Deep Drawing Process Using a Tractrix Die for Manufacturing Liners for a CNG High-Pressure Vessel (Type II)

The liner of a CNG pressure vessel is manufactured by a DDI (deep drawing and ironing) process for the cylinder part, which is a continuous process that includes a drawing process to reduce the diameter of the billet and a subsequent ironing process to reduce the thickness of the billet. A tractrix die used in the 1st deep drawing allows the blank to flow smoothly by decreasing the punch load and radial tensile stress occurring in the workpiece. It also increases the draw ratio compared to conventional dies, but it causes forming defects. In this study, a shape coefficient (Sc) is proposed for the tractrix die using the blank diameter (D0), inflow diameter of the workpiece (di), and inflow angle of the workpiece (θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta$$\end{document}) for design of the tractrix die. The effects of the thickness and inflow angle of the workpiece on wrinkling and folding were investigated through FEA. Also, a discriminant is proposed for the relative radial stress (σ~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\sigma }$$\end{document}) generated during the deep drawing process using the tractirx die and used to predict fracture. Based on the results, the blank thickness, the draw ratio, and the inflow of the workpiece angle in the 1st deep drawing process are suggested, and the number of operations in the DDI process was reduced from 6 to 4. This improves the productivity and reduces the manufacturing cost.


Introduction
The liner of a Type II CNG high-pressure vessel (34CrMo4) consists of a cylinder part obtained by DDI. DDI is a continuous process that includes deep drawing [1,2] to reduce the billet diameter and ironing [3,4] to decrease the thickness without change of the inner diameter. The manufacturing process for a liner obtained by spinning is composed of a 1 st stage (deep drawing with a tractrix die), a 2 nd stage (redrawing + ironing 1 + ironing 2), and a 3 rd stage (redrawing + ironing), as shown in Figure 1 [5][6][7].
Owing to the wrinkling and folding from compressive stress during the first deep drawing process [8,9], using a blank holder that forces the flange part of the sheet prevents from them [10][11][12]. But a CNG pressure vessel (Type II) is manufactured with a thick plate that has large resistance compared to a thin plate, so it is possible to form it by a simple tractrix die without a blank holder in a deep drawing process [13][14][15]. A simple tractrix die allows the blank to flow smoothly by decreasing the punch load and radial tensile stress applied to the blank wall, so it can increase the draw ratio compared to conventional dies [16], but care should be taken as forming defects may occur.
In a previous study on forming defects in a deep drawing process using a simple tractrix or conical die, Kesharwani designed a modified conical tractrix die to reduce the deformation force and to enhance the LDR (limit draw ratio). The formability of a tailor friction stir-welded sheet blank was improved by 27% [17]. Magrinho presented a new combined experimental and theoretical methodology for determining the formability limits by wrinkling. They did this by using experimentation and finite element modeling of cylindrical deep-drawing without a blank holder combined with the space of the effective strain vs. stress [18]. Narayanasamy studied the wrinkling behavior of steels and their suitability for a drawing process with the help of experimental work using conical and simple tractrix dies under lubricated and non-lubricated conditions [19]. Abdullah proposed a simple process for deep drawing of elliptical cups without a blank holder and investigated the effect of the die and punch geometry according to the half-cone angle and punch fillet radius on the LDR and drawing load for the optimization of the process [20,21]. Hezam developed a new process for increasing the drawability of square cups using a conical die with a square aperture at its end. The goal was to enable easy flow of the metal from the circular sections to the square aperture and the throat of the die [22].
The previous works have focused on a thin cylinder, and there is lack of study related to a thick blank. Forming defects in a deep drawing process using a tractrix die without a blank holder are associated with the initial blank thickness and shape of the die [20][21][22]. In order to improve and optimize the current DDI process, a study on the relation between blank thickness and the shape of the tractrix die is needed.
This study proposes a shape coefficient ( S C ) for a tractrix die using the blank diameter (D 0 ), inflow diameter of the workpiece ( d i ), and inflow angle of the workpiece ( θ ) for the design of a tractrix die, as shown in Figure 2. Effects of the thickness and inflow angle on wrinkling and folding were investigated through FEA. Also, a discriminant is proposed for the relative radial stress ( σ r ) generated during the deep drawing process using the tractrix die, and it was used to predict fracture. Based on the results, the minimum blank thickness, draw ratio, and inflow angle of the workpiece without forming defects are suggested. And flow chart of this study is shown in Figure 3. Figure 4 illustrates the geometric details of the liner of a CNG high-pressure vessel, and Table 1 shows the design  Rule 6 Draw ratio in 3 rd drawing process is smaller than 1.2.

Rule 7
Limit reduction ratio of cross-sectional area in ironing process (LRA ironing) is 35%.

Rule 8
When ironing processes are carried out two times in one stage, each reduction ratio of the crosssectional area (RA ironing) should be the same.

Rule 9
Limit reduction ratio of the cross-sectional area (LRA) in each stage is 50%. rules of the DDI process used in the actual field [23,24]. The volume of the CNG pressure vessel liner shown in Figure 5 is 10014808 mm 3 , and the blank thickness and diameter are determined according to rules 1-2. The punch diameters and the dies in the 1 st -3 rd stages are calculated by rules 3-9, where d p(m) and d i(m-n) are the punch diameters and inflow diameter of the workpiece in the m th stage and n th process.

Definition of Tractrix Curve
A tractrix is a curve along which a point (P) moves under the influence of friction when pulled on a vertical line (y) by a line segment attached to a tractor point (Q). The point moves at a right angle to the initial line between the object and the puller at a constant speed, as shown in Figure 6. A tractrix curve is defined by parameter t and shape coefficient S c , as shown in Eq. (1) and Eq. (2).

Calculation Method to Design Tractrix Curve
In the conventional design method for a tractrix curve, the shape coefficient (S c ) is calculated by Eq. (3) using the blank diameter (D 0 ) and inflow diameter of the workpiece (d i ). The tractrix curve is then obtained by Eqs. (1), (2), as shown in Figure 7.
Eq. (4) shows the slope of the tangent line at a point on the tractrix curve and is found by differentiating Eqs. (1) and (2) with respect to parameter t. It can then be rewritten as Eq. (5). From Eqs. (4) and (5), parameter t is expressed as the inflow angle of the material (θ), as shown in Eq. (6).  Eq. (7) is obtained by substituting Eq. (6) into Eq. (1), and the length of the tractrix curve in the x-direction is defined as the difference between the blank diameter (D 0 ) and the inflow diameter of the workpiece (d i ), as shown in Eq. (8). Therefore, by substituting Eq. (8) into Eq. (7), Eq. (9) is expressed to calculate the shape coefficient ( S c ), as illustrated in Figure 8.
Eq. (9) can design variable shapes of a tractrix die within the limit of the draw ratio with D 0 (blank diameter), d i (inflow diameter), and θ (inflow angle). It suggests the optimal inflow angle with no forming defects (wrinkling, folding, fracture). Figure 9 shows the variation of the tractrix curve according to the shape coefficients (S c ) obtained from Eq. (9) when the 1 st draw ratio (D 0 /d p ) is given as 1.84, which is used in the existing process.

Design of Initial Blank to Prevent Forming Defects
The forming analyses of the 1 st deep drawing process were conducted by varying the inflow angle of the workpiece to find out the onset and development of wrinkling and folding under different thicknesses using the . commercial software Forge NxT 2.0. The key factors affecting the wrinkling and folding are as follows.
• Friction and lubrication: The friction coefficient obtained by the friction test is 0.056, and its value is applied to the analysis of FEM. A Parkerizing process is also carried out for lubrication.
• Die shape: Die shape is controlled by the inflow angle ( θ ) obtained from Eq. (9) and has a large influence on preventing wrinkling and folding by reducing compressive stress in a deep drawing process without a blank holder.
• D 0 /d p , d p /t 0 : These are used to check the feasibility of the deep drawing with the tractrix die, and the first stage (deep drawing process) is carried out successfully to control inflow angles obtained from a variety of tractrix curves.

Analysis Model and Conditions
The analysis model consists of the tractrix die, punch, and blank, and its dimensions are a punch diameter (d p ) of 549.35 mm, inflow diameter of the workpiece (d i ) of 576.6 mm, initial blank thickness (t 0 ) of 12.5 mm, and blank diameter (D 0 ) of 1010 mm, as shown in Figure 10. The workpiece is set to a deformable domain with a volume mesh, which is periodically remeshed as the process progresses. Negligible deformation occurs in the die and  the punch during the analysis, so they are assumed as rigid bodies with a surface mesh [25]. The punch velocity (50 mm/s) was suggested from the actual field, and the friction coefficient (μ) was defined as 0.056 [1]. The material properties and the flow stress equation used in FEA were obtained through a room-temperature tensile test of the liner material, 34CrMo4, as shown in Table 2 and Eq. (10). Forge NxT 2.0 software defined the flow stress using the Hansel-Spittel model [26,27] shown in Eq. (11), which is based on a relation of the variables of strain, strain rate, and temperature. Therefore, the property coefficient of Eq. (11) should be derived using Eq. (10), as shown in Table 3 [28].

Design of Initial Blank Thickness
Wrinkling and folding caused by compressive hoop stress in the deep drawing process are associated with the blank thickness and shape of the tractrix die. Therefore, an analysis was carried out by varying the blank thicknesses   Table 4, and its results are expressed in Table 5.
The inflow angle of the workpiece was conducted up to 14°, in which the load corresponds to 832 t in the blank thickness (12 mm) when considering the rated load of the press (850 t). When the blank thickness was 7 mm, folding was observed at all inflow angles, as shown in Figure 11a, and when 8 mm, observed in the inflow angles below 7°. But when the angles were over 8°, the initial wrinkling decreased while the workpiece went through the die, as illustrated in Figure 11b.     During the deep drawing process, wrinkling can be removed at the end of the punch's stroke, as shown in Figure 12b, which causes deterioration of the formability and uneven distribution of the thickness along the drawing direction [23]. In order to verify the effect of the inflow angles on forming defects, lengths of the workpiece that contacted the die were compared at different inflow angles (5° and 14°), blank thickness (8 mm), and punch strokes (450 mm). The contact length was 261 mm at the inflow angle of 14° and was longer than that at 5° (183 mm). The results are shown in Figure 12. The contact length was longer with the increase of the inflow angles, and then friction resistance was greater, so the wrinkling was decreased.
Among the blank thicknesses (9-12 mm) not to occur wrinkling and folding determined, the initial minimum blank thickness (10 mm) was determined according to design rule (1) [t 0 (4.7 mm) ≥ 2 × t wall × 1.05] in Table 1 while considering the design specification of the bottom part and corrosion in heat treatment. The blank diameter (D 0 ) was calculated to be 1129 mm from the constant volume (10014808 mm 3 ), and the draw ratio (D 0 /d p ) should be within the LDR (2.2) in design rule (4).

The New Design of 1 st Deep Drawing Process
Forming analyses according to the inflow angles (5°-14°) were conducted on the blank diameter (1129 mm), blank thickness (10 mm), inflow diameter of the workpiece (576.6 mm), and the maximum forming loads obtained just before the workpiece passed through the inflow diameter of the workpiece, as shown in Figure 13. The tractrix die prevents wrinkling and folding by increasing the inflow angle of the workpiece, but it causes the forming load to increase, as shown in Figure 13, and then it leads to fracture of the workpiece.

Prediction of Fracture
The 1 st deep drawing should be conducted within the LDR to avoid fracture caused by the radial stress ( σ r ) of Eq. (12), as shown in Figure 14 [29,30]. The radial stress ( σ r ), friction coefficient (f) between the blank and die,  The fracture in a deep drawing process occurs from high tensile stress caused by the punch's feeding direction and is judged by the relative radial stress ( σ r = σ r /σ Y ), so it is expected by Eq. (15), which is derived from Eq. (12)-Eq. (14).
The tractrix curve shown in Figure 15 is expressed with the tractrix radius (r), which is the distance from a point C on the tractrix curve to the central axis, as well as the tractrix angle ( α ) between the central axis and the slope of the tangent line at point C.
r and dr at point C can be expressed as Eqs. (16) and (17).
Eq. (18) is derived by substituting Eqs. (16) and (17) into Eq. (15), and Eq. (19) can be obtained by substituting the dimensionless constant K (r i /a) into Eq. (18). σ r at the tractrix angles ( α ) of 0° to 90° can be obtained by integration of Eq. (19), and σ r ≥ 1 means that fracture occurs. (12) dσ r dr (16) r = r i + S c · sinα, The shape of the tractrix die using Eq. (9) is shown in Figure 16 and has a gap of S c · sech(t) in relation to the tractrix radius compared to the conventional shape shown in Figure 7. The tractrix radius is represented by Eq. (20), and Eq. (19) can be rewritten as Eq.

New Design of Tractrix Die
According to rule (4) in Table 1, a drawing process was carried out under a draw ratio of 2.2, which is used in the actual field. Now, reduction of the number of operations is possible due to the improved draw ratio. When the draw ratio was 2.3, a defect was not found at all inflow angles, when the 2.4 and 2.5, not observed under 9° and under 5°, respectively, but when 2.6, observed at all inflow angles. Therefore, the inflow angle should be    Table 6. Ironing process 2 in the 2 nd stage and redrawing process 1 in the 3 rd stage were removed when the draw ratio were 2.4 and 2.5. From Table 7, compared the draw ratio (2.5) and the inflow angle (5°) to the draw ratio (2.4) and the inflow angle (9°), the latter was chosen to have the minimum volume of the tractrix die, which has low manufacturing cost without forming defects. The punch and die dimensions according to the design rules are shown in Table 8.

Verification of the 1 st Deep Drawing Process suggested
Forming simulations with punch and die dimensions suggested in Table 8 were conducted, and forming defects did not occur, as shown in Figures 17 and 18. The new DDI process (draw ratio: 2.4, blank thickness: 10 mm, inner diameter of the workpiece: 491 mm) lets us remove the ironing process in the 2 nd stage and redrawing process in the 3 rd stage, as shown in Figure 19, unlike the existing process (draw ratio: 1.84 used in actual field, blank thickness: 12.5 mm, inner diameter of the workpiece: 550.4 mm). Therefore, improvement of the productivity and reduction of the production cost were possible.

Conclusions
This study has conducted the 1 st deep drawing process to manufacture a CNG high-pressure vessel (Type II) and has developed a theoretical approach to generate a tractrix curve and, according to it, predicted fracture of the workpiece. Also, effects of the thickness and inflow angle of the workpiece on forming defects (wrinkling and folding) were investigated through FEA. The summaries are as follows.
1) Effects of the thickness and inflow angle of the workpiece on wrinkling and folding were investigated through shapes of the tractrix die, which were calculated by using the initial blank thickness (t 0 ), inflow diameter of the workpiece (d i ), and inflow angle of the workpiece ( θ ) and were verified through FEA. 2) The contact length was longer with the increase of the inflow angles, and then friction resistance was greater, so the wrinkling was decreased. 3) A theoretical approach to predict fracture was proposed using the relative radial stress equation, and based on its result, the best draw ratio and inflow angle without fracture were determined as 2.4 and 9°, respectively. 4) Compared to the conventional 1 st deep drawing process, the new DDI process obtained from the study (draw ratio: 1.84 → 2.4, blank thickness: 12.5 mm → 10 mm, inner diameter of the workpiece: 549.35 mm → 491 mm) lets us remove ironing 2 process in the 2 nd stage and the redrawing process in the 3 rd stage, which contributes to improvement of the productivity and reduction of the production cost.
The results of the study could be widely adopted to manufacture pressure vessels for various size and uses.