Prediction Method of Void Distribution near Punched Surface of Medium-Carbon Steel Sheet using Scrap

Abstract

The objective of this study was to confirm the validity of the prediction method of void distribution near a punched surface of a blank (holder side) by observing the void distribution of scrap (hole side). It is important to know the exact void number density and void area fraction through an appropriate evaluation method such as that mentioned, just where a crack occurs owing to stretch-flange deformation in the same individual sample because the crack and the void behaviors fluctuate from sample to sample and with position, even under the same punching condition. This study investigated the correlation of void distribution near punched fracture surfaces of the blank and scrap in medium-carbon steel. The voids near the punched fracture surfaces of the blank and the scrap were measured using SEM images. The voids of the blank and the scrap were distributed point-symmetrically with respect to the center of the fracture surface. The equivalent plastic strain and the stress triaxiality that were analyzed with FE analysis were also distributed point-symmetrically. The void distribution of scrap was shifted to the sheared surface side, compared to that of the blank. To predict the void distribution of the blank using the scrap, the void distribution of the scrap with the burr side as a reference point was approximated by a cubic function. Furthermore, the void distribution of the scrap shifted to the burr side. The prediction method of void distribution near punched surface by scrap was validated by considering stress and strain during punching.

1. Introduction

Carbon steel is widely used for automobile suspension parts because of its excellent balance between strength, formability, and abrasion resistance. Many automotive suspension parts are made by stamping,¹⁾ which has good productivity. If the condition of the punched surface is poor, the secondary workability and fatigue strength^2,3,4) decrease. Of the various aspects of formability in sheet forming, stretch-flange formability is critical and determines the forming limits in the production of most parts, which depend on the nature of the punched surface before forming. Some studies have revealed the effect of voids near the fracture surface on stretch-flange-formability. Takashima et al. clarified that the hole-expanding ratio shows a linear relationship with void number density near the punched surface in dual-phase steel.⁵⁾ Ito et al. reported that voids near the fracture surface deteriorate the hole-expansion ratio in ferrite-pearlite steel.⁶⁾ These previous studies compared voids near the fracture surface of steel sheets with the hole-expansion ratio of another sample in the same steel sheet. In contrast, excellent stretch-flange formability of spheroidized, annealed, medium-carbon steel has been determined,⁷⁾ and the effects of spheroidized cementite dispersion on void initiation during tensile testing⁸⁾ and punching⁹⁾ have been clarified.

Stretch-flange formability is evaluated in terms of the hole-expansion ratio by a hole-expanding test. However, the hole-expansion ratio and edge-crack position have excessive uncertainty and variation across the sheet thickness. One of the major reasons for this is that it is difficult to obtain an identical punched surface owing to the sensitivity of the punched surface to slight fluctuations in punching conditions.^10,11) For this reason, there is a possibility that the void distribution—which is observed on a cross-section of a punched sheet after the punching test—and the void distribution before the hole-expanding test with another sample are different even with identical nominal punching conditions such as the clearance between the die and the punch, and the surface roughness. Therefore, non-destructive, radiographic, and ultrasonic measurements of voids near the punched surface before hole expansion test are required to confirm the effects of voids on the hole-expansion ratio in detail. However, detecting voids near the punched surface using non-destructive testing is difficult because many voids are smaller than the resolution of typical non-destructive testing apparatus. Hence, we propose a method to estimate void distribution (number density and area fraction) near a punched surface by measuring these quantities in scrap, which can be observed easily and yield exact information of the same individual sample.¹²⁾ However, the correspondence of void distribution between the blank and the scrap has not been confirmed by considering the stress and strain fields during the punching process.

The objective of this study was to confirm the validity of the prediction method of void distribution near a punched surface of a blank by measuring that of scrap. The stress triaxiality and equivalent plastic strain distributions obtained through FE analysis were compared with the void distribution to confirm the validity of this method.

2. Methods

2.1. Sample Material

A hot-rolled, medium-carbon steel sheet of 4-mm thickness was used. The major elements (mass percent) of the steel sheet were C (0.36), Si (0.20), Mn (0.71), and Cr (0.13). To precipitate spheroidized cementite at the ferrite grain boundaries, the steel sheet was annealed by holding at 710°C for 20 h. Figure 1 shows the microstructure of the medium-carbon steel sheet, which was composed of ferrite and spheroidized cementite. The anisotropy of the microstructure observed in the steel sheet was negligible, as shown in Fig. 1.

Fig. 1.

Microstructure of the cross-section of the medium-carbon steel sheet. (a) Parallel to the rolling direction. (b) Perpendicular to the rolling direction. The light and dark parts are cementite and ferrite, respectively (R.D.: rolling direction, T.D.: traverse direction, N.D.: normal direction).

2.2. Punching Test

The steel sheets were ground to 1 mm thickness by removing 0.5 mm from one side and 2.5 mm from the other side to remove center segregation of Mn, which usually forms during casting¹³⁾ and increases cracking on the punched surface.^14,15) Although there was no noticeable Mn center segregation in this steel sheet, this grinding was done to minimize the effect of the center segregation. A square specimen with a 50-mm side length was cut from the steel sheet. The specimen was used for the punching test. Two lines were drawn to intersect on the surface of the specimens, by which the same positions on the punched surfaces of both the scrap and blank could be easily identified. Figure 2 shows a schematic diagram of the punching test. A punch with a 10-mm diameter and a die with a 10.4-mm diameter hole were used to achieve a clearance of 20%. In this study, the clearance was set to 20% so that voids easily occur under large tensile deformation. In addition, we chose the clearance to have a typical punched surface composition. The punching test was conducted on a universal testing machine (Autograph^TM, AG-I-250kN, Shimadzu Corp.) at punching velocity of 500 mm/min. The punch force and displacement were also measured with the universal testing machine; the sampling period was set to 1.25 ms.

Fig. 2.

Schematic diagram of punching test (unit: mm).

2.3. Evaluation of Voids near Punched Surface after Punching Test

Figure 3 shows schematics of the observed area near the punched surface. To measure the void distribution in the thickness direction, the blank and the scrap were cut parallel to the rolling direction to obtain the cross-section of the punched surface. These cross-sections were polished to observe the voids with a scanning electron microscope (SEM, JSM-6060, JEOL Ltd). The acceleration voltage of the SEM was set to 15 kV. The observed area was within 50 μm of the fracture surfaces on the cross-sections. The observed area was divided into ten equal areas in the thickness direction. After observing the voids with a SEM, the surfaces were repolished and the voids were observed with a SEM. This process was repeated three times to reduce statistical errors. The thickness removed was approximately 20 μm per polish. The polished thickness was confirmed by measuring the size of the indentation made in the Vickers hardness test before and after polishing. The total number of voids, as well as the coordinates and areas of each void were measured with image analysis software (WinROOF^TM, Mitani Corporation). Voids with a diameter of 0.5 μm or more were analyzed with image analysis software, as shown in Fig. 3(d). This criterion was determined to distinguish between voids initiated after punching and those initiated before punching.⁸⁾ The void number density N_V was calculated using Eq. (1), and used as an index to represent void initiation,   

$$N_V=\frac{N}{S}$$(1)

where N is the number of voids and S the observed area. Similarly, the area fraction of voids S_f was calculated using Eq. (2); this was used as an index to represent void initiation and growth,   

$$S_f=\frac{S_V}{S}\times \text{100}$$(2)

where S_V is area of voids. The z-axis was determined as shown in Fig. 3. The z-origin of the blank and scrap was set at the burr. In addition, the distance from the z-origin was normalized using the length of the fractured surface.

Fig. 3.

Schematics and SEM images of observed area near the punched surface. (a) Scrap. (b) Blank. (c) SEM image (origin). (d) SEM image with marked voids.

2.4. Tensile Test

Tensile tests were carried out to obtain the material properties for the FE analysis explained in the following section. The specimens were tested with a universal testing machine (Autograph^TM, AG-100kNC, Shimadzu Corp.). The cross-head velocity was set to three levels; 3, 180, and 500 mm/min. The elongation was measured with a digital video extensometer (TRviewXTM, Shimadzu Corp.). Type-3 tensile test specimens were prepared in accordance with ISO 6892-1.¹⁶⁾ This specimens were ground to 1 mm in thickness using the method explained in section 2.2.

2.5. Material Model and Simulation Conditions for FE Analysis

In this simulation, it was assumed that the material was homogeneous, isotropic, elasto-plastic, and strain-rate dependent. The work-hardening characteristics of the material were approximated by Eq. (3), which is expressed as follows:   

$$\sigma \text{ = }K{\epsilon }^n{\dot{\epsilon }}^m$$(3)

where σ, K, ε, $\dot{\epsilon }$, n, and m are the true stress, stress amplitude, plastic strain, strain rate, work-hardening exponent, and strain-rate-sensitivity exponent, respectively. The values of K, n, and m for the specimen were determined from the true stress vs. plastic strain curves obtained in the tensile tests. The Young’s modulus of the specimen was also determined from the tensile tests.

Figure 4 shows the punching model of the FEM simulation. In this study, a two-dimensional axisymmetric analysis was performed using an elasto-plastic FE code STAMP 3D.¹⁷⁾ The punch, die, and holder were assumed to be rigid bodies. The radii of the cutting edges of the die and punch were 0.03 mm. The friction coefficient was 0.13. To simulate crack propagation during the punching process, the Cockcroft-Latham criterion was used,¹⁸⁾   

$${\int }_{\text{0}}^{\overline{{\epsilon }_{\text{f}}}}{\sigma }_{max}\text{d}\overline{\epsilon }=\text{C}$$(4)

where $\overline{{\epsilon }_{\text{f}}}$, σ_max, and C are the equivalent plastic strain at fracture, maximum normal stress, and ductile fracture criterion, respectively. C was defined using an interrupted punching test. This simulation was performed with using the element deletion method, which has been used in previous studies to simulate a crack propagation during punching.¹⁹⁾ The element deletion method is a method for setting the flow stress and the element rigidity to zero when a finite element fractures. The punching velocity was set at 1 mm/min. The punched depth at initiation of micro cracking around the punch corner was measured.

Fig. 4.

Initial arrangement of FE model of punching test. (a) Overall view. (b) Arrangement of elements in deformation area.

3. Results

3.1. Voids Distribution near Punched Surfaces of Blank and Scrap

Figure 5 shows the void distribution of the observed area divided into ten equal parts in the punching direction. The error bars show the standard errors. The void number density and the area fraction of the voids were a maximum near the burr. The void number density and area fraction of void decreased with increasing normalized distance from the burr, and were a minimum at the center of the fractured surface. In addition, the void number density and area fraction of the void increased again near the boundary (z = 1) between the fractured surface and the sheared surface. These void distribution tendencies are almost identical for the blank and scrap. In contrast, the void distribution of the scrap shifted to the sheared surface side compared to the void distribution of the blank. In addition, the void number density near the burr of the blank was smaller than that near the burr of the scrap.

Fig. 5.

Void distribution in thickness direction. (a) Void number density. (b) Area fraction of void.

3.2. Acquisition of Material Properties and Fracture Threshold

The true stress vs. plastic strain curves obtained by the tensile tests are shown in Fig. 6. The strain rates of each specimen were 8.33 × 10⁻⁴ s⁻¹, 5.00 × 10⁻² s⁻¹, and 0.139 s⁻¹. The fitted curves of each specimen are represented by dotted lines in Fig. 6. Equation (5) was obtained by logarithmic transformation of Eq. (3).   

$$\text{ln}\sigma =m\text{ln}\dot{\epsilon }+\text{ln}K+n\text{ln}\epsilon$$(5)

Fig. 6.

True stress vs. plastic strain curves of tensile tests obtained by experiments and fitting with σ = Kεⁿ$\dot{\epsilon }$^m. K: 926.1, n: 0.1898, m: 0.0211. (a) $\dot{\epsilon }$ = 8.33 × 10⁻⁴ s⁻¹. (b) $\dot{\epsilon }$ = 5.00 × 10⁻² s⁻¹. (c) $\dot{\epsilon }$ = 0.139 s⁻¹.

The procedure of the identification of the fitting parameters (K, n, m) was as follows:

1. The true stresses at plastic strains below 0.10 under each strain rate condition were obtained.

2. The strain rate sensitivity exponent m was obtained by linear approximation of the lnσ-ln$\dot{\epsilon }$ diagram, as shown in Eq. (5).

3. The work-hardening exponent n and stress amplitude K were obtained using the lnσ-lnε diagram for a strain rate of 8.33 × 10⁻⁴ s⁻¹.

The values of each material property are listed in Table 1. Swift’s hardening law corresponds reasonably well with the experimental data after the yield points. Although the yield point was not reproduced, it was not a major problem in the punching analysis.

Table 1. Material properties for FEM analyses.

Parameter	Medium-carbon steel sheet
K [Pa·sm]	926.1
n	0.1898
m	0.0211
Young’s modulus E [GPa]	195.3
Poisson’s ratio ν	0.30
Ductile fracture criterion C	1488

Figure 7 shows the micro cracking for a punch displacement of 0.46 mm, and the analyzed damage value distribution using FEM.

Fig. 7.

Observation of micro crack around the punch corner for determination of ductile fracture criterion. (a) Experiment (punch displacement of 0.46 mm). (b) FEM.

The ductile fracture criterion C was determined by an interrupted punching test. The punch displacement that initiated the micro crack at the cutting edge of the punch was 0.46 mm. To determine C, the punch displacement was set to 0.46 mm to analyze the damage distribution during punching, using FEM. The maximum ductile fracture criterion C at the punch displacement of 0.46 mm was 1488. Therefore, the ductile fracture criterion C was set to 1488. The elastic recovery of the indented surface was simulated with FE analysis. The elastic discovery of the punched surface was approximately 8 μm at a punched depth of 400 μm. Therefore, we considered that the elastic recovery of the punched surface was negligible.

3.3. Validity of FE Model of Punching Test

The punch experimental and analytical force-displacement diagrams are compared in Fig. 8 to check the validity of the FE analysis. Elastic deformation of the die and the testing machine occurred during punching; the experimental values were corrected using a method from previous research.²⁰⁾

Fig. 8.

Comparison of experimental and analytical punch force-displacement curves.

The increase tendency of the experimental punch force agrees approximately with the analytical tendency. However, the FE analysis exhibits three different sections: sharp punch force decrease, maximum punch force, and punch force decrease ratio after the maximum punch force. The punch displacement at the sharp punch force decrease coincides with the displacement at which re-meshing was performed. This result suggests that the shape of the material changed subtly when the material was re-meshed. The analytical punch force value is slightly larger than the experimental one. When the punch contacted the specimen, the strain rate was at least 100 s⁻¹, which is approximately three orders of magnitude higher than the strain rate in the tensile test. Therefore, the difference appears to be caused by the low strain rate in the tensile test. Furthermore, the punch force decrease rate is small. The punch force decreased as the crack propagated in the metal sheet. The punch force decrease ratio seems to depend on the crack propagation velocity. The stress triaxiality increased around the crack during punching. The Cockroft and Latham criterion model does not consider the effect of stress triaxiality. Hence, the rate of crack propagation determined analytically seems to be smaller than that determined experimentally. In contrast, the first crack initiated around the punch corner during punching. The crack position during punching was investigated using an FE model.²¹⁾ According to a previous study, cracks initiate preferentially in high stress triaxiality areas and high equivalent plastic strain areas. Yukawa et al. reported that the strain near the punch side is higher than that near the die side.²²⁾ Several punching simulations have shown that cracks initiate at the punch corner during punching.^23,24,25) The tendency of the results in Figs. 9 and 10 agrees with these previous studies. Therefore, this FE analysis is considered appropriate to qualitatively examine the tendency of the distribution of equivalent plastic strain and stress triaxiality during the punching process.

Fig. 9.

Distribution of equivalent plastic strain during punching at punch displacement of (a) 0.4 mm, (b) 0.5 mm, (c) 0.6 mm, and (d) 0.7 mm.

Fig. 10.

Distribution of stress triaxiality during punching process at punch displacement of (a) 0.4 mm, (b) 0.5 mm, (c) 0.6 mm, and (d) 0.7 mm.

3.4. Stress and Strain during Punching

Figures 9 and 10 show the equivalent plastic strain distribution and the stress triaxiality distribution during punching, respectively. The equivalent plastic strain and the stress triaxiality around the punch corner were larger than those at the die corner. The finite elements of the material near the punch corner exceeded the fracture threshold, as shown in Figs. 9(c) and 10(c). The equivalent plastic strain and the stress triaxiality are distributed approximately point-symmetrically with respect to the center of the fracture surface. The large equivalent plastic strain areas after punching were on the burr side and the boundary between the sheared surface and the fractured surface. This tendency appeared in both the blank and the scrap. The equivalent plastic strain near the burr of the scrap was larger than that near the burr of the blank.

4. Discussion

4.1. Relationship between Void Distribution and Ductile Fracture Factors

The equivalent plastic strain and the stress triaxiality were distributed approximately point-symmetrically with respect to the center of the fracture surface. The tendencies of void distribution in the blank were in good agreement with those of the scrap by setting the z-origin at each burr, as shown in Fig. 3. The voids were also distributed point-symmetrically with respect to the center of the fracture surface. Void initiation and growth are dominated by the equivalent plastic strain and the stress triaxiality, and the usefulness of the ductile fracture model has been confirmed.^26,27,28) Therefore, the voids were distributed approximately point-symmetrically with respect to the center of the fracture surface during the punching process, in a similar manner to the equivalent plastic strain distribution and stress triaxiality distribution. In addition, the equivalent plastic strain of the scrap was larger than that of the blank, especially near the burr, as shown in Fig. 9. These results agree with the fact that the void number density and the area fraction of the void of the scrap are larger than that of the blank, especially near the burr, as shown in Fig. 5. This result suggests that the stress and strain fields were not completely point-symmetrical with respect to the center of the fracture surface. The area of the sheet that touched the punch corner was presumed to be easily deformed because there was no holder on the back side of the scrap, as shown in Fig. 2. Furthermore, the void distribution of the scrap was shifted in the z-plus direction, compared to the void distribution of the blank in Fig. 5. This void distribution shift can be explained by the mechanism shown in Fig. 11.

Fig. 11.

Schematic diagram of the mechanism of void distribution shift. N_v: void number density, S_f: area fraction of voids.

The equivalent plastic strain and the stress triaxiality around the punch corner were larger than those around the die corner. The crack propagated in the area with the large plastic strain and high stress triaxiality. Therefore, the void distribution of the scrap shifted to the z-plus direction, compared to the blank.

4.2. Prediction Method to Estimate Void Distribution near Punched Surface by Scrap

From these observations, the void distribution was approximated by a cubic function to predict the void distribution in the blank. The cubic function was used to approximate the void in the scrap because the distribution had local minima and maxima.   

$$f\text{(}z\text{)}={\text{C}}_{\text{1}}{z}^{\text{3}}+{\text{C}}_{\text{2}}{z}^{\text{2}}+{\text{C}}_{\text{3}}z+{\text{C}}_{\text{4}}$$(6)

Here, f(z) and C_i(i=1−4) are the void distribution and coefficient of the cubic function, respectively. The void distribution of the scrap shifted slightly to the sheared surface, compared to that of the blank. Therefore, the void distribution of the scrap was corrected using Eq. (7).   

$$f\text{(}z\text{)}={\text{C}}_{\text{1}}{\left(z+{\mathrm{z}}_{\text{0}}\right)}^{\text{3}}+{\text{C}}_{\text{2}}{\left(z+{\mathrm{z}}_{\text{0}}\right)}^{\text{2}}+{\text{C}}_{\text{3}}\left(z+{\mathrm{z}}_{\text{0}}\right)+{\text{C}}_{\text{4}}$$(7)

Here, z₀ is the shift parameter, which was determined using the least squares method for the void distribution of the blank and the void distribution of the shifted scrap approximated with a cubic function. The shift parameter of the void number density and area fraction of voids were 0.061 and 0.049, respectively. The void number density of the blank varied significantly in proportion to the void number density of scrap. On the other hand, the amount of shift was defined as a fixed value because it did not vary significantly for each experiment. This tendency was also observed for area fraction of voids. Figure 12 shows a comparison between the void distribution of the blank and the estimated value of the scrap. The estimated value from the scrap is in good agreement with the measured void distribution of the blank. However, the void number density and area fraction of the voids near the burr side were overestimated because the equivalent plastic strain near the burr of the scrap was larger than that of the blank, as shown in Fig. 9. Therefore, the prediction method can be applied to fracture surfaces, excluding the area near the burr side.

Fig. 12.

Estimated void distribution of the blank corrected by approximating the void distribution of the scrap using a cubic function and shifting the void distribution of the scrap to the burr side. (a) Void number density: N_V(z) = 41830(z+z₀)³ – 23700(z+z₀)² –24500(z+z₀) + 27600. (b) Area fraction of void: S_f(z) = 6.22(z+z₀)³ – 8.67(z+z₀)² + 2.79(z+z₀) + 0.65.

5. Conclusions

In this study, the correlation of void distributions between scrap and a blank of a spheroidized, annealed, medium-carbon steel sheet was clarified to establish the prediction method of the void distribution of the blank by observing the scrap of the same sample.

(1) Prediction of the void distribution in the blank using the following method was suggested.

(i) Measurement of the void distribution in the thickness direction with each burr side as a reference point.

(ii) Approximation of the void distribution of the scrap using a cubic function.

(iii) Determination of the shift parameter using the least squares method for the void distribution of the blank and the void distribution of the shifted scrap approximated with a cubic function.

(iv) Shifting the void distribution of the scrap to the burr side.

(2) The equivalent plastic strain and the stress triaxiality were distributed approximately point-symmetrically with respect to the center of fracture surface. The equivalent plastic strain and the stress triaxiality near the punch corner were larger than those near the die corner. The void distribution coincided with the equivalent plastic strain distribution and the stress triaxiality distribution. The validity of this prediction method was confirmed by considering the stress triaxiality and the equivalent plastic strain during the punching process.

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