32) reported that an isothermal-adiabatic transition occurs in the material, leading to an increase in fracture strain with higher strain rates due to thermal softening. They found that the transition range for 590DP and 780TRIP steel is approximately 0.001/s to 1/s. Lim and Huh (Ref 33) observed that 980DP steel sheets show an upward trend in fracture strain with increase in strain rates within the range of 0.01/s to 1/s during UT tests.
Unlike general carbon steels, austenitic stainless steel and aluminum exhibit opposite rate sensitivity. Jin et al. (Ref 34) observed that the true failure strain of 304L stainless steel decreases as the strain rate increases in tensile tests in the range of 0.0001/s to 2550/s. Rahmaan (Ref 35) conducted hole tension experiments on AA7075-T6 aluminum alloy and found that the fracture strain decreases as the strain rate increases in the range of 0.01/s to 10/s. Cao et al. (Ref 36) proposed a strain rate-dependent fracture model for aluminum alloy 7050-T7451 and found that the fracture locus decreases as the strain rate increases to 0.001, 0.01, and 0.1/s for all deformation modes. Fietek et al. (Ref 25) presented research on the rate dependency of HE for austenitic stainless steel and aluminum, showing that the HE ratio decreases as the HE test speed increases. This opposite trend to our findings further strengthens the conclusion that the change in edge formability in HE with respect to strain rate has a strong correlation with fracture strain in the UT deformation mode.
However, this can also be seen as an indirect and qualitative inference regarding the correlation between strain rate and the HE ratio of the material. Therefore, there is a need to analyze the direct correlation between these two factors microstructurally to clearly elucidate their relationship.
Firstly, austenitic stainless steel is composed entirely of 100% austenite phase. When the material undergoes mechanical or thermal deformation, transformation-induced plasticity (TRIP) occurs, causing the austenite to transform into martensite. This phase transformation allows the material to achieve high strength and good ductility. The TRIP phenomenon is influenced by various factors, including strain rate. High strain rates increase the temperature in the material, stabilizing the austenitic phase and inhibiting martensitic phase transformation (Ref 37). Consequently, the TRIP effect decreases at high deformation rates, resulting in reduced work hardening and ultimately decreasing the HE ratio in austenitic stainless steel.
On the other hand, AHSS possesses a completely different microstructure. As noted in Section 2.1, DP steel consists mostly of a ferrite matrix with some martensite inclusion. The ductile ferrite phase exhibits an increase in material strength with increase in strain rate (Ref 38), while the high-strength martensite phase shows a decrease in strength with increase in strain rate (Ref 39). In multi-phase steels like DP steel, the HE ratio decreases as the strength difference between the two phases (hard martensite and softer ferrite) increases (Ref 40). This occurs because deformation is concentrated at the interface of the two phases, with more local deformation occurring in the softer ferrite phase. Consequently, a greater strength difference between ferrite and martensite leads to localized deformation in ferrite, potentially causing premature failure. However, when the strain rate increases in DP steel, the strength difference between ferrite and martensite decreases, resulting in an increased HE ratio compared to the original value. This is because the strain rate increase in the softer ferrite induces a higher strength gain than in martensite. In the case of CH steel, in addition to martensite, retained austenite is present within the ferrite matrix. Similar to austenitic stainless steel, it also exhibits a TRIP effect, but the amount of austenite in 980CH steel is relatively small, within 10%. Although there is a decrease in the TRIP effect with increase in strain rate, its overall impact is not significant. Therefore, similar to DP steel, the decrease in the strength difference between ferrite and martensite leads to an increase in the HE ratio in CH steel.
6 Conclusion
This study investigated the effect of strain rate variation on the HE ratio of two AHSS grades, 980CH and 980DP steel sheets. The punch speed in the HE test was varied to examine the influence of strain rate on the HE ratio. To eliminate variables other than strain rate, the holes were milled to ensure the same edge condition without any damage. Experimental results showed that both steel grades exhibited an approximately 10% increase in HE ratio when the strain rate increased from 0.01/s to 0.05/s. FE simulations were performed to investigate the material behavior at the hole edge during HE deformation. 3D solid elements and the Hill '48 model were used to consider material anisotropy. The material properties obtained from UT, which exhibited similar deformation modes, were utilized. Instead of introducing a rate-dependent model, the experimental UT properties corresponding to strain rates of 0.01/s and 0.05/s were applied in the FE simulations. The key findings are summarized below.
The stress–strain curve in the UT test did not show significant differences with a fivefold difference in strain rate, and the r value was also similar. Although different friction coefficients were applied in the HE test according to the deformation rate, no significant changes in edge formability were observed. There was no difference in thinning at the hole edge between the two speeds in the experimental and simulation results just before HE cracks occurred.
In both 980CH and 980DP, the local fracture strain increased as the strain rate increased in the UT test. By incorporating this fracture characteristic into the ductile fracture model, it was possible to predict that the HE ratio increases as the strain rate increases. The prediction accuracy of the HE ratio value was good, and the F–D curve, fracture shape, and location were well predicted. As the strain rate increases, the local fracture strain increases, delaying the occurrence of fracture in HE and leading to an increase in the HE ratio.
Other research (Ref 25) has shown that the HE ratio decreases as the strain rate increases for austenitic stainless steel and aluminum alloy. Both materials showed a decrease in UT fracture strain as the strain rate increased. This opposite result provides additional evidence of a strong correlation between strain rate-dependent changes in UT fracture strain and the HE ratio.
Microstructurally, the effect of strain rate on the HE ratio can be directly explained depending on the steel material. For austenitic stainless steel, the HE ratio decreases due to the decrease in the TRIP effect with increase in strain rate. In complex phase AHSS, the HE ratio increases due to the decrease in the strength difference between ferrite and martensite with increase in strain rate.
Acknowledgments
The authors are grateful to POSCO for generous support. MGL acknowledge the partial supports from National Research Foundation (NRF) of Korea (Grant No. 2022R1A2C2009315) and Institute of Engineering Research at Seoul National University.
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