Void-closing Behavior and Estimation Using Finite Element Analysis via Hydrostatic Integration in Hot Rolling of S10C Steel Plates

Abstract

The closure of large voids, whose thickness-to-void-height ratio exceeds 0.2, in S10C steel plates during hot rolling was investigated to determine whether hydrostatic integration (Q-value) can be used to predict the closing behavior of large voids. The steel plates with an open void along the rolling (RD), transverse (TD), and normal (ND) directions were hot rolled at 1000 and 1300°C with a target rolling reduction of 10% at each pass until 40% total target reduction. It was found that the effect of temperature on the closing behavior was negligibly small. RD and TD voids were almost entirely closed at a reduction of 40%, whereas ND voids could not be closed. The width of RD void was almost linearly decreased with reduction increase. TD void were closed at a lower reduction ratio than RD void. The thickness above and below the void was compressed after rolling in RD void but less reduced in TD void, which is presumable reason of the earlier closure of TD void. The FE analysis clarified that the void volume over initial volume (V/V₀) of the voids could be expressed as a function of the Q-value in the case of RD and TD voids. However, the closure behavior of the ND void cannot be expressed by the Q-value. These results indicate that the Q-value can be used to predict the closure of large voids in the RD and TD during rolling, although it cannot be used if the void shape is elongated in the compression direction.

1. Introduction

The shrinkage voids formed during solidification are unavoidable cast defects. These voids must be eliminated during the subsequent hot forging or hot rolling processes to prevent degradation of the mechanical properties. Therefore, it is necessary to predict void closure. As described later, numerous theoretical and experimental studies on void closures have been conducted.^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30, 31,32,33,34,35,36,37)} Recently, the behavior of void closure in actual specimens was investigated using finite element (FE) models.^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,33,34,35,36)} Many studies on void-closing behavior by hot rolling have mainly focused on the small porosity and influence of process parameters, such as pass reduction, roll diameter, and temperature gradient of the workpiece. A few studies on larger voids, whose ratio of the specimen to the void size exceeds 0.2, have been conducted.^2,3,36) If large voids are eliminated, alloys with a low castability or subjected to severe casting conditions can be used. During continuous casting, a relatively large shrinkage cavity may also form immediately below the bridged region.^38,39) In relation to this issue, we have been developing an early cutting process for continuously cast steel: bonding the solidified shell and cutting it before complete solidification. This process enables the use of latent heat to remove reheating process prior to hot rolling.⁴⁰⁾ In addition, we can shorten the length of the continuous casting equipment or increase the casting speed by adopting this cutting technique. However, cutting before complete solidification results in an extreme case of bridging; therefore, it is highly expected that a larger shrinkage cavity will be formed. For practical applications in the cutting process, the larger cavity must be closed by hot rolling. In this study, the closing behavior of large artificial channels formed in a laboratory-size plate by hot rolling was investigated. In addition, a comparison was made between the calculations obtained using the FE model and experiments.

The void closure process can be divided into two steps: mechanical closure of the void (contact of the void surfaces),¹⁸⁾ and bonding of the contacting surfaces (recovery of the original strength).⁶⁾ In this study, the mechanical closure of voids was investigated. Two numerical approaches are used to predict mechanical closure: microscopic analytical and macroscopic phenomenological. The microscopic analytical approach yields analytical or mathematical solutions for void closures. Typically, a single void in an infinite matrix is considered. In the macroscopic phenomenological approach, FE simulations are performed in the absence of voids because the location and shape of the voids are unknown a priori. Based on a comparison between the simulation and experimental results, several criteria for predicting void closure have been proposed, including the equivalent strain^2,4,5,7) and integration of the stress triaxiality ratio over the cumulated equivalent strain (Q-value).^{9,10,11,12,14,15)} Micromechanical analytical approaches have been investigated and compared with macroscopic models.^16,17) Recently, the dependence of void closure on the void shape^1,8,21,22) and Lode parameter¹³⁾ has also been reported. Among these criteria, the Q-value is widely used because of its universality and ease of use. Nakasaki et al. investigated the closing behavior of plasticine by cold rolling and cold forging and proposed a critical Q-value of 0.25 for void closure.⁹⁾ Kakimoto et al. investigated the void-closing behavior of aluminum by cold forging and proposed a critical Q-value of 0.21 for void closure.¹⁰⁾ Zhang et al. investigated the void-closing behavior of 42CrMo steel by hot forging and proposed a critical Q-value of 0.87 for void closure.¹²⁾ These results suggest that the critical Q-value depends on the material, void shape, and forming process.

In this study, three types of large artificial channels, whose ratio of the specimen to the void size exceeds 0.2, were formed in a laboratory-size plate. The effects of shape and rolling temperature on the closing behavior by hot rolling were investigated. Furthermore, the applicability of the Q-value to predict the closing behavior of large voids was evaluated.

2. Experimental Procedure

Two 130 mm × 50 mm × 12 mm JIS S10C steel plates were stacked in the thickness direction and arc-welded into a single plate with doubled thickness after introducing an artificial void with a cross-sectional area of 5 mm × 5 mm in the rolling (RD), transverse (TD), and normal (ND) directions through the center of the welded plates using a milling machine. The dimensions of the plates, artificial voids, and welded regions are shown in Fig. 1. The void size was chosen as the void-to-thickness ratio exceed 0.2. Our preliminary casting experiment show the void thickness can be over 5 mm. This selection does not affect the calculation of Q value since it is calculated in void-free model. The applicability of Q value can depend on the size of the void.

Fig. 1.

Dimensions of plates, void, and welded region. (a), (b), and (c) represent voids in the RD, TD, and ND, respectively. The red lines represent the arc-welded region, and the blue line represents the vent hole. (Online version in color.)

The welded plates were hot-rolled using a two-high rolling mill with a roll made of JIS SKD61 and a diameter of 200 mm at 1000 and 1300°C and a rolling speed of 6 rpm in an air atmosphere. The plate was heated in an electric furnace. Oxide layer was formed after heating but it was confirmed that its effect on the reduction was negligibly small. In addition, friction and heat transfer can be affect by the oxide layer, which should be taken into account when choosing the friction and heat transfer coefficients. The target reduction ratio was 10% for each pass. The RD void was closed entirely at a total target reduction of 50%. Hence, the shape of the void was measured after cutting the plate perpendicular to the void direction after the second, third, and fourth passes, which corresponds to 20, 30 and 40% total target reductions, respectively. After cutting, the remaining portion was transferred to the next pass. The TD voids were almost wholly closed after four passes (40% total target reduction). Therefore, two specimens were prepared for void measurements with two and three passes (20 and 30 total target reduction), and the shape of the void was measured after cutting the plate perpendicular to the void direction every 5 mm from the center in the TD. The ND voids could not be closed, and the shape of the ND voids was measured after the second, third, and fourth passes (20, 30 and 40% total target reductions). The void shape was evaluated at the surface and center of the void. The measured void widths and their labels are shown in Fig. 2. The largest and smallest widths in the RD, TD, and ND were evaluated as void shapes.

Fig. 2.

Schematic of void shape after rolling of (a) RD, (b) TD, and (c) ND voids. Measured void widths and their labels are shown.

3. Calculation Methodology

Hot rolling was simulated using the FORGE NxT 3.1^® software. A 1/4 analytical model (130 mm × 25 mm × 12 mm) without any voids was used. The constitutive equation for C10 in the FORGE database was used for the JIS S10C steel plate. The stress–strain relationship is expressed by Eq. (1) (the Hensel–Spittel equation:⁴¹⁾)   

$$\sigma =A{e}^{m_{1}T}{\epsilon }^{m_2}{\dot{\epsilon }}^{m_3}{e}^{\frac{m_4}{\epsilon }}{(1+\epsilon )}^{m_{5}T}{e}^{m_7\epsilon }{\dot{\epsilon }}^{m_{8}T}{T}^{m_9}$$(1)

where σ is the stress (MPa), T is the temperature (°C), ε is the strain, $\dot{\epsilon }$ is the strain rate, A, m₁, m₂, m₃, and m₄ are constants with values of 742.48676, −0.00219, −0.1523, 0.13792, and −0.0486, respectively. m₅, m₇, m₈, and m₉ are also constants whose values were set to zero. A tetragonal mesh of size less than 1.6 mm was used for the steel plate, and the initial temperature was 1000°C. Heat conductivity, specific heat, and density values were obtained from C10 data in the FORGE database and were 35.5 W/(mK), 778 J/(kg K), and 7850 kg/m³, respectively. The work roll was assumed rigid at a constant temperature of 20°C. A tetragonal mesh of size less than 1.5 mm was used for the roll. The roll was then rotated at 6 rpm. The atmospheric temperature was set to 20°C. The friction between the steel plate and rolls was set as “no lubrication” in FORGE, indicating that the Coulomb friction coefficient was 0.4, and the shear friction coefficient was 0.8 since we didn’t use any lubricant in our experiments. The Coulomb friction coefficient is consistent with the reported value of metal-oxide or oxide-oxide contact.⁴²⁾ The heat transfer coefficient between the steel plate and air was set to 10 W/(m² K), and that between the steel plate and the roll was set to 10000 W/(m² K). These values were taken from air and medium heat transfer at hot steel surface in FORGE database. The heat transfer coefficient between the steel plate and the roll is a little larger than the reported value for heat transfer through oxide layer, which is reported to be 3000 to 8000 W/(m² K).^43,44)

The roll gap was set as a reduction of 10%. The steel plate was reheated to 1000°C between the rolling passes. The Q-value was calculated using Eq. (2) using the user subroutine:   

$$Q={\int }_0^{\overline{\epsilon }}\left(-\frac{{\sigma }_m}{{\sigma }_{eq}}\right)\hspace{0.17em}d\overline{\epsilon }$$(2)

where $\overline{\epsilon }$ is the equivalent strain, σ_m is the hydrostatic stress, and σ_eq is the equivalent stress.

In FORGE NxT 3.1^®, a linear stress-triaxially based model was introduced to predict the volume per initial volume of void using Eq. (3):   

$$\frac{V}{V_0}=1+K_C{\int }_0^{\overline{\epsilon }}\frac{{\sigma }_m}{{\sigma }_{eq}}d\overline{\epsilon }=1+K_C\left(-Q\right)$$(3)

where V is the void volume, V₀ is the initial void volume, and K_C is the proportionality coefficient. The value of K_C was evaluated via the volume evolution of spherical voids during the compression of a cylindrical billet using FE simulation and proposed as K_C=5 as the initial entry value.¹⁸⁾ This K_C value is consistent with Kakimoto’s criterion¹⁰⁾ for void closure. The volume per initial void volume was calculated by averaging the V/V₀ inside the void location. Note that a value lower than zero was set to zero when averaging V/V₀.

4. Results and Discussion

4.1. Results of Experiments

Typical photographs of the cross-section of the RD void are shown in Fig. 3. When comparing the cases of 1000°C (Fig. 3(a)) and 1300°C (Fig. 3(b)), no significant difference was found. The void was almost completely closed at the target reduction of 40% (four passes). The width of the void when it closed was greater in the TD than in the ND. Therefore, the variation in the void width with the reduction ratio for TD and ND was measured and is shown in Fig. 4. The void widths decreased almost linearly as the reduction increased in the TD and ND. The void width in the ND is shown as circle plots in Fig. 4, and it decreased more than in the TD, which is shown as triangle plots. Figures 4(a) and 4(b) show the sample held at 1000 and 1300°C, respectively, and a comparison between Figs. 4(a) and 4(b) shows that the effect of the rolling temperature is slight in this temperature range. This trend is similar to that observed in Okumura’s results.³⁷⁾ Although it has been reported that the temperature gradient improves void closure efficiency,²³⁾ heat transfer to roll alone cannot make a difference between 1300 and 1000°C. The “Shell thickness” means the thickness of the upper and lower portions of the voids. This thickness was calculated as half of the thickness subtracted by the average of the void widths in the ND. The shell thickness was compressed to slightly less than the reduction ratio. This trend is presumably owing to the constraint in the TD of this thickness region; the sidewalls of the voids were elongated in the RD, and thus, the connected shell region must also elongate in the RD. This resulted in a compressed shell thickness that compensated for the positive strain in the RD to satisfy the volume conservation law. The compressed sidewalls also caused themselves to be extended in the TD as well as the RD, resulting in a smaller width in the TD than the initial width.

Fig. 3.

Cross-sections of RD void hot-rolled at (a) 1000 and (b) 1300°C. (Online version in color.)

Fig. 4.

Variations in void width with RD void hot-rolled at (a) 1000 and (b) 1300°C. The circular and triangular plots represent the void widths in the ND and TD, respectively. The open and filled plots show the minimum and maximum widths, respectively. The broken line shows the initial void width, and the open diamond plots are the shell thickness. Note that the shell thickness value is shown on the y-axis on the right.

Typical photographs of the cross-section of the TD void are shown in Fig. 5. Figures 5(a-1), 5(a-2), 5(b-1), and 5(b-2) show the void shapes after the second and third passes at 1000°C and the second and third passes at 1300°C, respectively. From left to right, the cross-sections of 0, 5, 10, 15 and 20 mm form the center are shown. At a target reduction of 30% (3 passes), the surfaces of the center of the void are in contact, as shown in Figs. 5(a-2) and 5(b-2). This result suggests that TD voids are more easily closed than RD voids. The area reduction of the void was larger in the central region than in the edge region in all cases. Therefore, the width and position of TD voids were measured. Besides, no significant difference was observed between the different rolling temperatures when comparing Figs. 5(a-1) 5(a-2) and 5(b-1) 5(b-2), similar to the results for the RD voids. Figure 6 shows the relationship between the void width and void position in the TD. Figures 6(a-1) and 6(a-2) show the void widths after the second and third passes at 1000°C, and Figs. 6(b-1) and 6(b-2) show the void widths after the second and third passes at 1300°C, respectively. The void widths in the RD and ND are shown as square and circular plots, respectively. The broken line shows the initial void width, and the open diamond plots are the shell thickness. The difference between 1000 and 1300°C was small. The void width in the RD in the center was close to the initial void width in any cases even after rolling by 30%. On the other hand, width of void in the RD increased to around 7 and 8 mm at the edges after reduction of 20 and 30%, respectively. Thus, the void was elongated in the RD at the edge after rolling. These were similar to the end flares during the rolling process. The void width in the ND was 0 to 3 mm and smaller than the initial width; that is, the void was compressed in the ND. Moreover, the void width in the ND at the edge was larger than at the center by around 1 mm except for ND Min. at the rolling reduction of 30%, which is almost zero. The thickness above and below the void (shell thickness) was slightly smaller than the initial value; that is, this part was less compressed than the RD void. Therefore, the TD void closed at a lower rolling reduction than the RD void, presumably because of the difference in the constraint conditions of the shell thickness region. In the TD void, the shell region was connected to the sidewalls in the RD. In this case, the strain in the shell region in the TD should be similar to that in the sidewall region, but the strain in the RD does not need to be. Consequently, the reduction in shell thickness in the TD void specimen was smaller than in the RD void, inducing a more significant reduction in the void width in the ND.

Fig. 5.

Cross-sections of TD void hot-rolled at (a-1, a-2) 1000 and (b-1, b-2) 1300°C after target reductions of (a-1, b-1) 20% (2 passes) and (a-2, b-2) 30% (3 passes). From left to right, the cross-sections of 0, 5, 10, 15 and 20 mm form the center are shown. (Online version in color.)

Fig. 6.

Position dependence of void width of TD void hot-rolled at (a-1, a-2) 1000 and (b-1, b-2) 1300°C after target reductions of (a-1, b-1) 20% and (a-2, b-2) 30%. The circular and square plots represent the void widths in the ND and RD, respectively. The open and filled plots show the minimum and maximum widths, respectively. The broken line shows the initial void width, and the open diamond plots are the shell thickness.

Photographs of the ND voids are shown in Fig. 7. Figures 7(a) and 7(b) show photographs after rolling at 1000 and 1300°C, respectively. The voids were elongated in the RD and slightly compressed in the TD as the reduction increased. Moreover, there was no significant difference between the hot-rolling temperatures similar to the RD and TD samples. Figure 8 shows the variation in reduction with the ND void widths. The widths in the RD and TD are shown as square and triangular plots, respectively. The change in void width at the surface of the plate is shown in Figs. 8(a-1) and 8(b-1). The width in the TD decreased slightly as the reduction increased, whereas the width in the RD increased. The plate is elongated in RD during rolling. Therefore, the width in the RD increased. On the other hand, the width is basically not changed in the TD during rolling. The reduction of the void width in the TD indicates that the metal flows not only in RD but also TD. The change in void width at the center of the plate is shown in Figs. 8(a-2) and 8(b-2). The width in the TD decreased as the reduction increased, and the width in the RD slightly increased as the reduction increased. This is presumably due to bulging when compression in the ND.

Fig. 7.

Photographs of ND voids hot-rolled at (a) 1000 and (b) 1300°C. (Online version in color.)

Fig. 8.

Reduction dependence of void width of ND void hot-rolled at (a-1, a-2) 1000 and (b-1, b-2) 1300°C at the (a-1, b-1) surface and (a-2, b-2) center of the void. The triangular and square plots represent the void widths in the TD and RD, respectively. The open and filled plots represent the minimum and maximum widths, respectively. The broken line represents the initial void width.

4.2. Comparison between Experimental and Calculation Results

The void volume over the initial void volume (V/V₀) was calculated by averaging the maximum and minimum widths, as shown in Figs. 4, 6, and 8 and multiplying the average widths. The measured V/V₀ was compared with the V/V₀ calculated using Eq. (3).

Figure 9 shows the calculated distribution of Q-value at the cross-section normal to the RD and the position of the center of the RD after (a) 10, (b), 20, (c) 30, (d) 40 and (e) 50% reduction. Q-value gradually increased with the increase of reduction. Q-value is higher at the center in the TD than edge.

Fig. 9.

The calculated distribution of Q-value at the cross-section normal to the RD and the position of the center of the RD after (a) 10, (b) 20, (c) 30, (d) 40 and (e) 50% reduction. (Online version in color.)

Figure 10 shows the measured and calculated V/V₀ values of the RD void. The black circle (●mark) is the FEM analysis result, and the white circle (〇mark) is the experimental result. The measured value was 0.4 at the reduction ratio of 20%, decreased continuously, and reached almost 0 at 40%. On the other hand, the calculated value was 0.3 at 20%, which is lower than the measured value, but it decreased gradually after that, and was slightly higher than 0 even at 50%. Therefore, the calculated V/V₀ values were consistent with the measured V/V₀ values. This result demonstrates that the linear stress-triaxially based model can be applied to large voids in the RD. However, the volume of voids was not completely zero at a reduction of 50%. In other words, the critical reduction for the complete closure of the void was overestimated using this calculation. In this case, if the void was completely closed in the calculation, the void was expected to be completely closed in the experiment.

Fig. 10.

Comparison between calculated and measured V/V₀ values of RD void. The open and filled plots represent measured and calculated V/V₀, respectively.

Comparisons between the calculated and measured V/V₀ values of the TD defects at target reductions of 20% and 30% are shown in Figs. 11(a) and 11(b), respectively. In the case of 20% of reduction ratio, the measured V/V₀ was approximately 0.2 at the center and gradually increased to 0.6 at 25 mm. On the other hand, The calculated V/V₀ showed data between 0.2 and 0.5 and scattered owing to the coarse mesh. However, it replicated the measured trend that V/V₀ was larger at the center than at the edge and increased with an increasing reduction ratio. The center of the plate is constrained by surrounding part. On the other hand, the edge is not constrained, causing bulging. As a result, the hydrostatic stress is smaller at the edge than center. Thus, the Q-value was smaller at the edge than at the center. This result reveals that the linear stress-triaxially based model can be applied to the significant defect in the TD even though it is calculated by defect-free model.

Fig. 11.

Comparison between calculated and measured V/V₀ values of TD void at target reduction ratios of (a) 20% and (b) 30%. The open and filled plots represent the measured and calculated V/V₀ values.

Figure 12 shows a comparison between the calculated and measured V/V₀ values of the ND void. In the central part, the measured value decreased from 1 at 0% to 0.6 at 40% of reduction ratio. On the other hand, the calculated value decreased as well as the increase in reduction ratio, and shows almost 0 at 30%. Furthermore, for the edges, the measured V/V₀ values were higher than or equal to 1 from 0 to 40% reduction, whereas the calculated reductions were lower than 1 and approached 0 as the reduction ratio increased, indicating that the calculated V/V₀ values were inconsistent with the measured V/V₀ values. This trend is because, presumably, there was no metal in the compression direction in the ND void, inducing a completely different behavior of the void closure from that of small spherical porosities in which the hydrostatic integration model is typically applied. The void closure of the ND void is mainly attributed to the bulging of the void surface in the TD. This mechanism is not thought to appear in the closure of small porosities, which probably causes a significant difference between the calculated and measured V/V₀ values in the ND void.

Fig. 12.

Comparison between calculated and measured V/V₀ values of ND void. The open and filled plots represent the measured and calculated V/V₀ values. The circular and triangular plots show the widths of the center and edge of the void, respectively.

5. Conclusions

In this study, the behavior of the closure of large voids in the RD, TD, and ND in the S10C steel plate was investigated. RD and TD voids were almost entirely closed by a rolling reduction of 40% at 1000 and 1300°C. The ND void could not be closed by hot rolling until a 40% reduction. The void shape was measured after the second, third, and fourth passes. V/V₀ was calculated using the measured shape as function of the rolling reduction and void position. The results show that TD voids are more easily closed than RD voids. In addition, FE analysis was conducted to clarify the applicability of the Q-value for predicting the closure of large voids, for which the thickness-to-void-height ratio was higher than 0.2. The results show that the closing behavior in terms of V/V₀ can be simulated using the linear stress-triaxially based model. However, the linear stress-triaxially based model ultimately failed to replicate the closing behavior of the ND void. These results indicate that the V/V₀ of large voids in the ND and TD can be predicted using the linear stress-triaxially based model, whereas the ND cannot. This is probably because the closing mechanism of the ND defect is entirely different from that of small spherical porosities, which is assumed to exist in hydrostatic integration models.

Acknowledgments

The authors acknowledge the support of H. Kamada for the experiments. We also thank Y. Kadowaki and R. Yamaguchi, Steel Plantech Co., Ltd., for their insightful discussions. Assistance in implementing the user subroutine provided by M. Hoshi, SCSK Corporation, is gratefully acknowledged.

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