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Influence of process parameters and viscosity on radial stresses in fluid assisted deep drawing process.

Introduction

Deep drawing process is a simple non-steady state metal forming process, it is widely used in industry for making seamless shells, cups and boxes of various shapes. In deep drawing a sheet metal blank is drawn over a die by a radiused punch. As the blank is drawn radially inwards the flange undergoes radial tension and circumferential compression [1]. The latter may cause wrinkling of the flange if the draw ratio is large, or if the cup diameter-to-thickness ratio is high. A blankholder usually applies sufficient pressure on the blank to prevent wrinkling [2]. Radial tensile stress on the flange being drawn is produced by the tension on the cup wall induced by the punch force. Hence, when drawing cups at larger draw ratios, larger radial tension are created on the flange and higher tensile stress is needed on the cup wall. Bending and unbending over the die radius is also provided by this tensile stress on the cup wall. In addition, the tension on the cup wall has to help to overcome frictional resistance, at the flange and at the die radius. As the tensile stress that the wall of the cup can withstand is limited to the ultimate tensile strength of the material, in the field of deep drawing process the special drawing processes such as hydro-forming [3], hydro-mechanical forming [4], counter-pressure deep drawing [5], hydraulic-pressure- augmented deep drawing [6].

The process is an automatic co-ordination of the punch force and blank holding force, low friction between the blank and tooling as the high pressure liquid lubricates these interfaces and elimination of the need for a complicated control system [7-12]. Hydraulic pressure can enhance the capabilities of the basic deep drawing process for making cups. Amongst the advantages of hydraulic pressure assisted deep drawing techniques, increased depth to diameter ratio's and reduces thickness variations of the cups formed are notable. In addition, the hydraulic pressure is applied on the periphery of the flange of the cup, the drawing being performed in a simultaneous push-pull manner making it possible to achieve higher drawing ratio's then those possible in the conventional deep drawing process. The pressure on the flange is more uniform which makes it easiest to choose the parameters in simulation.

In fluid assisted deep drawing process the radial stresses and hoop stresses are generated in the blank due to punch force is applied on it. These stresses are affected by blank geometry, fluid pressure, viscosity of fluid and process parameters. The effect of viscosity phenomenon and process parameters is considered for evaluation of the process.

In this paper the radial stresses are evaluated in terms of process parameters and viscosity of fluid for magnesium alloys and studied using above process theoretically.

Methodology

Determination of Radial Stress

The Fluid Assisted Deep Drawing Process as shown in fig. 1. In fluid assisted deep drawing Process, a high pressure is produced in the fluid by the punch penetration into the fluid chamber. This pressurized fluid is directed to the peripheral surface of the blank through the bypass holes and also this high pressure fluid leak out between the blank and both the blank holder and die. This creates a fluid film on upper and lower surface of the flange and subsequently reduces frictional resistance. During the process the shear stresses are acting by fluid on the both sides of semi drawn blank at a gap, which is provided between the blank holder and die surface and the semi drawn blank is taking place at middle of the gap. The height of the gap is more than the thickness of the blank. In this process radial stresses and hoop stresses are generated in the blank due to punch force. So these stresses are generated in circular blank material during in the fluid assisted deep drawing process. The various stresses acting on the blank element during the process is shown in fig.2.

For evaluation of radial stresses , let us consider a small element of blank 'dr' in between blank holder and die surface in radial direction at a distance ' r' from the job axis of the circular blank with in the fluid region (fig. 2.). The viscous fluid contact on the both sides of blank element, due to this, the viscous force is acted by fluid on the both sides of the blank element. The total shear stress acting by the fluid on the element = 2 [tau] i.e. shear stress [tau] is acting by the fluid on the both sides of element is same)

[FIGURE 1 OMITTED]

Then shear force is given by

Shear force [F.sub.1] = 2 [tau] x [A.sub.c]

Where [A.sub.c] = fluid contact area of element

But [A.sub.c] = rdr d[theta] + dr/2 drd [theta]

Apply the equilibrium condition in radial direction, i.e. Net forces acting on the element in the radial direction equal to zero.

=> [summation over ([??])] [F.sub.r] = 0, where [summation] [F.sub.r] = algebraic sum of the forces acting on the Element in radial direction

[FIGURE 2 OMITTED]

=> ([sigma].sub.r] - [sigma].sub.[theta]])dr + r d [[sigma].sub.r] = 2[tau]/t r dr (1)

As [[sigma].sub.r], [[sigma].sub.[theta]] are the two principle stresses, the equation is obtain by using Tresca's yield criteria

[[sigma].sub.r] - [[sigma].sub.[theta]] = [[sigma].sub.0] (2)

Combined eq. (2) and eq. (1)

=> ([sigma].sub.0] dr + rd [sigma].sub.r] = 2[tau]/t r x dr

Divide by ([sigma].sub.0] r on both sides

=> dr/r+d([sigma].sub.r]/[sigma].sub.0] = 2[tau]/[[sigma].sub.0]t dr

d([sigma].sub.r] = 2[tau]/t dr - [[sigma].sub.0] dr/r

Integrating

=> [integral] d[[sigma.sub.r] = [integral]2[tau]/t dr - [integral] [[sigma].sub.0] d/r => [[sigma.sub.r] = 2[tau]/t r - [[sigma].sub.0] lnr + c (3)

Where C is constant, it is obtained from boundary condition.

Now at r = [r.sub.j], [[sigma].sub.r] = 0 ([??][mu] = 0)

Where [mu] is the coefficient of friction between blank and both the blank holder and die surface

Sub. in eq. (3) we get

C = - 2[tau]/t [r.sub.j] + [[sigma].sub.0] ln [r.sub.j]

again component C is sub.in eq.(3)

=> [[sigma].sub.r] = [[sigma].sub.0] ln ([r.sub.j]/r) - 2[tau]/t ([r.sub.j] - r) (4)

This equation (4) represents distribution of radial stresses in the blank during the fluid assisted deep drawing process.

Radial stress at the beginning of die corner

([sigma].sub.rd])

Radius of die opening = [r.sub.d]

at r = [r.sub.d] => [[sigma].sub.r] = [[sigma].sub.rd], But [r.sub.d] = [r.sub.p] + c

We know that from eq.(4)

=> [[sigma].sub.r] = [[sigma].sub.0] ln ([r.sub.j]/r) - 2[tau]/t ([r.sub.j] - r)

=> [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[therefore] [[sigma].sub.rd] = [[sigma].sub.0] ln ([r.sub.j]/[r.sub.d]) - 2 [tau]/t ([r.sub.j] - [r.sub.d] (5)

In terms of [r.sub.p] and c

[[sigma].sub.rd] = [[sigma].sub.0] ln ([r.sub.j]/[r.sub.p]) + c) - 2[tau]/t [[r.sub.j] - ([r.sub.p] + c)] (6)

The equation (6) represents radial stress distribution in the blank at die corner during the drawing process

Viscosity Phenomenon

In this deep drawing process, the blank is interaction with the fluid, then the viscosity is comes into the picture. During the process the shear stresses and shear forces are acting by the fluid on the blank in the gap, which is the region between blank holder and die surface. During the fluid assisted deep drawing process, the blank is taking place at middle of the gap. The effect of viscosity phenomenon in this process as shown in below fig.3. Newton's law of viscosity is introduced to this process for evaluation of stresses in terms viscosity, then the study of effect of viscosity of influence on these stresses is incorporated.

Let us consider a small element of blank in between blank holder and die surface with in the fluid region i.e gap. as shown in fig.3.

But [(dy).sub.1] = [(dy).sub.2], because the blank element is taking place at middle of the gap

[therefore] [(dy).sub.1] = [(dy).sub.2] = (dy) => dy = h - t/2

[FIGURE 3 OMITTED]

but [[tau].sub.1] = [[tau].sub.2]

Because [(du/dy).sub.1] = [(du/dy).sub.2], According to Newton's

law of viscosity [[tau].sub.1] = [mu] [(du/dy).sub.1], [[tau].sub.2] = [mu] [(du/dy).sub.2]

Let us [[tau].sub.1] = [[tau].sub.2] = [tau]

The total shear stress acting by the fluid on the blank element

[[tau].sub.A] = [[tau].sub.1] + [[tau].sub.2] = 2 [[tau].sub.1] = 2 [tau]

[therefore] [[tau].sub.A] = 2 [tau]

But [tau] = [mu](du/dy), Where du = u - 0 = u

[therefore] [[tau].sub.A] = 2 [tau] = 2 [mu] (du/dy) = 2 [mu]u/(h - t/2)

= 4 [mu]u/h - t

[[tau].sub.A] = 2 [tau] = 4 [mu]u/h - t (7)

Now we have to determine the radial stresses in terms of viscosity and process parameters, then the study of viscosity influence on these stresses is incorporated.

Evaluation of Radial Stresses in Terms of Viscosity and Process Parameters

We know that radial stresses are produced in the blank at a radial distance 'r' is given by eq.4

[[sigma].sub.r] = [[sigma].sub.0] ln ([r.sub.j]/r) - 2[tau]/t ([r.sub.j] - r), and

2 [tau] = 4 [mu]u/h - t we get

[[sigma].sub.r] = [[sigma].sub.0] ln ([r.sub.j]/r) - 4 [mu]u/h - t x ([r.sub.j]/r)/t (8)

at the end of the blank , put r = [r.sub.j]

=> [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

The equation (8) represents the effect of process parameters and viscosity of fluid on the distribution of radial stresses in the blank during fluid assisted deep drawing process.

Magnesium Alloys--Description

Magnesium is the highest of the commercially important metals, having a density of 1.74 gm/[cm.sup.3] and specific gravity 1.74 (30% higher than aluminum alloys and 75% lighter than steel). Like aluminum, magnesium is relatively weak in the pure state and for engineering purposes is almost always used as an alloy. Even in alloy form, however, the metal is characterized by poor wear, creep and fatigue properties. Strength drops rapidly when the temperature exceeds 100[degrees]C, so magnesium should not be considered for elevated--temperature service. Its modulus of elasticity is even less than that of aluminum, being between one fourth and one fifth that of steel. Thick sections are required to provide adequate stiffness, but the alloy is so light that it is often possible to use thicker sections for the required rigidity and still have a lighter structure than can be obtained with any other metal. Cost per unit volume is low, so the use of thick sections is generally not prohibitive.

For engineering applications magnesium is alloyed mainly with aluminum, zinc, manganese, rare earth metals, and zirconium to produce alloys with high strength-to-weight ratios. Applications for magnesium alloys include use in aircraft, missiles, machinery, tools, and material handling equipment, automobiles and high speed computer parts.

On the other positive side, magnesium alloys have a relatively high strength- toweight ratio with some commercial alloys attaining strengths as high as 300 MPa. High energy absorption means good damping of noise and vibration. While many magnesium alloys require enamel or lacquer finishes to impart adequate connection resistance, this property has been improved markedly with the development of high purity alloys. For this analysis three types of Magnesium alloys considered namely AZ31B-0, AZ61A-F and HK31A- H24

Magnesium alloy AZ31B-0 : composition (%): 3.5 Al, 0.6 Mn , 1.0Zn and Tensile strength 240MPa, Yield strength 150MPa.

Magnesium alloy AZ61A-F: composition (%): 6.5Al, 1.0Zn and Tensile strength 248MPa, Yield strength 220Mpa.

Magnesium alloy HK31A-H24: composition (%): 3.2Th, 0.7Zr and Tensile strength 228MPa, Yield strength 205MPa

Results & Discussion

The radial stress distribution in the blank during the Fluid Assisted deep drawing is given by eq.8

[[sigma].sub.r] = [[sigma].sub.0] ln ([r.sub.j]/r) - 4 [mu]u/h - t x ([r.sub.j]/r)/t

The following process parameters and yield stress values of magnesium alloys are considered for evaluation of radial stresses of magnesium alloys with different fluids for successful formation of cup in fluid assisted deep drawing process.

[r.sub.p] = 25 mm, [r.sub.cp] = 3mm, [r.sub.d] = 30mm, [r.sub.cd] = 3mm, [r.sub.BH] = 3 mm, c = 5 mm, Radial pressure of fluid = P,

Punch speed (velocity of blank) u =9mm/sec, h =12 mm, thickness of blank t = 1.5mm, radius of blank [r.sub.j] = 90mm type of materials used: Magnesium alloys,

type of fluids used:

olive oil , viscosity [mu] = 0.081N-sec/[m.sup.2]

heavy machine oil , viscosity [mu] = 0.453 N-sec/[m.sup.2]

caster oil, viscosity [mu] = 0.985N-sec/[m.sup.2]

Yield stress values ([[sigma].sub.0]) of magnesium alloys:

AZ31B-0, [[sigma].sub.0] = 150 x [10.sup.6] N/[m.sup.2]

HK31A-H24, [[sigma].sub.0] = 205 x [10.sup.6] N/[m.sup.2]

AZ61A-F, [[sigma].sub.0] = 220 x [10.sup.6] N/[m.sup.2]

The evaluation of values of Radial stresses ([[sigma].sub.r]) in the blanks of magnesium alloys with different fluids at a radial distance from job axis for a given radius of blanks at constant thickness of blanks as follows.

Substitute the above values in above [[sigma].sub.r] equation we get generalized equation for evaluation of radial stresses during the process with respect to different viscosity of fluids for magnesium alloys are at constant thickness t =1.5mm

At [[mu].sub.olive oil] = 0.081N-sec/[m.sup.2]

[[sigma].sub.r] = [[sigma].sub.0] ln (90/r) - 1.035 [90 - r]

At [[mu].sub.heavy machine oil] = 0.453 N-sec/[m.sup.2]

[[sigma].sub.r] = [[sigma].sub.0] ln (90/r) - 1.035 [90 - r]

At [[mu].sub.caster oil] = 0.985N-sec/[m.sup.2]

[[sigma].sub.r] = [[sigma].sub.0] ln (90/r) - 2.25 [90 - r]

The radial stresses of magnesium alloys are presented in fig. 4,5, and 6 at t = 1.5mm with different fluids with in the range of r = 45mm to 85mm. From the figures, due to viscosity of fluids, the shear stresses and shear forces are acted on the blank surface during the fluid assisted deep drawing process so the radial stresses are decreasing with increasing of the radial distance of the blank from the job axis. Radial stresses are also depends up on process parameters, yield stress of alloys and fluid pressure. From fig.4. the magnesium alloys at olive oil , the range of radial stresses of AZ61A-F is 12574850.12N/[m.sup.2]-152492371.4N/[m.sup.2], in HK31A-H24 is 11717473.91 N/[m.sup.2]-1420952163.7 N/[m.sup.2] and AZ31B-0 is 8573761.151 N/[m.sup.2]-103972068.8 N/[m.sup.2]. The order of radial stresses AZ31B-0 < HK31A-H24 < AZ61A-F.Among these alloys, for a lower radial distance from the job axis of blank is 45mm,the radial stress is higher value in AZ61A-F and lower value in AZ31B-0. and medium values in HK31A-H24

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

From fig.5. the magnesium alloys at heavy machine oil , the range of radial stresses of AZ61A-F is12574845.87 N/[m.sup.2]-152492333.1N/[m.sup.2], in HK31AH24 is 11717469.66 N/[m.sup.2]-142095125.4 N/[m.sup.2] and AZ31B-0 is 8573756.901 N/[m.sup.2]-103972030.5N/[m.sup.2].The order of radial stresses as AZ31B-0 < HK31A-H24 < AZ61A-F. Among these alloys, for a lower radial distance from the job axis of blank is 45mm,the radial stress is higher value in AZ61A-F and lower value in AZ31B-0. and medium values in HK31A-H24

From fig.6. the magnesium alloys at caster oil the range of radial stresses of AZ61A-F is 12574839.79N/[m.sup.2]-152492278.5 N/[m.sup.2], in HK31A-H24 is 11717463.59N/[m.sup.2]-142095070.8 N/[m.sup.2] and AZ31B-0 is 8573750.826 N/[m.sup.2]-103971975.8 N/[m.sup.2], the order of radial stresses as AZ31B-0 < HK31A-H24 < AZ61A-F. Among these alloys, for a lower radial distance from the job axis of blank is 45mm,the radial stress is higher value in AZ61A-F and lower value in AZ31B-0 and medium values in HK31A-H24

Comparing the above results, the order of viscosity of fluids as [[mu].sub.caster oil] > [[mu].sub.heavy machine oil] > [[mu].sub.olive oil] then corresponding the order of radial stresses as

[[sigma].sub.r | caster oil] < [[sigma].sub.r | heavy machine oil] < [[sigma].sub.r | olive oil]

[FIGURE 6 OMITTED]

Conclusions

The Radial stresses are the function of process parameters, yield stress of magnesium alloys and viscosity of caster oil, heavy machine oil and olive oil. The radial stresses are decreasing with increasing the radius of blank of magnesium alloys. These effects are due to viscosity of oils acted on the blank of magnesium alloys during the forming process. The radial pressure of fluid acting on blank surface of alloys is equal to blank holding pressure is to for uniform deformation of blank during the process. The wrinkling is reduced in blank due to the blank supported by high pressurized viscous fluid.

Radial stresses of magnesium alloys are determined with in the range of r is 45mm-85mm with in the blank radius [r.sub.j] = 90mm.The highest value of radial stress occurred in AZ61A-F as 152492371.4 N/[m.sup.2] at r = 45mm with olive oil viscosity and lowest value occurred in AZ31B-0 as 8573750.826 N/[m.sup.2] at r = 85mm with caster oil viscosity and medium radial stress values are occurred in HK31A-H24 with heavy machine oil viscosity. The decreased amount of radial stress with olive oil in AZ31B-0 is 95398307.65 N/[m.sup.2], in HK31A-H24 is 130377689.8 N/[m.sup.2] and AZ61A-F is 139917521.3N/[m.sup.2]. Among these higher value in AZ61A-F and less value in AZ31B-0.

The decreased amount of radial stress with caster oil in AZ31B-0 is 95398224.97 N/[m.sup.2], in HK31A-H24 is 130377607.2N/[m.sup.2] and AZ61A-F is 139917438.7 N/[m.sup.2]. Among these higher value in AZ61A - F and less value in AZ31B-0.

The decreased amount of radial stress with heavy machine oil in AZ31B-0 is 95398273.6N/[m.sup.2], in HK31A-H24 is 130377655.7N/[m.sup.2] andAZ61A-F is 13991748.2N/[m.sup.2]. Among these higher value in AZ61A-F and less value in AZ31B-0.

Among this order of decrease amount in radial stresses of magnesium alloys as AZ31B-0 <HK31A-H24 < AZ61A- F

From these analysis the radial stresses are occurred in magnesium alloys is inversely proportional to the radial distance from vertical axis of job. Low viscous fluids gives higher radial stresses and high viscous fluids gives lower radial stresses. The highest values of radial stresses occurred in magnesium alloys with olive oil viscosity and lower values of radial stresses occurred with caster oil viscosity. The higher value of radial stresses is gives the minimizing the drawing time and higher in forming limits. The nature of graphs parabolic. The radial stresses are maximum at r is 45mm and minimum at r is 85mm and radial stresses are zero at r is equal to blank radius. These radial stresses are used to get good results of formability of magnesium alloys.
Nomenclature

[r.sub.p]              = Radius of punch
[r.sub.cp]             = corner radius on punch
[r.sub.d]              = radius of die opening
[r.sub.cd]             = corner radius on die
t                      = thickness of blank
[r.sub.j]              = radius of blank
[[sigma].sub.r]        = radial stress
[[sigma].sub.[theta]]. = hoop stress (circumferential compressive
                         stress)
[d[theta]]              = angle made by element at job axis
[P.sub.h]              = blank holder pressure
P                      = radial pressure of fluid
[tau]                  = Shear stress acting by the fluid on the two
                         sides of element
2[tau]                 = Total Shear stress acted by the fluid on the
                         Element
dr                     = width of element
r                      = radial distance of blank element from job axis
[sigma]0               = yield stress
[sigma]rd              = Radial stress at die corner.
c                      = clearance between die and punch =  [r.sub.d]
                         - [r.sub.p]
[(dy).sub.1]           = distance between upper surface of the blank
                         element and blank holder
[(dy).sub.2]           = distance between lower surface of the blank
                         element and die surface
dy                     = distance maintained by blank element from
                         both blank holder and die surface
[[tau].sub.1]          = shear stress acted by fluid on upper surface
                         of the blank element
[[tau].sub.2]          = shear stress acted by fluid on lower surface
                         of the blank element
du                     = velocity of the blank element relative to
                         blank holder and die surface
[mu]                   = dynamic viscosity or absolute viscosity or
                         Viscosity of fluid
[[tau].sub.A]          = 2 [tau], the total shear stress acting by
                         the fluid on the blank element
h                      = height of the gap = thickness of fluid


Acknowledgement

One of the authors (R.Uday Kumar) thanks the management, director and principal of Mahatma Gandhi Institute of Technology for encouraging and granting permission to carry out this work.

References

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[2] Eary, D.F., and Reed, E.A., 1974, "Techniques of Press-working Sheet Metal", prentice-Hall, New Jersey, pp. 100-172.

[3] Panknin, W., and "Mulhauser, W., 1957, Principles of the hydro form process," Mittleilungen der forschungrges Blechvererbeitung 24 ,pp.269-277.

[4] Larsen, B., 1977, "Hydromechanic forming of sheet metal", Sheet Met.Ind., pp. 162-166.

[5] Nakamura, K., (1987), "Sheet metal forming with hydraulic counter pressure" in Japan, Ann. CIRP 36 (1), pp.191-194.

[6] Thiruvarudchelvan, S.,1995, "A novel pressure augmented hydraulic- deepdrawing process for high draw ratios," J. Mater. Proc.Technol. 54 , pp.355-361.

[7] Lange, K., 1985, "Handbook of Metal forming," McGraw-Hill, New York, pp. 20.21-20.24.

[8] Yossifon, S.,and Tirosh, J., 1988 , "on the permissible fluid-pressure path in hydroforming deep drawing processes--analysis of failures and experiments," Trans. ASME J. Eng. Ind. 110 , pp.146-152.

[9] Thiruvarudchelvan, S.,and Lewis, W., 1999, "A note on hydro forming with constant fluid pressure," J. Mater. Process. Technol. 88, pp. 51-56.

[10] Oberlander, K., 1982, "The hydromechanical deep drawing of sheet metals- II," Blech Rohre Profile 4 , pp.161-164.

[11] Yang, D.Y., Kim, J.B.,and Lee, D.W., 1995, "Investigations into the manufacturing of very long cups by hydromechnical deep drawing and ironing with controlled radial pressure," Ann. CIRP 44 , pp.255-258.

[12] Zhang, S.H.,and Danckert, J., 1998, "Development of hydro-mechanical deep drawing", J. Mater. Process. Technol. 83, pp. 14-25.

R. Uday Kumar (1), P. Ravinder Reddy (2) and A. V. Sita Ramaraju (3)

(1) Assistant Professor, (2) Professor and HOD, (3) Professor

(1) Dept.of Mechanical Engg. Mahatma Gandhi Institute of Technology, Gandipet, Hyderabad . 500075. Andhra Pradesh. India.

(1) corresponding author E-mail: u_kumar2003 @yahoo.co.in

(2) Dept.of Mechanical Engg. Chaitanya Bharathi Institute of Technology, Gandipet, Hyderabad . 500075. Andhra Pradesh. India. E-mail:reddy_prr@yahoo.com

(3) Dept.of Mechanical Engg. JNTUH college of Engineering, Kukatpalli, Hyderabad. 500085 Andhra Pradesh. India. E-mail:avsrraju_2008@rediffmail.com
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Author:Kumar, R. Uday; Reddy, P. Ravinder; Ramaraju, A.V. Sita
Publication:International Journal of Applied Engineering Research
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Date:Oct 1, 2009
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