Research on the impact of mandrels in titanium tubes during tube continuous rolling

Tube continuous rolling (TCR) is a beneficial method for the production of titanium pipes due to its high efficiency and shorter process, with the mandrel playing a crucial role. The present study delved into the mechanism of the mandrel and its impact on the TCR process, through comprehensive analyses that included numerical simulations and experimental tests. The following conclusions were drawn: as the mandrel diameter coefficient increases, the area of the front slip zone gradually expands, and the tension changes from tensile stress to compressive stress. Additionally, the temperature exhibited an overall downward trend, while the strain distribution at the groove vertex and taper became more uneven. Concerning metallographic structure, instances where the mandrel diameter coefficient increases led to elongated, recrystallized grains at the groove vertex. The twinning quantity at the groove taper increased initially and then decreased, indicating a deformation mechanism shift from twinning to slip.


Introduction
Titanium and titanium alloys are characterized by high specific strength, long fatigue life, and excellent corrosion resistance to seawater and aqueous media. It is an attractive material for offshore use, especially in saltwater systems and other hostile environments [1][2][3]. Grade 2 (commercially pure titanium) pipe is by far the most used. It is readily cold bent, and easy to weld. This together with relatively low costs and good availability make it the preferred material for seawater and process fluid management systems and heat exchangers [4][5][6][7][8][9]. Currently, the production of pure titanium tube mostly employs the processes of hot extrusion+cold rolling+nnealing [10]. In comparison hot continuous rolling rolling has advantages such as multi-scale production, high production efficiency, low energy consumption, and shorter processing time [11]. Tube continuous rolling (TCR) is an ongoing thermal deformation process that employs rolls and mandrels as internal and external tools to adjust the outer diameter and wall thickness of a hollow billet. As a result, the state of the mandrel is crucial in this production procedure.
There is an abundance of literature available on the condition of the mandrel during the tube continuous rolling. Zhou [12] has conducted a study on the influence of mandrel limiting speed on the size accuracy of hollow shell and axial force of the mandrel. Yin et al [13] have employed a three-dimensional elastic-plastic thermal coupled finite element method to obtain the impact of mandrel velocity and friction coefficient on the force-energy parameters and deformation parameters during rolling. Zhao et al [14] have calculated the friction coefficient between the rolled piece and the mandrel in a Fully-floating mandrel mill and observed that this coefficient displayed a rising trend along the rolling direction. V. N. Danchenko and S. R. Rakhmanov [15] have established a mathematical model for the rod system in the mandrel-retention mechanism, which considers the Any further distribution of this work must maintain attribution to the author-(s) and the title of the work, journal citation and DOI.
oscillations conditions of the mandrel. Toporov et al [16] have used a finite element simulation program to study the impact of mandrel deviation from the rolling axis on the shape of pipes and metal fracture. Struin et al [17] have described the main developments and research aimed at extending the service life of mandrels of FQM-type continuous rolling mills.
The reduction of wall thickness depends on the diameter of the mandrel, however, only Chernykhu et al [18] have briefly described the impact of mandrel diameter on the shape and accuracy of the hollow shell, indicating a lack of deep understanding of its influence on the rolling process. Therefore, this manuscript employs a combination of numerical simulation and experimental methods to investigate the rolling forming mechanism under different mandrel diameters. It provides a detailed analysis of the slip, tension, stress-strain fields, metal flow, and temperature, while also discussing the impact of mandrel diameter on the microstructural evolution of pure titanium tube.
2. Finite element modeling (FEM) and experiments 2.1. Determining material constitutive relation Allowing for the significant impact of rolling temperature, strain and strain rate on rolling stress, straindependent hyperbolic sine constitutive model is used in this experiment [19].
where α, A and n are strain dependent material constants, e  is strain rate (s −1 ), σ is flow stress (MPa), Q is activation energy of deformation (kJ/mol), R is the gas constant (kJ/mol K −1 ) and T is absolute deformation temperature(K).
To demonstrate the reliability of the aforementioned constitutive model, flow stress prediction values are calculated under various deformation conditions using equation (1) and compared with experimental values, as shown in the figure 1. The average absolute relative error (AARE) is found to be 7.23% and the correlation coefficient (R) is determined to be 0.9889, demonstrating a high degree of predictive accuracy for the constitutive model.

Establishment of boundary conditions
Establish a TCR model of a hollow tube made from commercially pure titanium with the material designation of TA2 using the DEFORM software. The model comprises three roll stands with three identical rollers arranged symmetrically at an offset of 120°around the rolling centerline at each stand, as shown in figure 2(a). The corresponding position of the groove vertex and groove taper are shown in figure 2 The friction between the material and the tools is assumed to conform to the constant shear stress friction. Because of the water-based graphite lubricant daubed on the surface of the mandrel, the friction factor between the tube material and the roller is assumed at 0.7, and 0.08 between the tube material and the mandrel [20,21]. Prior to the TCR process, the hollow billet is heated at a temperature of 800°C, while the rollers and the mandrel are maintained at a constant room temperature of 20°C. The heat transfer coefficient between tube and roll is 20 MW/(mm2·K), and the heat transfer coefficient of thermal convection and thermal radiation is 0.15 MW/(mm2·K) [22]. The heat conversion coefficient caused by friction and plastic deformation is 0.7 and 0.9, respectively [23,24].
Prior to establishing the FEM of the TCR, the following assumptions are made: (1) Due to the minimal elastic deformation of the rollers and mandrel, they are considered rigid materials that only conduct heat during the modeling process.
(2) Compared to the plastic deformation of the hollow billet, its elastic deformation is negligible. Therefore, the hollow billet is considered a plastic model.
(3) In order to maintain a symmetrical rolling process and reduce the central processing unit processing time, only one-sixth of the model is simulated and the distance between the passes is appropriately shortened.
(4) The friction between the mandrel and hollow billet, as well as between the rollers and hollow billet, is assumed to have an ideal coefficient of friction that remains constant during the processing. The formula for the friction force in the shear friction model is (equation (3)): where fs is the frictional stress, k is the shear yield stress, and m is the friction factor.
A summary of the process parameters in the tube rolling process is presented in table 1. The schematic of the groove is shown in figure 3, and specific parameters are listed in table 2.

Experimental tests
The original microstructure of the hollow billet utilized in the experiment is depicted in figure 4. Experimental tests on the TCR were conducted using the Premium Quality Finishing (PQF) testing mill available at Taiyuan University of Science and Technology in Taiyuan, China. Figure 5(a) depicts the front view of the PQF where the mandrel is driven by a hydraulic trolley, and the utilization of medium-frequency induction furnaces for heating titanium tubes allows for online heating, reducing heat loss during mandrel insertion and transfer to the mill. Figure 5(b) illustrates the rear view of the PQF, show that the rollers are arranged at an angle of 120 degrees to each other and driven by three separate direct current motors. Furthermore, a torque sensor is mounted on the universal joint shaft corresponding to a specific rolling roller in each of the three passes. Assuming that the rolling torque is uniformly distributed among the three rollers in each pass, the torque measurement data are acquired using a multi-channel data acquisition card and recorded on a computer. To investigate the mechanism of a mandrel on TCR process, mandrel diameter coefficient is defined by:

Metal slip
The slip phenomenon has a critical impact on the plastic deformation and mechanical behavior of metals during rolling processes. In the case of TCR, unlike with sheet rolling, the calculation of slip must take into consideration not only the velocity increase reflected by the metal extension in the axial direction of the deformation zone, but also the variation in circumferential velocity of the rollers, along the width of the groove. Illustrated in figure 6, the direction of the arrow signifies the velocity vector's orientation, with its length indicating the difference in velocity magnitude. The angle indicates the circumferential position of the rolled tube, where 0°represents the groove vertex, and 60°represents the groove taper. The velocity vector corresponds to the rolling orientation, indicating that the metal's velocity surpasses the roller's line speed at that point -this area is called the front slip zone. Conversely, if the metal's velocity is slower than the roller line speed, it is referred to as the back slip zone. The diagram depicts the boundaries of the front slip zone and the contact zone. It should be noted that at the groove vertex, the metal's exit velocity exceeds the roller speed, causing front slip. In contrast, near the sidewall of the groove, the roller speed increases due to the larger roller diameter, causing the metal's exit velocity to be lower than the roller line speed, resulting in back slip.
As mentioned previously, when c is less than 0.3, the front slip zone is limited to a partial area at the groove vertex. At c = 0.3, the entire deformation area at the tube exit is encompassed by the front slip zone. Despite the increase in velocity difference within the front slip zone with increasing c up to 0.4, the area of the front slip zone does not exhibit significant enlargement. This is primarily attributed to the tension control mechanism that regulates front and back slip to achieve uniform mass flow. A detailed analysis of this specific mechanism will be presented in the ensuing section.  As mentioned in the previous paragraph, the forward slip zone is only present near the groove vertex area when c is below 0.3, and it continuously expands as c increases. When c is 0.3, the forward slip zone is distributed throughout the entire deformation zone at the tube exit, as shown in figure 6(c). By comparing the forward slip zones and their velocity magnitudes of the tubes in figures 6(c) and (d), it can be observed that although the arrow at the exit is longer for c = 0.4 than c = 3, the yellow area does not significantly increase. This means that although the difference in forward slip zone velocity continues to increase, the area of the zone does not significantly increase. This is primarily attributed to the tension control mechanism that regulates front and back slip to achieve uniform mass flow. A detailed analysis of this specific mechanism will be presented in the ensuing section.

Tension
Effective tension control is a crucial factor in ensuring the stability of the TCR. As shown in figure 7, tension varies with the mandrel diameter coefficient. With an increase in the mandrel diameter coefficient, tension shifts from positive to negative. At c = 0.3, both front and back tensions become negative, resulting in the transformation of tensile stress to compressive stress. Maximum tensile stress is achieved at c = 0.4, which explains why there is no significant increase in the area of the front slip zone at c = 0.4 compared to c = 0.3. Although an increase in the mandrel diameter coefficient leads to an increase in the exit speed of the workpiece, the amplified compressive stress in the front and back regions of the workpiece hinders expansion of the front slip zone area. This is a result of the interplay between the two factors, and it is clear that tension has the greatest impact on metal slip at c = 0.4.
Furthermore, in order to achieve steady-state rolling of the tube, the rate of change in tension decreases as a result of the decreasing rate of change of the cross-sectional area at the exit of the tube while maintaining constant rolling speed of the rollers. The conclusion can be inferred from the equation (5) T where T represents the tension, v i denotes the roll speed of the i-th pass, F i represents the cross-sectional area at the exit, and i-1 refers to the (i-1)-th pass. This equation is derived based on the principle of equal mass flow. Simultaneously, the magnitude of the front tension consistently exceeds that of the rear tension. Figure 8 shows the strain field at the exit of the second pass, where R, Theta, and Z denote the radial, circumferential, and axial directions of the tube, respectively. The radial strain in the groove vertex region is compressive, while that in the groove taper region has both compressive and tensile components. Meanwhile, the circumferential strain in the groove vertex region is tensile, whereas it is compressive in the groove taper region. As for axial strain, both regions experience tensile strains.

Stress and strain fields
It can be observed that the radial compressive strain, circumferential tensile strain, and axial tensile strain in the groove vertex region increase with an increase in the mandrel diameter. Additionally, as the mandrel diameter increases, the effect of the mandrel on the workpiece also increases, leading to greater metal flow in the circumferential and axial directions due to the greater radial compressive strain. This increase in metal flow leads to a more uneven distribution of strain in the groove vertex and groove taper regions. Figure 9 shows the stress field at the exit of the second pass. The radial, circumferential, and axial stresses in the groove vertex region are all compressive stresses. With an increase in the mandrel diameter coefficient, the three-dimensional stresses at the groove vertex location increase. As for the groove taper region, the radial stress has both tensile and compressive components, while the circumferential stress is mostly compressive, and the axial stress is entirely tensile, with a magnitude much larger than that of the radial and circumferential stresses.
Since the instantaneous axial stress cloud diagram cannot fully reveal the details of the tensile stress, a local magnification chart displays the trend of axial tensile stress at the groove taper region during the rolling process. It is found that the tensile stress increases with an increase in the mandrel diameter, and the tensile stress at the groove taper region is the main cause of cracking and tearing of the tube during the TCR. Therefore, in order to achieve good surface quality of the tube, the axial tensile stress should not be too large. Secondly, with an increase in the mandrel diameter coefficient, both the equivalent stress at the groove vertex and the groove taper regions increase.

Temperature field
Due to the consistent temperature distribution trends between the inner and outer surfaces of the workpiece [25], this article only analyzes the temperature distribution of the outer surface and central layer during rolling. Four nodes at the outer surface and center position of the groove vertex and groove taper were selected for analysis, as shown in figure 10(a). Specifically, P1 and P2 are located at the groove vertex of the first and third passes, while P3 and P4 are situated at the groove vertex of the second pass. Figure 10(b) illustrates the time history of rolling temperature for four different tracking points. It can be observed that there is a strong temperature non-uniformity in the groove vertex region of each stand, mainly due to the contact heat transfer between the tube and the mandrel/rolls, resulting in a rapid drop in the outer surface temperature followed by a slow rise, while the temperature in the central layer rises slightly due to plastic heat generation and then drops. As for the groove taper region of each stand, the temperature exhibits a slight increase, which is caused by the plastic heat generated by the tensile stress in the groove region, but the overall change is relatively small compared to the groove vertex region.
In figure 11, a comparison of rolling temperature history at the P1 and P2 tracking points during the TCR process using different mandrel diameter coefficient is presented. As the mandrel diameter coefficient increases, P2 experiences an increase in the amount of heat generated due to an increase in deformation, whereas P1 experiences an increase in temperature drop due to an increase in contact area with the roller. Additionally, the increase in core diameter leads to a decrease in pipe wall thickness, which causes a greater temperature drop due to heat exchange with the environment. The degree of temperature decrease observed in P1 is significantly greater than the temperature rise in P2. Furthermore, during the rolling detachment stage, both tracking points transfer heat to each other, resulting in an overall decreasing trend in the temperature of both points.

Material flow
Material flow has a significant impact on the quality of the workpiece. Figure 12(a) displays the initial cross-sectional mesh. The deformed cross-sectional mesh after the second pass is shown in figure 12(b). The material undergoes substantial radial compression at the groove vertex, with the amount of compression gradually decreasing towards the groove taper. Different circumferential flows are generated from the groove vertex to the groove taper, due to variations in contact conditions with the mandrel and rollers. These contact conditions, combined with the rolling temperature, determine the differences in microstructure at the groove vertex and the groove taper.
To analyse the impact of varying mandrel diameter coefficient on metal flow, all inner surface nodes were selected as depicted in figure 12(a). Figure 12(c) displays the angle deviation of the nodes after second pass of rolling, where in it can be observed that circumferential flow increases before decreasing. Furthermore, the circumferential flow of the metal varies with the mandrel diameter coefficient. As the mandrel diameter coefficient increases, the entire cross-sectional circumferential displacement also increases, leading to an even greater non-uniformity of metal flow.

Verification of the finite element model
The comparison between simulated and experimental data of the mechanical parameters is shown in figure 13, it can be observed that the trends in the simulated values and the experimental values for both rolling torque and rolling force are consistent. They both increase and then decrease, achieving the maximum value at the second rolling pass. At the same time, the simulated values are lower than the experimental values, up to a maximum of 16% for rolling torque and 14% for rolling force. This is due to the fact that the modeling did not consider the heat loss during the transfer of the tube from the furnace to the rolling mill, as well as some idealized modeling processes. However, overall, the simulation model exhibits minimal error and shows a high degree of reliability.

Microstructure evolution
The Grain morphology of the RD-ND in the tube after rolling observed by optical microscope (OM) is shown in figure 14. To better illustrate the differences in grain morphology changes and twinning states, the analysis is only conducted on c = 0.2, c = 0.3, and c = 0.4. For the area near the groove vertex, at c = 0.2, some grains showed a change in morphology, where their long axis formed a smaller angle with RD; at c = 0.3, many grains   displayed a wavy morphology as they continued to elongate at the RD direction. The formation of complex grain shapes in TCR may be due to two reasons: firstly, due to the three-axis compressive stress state in the deformation zone, as shown in figure 5, some grains bend to adapt to the compressive stress [26]. Secondly, under small deformation, local uneven stress fields in the thickness direction of the tube may lead to the formation of wavy grains. We also observed the appearance of recrystallized grains, which can be attributed to the increase in central temperature caused by large compression, and the promotion of dynamic recrystallization due to the large strain. When c = 0.4, with the increase of plastic deformation, the deformation became more uniform, thus most grains are further compressed and elongated along the RD direction, and the appearance of some dynamic recrystallization(DRX) grains recrystallized grains is also observed.
Different from the grain morphology at the groove vertex, a large number of twinning can be clearly observed at the groove taper. This is attributed to the one-way tensile stress state only present at the groove taper. Furthermore, with the increase of the mandrel diameter coefficient, the amount of twins first increases and then decreases at the groove taper. With the increase of deformation, the twins increases. However, as the deformation increases further, the deformation mechanism of the material deforms from twinning to slip, resulting in a decrease in the number of twins.

Conclusions
By conducting experiments and numerical analysis, this study investigated how the mandrel diameter influences the forming of tube continuous rolling and the mechanisms underlying microstructure evolution. The results reveal the following findings: 1. As the mandrel diameter coefficient increases, the front-slip region extends from the groove vertex to the entire deformation width. However, when the mandrel diameter coefficient reaches 0.4, there is no significant increase in the front-slip area compared to 0.3. This is mainly due to the change in tension from tensile stress to compressive stress, hindering the increase in the front-slip area.

2.
A larger mandrel diameter coefficient results in a greater magnitude of radial strain and stress, causing an increased circumferential flow of the metal and an even more non-uniform distribution of strain along the groove vertex and taper.
3. An increase in the mandrel diameter coefficient leads to a consequent elevation in temperature in the central layer. Nevertheless, the amplified contact area between the tube and the rollers engenders a more pronounced cooling effect, thereby resulting in an overall decrement in the temperature of the workpiece. 4. The elongation of grains at the groove vertex and the appearance of recrystallized grains are both observed with an increase in the mandrel diameter coefficient. In terms of the groove taper, the number of twins varies with the mandrel diameter coefficient, first increasing and then decreasing. This indicates a shift in deformation mechanism from twinning to slip.