Numerical simulation of void elimination in the billet during hot shape rolling processes based on the Gurson–Tvergaard–Needleman model Open Access

Due to the nonuniform solidification of materials during casting, the presence of void defects, including shrinkage cavities and porosities, is unavoidable in large ingots. The existence of voids will damage the strength and continuity of downstream products, lead to stress concentration and crack initiation, and eventually lead to loss of lifespan or even outright discarding.¹² Voids can be eliminated by severe plastic deformation of the billet during rolling or forging.^3,4 Consequently, investigating the void closure behavior of billets during the hot rolling process holds significant importance in enhancing product quality. In general, void elimination is divided into two main stages: closing the void volume until contact between the surfaces is achieved, and then connecting the contact surfaces at a sufficiently high temperature and compressive stress.^5,6 It is worth mentioning that only the void closure stage is involved in the present work.

Void elimination during the rolling processes has always been a research hotspot, which has been paid special attention by many researchers. Nakasaki et al.⁷ examined the impact of hydrostatic pressure on the closure of central porosity in slabs during the rolling and forging processes. They concluded that the cross-sectional area of the pore exhibited a proportional relationship with the hydrostatic integration during single-pass rolling. Through hot rolling experiments conducted on aluminum alloy at 400 °C, Chaijaruwanich et al.⁸ determined that a contact arc length-to-initial slab height ratio greater than 0.3 is necessary for achieving void closure. Nakasaki et al.⁷ discovered that increasing reduction is a fundamental approach to ensuring void closure.⁹ By using a triangular velocity field, Zhang et al.¹⁰ established the mechanical condition required for closing a rectangular defect in heavy plate rolling processes. Saby et al.^11,12 focused on the closure process of an elliptical void in hot metal forming processes. They proposed a model that considers the influences of void direction and size.

Gaosheng et al.¹³ investigated the evolution of central cracks in heavy plates using finite element modeling. Their study specifically focused on analyzing the impact of temperature gradient on pore closure. Joo et al.¹⁴ obtained pressure–time curves that highlighted the significance of pore surface contact time and deformed the surface length as a crucial parameter in pore welding through finite element model. Yu et al.¹⁵ investigated the effects of various factors, including the friction coefficient, crack size, edger roll shape, and fillet radius of grooved edger roll, on the closure and growth of cracks at the corner of thick plates by using three-dimensional finite element modeling. Experimental verification confirmed that larger reduction, thinner slabs, and larger roller diameters contribute to the improved closure of voids during rolling processes.¹⁶ 

To accurately investigate the void evolution problem, Zhang et al.¹⁷ introduced a stress triaxiality ratio and Lode stress parameter-based (STLB) model. The results demonstrated that the STLB model exhibited an average error of 2.35% and its accuracy was validated through comparisons with explicit simulations and experimental data. Wang and Dong¹⁸ presented a novel void prediction model that considered the influence of the Lode parameter, stress triaxiality, and effective strain. The model aimed to describe the variation in void relative volume for a wider range of stress states.

The aforementioned studies have provided valuable insights into void closure behavior under various conditions. However, many of these studies have primarily focused on the closure behavior of finite voids during forging or rolling processes, which may have limitations in terms of industrial applications. The Gurson–Tvergaard–Needleman (GTN) model has been extensively employed to investigate ductile damage in materials by describing the growth and coalescence of voids under stress.^19–23 However, its application in studying the issue of void elimination, where void shrinkage is considered as negative void growth,^24–26 has been relatively limited.

In this paper, the GTN model was introduced and the relevant parameters were obtained; then, the void elimination of a central porous free-cutting steel 1215 MS during a hot shape rolling process has been studied with a 3D finite element model. The central distribution of voids in the billet was considered in the finite element model. The relationships between the void elimination and the pressure stress in the billet were analyzed. The influences of rolling reduction, rotation speed, and friction between the work roller and billet on the void elimination were also discussed.

A 3D thermo-mechanical coupled simulation was performed using the ABAQUS/Explicit finite element software. The dimensions of the billet were 500 × 163 × 161 mm³, and the diameter of the work roller was 610 mm. To reduce computational time, only a quarter of the billet and half of the work rollers were modeled in the simulation due to the symmetric geometry of the billet, as shown in Fig. 1. The finite element calculation utilized the eight-node trilinear displacement and temperature-coupled element with reduced integration and hourglass control, known as C3D8RT. On the other hand, the work roller was treated as a rigid body and modeled as an analytical rigid surface.

In the model, both displacement and thermal boundary conditions were specified. On the central plane of the billet, which corresponds to the x–y and x–z symmetry planes, symmetrical displacement constraints were applied. In addition, an adiabatic condition was imposed on the temperature distribution. The heat transfer coefficient between the billet and work roller was set to 0.1 KW/(m² K),²⁷ while the heat transfer coefficient between the billet and air was set to 25 W/(m² K).²⁸ The initial temperature of the billet was 1100 °C.²⁹ The work roller was treated as a rigid body and rotated around its own axis at a specific tangential speed according to different working conditions, as shown in Table I,³⁰ and RPS means revolutions per second. Meanwhile, the other displacement constraints of the work roller were fixed. In addition, the billet had an initial velocity in the X direction to enter the rolling gap smoothly, which was less than the roller’s tangential velocity slightly.

Throughout the rolling process, the roller was modeled as 3D analytical rigid bodies and maintained surface-to-surface contact with the billet. The contact behavior was assumed to be controlled by the Coulomb friction law. To investigate the effect of friction on void elimination, three friction conditions were considered in our calculations and the corresponding Coulomb friction coefficients were set at 0.3, 0.4, and 0.5.³¹ 

The billet material used in the simulation was 1215 MS steel, which is a high-sulfur, free-cutting steel. The chemical composition of the 1215 MS steel was determined by using an inductively coupled plasma optical emission spectrometer and is provided in Table II.³² To investigate the plastic flow behavior of 1215 MS steel at elevated temperatures, cylindrical specimens with a diameter of 8 mm and a height of 12 mm were machined from extruded rods. These specimens were prepared for hot compression tests. Uniaxial hot compression tests were conducted up to a strain of 0.58 on the Gleeble 3800 system at four different stain rates (0.001s-1, 0.01s-1, 0.1s-1, and 1s-1) and four different temperatures (900, 1000, 1100, and 1200 °C). The flow stress curve of the material (1215 MS) at different temperatures and different strain rates is shown in Fig. 2.

The thermophysical properties of the billet employed in the finite element simulation are provided in Table III. As for the work roller, it was modeled as an analytical rigid surface, and only the heat capacity needed to be defined. For this purpose, a constant value of 460 J/(kg °C) was assumed.²⁵ 

The cracks in the billet produced by continuous casting are usually local and uneven. To observe the internal cracks of the billet, a section is taken from the 1215 MS billet for experimental detection, as shown in Fig. 3. The existing detection results show that the typical cracks of the 1215 MS billet mainly appear at the center, and these cracks are closed and not oxidized.¹² It is generally believed that these cracks are caused by solidification shrinkage during the casting process, and the cracks are more likely to appear at the central position. These non-oxidized cracks can be healed in the rolling process under appropriate conditions.

According to the pore distribution of the blank after casting, the center of only a section of the billet (white color) was assumed to have porous material properties, as shown in Fig. 4. The GTN model was applied to the porous section, and a finer mesh was used for accurate representation. The entire billet comprised a total of 69 300 elements, out of which 1176 elements were specifically allocated to the porous region.

The closure of voids typically happens when a specific stress state is reached, causing the surrounding material to undergo plastic deformation.³³ To gain a deeper understanding of the void elimination mechanism, it is essential to analyze and examine the hydrostatic stress state of the workpiece. Furthermore, the impact of rolling conditions on void elimination can be better comprehended by analyzing and discussing the effects of rolling reduction, the friction coefficient between the workpiece and roller, and the rolling speed.

The pressure distribution and evolution throughout the workpiece are shown in Fig. 5 as it is compressed through the roller. As the workpiece is rolled, more elements experience compression, and the contact area between the workpiece and the roller has a larger pressure than the other area of the workpiece. The pressure of the contact area between the workpiece and the roller trended to decrease when the roller moved forward. Void elimination of the central section in the workpiece occurred when the compression took place. If the compression is sufficient at some positions in the workpiece, the void closure of the central section may occur.

A random element was selected from the center of the workpiece to simplify the presentation of the results. The selected element number is 69 191, as shown in Fig. 6. The compressive stress of the represented element’s evolution over time is shown in Fig. 7. From the ordinate axis on the left of Fig. 7, the peak compressive stress of the element appeared in 0–1 s, when the selected element passed through the roller subjected to reduction, whereas the compressive stress of the element tends to be gentle during the following 1–3 s. This is because the selected element is at the front part of the workpiece, and it is in a high-pressure stress state when the roller starts rolling. The compressive stress of the selected element becomes gentle as the roller moves forward, indicating that the element has been out of the pass and the stress is stable.

The variation of void volume fraction (VVF) with time during rolling was obtained as shown in Fig. 7. The initial void volume fraction of the porous material section was taken as VVF₀ = 0.06. It should be noted that the value of the VVF dropped sharply when the pressure stress was in the unsettled state in 0–1 s, whereas the value of the VVF was almost not changed when the pressure stress tended to be in the gentle state in 1–3 s. A schematic of the evolution of void volume fraction, VVF, with the rolling time is shown in Fig. 8. It should be noted that, in Fig. 8, the void elimination of each element occurred as the rolling continues and these elements have a different value of VVF at the same time. This discrepancy might be due to the different element locations with the roller, resulting in variations in the pressure stress state during rolling. Since the stress state in the workpiece is nonuniform, the void elimination times are different for each element. The results of the simulation show that the void elimination of the element could occur under now-available conditions and are strongly related to the pressure stress borne by the element.

Under different rolling reduction rates, keeping the friction coefficient 0.4 and rolling speed 0.55 RPS, void elimination occurred in different degrees as illustrated in Fig. 9. In this study, five pass reduction rates were investigated, which are equal to 22%, 20%, 17%, 14%, and 12%. The results reported in Fig. 9 show that all void elimination occurred under different reduction rates, and at higher reduction rates, void elimination occurred earlier and more quickly during the rolling. One potential explanation is that a greater reduction might result in increased pressure stress within the workpiece, which serves as a key factor influencing void elimination.

In addition, it can be seen that the higher the rolling reduction rate, the lower the ultimate VVF value, which reflects the healing degree of the material. This finding is consistent with the result proposed in Ref. 26. The ultimate VVF value is below 4.8%, which is considered to be the critical value to guarantee the material’s self-healing,²⁵ when the reduction rates are equal to or greater than 17% in this case. As a consequence, the higher rolling reduction rate indicates a higher compressive stress state and, therefore, a higher void closure.

To examine the effect of the rolling speed on the void elimination in the billet during the hot shape rolling process, where the rolling reduction rate has been fixed to 20% and the friction coefficient has been fixed to 0.4, the VVF evolution as time for different roller rotation speeds is plotted in Fig. 10. It is seen that the VVF value drops sharply from the initial value of 0.06 during the 0–1 s when the elements selected pass through the roller subjected to reduction, and the higher the rolling speed, the greater the rate of decline of the VVF value. Figure 10 shows that the ultimate VVF values are less distinctive under different rolling speeds and that the ultimate VVF values are larger distinctive under the different rolling reductions, which suggests that the pass reduction has more influence on the material’s self-healing during the rolling than the roller speed.

Furthermore, it was also observed that although the larger roller rotation velocity causes the VVF curve to decrease faster with time, the final value of VVF is not the smallest form Fig. 10. When the rolling rotation velocity varies from 0.55 to 0.75 RPS, the smallest value of the final VVF exists at a certain rolling rotation velocity of 0.65 RPS. In other words, when the rolling rotation velocity varies from 0.55 to 0.65 RPS, the final VVF decreases. Otherwise, when the rolling rotation velocity varies from 0.65 to 0.75 RPS, the final VVF increases. The possible reasons are that the slower rolling speed means a slower loading rate and, therefore, it is not easy to obtain high stress for the workpiece, and that the workpiece will have not enough time to deform from high stress around the void under the condition of the higher rolling speed. The above findings suggest that the void closure occurs at a certain rolling speed, which results in the lowest VVF value. Thus, it is considered that there is a rational roller rotation velocity in the model, which is beneficial to the material self-healing during the shape rolling.

It is also of interest to see how the friction coefficient between the roller and workpiece affects the void elimination in the billet during hot shape rolling process. By fixing the rolling reduction rate as 20% and the speed as 0.65 RPS, the VVF evolution with time for different friction coefficients is investigated, as shown in Fig. 11. It can be noted that different coefficients of friction have little influence on the evolution of VVF with time. Furthermore, it can be seen that the larger the friction coefficient, the lower the ultimate VVF value, whereas the ultimate VVF values for different friction coefficients are so close. The friction at contact interfaces between the workpiece and roller contributes to the nonuniformity of the deformation and the density variation in the workpiece during the rolling, which results in the void elimination occurring in different degrees. It is clear that the larger coefficient of friction is helpful in void elimination of the material during the hot shape rolling.

The results of this paper described above and resumed in Figs. 9–11 show that the pass reduction rate plays an important role in the void elimination of the billets during the hot shape rolling, and the roller rotation velocity and friction coefficient have a certain effect on the ultimate value of the VVF. A higher pass reduction can significantly decrease the ultimate value of the VVF, which improves the material’s self-healing during the rolling. The lower roller rotation velocity and larger friction coefficient can decrease the ultimate value of the VVF to a certain extent, which is more beneficial for the material’s self-healing during the rolling process.

In this study, we proposed a scheme for predicting void elimination of the 1215 MS steel during the hot shape rolling based on the GTN model. The simulations were performed on a specimen with central porous material properties by using the finite element software ABAQUS. We considered the effects of the pass reduction, roller rotation velocity, and friction coefficient between the roller and the workpiece on the ultimate value of VVF. Based on our research results, we draw the following conclusions:

1. The pressure stress results in a decrease in the ultimate value of VVF, and the pressure stress value and the period of holding pressure appear to determine the degree of void elimination. The pass reduction has a significant influence on the ultimate value of VVF during the rolling. With an increase in the pass reduction, the ultimate value of VVF gradually decreases. Therefore, for our case studies, a pass reduction of greater than 17% should be chosen in hot shape rolling.

2. The roller rotation velocity and the friction coefficient have some impact on the void elimination, but they are not as significant as the pass reduction. Our results indicate that there is a critical rolling speed value for optimal void elimination. In addition, a higher friction coefficient is more beneficial for achieving better material void elimination during hot shape rolling. By employing computer simulations of hot shape rolling, it becomes possible to predict the void elimination in the workpiece and provide guidance for industrial production to a certain extent.

This research was funded by the Doctoral Research Fund of the HUAT (Grant No. BK202213).

The authors have no conflicts to disclose.

Shuiwen Zhu: Conceptualization (equal); Methodology (equal); Writing – original draft (equal). Yu Fu: Data curation (equal); Software (equal). Shunxin Wu: Resources (equal). Shuangxi Guo: Conceptualization (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.