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Abstract

Forming of flat sheets into shell conical parts is a complex manufacturing process. Hydrodynamic deep drawing process assisted by radial pressure is a new hydroforming technology in which fluid pressure is applied to the peripheral edge of the sheet in addition to the sheet surface. This technique results in higher drawing ratio and dimensional accuracy, better surface quality, and ability of forming more complex geometries. In this research, a new theoretical model is developed to predict the critical rupture pressure in production of cone cups. In this analysis, Barlat–Lian yield criterion is utilized and tensile instability is considered based on the maximum load applied on the sheet. The proposed model is then validated by a series of experiments. The theoretical predictions are in good agreement with the experimental results. The effects of geometrical parameters and material properties on critical rupture pressure are also studied. The critical pressure is increased with increase in the height ratio, strain hardening exponent, and anisotropy. Higher punch nose radius expands the safe zone. It is shown that the critical pressure decreases for drawing ratios higher than 4.

Keywords

Process analysis hydrodynamic deep drawing cone cup critical rupture pressure theoretical model

Introduction

Today, conical parts have a wide variety of uses in different industries such as aerospace, automotive, machinery, agricultural equipment, and kitchen appliances. Perfect forming of the sheet metal into cone cups without any defect is the main concern in all of the processes which are used in manufacturing of such products. Wrinkling and rupture are two common defects in these parts.¹ The small contact area between the punch tip and the blank at the beginning of deformation process and hence high stresses exerted on the sheet will cause premature rupture. Also, wrinkles easily appear on the conical walls and the flange area since the area between the punch and the blank holder is free. Hydrodynamic deep drawing process assisted by radial pressure (HDDRP) is a new technique in production of such parts.² In this method, the fluid pressure (oil or water) inside the die cavity is applied on both the surface and sides of the blank. In practice, it has a considerable effect on increasing the limiting drawing ratio (LDR) and surface quality of the product. Moreover, due to the hydrodynamic contact of the sheet flange on the die, the contact friction is reduced, and hence formability is improved.³ All of these advantages make this process suitable for production of cone cups.

In recent years, a few researches have been performed on this issue. Wan et al.⁴ used a theoretical approach to determine the fracture criteria and the LDR for conventional deep drawing of cone cups. Lang et al.^5,6 studied the forming of a cone cup by hydrodynamic deep drawing with uniform pressure. Effects of key process parameters including the punch surface roughness and the pressure variation on the finally formed parts were investigated with experiments and simulations. Fazli and Dariani⁷ analyzed the hydromechanical deep drawing process theoretically. Khandeparkar and Liewald^8,9 studied the hydromechanical deep drawing of conical–cylindrical cups on low carbon and stainless steels. They found that the transition zone between the cylindrical and cone sections is a very critical area. Gorji et al.¹⁰ discussed the forming of cylindrical–conical cups in HDDRP process numerically and experimentally. Azodi et al.¹¹ developed a theoretical model based on tensile instability to analyze the hydromechanical deep drawing of cylindrical cups. Their model showed a good agreement with experimental results. Bagherzadeh et al.¹² developed analytical models to investigate stress analysis and instability conditions in hydromechanical deep drawing of two-layered cylindrical cups. They also conducted several experiments to verify their model and predict the actual process window for this technique. Wang et al.¹³ proposed a new method to be able to apply a radial pressure higher than the chamber pressure in HDDRP by means of the inward flowing of the liquid during this process.

Process analysis of forming processes gives a good insight on the process design and determination of effective parameters.¹⁴ In this article, a new theoretical model is developed based on Barlat–Lian yield criterion to analyze the critical bursting pressure in hydrodynamic deep drawing of cone cups assisted by radial pressure. Experiments are carried out to verify the analytical results and validate the proposed model. The effects of material and geometrical parameters on critical rupture pressure are also studied.

Analytical study

Hydrodynamic deep drawing is a two-stage process. First, an initial pressure known as pre-bulge pressure is applied after filling of the die cavity with fluid. This stage causes sheet to make contact with the bottom of punch. At the second stage, punch moves downward and sheet takes the punch form. The schematic of this process is shown in Figure 1. In order to analyze the sheet deformation and determine the pressure equation at the second stage, blank is divided into three zones (I, II, III) as shown in Figure 2.

Figure 1.

The schematic of hydrodynamic deep drawing assisted by radial pressure process.

Figure 2.

Three zones in analysis of a conical section.

Zone I includes the flange part which is in contact with the blank holder from top and with the pressurized fluid from bottom. Zone II is the curved part that has no contact with any of the die components. Zone III is the area completely in contact with the punch under the fluid pressure.

The simplifying assumptions are as follows:

Punch, die, and blank holder are rigid.

The radial and peripheral directions are considered as principals.

The thickness at Zones I and II remains constant during deformation. This assumption is acceptable in this process due to small thickness changes compared to the conventional deep drawing.

Elastic behavior of the sheet is ignored and the material is expressed as power law hardening.

In this section, the hydromechanical deep drawing process has been analyzed using the Barlat–Lian quadratic yield criterion with consideration of normal anisotropy of the sheet. By investigating the status of stress and strain of deformation areas, the conditions of tensile instability during the process will be obtained based on plane strain tensile instability.¹⁵ Barlat–Lian¹⁶ anisotropic non-quadratic yield criterion is expressed based on the stress principal components as follows:

{| k_{1} + k_{2} |}^{a} + {| k_{1} - k_{2} |}^{a} + \frac{c}{2 - c} {| 2 k_{2} |}^{a} = \frac{2}{2 - c} {\bar{σ}}^{a}

(1)

where a is Barlat–Lian yield function exponent

k_{2} = \frac{σ_{1} - u σ_{2}}{2}, k_{1} = \frac{σ_{1} + u σ_{2}}{2}

(2)

c = 2 \sqrt{\frac{R_{0}}{1 + R_{0}} \frac{R_{90}}{1 + R_{90}}}, u = \sqrt{\frac{R_{0}}{1 + R_{0}} \frac{1 + R_{90}}{R_{90}}}

(3)

Assuming plane isotropic state $(R_{0} = R_{90})$ , yield criterion can be written as follows

{| σ_{1} |}^{a} + {| σ_{2} |}^{a} + \frac{c}{2 - c} {| σ_{1} - σ_{2} |}^{a} = \frac{2}{2 - c} {| \bar{σ} |}^{a}

(4)

where

c = \frac{2 R}{1 + R}

(5)

Associated flow rule is expressed as follows

d ε_{ij} = d λ \frac{\partial f}{\partial σ_{ij}}

(6)

Using the yield criterion (a = 2) and the associated flow rule, the following relation is obtained

\begin{array}{l} \frac{d ε_{1}}{σ_{1} + \frac{c}{2 - c} (σ_{1} - σ_{2})} = \frac{d ε_{2}}{σ_{2} - \frac{c}{2 - c} (σ_{1} - σ_{2})} \\ = \frac{- d ε_{3}}{σ_{1} + σ_{2}} = \frac{d \bar{ε}}{\frac{2}{2 - c} \bar{σ}} \end{array}

(7)

Zone I (flange zone)

To analyze the process in flange zone, a radial element in Figure 3 is shown. Due to axial symmetry, polar equilibrium equation of the element on the edge zone in radial direction is as follows

\frac{d}{dr} (t σ_{r}) + \frac{t}{r} (σ_{r} - σ_{θ}) + F_{f} = 0

(8)

Figure 3.

Radial element at Zone I.

where $F_{f}$ , the coulomb friction force exerted on the flange, is

F_{f} = μ P

(9)

Considering the radial and peripheral directions as principal directions, since the radial and environmental stresses at Zone I are tensile and compressive, respectively, equation (10) can be written from equation (7)

{\begin{matrix} d ε_{r} = \frac{d \bar{ε}}{2 \bar{σ}} (2 σ_{r} - c σ_{θ}) \\ d ε_{θ} = \frac{d \bar{ε}}{2 \bar{σ}} (2 σ_{θ} - c σ_{r}) \end{matrix}

(10)

σ_{r} - σ_{θ} = \frac{2}{2 + c} \frac{\bar{σ}}{d \bar{ε}} (d ε_{r} - d ε_{θ})

(11)

The effective stress from equation (4) can be written as follows

σ_{r}^{2} + σ_{θ}^{2} - c σ_{r} σ_{θ} = {\bar{σ}}^{2}

(12)

Component of the effective strain will be obtained using the recent equation and equation (7) based on radial and tangential strain increments

d \bar{ε} = 2 {[\frac{1}{4 - c^{2}} (d ε_{r}^{2} + d ε_{θ}^{2} + cd ε_{r} d ε_{θ})]}^{\frac{1}{2}}

(13)

Since the volume is assumed to be constant during the deformation, regardless of the flange thickness changes, it can be written as

d ε_{r} = - d ε_{θ}

(14)

d \bar{ε} = 2 \sqrt{\frac{1}{2 + c}} d ε_{r}

(15)

Radial strain at any point is obtained by equation (16)

ε_{r} = ε_{θ} = \ln (\frac{r_{o}}{r})

(16)

Thus, the effective strain in terms of the radial strain is obtained as follows

\bar{ε} = 2 \sqrt{\frac{1}{2 + c}} ε_{r}

(17)

\bar{σ} = K {\bar{ε}}^{n}

(18)

Due to the assumption of hardening behavior of the material as power rule in equation (18), the result of relations (11), (14), and (17) will be

σ_{r} - σ_{θ} = K \frac{2^{(n + 1)}}{{(2 + c)}^{(\frac{n + 1}{2})}} ε_{r}^{n}

(19)

Applying the equilibrium equation in the radial direction to the assumed element gives

(σ_{r} + d σ_{r}) (r + d r) d θ \cdot t - σ_{r} r \cdot d θ \cdot t - 2 σ_{θ} \cdot \frac{1}{2} d θ \cdot d_{r} t + μ P \cdot d A_{1} = 0

(20)

d A_{1} = \frac{π d θ}{2 π} ({(r + dr)}^{2} - r^{2}) ≅ rdrd θ

(21)

where dA₁ is the element area. Simplifying and using the above equations

t \frac{d}{dr} (σ_{r}) + \frac{t}{r} K \frac{2^{n + 1}}{{(2 + c)}^{\frac{n + 1}{2}}} ε_{r}^{n} + μ P = 0

(22)

σ_{r}^{I} = \int_{r}^{b} \frac{1}{r} (K \frac{2^{n + 1}}{{(2 + c)}^{\frac{n + 1}{2}}} ε_{r}^{In}) dr + μ P \frac{(b - r)}{t} + c 1

(23)

And integrating the obtained result with regard to the boundary condition at the end of the rim gives

c 1 = {σ_{r}}^{I} (b) = - P

(24)

Therefore, the radial stress at Zone I will be as the follows

σ_{r}^{I} = \int_{r}^{b} \frac{1}{r} (K \frac{2^{n + 1}}{{(2 + c)}^{\frac{n + 1}{2}}} ε_{r}^{In}) dr + \frac{[μ (b - r) - t] P}{t}

(25)

According to Figure 3

R_{2} = R_{1} + [h - (ρ + r_{p}) (1 - \sin α)] \tan α

(26)

Radial strain at Zone I is calculated by equation (27)

ε_{r}^{I} = \ln (\frac{r_{o}^{I}}{r}), (R_{2} + ρ \cos α) \leq r \leq b

(27)

Now with respect to Figures 3 and 4 and assuming the volume constancy, the relation between $r_{o}$ and $r$ at Zone I can be defined as equation 28

\begin{matrix} r_{0}^{I^{2}} = r^{2} - {(R_{2} + ρ \cos α)}^{2} \\ + 2 ρ [(\frac{π}{2} - α) (R_{1} + ρ \cos α) - ρ \cos α] \\ + (R_{2} + R_{1}) \sqrt{{(R_{2} - R_{1})}^{2} + {(h - ρ (1 - \sin α))}^{2}} \\ + 2 r_{p} (\frac{π}{2} - α) [R_{1} - r_{p} \cos α (1 - \frac{1}{(\frac{π}{2} - α)})] \\ + {(R_{1} - r_{p})}^{2} \end{matrix}

(28)

Figure 4.

Dimensional parameters of the cone cup at Zone II.

Zone II (curved zone)

At this zone, there may be two different situations depending on the inside pressure: First, the pressure is high enough so that there is no frictional contact between the die entrance radius and sheet. Second, the chamber pressure is low so the sheet takes place on the die entrance radius. Assuming the first situation, friction force at this area is zero. Thus, the equilibrium equations are as follows. Dimensional parameters of the cone cup at Zone II are presented in Figure 4

\frac{d}{dr} (t σ_{r}) + \frac{t}{r} (σ_{r} - σ_{θ}) = 0

(29)

t \frac{d}{dr} (σ_{r}) \frac{t}{r} K \frac{2^{n + 1}}{{(2 + c)}^{\frac{n + 1}{2}}} ε_{r}^{n} = 0

(30)

Similar to the presented method for Zone I, the radial stress equation at Zone II will be as follows (see Figure 5)

σ_{r}^{II} = \int_{r}^{R_{2} + t + ρ \cos α} \frac{1}{r} K \frac{2^{n + 1}}{{(2 + c)}^{\frac{n + 1}{2}}} ε_{r}^{IIn} dr + c 2

(31)

Figure 5.

Radial element at Zone II.

The constant of integration is obtained based on the continuity of stress at boundaries of Zones I and II

c 2 = σ_{r}^{I} |_{r = R_{2} + t + ρ \cos α}

(32)

Radial strain at Zone II is calculated by equation (33)

ε_{r}^{II} = \ln (\frac{r_{o}^{II}}{r})

(33)

So, the radial stress equation at Zone II will be as shown in equation (34)

σ_{r}^{I I} = \int_{r}^{R_{2} + t + ρ c o s α} \frac{1}{r} K \frac{2^{n + 1}}{{(2 + c)}^{\frac{n + 1}{2}}} ε_{r}^{I I n} d r + σ_{r}^{I} | r = R_{2} + t + ρ c o s α |

(34)

The ratio of initial and current radii at one point of the sheet at Zone II is as follows (assuming the constant volume) according to Figure 4

\begin{matrix} r_{0}^{I I^{2}} = 2 ρ (φ - α) \\ \cdot (R_{2} + ρ \cos α - (\frac{ρ \sin (φ - α)}{φ - α})) + 2 (R_{2} + R_{1}) \\ \cdot \sqrt{{(R_{2} - R_{1})}^{2} + {(h - (ρ + r_{p}) (1 - \sin α))}^{2}} \\ + 2 r_{p} (\frac{π}{2} - α) \cdot [R_{1} - r_{p} \cos α (1 - \frac{1}{(\frac{π}{2} - α)})] \\ + {(R_{1} - r_{p} \cos α)}^{2} \end{matrix}

(35)

φ = \cos^{- 1} [\frac{(R_{2} + ρ \cos α - r)}{ρ}], (R_{2} < r^{(I I)} < R_{2} + ρ \cos α)

(36)

The curvature of sheet for the conical parts in hydroforming process is determined based on the equilibrium of vertical forces according to Figure 5

\frac{ρ}{R_{2}} = \frac{1}{\cos α} ({(1 + \frac{2 t {σ_{r}^{II}}_{(r = R_{2})}}{P R_{2}})}^{0.5} - 1)

(37)

Determination of critical rupture pressure

In this process, two possible failure states may occur depending on the pressure of the chamber fluid: first, at low pressure condition, the sheet curvature increases and wrinkling happens; second, at high pressure condition, sheet is bulged enough and floating condition is established. In this situation, plastic instability occurs at the boundary of Zones II and III.¹¹ Using the equilibrium of the forces in the boundary of Zones II and III, the amount of force applied on the sheet is as follows

F = 2 π R_{2} t {σ_{z}^{I I I} |}_{z = h - ρ (1 - \sin α)} + μ P π (R_{1} + R_{2}) [h - (ρ + r_{p}) (1 - \sin α)] + P π R_{1}^{2}

(38)

In this equation, $σ_{z}^{III}$ is the stress induced on the wall of the cone. According to the Swift’s instability criterion, the condition for necking occurs when the punch force reaches its maximum value, consequently⁸

dF = 0

(39)

Applying this rupture instability criterion to equation (38) and using the definition of thickness strain ( $d ε_{t} = dt / t$ ), the following relation can be obtained

\frac{d σ_{z}}{d ε_{z}} = σ_{z}

(40)

One of the major advantages of sheet hydroforming is the uniform compressive loading that makes the sheet and punch stick together. Thus, the peripheral strain component is zero and plane strain condition is established

d ε_{θ} = 0

(41)

Plane strain and constant volume conditions give

d ε_{z} = - d ε_{t}

(42)

γ = \frac{σ_{θ}}{σ_{z}}

(43)

Assuming the ratio between the tangential and axial stress components ( $γ$ ) is constant at the instability time and by using equations (7), (12), and (40), one can obtain

\frac{d \bar{σ}}{d \bar{ε}} = \bar{σ} (\frac{(2 - c γ)}{2 {(1 - c γ + γ^{2})}^{\frac{1}{2}}})

(44)

Using the power hardening rule, critical effective plane strain according to Barlat–Lian yield criterion is obtained as¹⁰

\bar{ε} = 2 n \frac{{(1 - c γ + γ^{2})}^{\frac{1}{2}}}{(2 - c γ)}

(45)

Accordingly, the critical axial stress is

σ_{z_{cr}} = K {(\frac{2 n}{2 - c γ})}^{n} {(1 - c γ + γ^{2})}^{\frac{n - 1}{2}}

(46)

With regard to the continuity of stress at the intersection point of Zones II and III in instability conditions, the fluid critical pressure is determined by equalizing the stresses of Zones II and III ( $σ_{r}^{II}$ and $σ_{r}^{III}$ ) at the point $z = h - ρ (1 - \sin α)$

\begin{matrix} {\bar{P}}_{cr} = \frac{P_{cr}}{K} = \frac{1}{(μ / t) (b - (R_{2} + t + ρ \cos α)) - 1} \\ {n^{n} {(\frac{4}{4 - c^{2}})}^{\frac{n + 1}{2}} - \frac{2^{(n + 1)}}{{(2 + c)}^{(\frac{n + 1}{2})}} \\ (\int_{R_{2}}^{R_{2} + t + ρ \cos α} \frac{ε_{r}^{I I^{n}}}{r} dr + \int_{R_{2} + t + ρ \cos α}^{b} \frac{ε_{r}^{I^{n}}}{r} dr)} \end{matrix}

(47)

Also, the radius of blank curvature at the point of instability $z = h - ρ (1 - \sin α)$ can be obtained using equations (37), (45), and (46)

\frac{ρ}{R_{2}} = \frac{1}{\cos α} ({(1 + \frac{2 tK n^{n} {(\frac{4}{4 - c^{2}})}^{\frac{n + 1}{2}}}{P_{cr} R_{2}})}^{0.5} - 1)

(48)

Combining equations (47) and (48) with the related equations, the critical fluid pressure can be calculated with the Newton’s numerical method by means of MATLAB software. Figure 6 shows the flowchart of the algorithm written in MATLAB to calculate equation (48).

Figure 6.

The algorithm flowchart.

Experimental study

In order to verify the theoretical model developed in the present research and evaluate the critical fluid pressure in the HDDRP of cone cups, experimental tests were performed on 1 mm thick metal sheets with the system shown in Figure 7. The fluid pressure in the chamber was controlled by a relief valve and measured by a pressure gauge. The depth of each failed cup under its corresponding loading path was recorded and drawn along with the fluid pressure as one point of critical pressure versus the height ratio (H/R₂). The geometrical parameters and material properties are shown in Table 1.

Figure 7.

HDDRP experimental setup.

Table 1.

Material properties and process parameters.

Parameters	Values
Material	St12
Thickness, t (mm)	1
Strength coefficient, K (MPa)	515
Strain hardening exponent, n	0.23
Normal anisotropy ratio, R	1
Initial blank diameter blank, D₀ (mm)	110
Punch nose radius, r_p (mm)	5
Half apex angle of cone, α	22.5
Cone head flat radius, r₁ (mm)	15
Friction coefficient (blank and blank holder), µ	0.08

Results and discussion

Comparison of experimental and theoretical results

In Figure 8, the experimental and theoretical critical pressures versus the height ratio have been compared. By increasing the height ratio, the critical pressure increases, and boundary of the safe forming area tends toward higher pressures. This phenomenon is mainly due to the increase in material’s strength and its hardening behavior under plastic deformation during the process. It is observed that there is a good agreement between theoretical and experimental results in the range of experiments. This issue indicates the good precision of hydrodynamic deep drawing process in one step manufacturing of cone geometries.

Figure 8.

Comparison of analytical and experimental critical pressure.

Effective parameters on critical pressure

In this section, influential parameters on occurrence of instability and fracture in a cone cup are discussed.

Sheet thickness

The obtained critical pressures for sheets with different thicknesses in hydrodynamic deep drawing are compared in Figure 9. According to the results, the critical pressure grows in an exponential manner when the thickness of the sheet increases. In other words, in production of industrial parts with the same material but less thickness, rupture occurs under lower pressures. So, the accurate controlling of the pressure loading path becomes more important.

Figure 9.

Effect of sheet thickness on critical pressure.

Drawing ratio

Drawing ratio (DR) in cone forming is defined as D₀/2r₁. As shown in Figure 10, the higher the DR, the lower will be the critical pressure in this process. The important point is that a significant reduction in critical pressure occurs with increase in this ratio. It indicates that for production of the parts with higher DRs, the pressure range applicable to sheet during the forming process has to be lower in order to prevent instability and rupture. Therefore, design and manufacturing of sound parts with high DRs has more complexity and should be dealt with carefully. Then again the importance of precise determination and control of the pressure loading path is observed.

Figure 10.

Effect of drawing ratio on critical pressure.

Strain hardening exponent

Figure 11 shows that with increase in the strain hardening exponent, the rupture pressure increases. This increase is gradually reduced in final stages of forming.

Figure 11.

Effect of strain hardening exponent on critical pressure.

Normal anisotropy coefficient

In Figure 12, the effect of normal anisotropy coefficient on critical rupture pressure is shown. Anisotropy property is an important factor in determination of the behavior of sheet and its rupture limits during the forming process. As can be seen, normal anisotropy coefficient has a significant effect on the fluid critical pressure and sheet rupture. Normal anisotropy coefficient is defined as the ratio of transverse to thickness strains of the sample. Higher R value means less thickness strain during the formation of cone part. In other words, sheets with higher anisotropy encounter less thinning, and thus later rupture. This issue justifies the increase in rupture pressure and extension of the safe process window for higher anisotropic sheets.

Figure 12.

Effect of normal anisotropy on critical pressure.

Punch radius

Figure 13 and the analytical model show that the punch nose radius is important in determination of the critical rupture pressure. It is shown that the increase in the punch nose radius expands the range of allowable pressures.

Figure 13.

Effect of the punch nose radius on critical pressure.

Conclusion

A new analytical model is developed for hydrodynamic deep drawing of cone cups using the Barlat–Lian criterion. The allowable pressure range is obtained assuming plane strain condition. The model is validated using a laboratory HDDRP system. The results show a good agreement between the experiments and the newly developed theoretical model. The effects of process parameters and material properties are also studied. Several design guidelines are provided as follows:

The critical pressure increases with increase in the height ratio.

Lower pressures should be applied in high drawing ratios to avoid fracture.

Critical rupture pressure increases with increase of the strain hardening exponent.

Critical pressure is higher for more anisotropic materials.

Increase in the punch nose radius expands the allowed pressure range (safe zone).

For DRs higher than 4, the critical pressure decreases with a slow slope.

Footnotes

Appendix 1 Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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Process analysis of hydrodynamic deep drawing of cone cups assisted by radial pressure

Abstract

Keywords

Introduction

Analytical study

Zone I (flange zone)

Zone II (curved zone)

Determination of critical rupture pressure

Experimental study

Results and discussion

Comparison of experimental and theoretical results

Effective parameters on critical pressure

Sheet thickness

Drawing ratio

Strain hardening exponent

Normal anisotropy coefficient

Punch radius

Conclusion

Footnotes

Appendix 1

Declaration of Conflicting Interests

Funding

References