Analysis and Control of Cracking and Wrinkling at the End of Seamless Steel Tube with Multi-Pass Large Deformation Diameter-Reducing

: Axial cracking and circumferential wrinkling are found at the end of seamless steel tubes during multi-pass large deformations pushing diameter-reducing (PDR), which seriously affects product quality. However, the cracking and wrinkling mechanism of PDR has not been elucidated yet. In this paper, the Equation of circumferential residual stress at the end was deduced from the warping deformation and shear stress. It is revealed that the circumferential residual stress in the end warping area from the inner to outer surface is tensile, and the generation mechanism of circumferential wrinkling on the inner wall at the end was revealed through the deformation analysis of PDR. The geometric model of the tube with periodic alternating variation of wall thickness was established to reveal the generation and development of circumferential wrinkling. In addition, the four-pass PDR experiments and simulations were developed to reveal the inﬂuence of reducing pass and wall thickness deviation on the end warpage, unevenness and circumferential residual tensile stress. The pushing-pulling diameter-reducing (PPDR) method was proposed to control the wrinkling and cracking. The simulation and experimental results showed that the end warpage, unevenness and circumferential residual tensile stress are all greatly decreased, and the risk of axial cracking and circumferential wrinkling is eliminated. On the shear stress τ n bottom and there is axial tensile stress σ ρ o and compressive stress σ ρ i on the outer and inner layer, respectively, the axial resultant force is zero. The circumferential stress on the front and rear surface is σ θ m and σ θ n , respectively.


Introduction
PDR refers to the cold forming method that consists of reducing the tube diameter by a conical die [1,2]. It has the advantages of high production efficiency, low manufacturing cost and simple die structure and has been widely used in the manufacturing of reducing tubes in automobiles, engineering machinery and aerospace [3][4][5].
There are many studies on diameter-reducing deformation and force. Rumiński et al. [6] presented the influence of different die shapes on the strain field distribution of the reducing tube by simulation and hardness measurements. Lu [7] obtained the approximate calculation method of the reducing ratio and loading rate by using the volume invariant condition and Mises equation. Zhao et al. [8] established the strain velocity field of tube drawing, and the drawing force was solved based on the inner product integral of strain rate vector. Ogbeyemi et al. [9] revealed the stress and pressure fields of the reducing tube by using Bubnov-Galerkin finite element model. Sadok et al. [10] analyzed the strain state of the reducing tube by finite element simulation and carried out the hardness measurement; the results showed that the reducing deformation is uneven, and the maximum effective strain is on the tube's inner surface.
Due to the uneven reducing deformation, the residual stress exists in the reduced tube [11]. Vollert et al. [12] researched the residual stress of medium carbon tube with hollow and fixed mandrel drawing and revealed that the circumferential residual stress on the inner and outer surface is compressive and tensile, respectively. Gattmah et al. [13] developed the fixed mandrel drawing residual stress of AISI 1010 tube with the different reduction of the area by an X-ray method. The results showed that the circumferential residual stress near the outer and inner surface at the small reduction of the area is compressive and tensile, respectively. Kishimoto et al. [14] elucidated the mechanism causing the excessive thinning of the outer diameter during microscale hollow drawing by considering the size effect and plastic anisotropy.
Different from the tube drawing, the tube is subjected to the pushing force during the PDR, and it is not affected by external force after reducing it, while the end of the reduced tube is warped. Liu et al. [15] studied the residual stress of one-pass PDR tube without wall thickness deviation by the X-ray method and simulation, the results showed that the circumferential residual stress is tensile from the inner to outer surface at the end, and it is compressive and tensile at the inner and outer surface of sizing area, respectively.
The axial buckling and wrinkling may occur in the middle area of the PDR tube due to the excessive reducing force. Liu et al. [16] investigated the influence of temperature on the axial instability of the hot extrusion transmission shaft and provided the optimal forming parameters without axial instability. Teng et al. [17] studied the influence of die cone angle on the wrinkling of reducing thin-walled cylindrical cup, and presented that the larger the cone angle, the easier the wrinkling.
Different from the PDR, the circumferential buckling and wrinkling exist in the middle deformation area of the tube with outer pressure necking in a viscous or solid particle medium. Many studies have researched the circumferential wrinkling of thin-walled alloy tube with outer pressure necking. Zhao et al. [18] analyzed the wrinkling of thin-walled tube with outer pressure necking and researched the influence of forming factors on the anti-wrinkle of outer pressure necking tube. Zhang et al. [19] revealed the effect of the buckling mode and uneven thickness defect on the wrinkling of outer pressure necking tube by finite element simulation.
In the engineering practice of the multi-pass large deformation PDR of a seamless steel tube, it is found that the end of reduced tube is warped, and there is axial cracking and circumferential wrinkling at the end of individual reduced tube, which seriously affects the quality of the products, as shown in Figure 1. However, the cracking and wrinkling mechanism of PDR has not been reported yet.
Metals 2021, 11, 1438 2 of 23 measurement; the results showed that the reducing deformation is uneven, and the maximum effective strain is on the tube's inner surface. Due to the uneven reducing deformation, the residual stress exists in the reduced tube [11]. Vollert et al. [12] researched the residual stress of medium carbon tube with hollow and fixed mandrel drawing and revealed that the circumferential residual stress on the inner and outer surface is compressive and tensile, respectively. Gattmah et al. [13] developed the fixed mandrel drawing residual stress of AISI 1010 tube with the different reduction of the area by an X-ray method. The results showed that the circumferential residual stress near the outer and inner surface at the small reduction of the area is compressive and tensile, respectively. Kishimoto et al. [14] elucidated the mechanism causing the excessive thinning of the outer diameter during microscale hollow drawing by considering the size effect and plastic anisotropy.
Different from the tube drawing, the tube is subjected to the pushing force during the PDR, and it is not affected by external force after reducing it, while the end of the reduced tube is warped. Liu et al. [15] studied the residual stress of one-pass PDR tube without wall thickness deviation by the X-ray method and simulation, the results showed that the circumferential residual stress is tensile from the inner to outer surface at the end, and it is compressive and tensile at the inner and outer surface of sizing area, respectively.
The axial buckling and wrinkling may occur in the middle area of the PDR tube due to the excessive reducing force. Liu et al. [16] investigated the influence of temperature on the axial instability of the hot extrusion transmission shaft and provided the optimal forming parameters without axial instability. Teng et al. [17] studied the influence of die cone angle on the wrinkling of reducing thin-walled cylindrical cup, and presented that the larger the cone angle, the easier the wrinkling.
Different from the PDR, the circumferential buckling and wrinkling exist in the middle deformation area of the tube with outer pressure necking in a viscous or solid particle medium. Many studies have researched the circumferential wrinkling of thin-walled alloy tube with outer pressure necking. Zhao et al. [18] analyzed the wrinkling of thinwalled tube with outer pressure necking and researched the influence of forming factors on the anti-wrinkle of outer pressure necking tube. Zhang et al. [19] revealed the effect of the buckling mode and uneven thickness defect on the wrinkling of outer pressure necking tube by finite element simulation.
In the engineering practice of the multi-pass large deformation PDR of a seamless steel tube, it is found that the end of reduced tube is warped, and there is axial cracking and circumferential wrinkling at the end of individual reduced tube, which seriously affects the quality of the products, as shown in Figure 1. However, the cracking and wrinkling mechanism of PDR has not been reported yet.
Axial cracking In this paper, for the seamless steel tube with the multi-pass large deformation PDR, the Equation of circumferential residual tensile stress at the end was deduced, and the generation mechanism of circumferential wrinkling on the inner wall at the end was re- In this paper, for the seamless steel tube with the multi-pass large deformation PDR, the Equation of circumferential residual tensile stress at the end was deduced, and the generation mechanism of circumferential wrinkling on the inner wall at the end was revealed through theoretical analysis. The influence of wall thickness deviation and reducing pass on circumferential residual stress, end warpage and unevenness were revealed by finite element analysis (FEA) and the experiment (EXP). In order to control the axial cracking and circumferential wrinkling, a new method of PPDR was proposed, which improves the forming quality of the reduced tube and provides a reference for engineering application.

Wall Thickness Measurement
Hot-rolled seamless steel tube is generally adopted in large deformation PDR processes. Due to the run-out and wear of rolling mandrel and the uneven heating of the tube, the unavoidable wall thickness deviation defects exist in the tube [20,21]. The hot-rolled seamless steel tube Q345B selected for the bulging-pressing-formed automobile axle housing [22] with a load of 6.5 tons is taken as the research object. For the initial tube, the length L 0 is 1350 mm, the outer diameter d 0 is 219 mm and the nominal wall thickness is 7.5 mm. The chemical composition and mechanical properties of the Q345B seamless steel tube is shown in Tables 1 and 2, respectively. The true stress-strain curve of the initial tube is shown in Figure 2. vealed through theoretical analysis. The influence of wall thickness deviation and reducing pass on circumferential residual stress, end warpage and unevenness were revealed by finite element analysis (FEA) and the experiment (EXP). In order to control the axial cracking and circumferential wrinkling, a new method of PPDR was proposed, which improves the forming quality of the reduced tube and provides a reference for engineering application.

Wall Thickness Measurement
Hot-rolled seamless steel tube is generally adopted in large deformation PDR processes. Due to the run-out and wear of rolling mandrel and the uneven heating of the tube, the unavoidable wall thickness deviation defects exist in the tube [20,21]. The hot-rolled seamless steel tube Q345B selected for the bulging-pressing-formed automobile axle housing [22] with a load of 6.5 tons is taken as the research object. For the initial tube, the length L0 is 1350 mm, the outer diameter d0 is 219 mm and the nominal wall thickness is 7.5 mm. The chemical composition and mechanical properties of the Q345B seamless steel tube is shown in Table 1 and Table 2, respectively. The true stress-strain curve of the initial tube is shown in Figure 2.  On the tube's outer surface, the circles are drawn with equal spacing of 145 mm and the axial lines are drawn with an equal circumferential interval angle of 15°. Accordingly, the intersection of the circle and axial lines is the wall thickness measuring point, the sections where the circles are located are designated as cross-section P1 to P10, respectively, as shown in Figure 3. On the tube's outer surface, the circles are drawn with equal spacing of 145 mm and the axial lines are drawn with an equal circumferential interval angle of 15 • . Accordingly, the intersection of the circle and axial lines is the wall thickness measuring point, the sections where the circles are located are designated as cross-section P 1 to P 10 , respectively, as shown in Figure 3. vealed through theoretical analysis. The influence of wall thickness deviation and reducing pass on circumferential residual stress, end warpage and unevenness were revealed by finite element analysis (FEA) and the experiment (EXP). In order to control the axial cracking and circumferential wrinkling, a new method of PPDR was proposed, which improves the forming quality of the reduced tube and provides a reference for engineering application.

Wall Thickness Measurement
Hot-rolled seamless steel tube is generally adopted in large deformation PDR processes. Due to the run-out and wear of rolling mandrel and the uneven heating of the tube, the unavoidable wall thickness deviation defects exist in the tube [20,21]. The hot-rolled seamless steel tube Q345B selected for the bulging-pressing-formed automobile axle housing [22] with a load of 6.5 tons is taken as the research object. For the initial tube, the length L0 is 1350 mm, the outer diameter d0 is 219 mm and the nominal wall thickness is 7.5 mm. The chemical composition and mechanical properties of the Q345B seamless steel tube is shown in Table 1 and Table 2, respectively. The true stress-strain curve of the initial tube is shown in Figure 2.  On the tube's outer surface, the circles are drawn with equal spacing of 145 mm and the axial lines are drawn with an equal circumferential interval angle of 15°. Accordingly, the intersection of the circle and axial lines is the wall thickness measuring point, the sections where the circles are located are designated as cross-section P1 to P10, respectively, as shown in Figure 3. The maximum and minimum wall thickness are expressed as t m0 and t n0 , respectively; the average wall thickness is expressed as t 0 , the difference between the t m0 and t 0 is defined as the positive deviation T m , the difference between the t n0 and t 0 is defined as the negative deviation T n and the difference between the t m0 and t n0 is defined as the initial wall thickness deviation T 0 . Ten tubes are randomly selected, and the wall thickness is measured by MT-160 ultrasonic thickness gauge. The measurement results of the half partial cross-section are shown in Figure 4, and the wall thickness distribution of the other half is symmetrical. The measurement results are as follows: Metals 2021, 11,1438  The maximum and minimum wall thickness are expressed as tm0 and tn0, respe the average wall thickness is expressed as t0, the difference between the tm0 and fined as the positive deviation Tm, the difference between the tn0 and t0 is defined negative deviation Tn and the difference between the tm0 and tn0 is defined as th wall thickness deviation T0. Ten tubes are randomly selected, and the wall thick measured by MT-160 ultrasonic thickness gauge. The measurement results of the h tial cross-section are shown in Figure 4, and the wall thickness distribution of th half is symmetrical. The measurement results are as follows:  (1). The wall thickness along the circumferential direction is alternately thin an such as the wall thickness of P3 and P5 cross-section of No.2 tube, P2 and P section of No.5 tube, P4 and P7 cross-section of No.9 tube all varies alternate 0° to 180°. (2). The absolute value of the maximum positive and negative deviation wall th on the same cross-section is close. Such as the maximum positive and negativ ation wall thickness of P3 cross-section of No.2 tube is 0.35 mm and −0.33 mm, tively, P2 cross-section of No.5 tube is 0.62 mm and −0.65 mm, respectively, P section of No.9 tube is 0.40 mm and −0.42 mm, respectively.

Geometric Model
Based on the measurement results of seamless steel tube wall thickness, the a tions are made as follows: (1) The tube's outer surface shape is a circle, and the inner surface shape is a p sinusoidal variation and the wall thickness on the same axis are equal. (2) The absolute value of the maximum positive deviation Tm and the maximum tive deviation Tn of the same cross-section wall thickness are equal. Based on the above assumptions, the geometric model of tube with wall th deviation is established, as shown in Figure 5. The trough of the inner wall's sine the minimum wall thickness tn0, and the crest is the maximum wall thickness tm0. T plitude is T0/2, the period angle is θn, the wave number is n. (1) The wall thickness along the circumferential direction is alternately thin and thick, such as the wall thickness of P 3 and P 5 cross-section of No.2 tube, P 2 and P 9 crosssection of No.5 tube, P 4 and P 7 cross-section of No.9 tube all varies alternately from 0 • to 180 • . (2) The absolute value of the maximum positive and negative deviation wall thickness on the same cross-section is close. Such as the maximum positive and negative deviation wall thickness of P 3 cross-section of No.2 tube is 0.35 mm and −0.33 mm, respectively, P 2 cross-section of No.5 tube is 0.62 mm and −0.65 mm, respectively, P 7 cross-section of No.9 tube is 0.40 mm and −0.42 mm, respectively.

Geometric Model
Based on the measurement results of seamless steel tube wall thickness, the assumptions are made as follows: (1) The tube's outer surface shape is a circle, and the inner surface shape is a periodic sinusoidal variation and the wall thickness on the same axis are equal. (2) The absolute value of the maximum positive deviation T m and the maximum negative deviation T n of the same cross-section wall thickness are equal.
Based on the above assumptions, the geometric model of tube with wall thickness deviation is established, as shown in Figure 5. The trough of the inner wall's sine curve is the minimum wall thickness t n0 , and the crest is the maximum wall thickness t m0 . The amplitude is T 0 /2, the period angle is θ n , the wave number is n.

Unevenness T
The difference between the maximum wall thickness tm and the minimum wa ness tn of the same cross-section is defined as unevenness T, which is expressed a n m t t T  

Relative Unevenness K
The ratio of unevenness T to the average inner diameter di is defined as relat venness K, which is expressed as The relative unevenness K is used as an evaluation index for whether the cir ential wrinkling produced.

Critical Relative Unevenness Kc
The maximum wall thickness deviation of seamless steel tube allowed by GB 2018 (China Standard) [23] is Tx. The ratio of Tx to the initial tube inner diame defined as the critical relative unevenness Kc, which is expressed as When the relative unevenness K of reduced tube is greater than the critical unevenness Kc, it is evaluated as the circumferential wrinkling.

Mechanical Analysis
During the PDR, the upper and lower clamping die is used to fix the middl the tube, and the reducing die is reduced from the tube's end to the inner side. Th of the reducing die is divided into the reducing area, the rounded corner area and area, as shown in Figure 6. Based on the deformation characteristics in each are reducing die, the PDR deformation process can be divided into reducing stage, stage and warping stage. The mechanical model of the PDR tube with a wall t

Unevenness T
The difference between the maximum wall thickness t m and the minimum wall thickness t n of the same cross-section is defined as unevenness T, which is expressed as

Relative Unevenness K
The ratio of unevenness T to the average inner diameter d i is defined as relative unevenness K, which is expressed as The relative unevenness K is used as an evaluation index for whether the circumferential wrinkling produced.

Critical Relative Unevenness K c
The maximum wall thickness deviation of seamless steel tube allowed by GB/T8162-2018 (China Standard) [23] is T x . The ratio of T x to the initial tube inner diameter d i0 is defined as the critical relative unevenness K c , which is expressed as When the relative unevenness K of reduced tube is greater than the critical relative unevenness K c , it is evaluated as the circumferential wrinkling.

Mechanical Analysis
During the PDR, the upper and lower clamping die is used to fix the middle area of the tube, and the reducing die is reduced from the tube's end to the inner side. The cavity of the reducing die is divided into the reducing area, the rounded corner area and the exit area, as shown in Figure 6. Based on the deformation characteristics in each area of the reducing die, the PDR deformation process can be divided into reducing stage, bending stage and warping stage. The mechanical model of the PDR tube with a wall thickness deviation is established, the upper and lower sides of the tube are defined as the thick and thin wall side, respectively. For the reducing die, the half-cone angle is α, the inner diameter of the exit area is d m and the rounded corner radius is R. For the initial tube, the diameter is d 0 , the intermediate layer radius of the thick and thin wall side is R m0 and R n0 , respectively, and the corresponding wall thickness is t m0 and t n0 , respectively.

Reducing Stage
After the tube is deformed in the reducing stage, the intermediate layer radi thick and thin wall sides decreases to Rm1 and Rn1, respectively, and the corres wall thickness increases to tm1 and tn1, respectively, as shown in Figure 6.
In the reducing stage, the tube wall thickness is increased while the diame duced, and the length is elongated along the axial direction. The circumferential c sive strain is the main deformation, so that the thick wall side metal flows to the thin wall side, which is beneficial to decrease unevenness. The normal compressi is smaller than the circumferential compressive stress. The axial stress of the en proximately zero, and on the inner side of the end it is compressive.

Bending Stage
After the tube is deformed in the reducing stage, it flows into the rounded area of the reducing die and continues to undergo axial bending deformation, a in Figure 7. Furthermore, after the tube is deformed in the bending stage, the inter layer radius of the thick and thin wall side is Rm2 and Rn2, respectively, the corres wall thickness is tm2 and tn2, respectively, and the tube end diameter d02 is less inner diameter dm of the reducing die exit area.

Reducing Stage
After the tube is deformed in the reducing stage, the intermediate layer radius of the thick and thin wall sides decreases to R m1 and R n1 , respectively, and the corresponding wall thickness increases to t m1 and t n1 , respectively, as shown in Figure 6.
In the reducing stage, the tube wall thickness is increased while the diameter is reduced, and the length is elongated along the axial direction. The circumferential compressive strain is the main deformation, so that the thick wall side metal flows to the adjacent thin wall side, which is beneficial to decrease unevenness. The normal compressive stress is smaller than the circumferential compressive stress. The axial stress of the end is approximately zero, and on the inner side of the end it is compressive.

Bending Stage
After the tube is deformed in the reducing stage, it flows into the rounded corner area of the reducing die and continues to undergo axial bending deformation, as shown in Figure 7. Furthermore, after the tube is deformed in the bending stage, the intermediate layer radius of the thick and thin wall side is R m2 and R n2 , respectively, the corresponding wall thickness is t m2 and t n2 , respectively, and the tube end diameter d 02 is less than the inner diameter d m of the reducing die exit area.

Reducing Stage
After the tube is deformed in the reducing stage, the intermediate layer radiu thick and thin wall sides decreases to Rm1 and Rn1, respectively, and the corresp wall thickness increases to tm1 and tn1, respectively, as shown in Figure 6.
In the reducing stage, the tube wall thickness is increased while the diamet duced, and the length is elongated along the axial direction. The circumferential co sive strain is the main deformation, so that the thick wall side metal flows to the a thin wall side, which is beneficial to decrease unevenness. The normal compressiv is smaller than the circumferential compressive stress. The axial stress of the en proximately zero, and on the inner side of the end it is compressive.

Bending Stage
After the tube is deformed in the reducing stage, it flows into the rounded area of the reducing die and continues to undergo axial bending deformation, as in Figure 7. Furthermore, after the tube is deformed in the bending stage, the interm layer radius of the thick and thin wall side is Rm2 and Rn2, respectively, the corresp wall thickness is tm2 and tn2, respectively, and the tube end diameter d02 is less t inner diameter dm of the reducing die exit area.
Thin wall side

Deformation Analysis
The element body including inner and outer surface of the tube is intercepted in the bending deformation area, as shown in Figure 8. The axial length is dl, the circumferential length is dy and the wall thickness is t e . On the right surface, the axial equivalent compressive stress on the outside of intermediate layer is σ ρe1 , the axial equivalent tensile stress on the inside of intermediate layer is σ ρe2 and the value of σ ρe1 and σ ρe2 is equal. The torque dM on the right surface is etals 2021, 11,1438 (3) On the same cross-section, the torque caused by the axial stress on th outside of intermediate layer on circumferential unit angle micro-plane

Deformation Analysis
The element body including inner and outer surface of the tube is inter bending deformation area, as shown in Figure 8. The axial length is dl, the cir length is dy and the wall thickness is te. On the right surface, the axial equival sive stress on the outside of intermediate layer is σρe1, the axial equivalent ten the inside of intermediate layer is σρe2 and the value of σρe1 and σρe2 is equa dM on the right surface is In the bending stage, the outside metal of the intermediate layer is sho the axial direction and the thickness is increased, while the inside metal of th ate layer is elongated, and the thickness is decreased. Based on the basic assu the wall thickness of the thin and thick wall side, it can be known from the that the axial tensile stress on the inside of the intermediate layer of the thi greater than that of the thick wall side. So, the axial deformation and thinning of the intermediate layer of the thin wall side is larger than that of the th which results in the unevenness is increased on the inner wall of the end.

Warping Stage
After the tube end is deformed in the bending stage, the axial elongation metal of the intermediate layer is greater than that of the outside, so that the upward to form end warping area. As shown in Figure 9, there is shear de the end warping area, which results in the shear stress τnρ on the lower su body pointing from left to right, and the shear stress τρn on the right surface pointing from top to bottom. After the tube end deformed in the warping sta mediate layer radius of the thick and thin wall side is increased to Rm3 and tively, the corresponding wall thickness is decreased to tm3 and tn3, respectiv Upper clamping die In the bending stage, the outside metal of the intermediate layer is shortened along the axial direction and the thickness is increased, while the inside metal of the intermediate layer is elongated, and the thickness is decreased. Based on the basic assumptions and the wall thickness of the thin and thick wall side, it can be known from the Equation (4) that the axial tensile stress on the inside of the intermediate layer of the thin wall side is greater than that of the thick wall side. So, the axial deformation and thinning at the inside of the intermediate layer of the thin wall side is larger than that of the thick wall side, which results in the unevenness is increased on the inner wall of the end.

Warping Stage
After the tube end is deformed in the bending stage, the axial elongation of the inside metal of the intermediate layer is greater than that of the outside, so that the end warped upward to form end warping area. As shown in Figure 9, there is shear deformation in the end warping area, which results in the shear stress τ nρ on the lower surface of unit body pointing from left to right, and the shear stress τ ρn on the right surface of unit body pointing from top to bottom. After the tube end deformed in the warping stage, the intermediate layer radius of the thick and thin wall side is increased to R m3 and R n3 , respectively, the corresponding wall thickness is decreased to t m3 and t n3 , respectively. the end warping area, which results in the shear stress τnρ on the lo body pointing from left to right, and the shear stress τρn on the right s pointing from top to bottom. After the tube end deformed in the warp mediate layer radius of the thick and thin wall side is increased to R tively, the corresponding wall thickness is decreased to tm3 and tn3, res Sizing area

Basic Assumptions
(1) Due to the PDR deformation is axisymmetric, the circumferential stress is main stress, the shear stress components related to the circumferential direction are all zero. (2) Axial residual stress on the cross-section of reduced tube is self-balanced from the inner to outer surface.

Analysis of End Residual Stress
The element body including the inner and outer surface of the tube is intercepted in the end warping area, and the left surface of the element body is the interface between warping area and sizing area, as shown in Figure 10. For the element body, the front side is thin wall side with the wall thickness t n , the rear side is thick wall side with the wall thickness t m , the circumferential angle between the front and rear surface is dθ, the outer circumferential curvature radius of the left and right side is d 1 /2 and d 2 /2, respectively, the axial curvature radius and included angle of the upper side is r ρ and β, respectively. The right, upper and lower surface are all free surfaces. On the left surface, the shear stress τ ρn pointing from bottom to top, and there is axial tensile stress σ ρo and compressive stress σ ρi on the outer and inner layer, respectively, the axial resultant force is zero. The circumferential stress on the front and rear surface is σ θm and σ θn , respectively. (1) Due to the PDR deformation is axisymmetric, the circumferential stress is main the shear stress components related to the circumferential direction are all ze (2) Axial residual stress on the cross-section of reduced tube is self-balanced fr inner to outer surface.

Analysis of End Residual Stress
The element body including the inner and outer surface of the tube is interce the end warping area, and the left surface of the element body is the interface b warping area and sizing area, as shown in Figure 10. For the element body, the fro is thin wall side with the wall thickness tn, the rear side is thick wall side with th thickness tm, the circumferential angle between the front and rear surface is dθ, th circumferential curvature radius of the left and right side is d1/2 and d2/2, respectiv axial curvature radius and included angle of the upper side is rρ and β, respective right, upper and lower surface are all free surfaces. On the left surface, the shear st pointing from bottom to top, and there is axial tensile stress σρo and compressive st on the outer and inner layer, respectively, the axial resultant force is zero. The circ ential stress on the front and rear surface is σθm and σθn, respectively. According to the equilibrium condition of force in the wall thickness direction be obtained as According to the equilibrium condition of force in the wall thickness direction, it can be obtained as Since dθ is very small, and sin (dθ/2) = dθ/2 is approximated, the above Equation (5) is simplified as Based on the geometric analysis, the approximate Equation is Substituting the Equation (7) into (6) to obtain According to the equilibrium condition of force in the tangential direction, it can be obtained as Substituting the Equation (9) into (8) to obtain According to the Equation (10), the circumferential residual stress σ θ in the end warping area is related to end warping deformation parameters and shear stress τ ρn . Since the direction of the τ ρn is upward, the σ θ is characterized as tensile stress from the inner to outer surface and the σ θ is verified as tensile stress by the X-ray measurement and FEA in the literature [15]. Moreover, the σ θn of the thin wall side is greater than the σ θm of the thick wall side. When the circumferential residual tensile stress of the end is greater than the ultimate tensile strength, axial cracking may occur and the axial cracking is more likely occur on the thin wall side.

Unevenness Analysis
The tube end is warped in the warping stage, which results in the end diameter expansion and wall thickness thinning. Due to the σ θn is greater than the σ θm , the circumferential deformation and thinning of the thin wall side is larger than that of the thick wall side, which results in the end unevenness is further increased. When the relative unevenness exceeds the critical value, circumferential wrinkling will be generated. However, the inner side of the end is no longer warped, the unevenness is no longer changed as well, and the sizing area is formed. The outer diameter of the sizing area is less than the inner diameter of the exit area of the reducing die.

Research Objects
The hot-rolled seamless steel tube Q345B is selected as the initial tube with the length L 0 of 1350 mm, the outer diameter d 0 of 219 mm and the nominal wall thickness of 7.5 mm. According to the GB/T8162-2018 [23], the maximum allowable wall thickness deviation is 2.25 mm, so the critical relative unevenness K c is 1.10%. During diameter-reducing, the middle area of the tube is clamped by clamping die, which is to keep the length of 456 mm without deformation, as shown in Figure 11. Moreover, the radius of upper and lower clamping die is 109.5 mm.
L0 of 1350 mm, the outer diameter d0 of 219 mm and the nominal wall thickness of 7.5 mm. According to the GB/T8162-2018 [23], the maximum allowable wall thickness deviation is 2.25 mm, so the critical relative unevenness Kc is 1.10%. During diameter-reducing, the middle area of the tube is clamped by clamping die, which is to keep the length of 456 mm without deformation, as shown in Figure 11. Moreover, the radius of upper and lower clamping die is 109.5 mm.

PDR Experiments
The initial unevenness T0 of 0.4 mm, 0.8 mm and 1.2 mm was selected, respectively. Four-pass PDR was carried out on the THP63-200/300/100 × 2 hydraulic press. The reducing die of each pass is shown in Figure 12, the half-cone angle α is 23°, the rounded corner radius R is 50 mm and the inner diameter of the exit area dm is 190 mm, 162 mm, 138 mm and 115 mm, respectively.

PDR Experiments
The initial unevenness T 0 of 0.4 mm, 0.8 mm and 1.2 mm was selected, respectively. Four-pass PDR was carried out on the THP63-200/300/100 × 2 hydraulic press. The reducing die of each pass is shown in Figure 12, the half-cone angle α is 23 • , the rounded corner radius R is 50 mm and the inner diameter of the exit area d m is 190 mm, 162 mm, 138 mm and 115 mm, respectively. Before diameter-reducing, the outer wall of both ends of the tube and the inner wall of the reducing die are lubricated by lubricating oil, then the tube is placed into the lower clamping die and the center is aligned. During diameter-reducing, the upper clamping die is driven by the upper slider of hydraulic press to move downward to clamp the middle area of the tube, and the left and right reducing die are respectively pushed by the left and right slider of hydraulic press from the tube end to the inner side at the speed of 10 mm/s.

Stress-Strain Relationship
The tensile specimens with ASTM-E8 standard were obtained by cutting in the initial tube and the sizing area of reduced tubes along the axial direction, the geometry and dimensions of each pass tensile specimen is shown in Figure 13 and Table 3. The uniaxial tensile tests were carried out on the Inspekt-Table100 (Hegewald & Peschke, Nossen, Germany) electronic universal testing machine at room temperature, and the test speed was 5.0 mm/min. The contact extensometer was used, and the original gauge length of the extensometer is 25 mm. The true stress-strain curve is obtained, as shown in Figure 14.   Before diameter-reducing, the outer wall of both ends of the tube and the inner wall of the reducing die are lubricated by lubricating oil, then the tube is placed into the lower clamping die and the center is aligned. During diameter-reducing, the upper clamping die is driven by the upper slider of hydraulic press to move downward to clamp the middle area of the tube, and the left and right reducing die are respectively pushed by the left and right slider of hydraulic press from the tube end to the inner side at the speed of 10 mm/s.

Stress-Strain Relationship
The tensile specimens with ASTM-E8 standard were obtained by cutting in the initial tube and the sizing area of reduced tubes along the axial direction, the geometry and dimensions of each pass tensile specimen is shown in Figure 13 and Table 3. The uniaxial tensile tests were carried out on the Inspekt-Table100 (Hegewald & Peschke, Nossen, Germany) electronic universal testing machine at room temperature, and the test speed was 5.0 mm/min. The contact extensometer was used, and the original gauge length of the extensometer is 25 mm. The true stress-strain curve is obtained, as shown in Figure 14. Before diameter-reducing, the outer wall of both ends of the tube and the inn of the reducing die are lubricated by lubricating oil, then the tube is placed into th clamping die and the center is aligned. During diameter-reducing, the upper cl die is driven by the upper slider of hydraulic press to move downward to clamp t dle area of the tube, and the left and right reducing die are respectively pushed by and right slider of hydraulic press from the tube end to the inner side at the spee mm/s.

Stress-Strain Relationship
The tensile specimens with ASTM-E8 standard were obtained by cutting in th tube and the sizing area of reduced tubes along the axial direction, the geometry mensions of each pass tensile specimen is shown in Figure 13 and Table 3. The u tensile tests were carried out on the Inspekt-Table100 (Hegewald & Peschke, Nosse many) electronic universal testing machine at room temperature, and the test spe 5.0 mm/min. The contact extensometer was used, and the original gauge length extensometer is 25 mm. The true stress-strain curve is obtained, as shown in      Initial tube The third pass reduced tube The second pass reduced tube The first pass reduced tube The uniaxial tensile test results are shown in Table 4, and it can be found as follows: (1) The yield strength and ultimate tensile strength of each pass reduced tube are significantly higher than that of the initial tube. Both the yield strength and ultimate tensile strength are increased with the increase of reducing pass. (2) Both the elongation and section shrinkage are gradually decreased with the increase of reducing pass, which indicates that the plastic deformation ability of the reduced tube is gradually decreased, and axial cracking may occur.

FEA of PDR
The geometric models of tube with the wall thickness deviation T0 of 0 mm, 0.4 mm, 0.8 mm, 1.2 mm, 1.6 mm, 2.0 mm were established, respectively. The diameter d0, average wall thickness t0 and wave number n of the tube is 219 mm, 7.5 mm and 15, respectively. The software ABAQUS is used to simulate the four-pass PDR of wall thickness deviation tube. Since the tube and reducing dies are all axially symmetric structures, in order to facilitate calculation, the 1/4 finite element model was established, as shown in Figure 15.

Clamping die
The first pass reducing die Wall thickness deviation tube The second pass reducing dieThe third pass reducing die The fourth pass reducing die Figure 15. Finite element model.
The stress-strain curves as shown in Figure 14 were used in the corresponding reducing pass. For the boundary conditions, symmetrical constraints were set on longitudinal center-section and middle cross-section of the tube, respectively. The middle area of the The uniaxial tensile test results are shown in Table 4, and it can be found as follows: (1) The yield strength and ultimate tensile strength of each pass reduced tube are significantly higher than that of the initial tube. Both the yield strength and ultimate tensile strength are increased with the increase of reducing pass. (2) Both the elongation and section shrinkage are gradually decreased with the increase of reducing pass, which indicates that the plastic deformation ability of the reduced tube is gradually decreased, and axial cracking may occur.

FEA of PDR
The geometric models of tube with the wall thickness deviation T 0 of 0 mm, 0.4 mm, 0.8 mm, 1.2 mm, 1.6 mm, 2.0 mm were established, respectively. The diameter d 0 , average wall thickness t 0 and wave number n of the tube is 219 mm, 7.5 mm and 15, respectively. The software ABAQUS is used to simulate the four-pass PDR of wall thickness deviation tube. Since the tube and reducing dies are all axially symmetric structures, in order to facilitate calculation, the 1/4 finite element model was established, as shown in Figure 15.
The stress-strain curves as shown in Figure 14 were used in the corresponding reducing pass. For the boundary conditions, symmetrical constraints were set on longitudinal centersection and middle cross-section of the tube, respectively. The middle area of the tube was fixed by upper and lower clamping die, the clamping die and reducing die of each pass were set as a rigid body, while the tube was set as a deformed body. The grid type of the tube is C3D8R, and the number of grid cells along the wall thickness direction is three.
With the coulomb friction model, the rigid-flexible contact was established between the clamping die and the tube, and between the tube and each reducing die, respectively, and the dynamic friction coefficient of the contact was set as 0.15 and 0.10, respectively [24]. of reducing pass, which indicates that the plastic deformation ability of the tube is gradually decreased, and axial cracking may occur.

FEA of PDR
The geometric models of tube with the wall thickness deviation T0 of 0 mm, 0.8 mm, 1.2 mm, 1.6 mm, 2.0 mm were established, respectively. The diameter d0, wall thickness t0 and wave number n of the tube is 219 mm, 7.5 mm and 15, resp The software ABAQUS is used to simulate the four-pass PDR of wall thickness d tube. Since the tube and reducing dies are all axially symmetric structures, in facilitate calculation, the 1/4 finite element model was established, as shown in Fi Clamping die The first pass reducing die Wall thickness deviation tube The second pass reducing dieThe third pass reducing die The fourth pass reducing die The stress-strain curves as shown in Figure 14 were used in the correspondin ing pass. For the boundary conditions, symmetrical constraints were set on long center-section and middle cross-section of the tube, respectively. The middle are tube was fixed by upper and lower clamping die, the clamping die and reducin each pass were set as a rigid body, while the tube was set as a deformed body. T

End of Reduced Tubes
After each pass PDR, the maximum and minimum wall thickness of the end were measured, and the end warpage (the difference between the radius of the end and sizing area) was measured. The maximum and minimum wall thickness of the sizing area were measured by transverse cutting at the distance of 50 mm from the end. The reduced tube with T 0 = 1.2 and 0.4 are shown in Figure 16. It can be observed that the end of the reduced tube is warped. There is no obvious circumferential wrinkling at the end of the fourth pass reduced tube with T 0 = 0.4. There is no obvious circumferential wrinkling at the end of the first pass reduced tube with T 0 = 1.2, but obvious circumferential wrinkling on the inner wall at the end of the fourth pass, with the end unevenness of 2.47 mm and the sizing area unevenness of 0.84 mm. type of the tube is C3D8R, and the number of grid cells along the wall thickness direction is three. With the coulomb friction model, the rigid-flexible contact was established between the clamping die and the tube, and between the tube and each reducing die, respectively, and the dynamic friction coefficient of the contact was set as 0.15 and 0.10, respectively [24].

End of Reduced Tubes
After each pass PDR, the maximum and minimum wall thickness of the end were measured, and the end warpage (the difference between the radius of the end and sizing area) was measured. The maximum and minimum wall thickness of the sizing area were measured by transverse cutting at the distance of 50 mm from the end. The reduced tube with T0 = 1.2 and 0.4 are shown in Figure 16. It can be observed that the end of the reduced tube is warped. There is no obvious circumferential wrinkling at the end of the fourth pass reduced tube with T0 = 0.4. There is no obvious circumferential wrinkling at the end of the first pass reduced tube with T0 = 1.2, but obvious circumferential wrinkling on the inner wall at the end of the fourth pass, with the end unevenness of 2.47 mm and the sizing area unevenness of 0.84 mm. Half of the sum of the maximum and minimum wall thicknesses is taken as the average wall thickness. The measurement results of diameter and wall thickness are shown in Table 5. It can be found that the average wall thickness of the end is less than that of the sizing area. While the diameter of the end is greater than that of the sizing area, and the latter is less than the inner diameter of the exit area of reducing die. Table 5. Geometric parameters of reduced tube (mm).

End Warpage
The end warpage of each pass reduced tube is shown in Table 6. It can be found that the end warpage is slowly increased with the increase of reducing pass (that is, the in- Half of the sum of the maximum and minimum wall thicknesses is taken as the average wall thickness. The measurement results of diameter and wall thickness are shown in Table 5. It can be found that the average wall thickness of the end is less than that of the sizing area. While the diameter of the end is greater than that of the sizing area, and the latter is less than the inner diameter of the exit area of reducing die. Table 5. Geometric parameters of reduced tube (mm).

End Warpage
The end warpage of each pass reduced tube is shown in Table 6. It can be found that the end warpage is slowly increased with the increase of reducing pass (that is, the increase of reducing deformation), but the increase value is small, while the wall thickness deviation has little effect on the end warpage. The increased value of the end warpage of the experimental four-pass reduced tubes with T 0 = 1.2 is 0.03 mm, 0.10 mm and 0.11 mm, respectively. While the end warpage of the experimental fourth pass reduced tube with T 0 of 0.4 mm, 0.8 mm and 1.2 mm is 1.75 mm, 1.73 mm and 1.76 mm, respectively. The maximum difference between the FEA and EXP value of the end warpage is 7.34%, which is in the third pass reduced tube with T 0 = 0.8.

Unevenness
The end unevenness T d is greater than the initial unevenness T 0 , while the sizing area unevenness T i is less than the T 0 , as shown in Figure 17. Further, it can be found that the T d is sharply increased with the increase of wall thickness deviation and reducing pass, while the T i is slowly increased. For example, the T d of the experimental four-pass reduced tubes with T 0 = 1.2 is 1.39 mm, 1.69 mm, 1.90 mm and 2.47 mm, respectively, while the T i is 0.69 mm, 0.73 mm, 0.80 mm and 0.84 mm, respectively. The maximum difference between the FEA and EXP value of unevenness is 11.91%, which is in the sizing area of the fourth pass reduced tube with T 0 = 1.2. maximum difference between the FEA and EXP value of the end warpage is 7.34%, which is in the third pass reduced tube with T0 = 0.8.

Unevenness
The end unevenness Td is greater than the initial unevenness T0, while the sizing area unevenness Ti is less than the T0, as shown in Figure 17. Further, it can be found that the Td is sharply increased with the increase of wall thickness deviation and reducing pass, while the Ti is slowly increased. For example, the Td of the experimental four-pass reduced tubes with T0= 1.2 is 1.39 mm, 1.69 mm, 1.90 mm and 2.47 mm, respectively, while the Ti is 0.69 mm, 0.73 mm, 0.80 mm and 0.84 mm, respectively. The maximum difference between the FEA and EXP value of unevenness is 11.91%, which is in the sizing area of the fourth pass reduced tube with T0 = 1.2.

Relative Unevenness of the End
The relative unevenness Kd of the end is sharply increased with the increase of wall thickness deviation and reducing pass, as shown in Figure 18. According to the judgment condition of the critical relative unevenness Kc of 1.10%, the Kd of the four-pass reduced tubes with T0 = 0.4 are all less than the Kc, which indicates that there is no circumferential wrinkling. The Kd of the experimental fourth pass reduced tube with T0 = 0.8 is 1.60% and the Kd of the second, third, fourth pass reduced tube with T0 = 1.2 is 1.15%, 1.61% and 2.57%, respectively, which are all larger than the Kc and indicate that the circumferential wrinkling at the end are all generated. The maximum difference between the FEA and EXP value of the Kd is 12.84%, which is in the first pass reduced tube with T0 = 0.8.

Relative Unevenness of the End
The relative unevenness K d of the end is sharply increased with the increase of wall thickness deviation and reducing pass, as shown in Figure 18. According to the judgment condition of the critical relative unevenness K c of 1.10%, the K d of the four-pass reduced tubes with T 0 = 0.4 are all less than the K c , which indicates that there is no circumferential wrinkling. The K d of the experimental fourth pass reduced tube with T 0 = 0.8 is 1.60% and the K d of the second, third, fourth pass reduced tube with T 0 = 1.2 is 1.15%, 1.61% and 2.57%, respectively, which are all larger than the K c and indicate that the circumferential wrinkling at the end are all generated. The maximum difference between the FEA and EXP value of the K d is 12.84%, which is in the first pass reduced tube with T 0 = 0.8. From the above analysis, it can be found that the trend of the FEA and EXP results is the same, but there is a deviation in the value. The deviation value is mainly due to the difference between the finite element model and experimental conditions, such as the wall thickness deviation distribution, boundary condition setting and so on, and there are also some deviations in the measurement. However, the overall trend is reliable, which verifies the reliability of the geometric model of the tube with wall thickness deviation and the FEA results.

Generation and Development of End Wrinkling by the FEA
The stress and deformation of PDR with T0 = 1.2 was analyzed. The upper side of the tube is designated as the thin wall side, and the lower side is designated as the thick wall side. The measuring points A1, A, and A3, A4 were respectively selected on the outer and inner surface of the thin and thick wall side at the end. The measuring points B1, B2 and B3, B4 were respectively selected on the outer and inner surface of the thin and thick wall side at 25 mm away from the end. For the stress values, the "+" is signed for tensile stress, while the "−" is signed for compression stress.

Reducing Stage
During the reducing stage, the circumferential stress σθ is compressive, as shown in Figure 19, the σθ of the point B1 is −806.35 MPa. Moreover, the normal stress σn is compressive, the σn of the point B1 is −85.48 MPa. After the reducing stage deformation, the diameter of the end is decreased to 185.35 mm, the end wall thickness of the thin and thick wall side is increased to 7.42 mm and 8.65 mm, respectively, the end unevenness is 1.23 mm. While the diameter of the inner side of the end is decreased to 190.09 mm, the wall thickness of the thin and thick wall side is increased to 7.68 mm and 8.63 mm, respectively, and the unevenness is decreased to 0.95 mm.   From the above analysis, it can be found that the trend of the FEA and EXP results is the same, but there is a deviation in the value. The deviation value is mainly due to the difference between the finite element model and experimental conditions, such as the wall thickness deviation distribution, boundary condition setting and so on, and there are also some deviations in the measurement. However, the overall trend is reliable, which verifies the reliability of the geometric model of the tube with wall thickness deviation and the FEA results.

Generation and Development of End Wrinkling by the FEA
The stress and deformation of PDR with T 0 = 1.2 was analyzed. The upper side of the tube is designated as the thin wall side, and the lower side is designated as the thick wall side. The measuring points A 1 , A, and A 3 , A 4 were respectively selected on the outer and inner surface of the thin and thick wall side at the end. The measuring points B 1 , B 2 and B 3 , B 4 were respectively selected on the outer and inner surface of the thin and thick wall side at 25 mm away from the end. For the stress values, the "+" is signed for tensile stress, while the "−" is signed for compression stress.

Reducing Stage
During the reducing stage, the circumferential stress σ θ is compressive, as shown in Figure 19, the σ θ of the point B 1 is −806.35 MPa. Moreover, the normal stress σ n is compressive, the σ n of the point B 1 is −85.48 MPa. After the reducing stage deformation, the diameter of the end is decreased to 185.35 mm, the end wall thickness of the thin and thick wall side is increased to 7.42 mm and 8.65 mm, respectively, the end unevenness is 1.23 mm. While the diameter of the inner side of the end is decreased to 190.09 mm, the wall thickness of the thin and thick wall side is increased to 7.68 mm and 8.63 mm, respectively, and the unevenness is decreased to 0.95 mm. From the above analysis, it can be found that the trend of the FEA and EXP results is the same, but there is a deviation in the value. The deviation value is mainly due to the difference between the finite element model and experimental conditions, such as the wall thickness deviation distribution, boundary condition setting and so on, and there are also some deviations in the measurement. However, the overall trend is reliable, which verifies the reliability of the geometric model of the tube with wall thickness deviation and the FEA results.

Generation and Development of End Wrinkling by the FEA
The stress and deformation of PDR with T0 = 1.2 was analyzed. The upper side of the tube is designated as the thin wall side, and the lower side is designated as the thick wall side. The measuring points A1, A, and A3, A4 were respectively selected on the outer and inner surface of the thin and thick wall side at the end. The measuring points B1, B2 and B3, B4 were respectively selected on the outer and inner surface of the thin and thick wall side at 25 mm away from the end. For the stress values, the "+" is signed for tensile stress, while the "−" is signed for compression stress.

Reducing Stage
During the reducing stage, the circumferential stress σθ is compressive, as shown in Figure 19, the σθ of the point B1 is −806.35 MPa. Moreover, the normal stress σn is compressive, the σn of the point B1 is −85.48 MPa. After the reducing stage deformation, the diameter of the end is decreased to 185.35 mm, the end wall thickness of the thin and thick wall side is increased to 7.42 mm and 8.65 mm, respectively, the end unevenness is 1.23 mm. While the diameter of the inner side of the end is decreased to 190.09 mm, the wall thickness of the thin and thick wall side is increased to 7.68 mm and 8.63 mm, respectively, and the unevenness is decreased to 0.95 mm.  Figure 19. Reducing stage. Figure 19. Reducing stage.

Bending Stage
After the end deformed in the bending stage, the circumferential stress of the point A 1 , A 2 , A 3 and A 4 is +464.76 MPa, +69.89 MPa, +435.95 MPa and +67.48 MPa, respectively, as shown in Figure 20. The axial stress of the outer surface point B 1 and B 3 is −564.81 MPa and −595.18 MPa, respectively. The axial stress of the thin wall side point B 2 is +471.47 MPa, which is larger than the thick wall side point B 4 of +442.55 MPa. The results are in good agreement with the theoretical analysis.

Bending Stage
After the end deformed in the bending stage, the circumferential stress of the point A1, A2, A3 and A4 is +464.76 MPa, +69.89 MPa, +435.95 MPa and +67.48 MPa, respectively, as shown in Figure 20. The axial stress of the outer surface point B1 and B3 is −564.81 MPa and −595.18 MPa, respectively. The axial stress of the thin wall side point B2 is +471.47 MPa, which is larger than the thick wall side point B4 of +442.55 MPa. The results are in good agreement with the theoretical analysis. The end wall thickness of the thin and thick wall side is decreased from 7.42 mm and 8.65 mm to 7.35 mm and 8.63 mm, respectively. The thinning of the thin wall side is 0.07 mm, which is greater than the thick wall side of 0.02 mm, and the end unevenness is increased to 1.28 mm. While the diameter of the inner side of the end is decreased to 187.56 mm, the wall thickness of the thin and thick wall side is increased to 7.92 mm and 8.69 mm, respectively, and the unevenness is decreased to 0.77 mm.

Warping Stage
In warping stage, the end diameter is expanded to 190.36 mm. The circumferential stress σθ of the thin wall side point A1 is +643.36 MPa, which is greater than the thick wall side point A4 of +639.79 MPa. The σθ of the thin wall side point A2 is +625.74 MPa, which is greater than the thick wall side point A3 of +605.49 MPa. The FEA results are in good agreement with the theoretical analysis. There is the shear stress τρn in the end warping area, as shown in Figure 21; the maximum shear stress of the thin and thick wall side is 81.02 MPa and 75.13 MPa, respectively. The end wall thickness of the thin and thick wall side is decreased from 7.35 mm and 8.63 mm to 7.28 mm and 8.58 mm, respectively. The thinning of the thin wall side is 0.07 mm, which is greater than the thick wall side of 0.05 mm. Moreover, the end unevenness is further increased to 1.30 mm. While the inner side of the end is no longer warped, and the unevenness is no longer changed as well.

Multi-Pass Diameter-Reducing
The FEA results of the reduced tubes were mirrored for convenient observation, as shown in Figure 22. It can be observed that the end unevenness Td is gradually increased The end wall thickness of the thin and thick wall side is decreased from 7.42 mm and 8.65 mm to 7.35 mm and 8.63 mm, respectively. The thinning of the thin wall side is 0.07 mm, which is greater than the thick wall side of 0.02 mm, and the end unevenness is increased to 1.28 mm. While the diameter of the inner side of the end is decreased to 187.56 mm, the wall thickness of the thin and thick wall side is increased to 7.92 mm and 8.69 mm, respectively, and the unevenness is decreased to 0.77 mm.

Warping Stage
In warping stage, the end diameter is expanded to 190.36 mm. The circumferential stress σ θ of the thin wall side point A 1 is +643.36 MPa, which is greater than the thick wall side point A 4 of +639.79 MPa. The σ θ of the thin wall side point A 2 is +625.74 MPa, which is greater than the thick wall side point A 3 of +605.49 MPa. The FEA results are in good agreement with the theoretical analysis. There is the shear stress τ ρn in the end warping area, as shown in Figure 21; the maximum shear stress of the thin and thick wall side is 81.02 MPa and 75.13 MPa, respectively.

Bending Stage
After the end deformed in the bending stage, the circumferential stress of the point A1, A2, A3 and A4 is +464.76 MPa, +69.89 MPa, +435.95 MPa and +67.48 MPa, respectively, as shown in Figure 20. The axial stress of the outer surface point B1 and B3 is −564.81 MPa and −595.18 MPa, respectively. The axial stress of the thin wall side point B2 is +471.47 MPa, which is larger than the thick wall side point B4 of +442.55 MPa. The results are in good agreement with the theoretical analysis. The end wall thickness of the thin and thick wall side is decreased from 7.42 mm and 8.65 mm to 7.35 mm and 8.63 mm, respectively. The thinning of the thin wall side is 0.07 mm, which is greater than the thick wall side of 0.02 mm, and the end unevenness is increased to 1.28 mm. While the diameter of the inner side of the end is decreased to 187.56 mm, the wall thickness of the thin and thick wall side is increased to 7.92 mm and 8.69 mm, respectively, and the unevenness is decreased to 0.77 mm.

Warping Stage
In warping stage, the end diameter is expanded to 190.36 mm. The circumferential stress σθ of the thin wall side point A1 is +643.36 MPa, which is greater than the thick wall side point A4 of +639.79 MPa. The σθ of the thin wall side point A2 is +625.74 MPa, which is greater than the thick wall side point A3 of +605.49 MPa. The FEA results are in good agreement with the theoretical analysis. There is the shear stress τρn in the end warping area, as shown in Figure 21; the maximum shear stress of the thin and thick wall side is 81.02 MPa and 75.13 MPa, respectively. The end wall thickness of the thin and thick wall side is decreased from 7.35 mm and 8.63 mm to 7.28 mm and 8.58 mm, respectively. The thinning of the thin wall side is 0.07 mm, which is greater than the thick wall side of 0.05 mm. Moreover, the end unevenness is further increased to 1.30 mm. While the inner side of the end is no longer warped, and the unevenness is no longer changed as well.

Multi-Pass Diameter-Reducing
The FEA results of the reduced tubes were mirrored for convenient observation, as shown in Figure 22. It can be observed that the end unevenness Td is gradually increased The end wall thickness of the thin and thick wall side is decreased from 7.35 mm and 8.63 mm to 7.28 mm and 8.58 mm, respectively. The thinning of the thin wall side is 0.07 mm, which is greater than the thick wall side of 0.05 mm. Moreover, the end unevenness is further increased to 1.30 mm. While the inner side of the end is no longer warped, and the unevenness is no longer changed as well.

Multi-Pass Diameter-Reducing
The FEA results of the reduced tubes were mirrored for convenient observation, as shown in Figure 22. It can be observed that the end unevenness T d is gradually increased with the increase of reducing pass, which is consistent with the trend of the EXP results. For each pass reduced tube, the T d is 1.30 mm, 1.63 mm, 1.77 mm and 2.21 mm, respectively, the sizing area unevenness is 0.77 mm, 0.80 mm, 0.86 mm and 0.94 mm, respectively, and the relative unevenness of the end is 0.76%, 1.13%, 1.56%, 2.34%, respectively. In addition, circumferential wrinkles are generated on the inner wall at the end warping area in the second, third and fourth pass. with the increase of reducing pass, which is consistent with the trend of the EXP results. For each pass reduced tube, the Td is 1.30 mm, 1.63 mm, 1.77 mm and 2.21 mm, respectively, the sizing area unevenness is 0.77 mm, 0.80 mm, 0.86 mm and 0.94 mm, respectively, and the relative unevenness of the end is 0.76%, 1.13%, 1.56%, 2.34%, respectively. In addition, circumferential wrinkles are generated on the inner wall at the end warping area in the second, third and fourth pass.

Evolution Law of Unevenness
The end unevenness Td is greater than the initial unevenness T0, while the sizing area unevenness Ti is less than the T0, as show in Figure 23. It can be found that the Td is gradually increased from the reducing stage to warping stage. While the Ti is gradually decreased from the reducing stage to bending stage, and no longer changed in the warping stage.

Circumferential Residual Stress of the End
In the thin wall side, the average value of the sum of circumferential residual tensile stress of each depth layer at the end is taken as the circumferential residual tensile stress σθ. The σθ of each pass reduced tube is shown in Figure 24. It can be observed that the σθ is increased with the increase of reducing pass, while the wall thickness deviation has little effect on the σθ. When the σθ is greater than the ultimate tensile strength, the end may be cracked axially.

Evolution Law of Unevenness
The end unevenness T d is greater than the initial unevenness T 0 , while the sizing area unevenness T i is less than the T 0 , as show in Figure 23. It can be found that the T d is gradually increased from the reducing stage to warping stage. While the T i is gradually decreased from the reducing stage to bending stage, and no longer changed in the warping stage.

Evolution Law of Unevenness
The end unevenness Td is greater than the initial unevenness T0, while the sizing area unevenness Ti is less than the T0, as show in Figure 23. It can be found that the Td is gradually increased from the reducing stage to warping stage. While the Ti is gradually decreased from the reducing stage to bending stage, and no longer changed in the warping stage.

Circumferential Residual Stress of the End
In the thin wall side, the average value of the sum of circumferential residual tensile stress of each depth layer at the end is taken as the circumferential residual tensile stress σθ. The σθ of each pass reduced tube is shown in Figure 24. It can be observed that the σθ is increased with the increase of reducing pass, while the wall thickness deviation has little effect on the σθ. When the σθ is greater than the ultimate tensile strength, the end may be cracked axially.

Circumferential Residual Stress of the End
In the thin wall side, the average value of the sum of circumferential residual tensile stress of each depth layer at the end is taken as the circumferential residual tensile stress σ θ . The σ θ of each pass reduced tube is shown in Figure 24. It can be observed that the σ θ is increased with the increase of reducing pass, while the wall thickness deviation has little effect on the σ θ . When the σ θ is greater than the ultimate tensile strength, the end may be cracked axially.

PPDR Process
From the above analysis, it can be known that the end warping and wrinkling are mainly caused by the bending deformation in the rounded corner area of reducing die, which results in the diameter of sizing area being less than that of the end and the reducing die exit area and leads to the circumferential residual tensile stress at the end. In order to control the end circumferential wrinkling and axial cracking, it is necessary to decrease the bending deformation, so as to decrease the deformation difference between the inner and outer side metals, and then decrease the end warping.
The PPDR process is proposed based on the above control ideas. The mandrel is pushed into the tube inside before reducing. During the reducing die moves inward from the tube end with the speed V0 under the pushing force Fr, the mandrel moves from the tube inside to outside with the speed Vm under the pulling force Fm. Moreover, the pulling speed Vm is greater than the tube elongation speed V1, as shown in Figure 25. After the tube deformed in the reducing stage, the inner surface of the tube is in contact with the outer surface of the mandrel, and the tube is subjected to the normal pressure Fn2 and the tangential friction force Ft2. The direction of the Ft2 is opposite to the speed V0 of the reducing die, which is equivalent to the drawing force exerted on the tube inner wall. Compared with the fixed mandrel, the possibility of axial instability in the tube middle force transfer area is effectively decreased by the pulling mandrel. Under the combined action of the reducing die and mandrel, the bending deformation is decreased, the deformation difference between the inner and outer side metals is decreased and then the end warpage and unevenness are decreased.

PPDR Process
From the above analysis, it can be known that the end warping and wrinkling are mainly caused by the bending deformation in the rounded corner area of reducing die, which results in the diameter of sizing area being less than that of the end and the reducing die exit area and leads to the circumferential residual tensile stress at the end. In order to control the end circumferential wrinkling and axial cracking, it is necessary to decrease the bending deformation, so as to decrease the deformation difference between the inner and outer side metals, and then decrease the end warping.
The PPDR process is proposed based on the above control ideas. The mandrel is pushed into the tube inside before reducing. During the reducing die moves inward from the tube end with the speed V 0 under the pushing force F r , the mandrel moves from the tube inside to outside with the speed V m under the pulling force F m . Moreover, the pulling speed V m is greater than the tube elongation speed V 1 , as shown in Figure 25.

PPDR Process
From the above analysis, it can be known that the end warping and wrinkling are mainly caused by the bending deformation in the rounded corner area of reducing die, which results in the diameter of sizing area being less than that of the end and the reducing die exit area and leads to the circumferential residual tensile stress at the end. In order to control the end circumferential wrinkling and axial cracking, it is necessary to decrease the bending deformation, so as to decrease the deformation difference between the inner and outer side metals, and then decrease the end warping.
The PPDR process is proposed based on the above control ideas. The mandrel is pushed into the tube inside before reducing. During the reducing die moves inward from the tube end with the speed V0 under the pushing force Fr, the mandrel moves from the tube inside to outside with the speed Vm under the pulling force Fm. Moreover, the pulling speed Vm is greater than the tube elongation speed V1, as shown in Figure 25. After the tube deformed in the reducing stage, the inner surface of the tube is in contact with the outer surface of the mandrel, and the tube is subjected to the normal pressure Fn2 and the tangential friction force Ft2. The direction of the Ft2 is opposite to the speed V0 of the reducing die, which is equivalent to the drawing force exerted on the tube inner wall. Compared with the fixed mandrel, the possibility of axial instability in the tube middle force transfer area is effectively decreased by the pulling mandrel. Under the combined action of the reducing die and mandrel, the bending deformation is decreased, the deformation difference between the inner and outer side metals is decreased and then the end warpage and unevenness are decreased. After the tube deformed in the reducing stage, the inner surface of the tube is in contact with the outer surface of the mandrel, and the tube is subjected to the normal pressure F n2 and the tangential friction force F t2 . The direction of the F t2 is opposite to the speed V 0 of the reducing die, which is equivalent to the drawing force exerted on the tube inner wall. Compared with the fixed mandrel, the possibility of axial instability in the tube middle force transfer area is effectively decreased by the pulling mandrel. Under the combined action of the reducing die and mandrel, the bending deformation is decreased, the deformation difference between the inner and outer side metals is decreased and then the end warpage and unevenness are decreased.

Finite Element Model
For the initial tube with T 0 = 1.2, the software ABAQUS is used to simulate the four-pass PPDR. Based on the axial stability of the tube force transfer area and the end without cracking [25], the diameter of each pass mandrel is respectively determined as 173.8 mm, 144.8 mm, 118.6 mm and 93.7 mm. The finite element model of PPDR is shown in Figure 26. The setting of tube, clamping die and reducing die is the same as that of the PDR finite element model, and the mandrel is set as rigid body, the friction coefficient between mandrel and tube is set as 0.15, the pushing speed of reducing die is 10 mm/s and the pulling speed of mandrel is 12 mm/s.

Finite Element Model
For the initial tube with T0 = 1.2, the software ABAQUS is used to simulate the fourpass PPDR. Based on the axial stability of the tube force transfer area and the end without cracking [25], the diameter of each pass mandrel is respectively determined as 173.8 mm, 144.8 mm, 118.6 mm and 93.7 mm. The finite element model of PPDR is shown in Figure  26. The setting of tube, clamping die and reducing die is the same as that of the PDR finite element model, and the mandrel is set as rigid body, the friction coefficient between mandrel and tube is set as 0.15, the pushing speed of reducing die is 10 mm/s and the pulling speed of mandrel is 12 mm/s.

Clamping die
Wall thickness deviation tube The first pass mandrel The fourth pass mandrel The second pass mandrel The third pass mandrel The first pass reducing die The second pass reducing die The third pass reducing die The fourth pass reducing die Figure 26. Finite element model.

FEA Results and Discussions
(1) End warping A comparison of the end warpage and sizing area diameters between PDR and PPDR is shown in Table 7. It can be found that the end warpage is greatly decreased by the PPDR, while the sizing area diameter is increased, which indicates that the deformation difference in the bending stage is decreased. In addition, the end warpage of each pass PPDR is decreased by 85.89%, 86.83%, 82.76% and 82.07%, respectively, and the forming quality of reduced tube is better. (2) Circumferential residual tensile stress of the end Comparison of the circumferential residual tensile stress at end in the first pass between PDR and PPDR is shown in Figure 27. The comparison results were as follows:

FEA Results and Discussions
(1) End warping A comparison of the end warpage and sizing area diameters between PDR and PPDR is shown in Table 7. It can be found that the end warpage is greatly decreased by the PPDR, while the sizing area diameter is increased, which indicates that the deformation difference in the bending stage is decreased. In addition, the end warpage of each pass PPDR is decreased by 85.89%, 86.83%, 82.76% and 82.07%, respectively, and the forming quality of reduced tube is better. (2) Circumferential residual tensile stress of the end Comparison of the circumferential residual tensile stress at end in the first pass between PDR and PPDR is shown in Figure 27. The comparison results were as follows: (1) The circumferential residual tensile stress of the end can be effectively controlled by the PPDR. The circumferential residual stress of the outer and inner surface at the end is tensile and compressive, respectively. (2) The circumferential residual tensile stress can be greatly decreased by the PPDR. In (1) The circumferential residual tensile stress of the end can be effectively controlled by the PPDR. The circumferential residual stress of the outer and inner surface at the end is tensile and compressive, respectively. (2) The circumferential residual tensile stress can be greatly decreased by the PPDR. In the first pass, the circumferential residual tensile stress at the outer surface of end is decreased by 71.67%, which effectively eliminates the risk of axial cracking at end. (3) End unevenness Comparison of end unevenness and relative unevenness between PDR and PPDR is shown in Table 8. It can be found that the end unevenness and relative unevenness can be greatly decreased by the PPDR. The end unevenness of each pass PPDR is decreased by 65.37%, 79.14%, 81.92%, 84.16%, respectively. The end's relative unevenness is less than the initial and critical relative unevenness, which effectively controls the circumferential wrinkling at end.

Experimental Verification
The initial tube with T 0 = 1.2 was selected to carry out the four-pass PPDR on the THP63-200/300/100 × 2 hydraulic press. The reducing die of each pass is the same as the PDR, and the mandrel diameter of each pass is the same as the FEA.
The dies of PPDR are shown in Figure 28. The left and right reducing die are respectively fixed on the left and right support cylinder, which are respectively connected with the left and right slider of hydraulic press. The left and right mandrel are respectively connected with the left and right central cylinder of hydraulic press. The upper clamping die is fixed on the upper die base, which is connected with the upper slider of hydraulic press. The lower clamping die is fixed on the lower die base, which is fixed on the working platform of hydraulic press. Before diameter-reducing, the upper slider of hydraulic press drives the upper die base and upper clamping die to move downward to clamp the middle area of the tube, and the left and right mandrel are respectively pushed into the tube inside by the left and right central cylinder. During diameter-reducing, the left and right reducing die are respectively pushed inward from the tube end with the speed 10 mm/s by the left and right slider, while the left and right mandrel are respectively pulled from the tube inside to outside with the speed 12 mm/s by the left and right central cylinder. For the convenience of measurement and observation, the reduced tube is cut along the 1/4 longitudinal section, as shown in Figure 29. It can be observed that there is no obvious warping and cir- Before diameter-reducing, the upper slider of hydraulic press drives the upper die base and upper clamping die to move downward to clamp the middle area of the tube, and the left and right mandrel are respectively pushed into the tube inside by the left and right central cylinder. During diameter-reducing, the left and right reducing die are respectively pushed inward from the tube end with the speed 10 mm/s by the left and right slider, while the left and right mandrel are respectively pulled from the tube inside to outside with the speed 12 mm/s by the left and right central cylinder. For the convenience of measurement and observation, the reduced tube is cut along the 1/4 longitudinal section, as shown in Figure 29. It can be observed that there is no obvious warping and circumferential wrinkling at the end. drel; 5. Upper clamping die; 6. Lower clamping die; 7. Lower die base; 8. Right mandrel; 9. Right reducing die; 10. Right support cylinder.
Before diameter-reducing, the upper slider of hydraulic press drives the upper die base and upper clamping die to move downward to clamp the middle area of the tube, and the left and right mandrel are respectively pushed into the tube inside by the left and right central cylinder. During diameter-reducing, the left and right reducing die are respectively pushed inward from the tube end with the speed 10 mm/s by the left and right slider, while the left and right mandrel are respectively pulled from the tube inside to outside with the speed 12 mm/s by the left and right central cylinder. For the convenience of measurement and observation, the reduced tube is cut along the 1/4 longitudinal section, as shown in Figure 29. It can be observed that there is no obvious warping and circumferential wrinkling at the end. Comparison of the EXP and FEA value of end warpage and unevenness between PDR and PPDR is shown in Table 9. The comparison results showed as follows: Table 9. Comparison results (mm).

Reducing
Pass. The above research results indicate that the end warping and circumferential wrinkling can be greatly controlled by the PPDR, the risk of axial cracking at end can be effectively eliminated and the forming quality of reduced tube can be improved. Comparison of the EXP and FEA value of end warpage and unevenness between PDR and PPDR is shown in Table 9. The comparison results showed as follows: Table 9. Comparison results (mm).

Reducing
Pass. The above research results indicate that the end warping and circumferential wrinkling can be greatly controlled by the PPDR, the risk of axial cracking at end can be effectively eliminated and the forming quality of reduced tube can be improved.

Conclusions
In this paper, the mechanical model of PDR and the geometric model of the tube with the wall thickness deviation were established to reveal the axial cracking and circumferential wrinkling mechanism of PDR. The experiments and simulations were developed to validate the theoretical analysis and reveal the influence of process parameters on the wrinkling and cracking. In order to solve the above problems, the method of PPDR was proposed. The main conclusions could be summarized as follows: (1) The Equation of circumferential residual stress at end was deduced from the warping deformation and shear stress, it reveals that the circumferential residual stress in the end warping area from the inner to outer surface is tensile. The circumferential residual tensile stress of the thin wall side is greater than that of the thick wall side. (2) The generation mechanism of circumferential wrinkling on the inner wall at the end was revealed. The geometric model of tube with wall thickness deviation was proposed, in which the outer surface shape is a circle and the inner surface shape is a periodic sinusoidal variation. On the base of it, the generation and development V 0 Speed of reducing die (mm/s)