Introduction

Seamless tubes are fundamental materials for many industries, with numerous mechanical applications [1,2,3]. The fabrication process for seamless tubes begins with roll-piercing [4], extrusion [2, 5], and rolling [6] of the initial material, followed by cold drawing to create the tube. Preform materials may be geometrically and mechanically poor because of thickness non-uniformity, shape ovality, and non-homogeneous residual stress.

In industrial applications, some eccentricity is allowed; however, the consumption of the material becomes inefficient because the thickness specification is determined by the minimum thickness. The eccentricities of industrial tubes are between 1 and 3% for aluminum, up to 5% for copper, and up to 10% for steel [2, 5, 7] as defined by the difference between the maximum and minimum external diameters of the tube per its mean external diameter [2]. These geometrical and mechanical features are inevitable during fabrication because the tool (e.g., the plug) does not allow flexible control during pre-work step.

Additionally, non-homogeneous residual stresses can cause a decrease in structural strength and local metallurgical weakness during service and tube drawing. Residual stress may induce buckling of cylindrical tubes [8], and severe residual stresses may increase manufacturing costs during post-work surface or heat treatments intended to improve the formability and crack resistance of the final product. In contrast, improvements in non-homogeneity of the drawn tubes are useful to process engineers because consumer demand steadily pushes tube manufacturers to reduce tube weight and cost. This demand is expected to grow with existing efforts to combat global warming and improve material strength and use efficiency.

Tube drawings with various types of plugs have thus been studied in multiple ways [8,9,10,11]. Neves et al. [9] attempted to identify the best die and plug geometry to reduce the drawing force using a finite element method (FEM). Kuboki et al. [10] studied the effect of a plug on residual tube stress by drawing tubes to determine the minimum bare reduction in thickness that is needed to level out and reduce residual stress. Yoshida et al. [11] experimentally and numerically studied fine tube drawing, with an emphasis on fixed and floating plugs. Linardon et al. [12] conducted experimental analyses of a tube drawing process with a long conical plug, then presented a numerical model with a pressure-dependent friction model. However, it is difficult to address geometrical and mechanical insufficiencies with conventional cold tube drawing because the plug is nearly always free to move.

Most research investigations have been based on the idealization of symmetry or perfection of the starting materials and dies or tools. Yoshida et al. [13] experimentally and numerically studied the tube drawing of Ni–Ti shape-memory alloy for medical applications, emphasizing the usage of a plug to improve the surface quality of the drawn tube. Palengat et al. [14] experimentally and numerically studied the drawing process of thin-walled 316L stainless steel tubes for medical applications. Alexandrova [15] analytically studied the failure of incompressible, rigid perfectly plastic material during drawing processes using Hill’s general method and a fracture criterion based on a workability diagram. Rubio [16] conducted an analytical study of tube drawing using various theoretical approaches. Lee et al. [17] studied the causes of material fracture during cold tube drawing using a rigid-plastic FEM and the theory of ductile fracture.

Palengat et al. [18] presented an analysis model to accurately predict the drawing process, with an emphasis on hypothesis, physical modeling (flow behavior, tool behavior, and friction), and numerical modeling (load, meshing, friction, and convergence). Rubio et al. [19] analytically studied the tube drawing process, with an emphasis on the effects of input parameters (e.g., material flow behavior, die conical angle, area reduction, and friction conditions) on the process and product using an upper bound method. Boutenel et al. [20] presented a model for predicting the final dimensions of the tube with high accuracy in the cold tube drawing process. They provided some detailed information regarding the finite element (FE) model, including geometry, materials, boundary conditions, mesh, and numerical factors.

Pirling et al. [2] measured the residual strain of a cold-drawn copper tube using neutron diffraction to resolve the interconnection between the geometrical changes in the tubes caused by deformation non-homogeneity and the resulting residual stresses. Gattmah et al. [21] developed a technique for the measurement of residual stress for drawn tubes; the approach was validated by a comparison of FE predictions and X-ray diffraction results.

Béland et al. [22] conducted process design optimization of a drawing process in terms of the drawing force or maximum stress level to reduce the number of drawing stages (passes). Sheu et al. [23] optimized the profiles of the drawing die orifice and the plug, in terms of both drawing force and strain non-homogeneity, using the Taguchi method with the FEM. Hosseinzadeh and Mouziraji [24] attempted to identify optimal combinations of drawing parameters in terms of drawing force and thickness distribution in the round-to-square tube drawing. They also studied the optimization of process design parameters using an FEM with an optimization technique. Jafari et al. [25] presented an optimization approach that combined the Taguchi method and FEM for tube drawing; their approach identified the minimum forming force necessary to produce the minimum dimensional error. Farahani et al. [26] presented an optimized curved die profile for tube drawing that used a fixed conical plug to minimize the necessary tensile stress or drawing force and to reduce the stresses exerted on the die and plug; these effects were based on the appropriate determination of flow behavior of the material, area reduction, plug angle, and friction conditions.

However, few researchers have studied tube drawing via realistic tube drawing processes with non-uniform thicknesses and/or non-idealized tools or dies. Tetley [27] reported that the drawn tube could be significantly improved during block tube drawing, depending on the initial eccentricity of the tube and reduction in the area of the drawing process. Pirling et al. [3] studied the effects of wall thickness, eccentricity, ovality, and residual stress in tube drawing on tube qualities using a three-dimensional FEM. Foadian et al. [7] experimentally and numerically studied the effects of tilting the die and shifting the tube on the resulting tube eccentricity and residual stresses, using as-received tubes with eccentricity; they showed that a tilted die improves product quality with respect to residual stresses. Al-Hamdany et al. [5] conducted experimental analyses of the effect of a tilted die in tube drawing on the texture, dislocation density, and mechanical properties of the drawn tube; they concluded that the method of tilting the drawing die is effective for controlling tube eccentricity.

This study presented a finite element analysis model to investigate the tube drawing process characterized by a tilted drawing die and non-uniform raw tube thickness using an implicit elastoplastic FEM and a multibody treatment scheme (MBTS [28]). MBTS treated the contact condition between the tube and plug using the penalty method. The plug was regarded as a laterally movable and deformable body to consider its movement and vibration inside the tube. However, its rotational motion was constrained. The thickness distribution of the prediction was compared with experimental results found in the literature, and the effects of bearing length were analyzed. Our analysis results showed that bearing length and tilted angle strongly affect the eccentricity of the tube produced by tube drawing.

FE modelling

Definition of process

Al-Hamdany et al. [5] experimentally studied the drawing process of a copper tube with a ± 5° tilted die; a schematic diagram of the + 5° tilted die case is shown in Fig. 1. The extruded and annealed copper tube (outer diameter: 65.0 mm; mean thickness: 5.5 mm) was investigated. In this study, we used the tube drawing process described by Al-Hamdany et al.; our results were numerically studied and compared with their experimental results. The non-uniform thickness of the initial tube only at the cross-section was considered. Following Al-Hamdany et al.'s work, the minimum and maximum thicknesses of the initial tube were purposely fixed at 5.14 and 5.78 mm, respectively, which were defined by the eccentricity, i.e., the gap between the central lines of inner and outer circles of the tube cross-section. Importantly, the maximum thickness is located on the opposite side of the x-direction for all figures presented in this paper.

Fig. 1
figure 1

Schematic diagram of tube drawing process studied

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The tube was assumed to behave as an elastoplastic material. A three-dimensional FE model (i.e., a half model) was developed for the tube drawing process by adaptation of an implicit elastoplastic FEM with the MBTS. In the work of Al-Hamdany et al., three main parameters were not described or considered: bearing length of the die, denoted as \({l}_{b}\); fillet radius of the die’s inner corner, denoted as \({R}_{f}\); and friction coefficients. Point O on the drawing die in Fig. 1 is the center of rotation when the die is tilted clockwise or counterclockwise. In this study, the drawing velocity \({V}_{d}\) was fixed at 1.0 mm/s because the material was assumed to be rate-independent. Additionally, the friction coefficient between the tube and plug, and between the tube and die, was assumed to be 0.05 [29,30,31]. Dies with different bearing lengths and fillet radii, lb and \({R}_{f}\), varying from 5–20 mm and 0.5–30 mm, respectively, were used to investigate the effects of these parameters on the thickness distribution after tube drawing. Table 1 summarizes the key dimensions of the process studied.

Table 1 Key dimensions of initial tube drawing process [5]
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Table 2 shows the material and flow properties of the tube and plug. The flow function of the copper tube was characterized using a generalized blending flow model [32] from the reference flow curve (RFC) described by the selected points from O to T, as shown in Fig. 2. The RFC was obtained using the method of Joun et al. [30] from the experimental tensile test curve, as shown in the lower panel of Fig. 2, given by Al-Hamdany et al. [5]. The dashed line in the lower panel of Fig. 2 is the theoretical tensile test curve predicted using the generalized blending flow model, which implies that it is accurate. Note that point N (\({\varepsilon }_{t}=0.41\)) on the flow curve (true stress–strain curve) is corresponds to the necking point \({{\text{N}}}_{e}\) (\({\varepsilon }_{e}=0.50\)) on the engineering stress–strain curve. The experimental flow curve from N to T describes the post-necking strain hardening. It is also noted that the predicted engineering stress–strain curve is in good agreement with the experiments, as can be shown in the lower panel of Fig. 2. However, it can be seen that the flow curve is remarkably different from the engineering stress–strain curve, compared with the forgeable steels including SCM415, SCM435, and SWCH10A [33]. The reason for such a big difference lies in the significant elongation of the material, like stainless steel [34]. The initial effective strain of the tube was assumed to be 0.1 to satisfy the value of 124 MPa for its initial flow stress.

Table 2 Material properties of tube material and plug [5]
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Fig. 2
figure 2

Flow information of the copper

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Because of the difficulty of placing the plug in the correct position, the material and plug were assumed to simultaneously move towards the orifice until the process reaches the steady-state condition. The bottom side of the material and plug were regarded as velocity-prescribed boundaries in the drawing direction until the plug reached its target position.

The floating plug has a significant effect on correction for misalignment errors that affect tube quality. Figure 3 shows the FE model; here, the importance of the modeling technique of the plug is emphasized under the conditions of non-axisymmetric tube drawing, along with die misalignment and tube thickness non-uniformity. The drawing die was assumed to be rigid, whereas the plug was regarded as a laterally movable and deformable (elastic–plastic) body. However, the plug was regarded as a laterally fixed and rigid body only as an exceptional case in Fig. 3a. The predicted histories of the deformation and the effective strain of the tube drawing process for both rigid and deformable plugs are shown in Fig. 3a and b, respectively.

Fig. 3
figure 3

History of deformation with effective strain (+ 5° tilted die case)

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In Fig. 4, the laterally movable and deformable plug moves towards the right side because of the need for force equilibrium. This demonstrates that the treatment scheme of the plug is a critical consideration during non-axisymmetric tube drawing. Notably, there is no actual axisymmetric drawing because of the eccentricity of the initial tube.

Fig. 4
figure 4

Nodal velocity components of tube material and plug [mm/s] (+ 5° tilted die case)

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FE predictions of tube drawing processes with a tilted die

Numerical effects on predictions

We assumed a plane of symmetry for the tube drawing process. First, the numerical effects were assessed using various FE mesh systems [35]. The same die and process conditions were used. For this purpose, the effects of the number of tetrahedral elements on tube reference diameter and eccentricity were investigated. The reference diameter, denoted as \({\varnothing }_{f}\) as shown in Fig. 1b, is the predicted distance of the two external surfaces of the drawn tube on the symmetric plane. The eccentricity, denoted as e, is defined as follows:

$$e=\frac{{t}_{max}-{t}_{min}}{{t}_{max}+{t}_{min}},$$
(1)

where \({t}_{max}\) and \({t}_{min}\) are the maximum and minimum thicknesses of the drawn tube, respectively.

The tube drawing process of Al-Hamdany et al. [5] was conducted using the initial tube diameter of 65.0 mm, and minimum and maximum thicknesses of 5.14 and 5.78 mm, respectively, as summarized in Table 1. The initial tube eccentricity was 5.86%. All predicted tube thicknesses and diameters used in this study were obtained after springback analyses. Figure 5 shows the FE mesh systems constructed with the lowest (39,000) and highest (142,000) tetrahedral elements tested for both the plug and tube material.

Fig. 5
figure 5

FE mesh systems of half FE model (+ 5° tilted die case)

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In Fig. 6, the e-values slightly decreased as the number of tetrahedral FEs increased, up to approximately 90,000; subsequently, the e-values generally remained constant. When 39,000 tetrahedral elements were used, the maximum predicted eccentricity was 4.41%; when 89,000 tetrahedral elements were used, the minimum predicted eccentricity was 3.14%. Figure 6 shows that as the number of tetrahedral FEs approached 90,000, the predicted e-values became more stable and reliable, as revealed in a previous study [36] regarding the FE simulation of a rotary tube-draw bending process. In contrast, the reference diameter \({\varnothing }_{f}\) became stable beginning at 60,000 tetrahedral elements. An FE mesh system with approximately 100,000 tetrahedral elements was thus used for all FE models in this study.

Fig. 6
figure 6

Effects of the number of tetrahedral elements on the eccentricity and reference diameter of the drawn tube

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Effects of bearing length and fillet radius on eccentricity

The bearing length, \({l}_{b},\) and the fillet radius of the die’s inner corner,\({R}_{f}\), have significant influences on the drawing phenomena for non-axisymmetric tube drawing, particularly when a tilted drawing die is involved. These effects occur because the tilted angle directly affects the reduction of the area.

In accordance with the work of Al-Hamdany et al. [5], \({t}_{max}\) and \({t}_{min}\) after the first pass, in the case of die tilted + 5°, were 4.63 and 4.35 mm, respectively. Therefore, the e-value experimentally determined from the literature was 3.42%; this value was used in our numerical study. However, because Al-Hamdany et al. did not provide any details regarding \({l}_{b}\) and \({R}_{f}\), we selected four \({l}_{b}\) values (5, 10, 15, and 20 mm) and three \({R}_{f}\) values (0.5, 10, and 30 mm) for our simulation of the non-axisymmetric tube drawing process. Figures 7 and 8 show two examples of assumed dies using the selected \({l}_{b}\) and \({R}_{f}\) values, respectively, with an emphasis on the projected die orifice distance on the symmetric plane, denoted as \({\varnothing }_{p}\); importantly, \({\varnothing }_{p}\) is similar to \({\varnothing }_{f}\) in the figures.

Fig. 7
figure 7

Schematic diagrams of drawing dies using different \({{\varvec{l}}}_{{\varvec{b}}}\) values, with an emphasis on \({\boldsymbol{\varnothing }}_{{\varvec{p}}}\) [mm]

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Fig. 8
figure 8

Schematic diagrams of drawing dies using different \({{\varvec{R}}}_{{\varvec{f}}}\) values, with an emphasis on \({\boldsymbol{\varnothing }}_{{\varvec{p}}}\) [mm]

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Figure 9 shows the variation in the reference diameter of the drawn tubes with \({l}_{b}\) and \({R}_{f}\), indicating that the mean outer diameter of a tube drawn by the tilted die may decrease when the bearing length increases or the fillet radius decreases.

Fig. 9
figure 9

Predicted reference diameters, \({\varnothing }_{f}\) with different \({l}_{b}\) and \({R}_{f}\) for the + 5° tilted die cases

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Figure 10 shows the bearing length–eccentricity curve of the process with the + 5° tilted die for fillet radii of 0.5, 10, and 30 mm. Notably, it was calculated after the prediction of springback because the tube drawing process is considerably affected by the springback phenomenon [23, 24]. Eccentricity linearly decreased with bearing length, except for the shortest bearing length, and a distinct minimum point at the bearing length of approximately 10 mm was observed for all fillets.

Fig. 10
figure 10

Bearing length–eccentricity curve with different \({l}_{b}\) and \({R}_{f}\) values for the + 5° tilted die case

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Eccentricity is primarily affected by bearing length when a tilted die is used because it determines the \({\varnothing }_{p}\) value or the area reduction. This relationship is depicted in Figs. 7 and 8, implying that a tilted die with an appropriate bearing length should be used to reduce the eccentricity of drawn tubes. Figure 11 shows the changes in traction and material–die interface according to bearing length in the + 5° tilted die case, indicating that contact discontinuity varies according to bearing length. Notably, contact on the thicker side is not affected by the bearing length, whereas contact on the thinner side is affected. These findings suggest that contact discontinuity is a primary factor influencing thickness uniformity.

Fig. 11
figure 11

Tube traction for different \({{\varvec{l}}}_{{\varvec{b}}}\)(\({{\varvec{R}}}_{{\varvec{f}}}\) = 10 mm, + 5° tilted die)

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Considering that the tube was initially in contact with the die at the left side (i.e., thicker wall), the tube tended to move towards the x-direction. Inwardly, the plug used for this process was being pushed in the same direction. This relationship explains the higher value of traction force and effective strain on the right side in all tubes in Figs. 11 and 12, respectively.

Fig. 12
figure 12

Effective strain of drawn tube for different \({{\varvec{l}}}_{{\varvec{b}}}\)(\({{\varvec{R}}}_{{\varvec{f}}}\) = 10 mm, + 5° tilted die)

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In Fig. 12, the material in the entry side was bent to the left in the case of a 5-mm bearing length and to the right in cases of 15- and 20-mm bearing lengths when it was observed towards the negative z-axis; material alignment was maintained in the case of a 10-mm bearing length.

Furthermore, all bearing lengths produced similar effective strain distribution patterns: larger effective strain of the drawn tube on the thinner side than on the thicker side, and minimum effective strain at the middle between the two sides (except in the case with the lowest bearing length). The larger effective strain on the thinner side presumably led to improved thickness uniformity (i.e., reduction in eccentricity). As indicated by the effective strain rate distribution in Fig. 13, the material point passed through a longer distance on the thinner side than on the thicker side because the material in the exit side leaned to the right. The 15- and 20-mm bearing length cases exhibited a large difference in effective strain rate between the middle point and the thicker side, potentially leading to deterioration of the thickness uniformity of the drawn tube, compared with the 10-mm bearing length case.

Fig. 13
figure 13

Effective strain rate of drawn tube for different \({{\varvec{l}}}_{{\varvec{b}}}\)(\({{\varvec{R}}}_{{\varvec{f}}}\) = 10 mm, + 5° tilted die)

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Notably, the projected lengths of the material–die contacts on the symmetric plane varied according to bearing length, indicating that the 10-mm bearing length case exhibited minimum and maximum projected lengths on the left and right sides, respectively, when they were observed in the negative z-axis direction. These findings are related to the minimum eccentricity at the bearing length of 10 mm, as shown in Fig. 10.

Experimental validation of predictions

In accordance with the work of Al-Hamdany et al. [5], the drawing die was tilted in two directions: clockwise and counterclockwise, denoted as − 5° and + 5°, respectively. The experiments by Al-Hamdany et al. indicated that the direction of the tilted die significantly affects the thickness distribution. When the die angle was reduced on the thinner side, as shown in Fig. 14a, the eccentricity worsened. In contrast, when the die angle was reduced on the thicker side, as shown in Fig. 14b, the eccentricity improved. As the angle on one side decreased, the angle on the other side increased.

Fig. 14
figure 14

Predicted history of plastic deformation during tube drawing using the ± 5° tilted angle

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Bearing length and fillet radius have a direct influence on the outer diameter of the drawn tube because the \({\varnothing }_{p}\) value affects the reduction of the area. However, there was no detailed information regarding die geometries, such as die bearing length and fillet radius, in the work by Al-Hamdany et al. Identical values were provided for both the initial orifice diameter and outer diameter of the drawn tube in the misaligned tube drawing process with the ± 5° tilted die. Because the present study was intended to determine the mechanism underlying reduction of the eccentricity of the drawn tube, the initial orifice diameter and fillet radius were assumed to be 60.8 and 10 mm, respectively, which produced an outer diameter nearly identical to the experiments.

Table 3 summarizes and compares the predicted and experimental results for the cases mentioned above. The bearing lengths and inner corner radii used for both cases were 10 mm. Table 3 shows that the e-value of the tube drawn by the − 5° tilted die increased by approximately 2.3% from the initial tube (i.e., the eccentricity worsened). In contrast, the e-value of the tube drawn by the + 5° tilted die decreased by approximately 2.4%. Surprisingly, the maximum percentage error between the predictions and experiments of the wall thickness and eccentricity was 3.31%, indicating that the predictions were accurate.

Table 3 Experiment and prediction results
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Conclusions

We conducted a numerical study of the tube drawing process, in which die misalignment was assumed because of a tilted die and thickness non-uniformity in the initial tube. An implicit elastoplastic FEM was used, along with an MBTS. We simulated an example tube drawing process in the literature, characterized by its initial tube with 5.86% eccentricity and ± 5° tilted die angle, and compared the predicted results with experimental findings.

Because a tilted die causes an extensive change in the lateral force exerted on the plug, a novel FE model was presented based on the MBTS in which the plug was modeled as an elastically deformable and rigidly movable material. The numerical effects of FE mesh systems on the predictions were assessed, revealing that the half FE model with approximately 100,000 tetrahedral elements for both the plug and tube material is robust in terms of numerical effects.

The bearing length \({l}_{b}\) and the fillet radius \({R}_{f}\) played essential roles in controlling the thickness distribution, as well as the reference diameter of the tube \({\varnothing }_{f}\) in non-axisymmetric tube drawing using the tilted die and non-uniform initial tube. Numerical analysis revealed that the predicted \({\varnothing }_{f}\) linearly decreases as the bearing length increases when the bearing length exceeds a low threshold value. Moreover, an optimal bearing length minimizes the eccentricity, depending on the initial eccentricity of the tube material and the degree of die misalignment. In this study, the optimal \({l}_{b}\) and \({R}_{f}\) for the process studied were both 10 mm when the + 5° tilted die was used. The maximum errors between predictions and experiments for wall thicknesses and eccentricity were very small.

Researchers often overlook tube non-uniformity and eccentricity when conducting numerical analyses of tube drawing processes. However, these issues are becoming increasingly crucial in minimizing material consumption and protecting the environment. This study provides a deeper understanding of the non-axisymmetric tube drawing process with die misalignment and initial tube thickness non-uniformity. It isn't easy to improve thickness uniformity during tube drawing because of the characteristics of the tube drawing process. This study provides insights into the development of novel drawing processes and technologies for the fabrication of high-quality tubes, while contributing to green manufacturing practices in an effort towards net-zero emissions.