Mathematical modeling and analysis of hoop stresses in hydroforming deep drawing process
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Mathematical modeling and analysis of hoop stresses in hydroforming deep drawing process

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    Mathematical modeling and analysis of hoop stresses in hydroforming deep drawing process Mathematical modeling and analysis of hoop stresses in hydroforming deep drawing process Document Transcript

    • International Journal of Advanced Research in ADVANCED RESEARCH IN INTERNATIONAL JOURNAL OF Engineering and Technology (IJARET), ISSN 0976 – ENGINEERING AND TECHNOLOGY Number 2, July-December 6480(Print), ISSN 0976 – 6499(Online) Volume 3, (IJARET) (2012), © IAEMEISSN 0976 - 6480 (Print)ISSN 0976 - 6499 (Online) IJARETVolume 3, Issue 2, July-December (2012), pp. 43-51© IAEME: www.iaeme.com/ijaret.htmlJournal Impact Factor (2012): 2.7078 (Calculated by GISI) ©IAEMEwww.jifactor.comMATHEMATICAL MODELING AND ANALYSIS OF HOOP STRESSES IN HYDROFORMING DEEP DRAWING PROCESS Dr.R.Uday Kumar Assistant professor, Dept.of Mechanical Engineering Mahatma Gandhi Institute of Technology, Gandipet, Hyderabad. 500075. Andhra Pradesh. India. E-mail: u_kumar2003 @yahoo.co.in Cell No# 09000666596ABSTRACT This paper presents evaluation of hoop stresses of magnesium alloys throughmathematical formulations in hydroforming deep drawing process. In hydroforming deepdrawing process, applying the hydraulic pressure on blank periphery in radial direction. It isobtained through the punch movement within the fluid chamber, which is provided in punchand die chambers. These two chambers are connected with the bypass path and it is providedin the die. During the process punch movement within the fluid chamber the pressure isgenerated in fluid and it is directed through the bypass path to blank periphery, the fluid filmis created on the upper and lower surfaces of the blank and subsequently reduces frictionalresistance and is to reduce tensile stresses acting on the wall of the semi drawn blank. Theblank is taking at centre place in between blank holder and die surface with supporting ofhigh pressurized viscous fluid. The hoop stresses are produced in the blank in circumferentialdirection due to punch force applied on it, the shear stresses acted by viscous fluid on theboth sides of blank, so apply viscosity phenomenon to this analysis. The blank holderpressure is controlled by the radial pressure of fluid and these are equal for uniformdeformation of blank to obtained required shape and also elimination of failure of blank indeformation. The hoop stresses are determined in terms of viscosity of castor oil, blankgeometry and process parameters for magnesium alloys. Keywords: hoop stress, viscosity, shear stress, deep drawing process, hydro forming 1. INTRODUCTION Hydroforming deep drawing is one of sheet metal forming process to produce seamlessshells, cups and boxes of various shapes. In this forming process, an additional element such asfluid pressure is to be contributes positively in several ways. Amongst the advantages of hydraulicpressure forming deep drawing techniques, increased depth to diameter ratio’s and reduces 43
    • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July-December (2012), © IAEMEthickness variations of the cups formed are notable. Apart from that, the hydraulic pressure isapplied on the periphery of the flange of the cup, the drawing being performed in a simultaneouspush-pull manner making it possible to achieve higher drawing ratio’s then those possible in theconventional deep drawing process. In deep drawing a sheet metal blank is drawn over a die by aradiuses punch. As the blank is drawn radially inwards the flange undergoes radial tension andcircumferential compression [1]. The latter may cause wrinkling of the flange if the draw ratio islarge, or if the cup diameter-to-thickness ratio is high. A blank-holder usually applies sufficientpressure on the blank to prevent wrinkling [2]. Radial tensile stress on the flange being drawn isproduced by the tension on the cup wall and hoop stresses are induced by the punch force. Hence,when drawing cups at larger draw ratios, larger radial tension are created on the flange and highertensile stress is needed on the cup wall. Bending and unbending over the die radius is alsoprovided by this tensile stress on the cup wall. In addition, the tension on the cup wall has to helpto overcome frictional resistance, at the flange and at the die radius. As the tensile stress that thewall of the cup can withstand is limited to the ultimate tensile strength of the material, in the fieldof hydro form deep drawing process the special drawing processes such as hydro-forming [3],hydro-mechanical forming [4], counter-pressure deep drawing [5], hydraulic-pressure- augmenteddeep drawing [6] . The process is an automatic co-ordination of the punch force and blank holding force, lowfriction between the blank and tooling as the high pressure liquid lubricates these interfaces andelimination of the need for a complicated control system [7-12]. The pressure on the flange ismore uniform which makes it easiest to choose the parameters in simulation. The pressure in thedie cavity can be controlled very freely and accurately, with the approximate liquid pressure as afunction of punch position, the parts can drawn without any scratches on the outside of the partand also obtained in good surface finish, surface quality, high dimensional accuracy andcomplicated parts. In the hydroforming deep drawing process the pressurized fluid serves severalpurposes are supports the sheet metal from the start to the end of the forming process, thusyielding a better formed part, delays the on set of material failure and reduces the wrinklesformation. In this paper the hoop stresses are evaluated in terms of viscosity of fluid, blank geometry,and process parameters for magnesium alloys and studied using above process theoretically. Theviscosity phenomenon is considered for evaluation of the process.2. NOTATIONrp = Radius of punch rcp = corner radius on punchrd = radius of die opening rcd = corner radius on diet = thickness of blank rj = radius of blankσr = radial stress σө = hoop stressdθ = angle made by element at job axis Ph = blank holder pressureP = radial pressure of fluidτ = Shear stress acting by the fluid on each side of element2τ = Total Shear stress acted by the fluid on the Elementdr = width of elementr = radial distance of blank element from job axisσo = yield stressσrd = Radial stress at die corner.C = clearance between die and punch = rd − rp(dy)1 = distance between upper surface of the blank element and blank holder 44
    • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July-December (2012), © IAEME(dy)2 = distance between lower surface of the blank element and die surface dy = distance maintained by blank element from both blank holder and die surfaceτ1 = shear stress acted by fluid on upper surface of the blank elementτ2 = shear stress acted by fluid on lower surface of the blank elementdu = velocity of the blank element relative to blank holder and die surfaceµ = dynamic viscosity or absolute viscosity or Viscosity of fluidτA = 2 τ , the total shear stress acting by the fluid on the blank elementh = height of the gap = thickness of fluid 3. MATHEMATICAL MODELING 3.1 Evaluation of Hoop Stress The Hydroforming deep drawing Process as shown in fig. 1. In this 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. The radial stresses are generated in the blank in radial direction and hoop stresses are produced in the blank in circumferential direction due to punch force applied on it. So these stresses are generated in circular blank material during in the hydroforming deep drawing process. The various stresses acting on the blank element during the process is shown in fig.2. For mathematical modeling and evaluation of hoop 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 τ (i.e. shear stress τ is acting by the fluid on the each sides of element and it is same).Then shear force F1 is given by, F1 = 2 τ x Ac Where Ac = fluid contact area of element But Ac = dr rdr dθ + dr dθ 2 45
    • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July-December (2012), © IAEME July DecemberFig.1 Hydroforming Deep Drawing process Fig.2. Stresses acting on the element during ing Drawing process Apply the equilibrium condition in radial direction, i.e. Net forces acting on the element in the radial direction equal to zero. ⇒ ∑F r = 0 + → 2τ ⇒ ( σ r − σ θ ) dr + r dσ r = r dr (1) t As σ r , σ θ are the two principle stresses, the equation is obtain by using Tresca’s yield criteria stresses, σ r − σθ = σ0 (2) ⇒ σ r = σ 0 +σθ ⇒ dσ r = dσ 0 + d σ θ (3) Combining eq. (2) , (3) and eq.(1) 2τ ⇒ σ 0 dr + r dσ 0 + r dσ θ = r dr t Dividing by σ 0 r on both sides, we get 2τ dr ⇒ dσ θ = dr − dσ 0 − σ 0 (4) t r 2τ dr Integrating ⇒ ∫ dσ θ = ∫ t dr − ∫ dσ 0 − ∫ σ 0 r 46
    • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July December (2012), © IAEME July-December 2τ ⇒ σθ = r − σ 0 [1 + ln r ] + C (5) t Where C is constant, it is obtained from boundary condition.That boundary condition : at r = r j , σ θ = − σ 0 and σ r = 0 (Q µ = 0 ) where µ is thecoefficient of friction between blank and both the blank holder and die surface riction 2τThe boundary condition is sub. in eq. (5 ) we get , C = − ub. r j + σ 0 ln r j tComponent C is sub. in eq.(5)   rj    2τ  ⇒ σ θ = σ 0 ln   − 1 − ( rj − r) (6)     r     tThis equation (6) represents mathematical modeling and distribution of hoop stresses in the blankduring the hydroforming deep drawing process.4. PHENOMENA OF VISCOSITYIn this hydroforming deep drawing process, the blank is interaction with the fluid, then the interactionviscosity is comes into the picture. During the process the shear stresses and shear forces areacting by the fluid on the blank in the gap, which is the region between blank holder and diesurface. During the hydroformin deep drawing process, the blank is taking place at middle of the hydroforminggap. The effect of viscosity phenomenon in this process as shown in below fig.3. Newton’s law ofviscosity is introduced to this process for evaluation of hoop stresses in terms viscosity, then thestudy 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 fluidregion i.e gap. as shown in fig.3.But (dy )1 = (dy )2 , because the blank element is taking place at middle of the gap ecause Fig.3.The blank element between bl blank holder and die surface within the fluid region in h−t ∴ (dy )1 = (dy )2 = (dy ) ⇒ dy = 2 but τ 1 = τ 2 47
    • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July-December (2012), © IAEME  du   du   du Because   =   , According to Newton’s law of viscosity τ 1 = µ   ,  dy   dy   dy  τ2=  1   2  1  du µ     dy  2 Let us τ 1 = τ 2 = τThe total shear stress acting by the fluid on the blank element τ A = τ 1 + τ 2 = 2τ 1 = 2τ ∴ τA = 2 τ  du  But τ = µ  ,  dy  Where du = u – 0 = u    du  µu 4µ u ∴ τA = 2τ = 2 µ   dy  = 2 =   h−t  h−t    2  4µ u τA = 2 τ = (7) h−tNow we have to determine the hoop stresses in terms of viscosity and then the study of effect ofviscosity of influence on these stresses is incorporated.5. HOOP STRESSES EXPRESSED IN TERMS OF VISCOSITYWe know that hoop stresses distribution in magnesium alloys blanks during the process is givenby eq.6.   rj    2τ  ⇒ σ θ = σ 0 ln   − 1 − ( rj − r) and substitute 2 τ components from eq. ( 7 ) we  r       tget   rj   4µ u  rj − r  ⇒ σ θ = σ 0 ln    − 1 −      (8)  r   h−t  t The equation (8) represents the effect of viscosity of fluid on the mathematical modeling anddistribution of hoop stresses in the blank during hydroforming deep drawing process.6. DESCRIPTION OF MAGNESIUM ALLOYS Magnesium is the highest of the commercially important metals, having a density of 1.74gm/cm3 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 isalmost always used as an alloy. Even in alloy form, however, the metal is characterized by poorwear, creep and fatigue properties. Strength drops rapidly when the temperature exceeds 1000C,so magnesium should not be considered for elevated – temperature service. Its modulus ofelasticity is even less than that of aluminum, being between one fourth and one fifth that of steel. 48
    • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July-December (2012), © IAEMEThick sections are required to provide adequate stiffness, but the alloy is so light that it is oftenpossible to use thicker sections for the required rigidity and still have a lighter structure than canbe obtained with any other metal. Cost per unit volume is low, so the use of thick sections isgenerally not prohibitive.For engineering applications magnesium is alloyed mainly withaluminum, 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 otherpositive side, magnesium alloys have a relatively high strength-to-weight ratio with somecommercial alloys attaining strengths as high as 300 MPa. High energy absorption means gooddamping of noise and vibration. While many magnesium alloys require enamel or lacquer finishesto impart adequate connection resistance, this property has been improved markedly with thedevelopment of high purity alloys. For this analysis two types of Magnesium alloys considerednamely AZ31B-O and AZ61A-FMagnesium alloy AZ31B-O : composition (%): 3.5 Al, 0.6 Mn , 1.0Zn and Tensile strength240MPa, Yield strength 150MPa.Magnesium alloy AZ61A-F: composition (%): 6.5Al, 1.0Zn and Tensile strength 248MPa, Yieldstrength 220Mpa.7. RESULTS & DISCUSSION7.1 Hoop stressThe hoop stress distribution in the blank during the hydroforming deep drawing is given by eq .8   rj   4µ u  rj − r  ⇒ σ θ = σ 0 ln   − 1 −  r          h−t  t The following process parameters and yield stress values of magnesium alloys are considered forevaluation of hoop stresses of magnesium alloys with given fluid for successful formation of cupin hydroforming deep drawing process.rp = 30 mm, rcp = 4mm, rd = 35mm , rcd = 4mm, rBH = 4 mm, c = 5 mm,Radial pressure of fluid = P , Punch speed (velocity of blank) u = 12mm/sec, h = 14 mm,thickness of blank t = 3mm, radius of blank rj = 105mm ,type of materials used: Magnesium alloystype of fluid used: castor oil , viscosity µ = 0.985N–sec/ m2Yield stress values ( σ 0 ) of magnesium alloys: AZ31B-O σ 0 = 150 x 106 N/m2 AZ61A-F σ 0 = 220 x 106 N/m2 The evaluation of values of hoop stresses ( σ θ ) in the blanks of magnesium alloys with agiven fluid at a radial distance from job axis for a given radius of blanks at constant thickness ofblanks as follows.Substitute the above values in above σ θ equation we get generalized equation for evaluation ofhoop stresses during the process with respect to different radial distance of blank from job axis forblanks of magnesium alloys with castor oil medium are at constant thickness of blanks t = 3mm. At rj = 105mm   105   ⇒ σ θ = σ 0 ln  − 1 − 1.43 [ 105 − r ]   r   49
    • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July-December (2012), © IAEME The hoop stresses of magnesium alloys are presented in fig. 4 at t = 3mm with radius ofblanks rj = 105mm within the range of radial distance r = 65mm to 95mm. From the figure, dueto viscosity of fluids, the shear stresses and shear forces are acted on the blank surface during thehydroforming deep drawing process. 8 2.2x10 AZ31B - O 8 2.0x10 AZ61A - F 8 1.8x10 Hoop stress σ θ ( N/m ) 2 8 1.6x10 8 1.4x10 8 1.2x10 8 1.0x10 7 8.0x10 7 6.0x10 60 65 70 75 80 85 90 95 Radial distance from job axis r (mm) Fig.4. Hoop stress distribution in Magnesium alloys at rj = 105mm So the Hoop stresses are increases with increasing of the radial distance of the blank fromthe job axis. Hoop stresses are also depends up on process parameters, yield stress of alloys andfluid pressure. From fig.4. the magnesium alloys at rj = 105mm with castor oil viscosity, the rangeof radial stresses of AZ61A-F is 114493979.5N/m2 – 197981653.4 N/m2 and AZ31B–O is78064095.16 N/m2 – 134987481.2 N/m2 , the order of hoop stresses of magnesium alloys asAZ31B-O < AZ61A-F.Among these alloys, for a high radial distance from the job axis of blank is95mm, the hoop stress is higher value in AZ61A-F and lowest value in AZ31B-O8. CONCLUSIONS The Hoop stresses are the function of process parameters, yield stress of magnesium alloysand viscosity of castor oil. The hoop stresses are increseses with increasing of the radial distanceof the blank from the vertical job axis. These effects are due to viscosity of castor oil acted on theblanks of magnesium alloys during the forming process. The radial pressure of fluid acting onblank surface of alloys is equal to blank holding pressure is to for uniform deformation of blankduring the process. The wrinkling is reduced in blank due to the blank supported by highpressurized viscous fluid. Hoop stresses of magnesium alloys are determined with in the range ofr is 65mm – 95mm with castor oil. For rj = 105mm, at r = 95mm the highest value of hoop stressoccurred in AZ61A-F is 197981653.4 N/m2 , lowest value occurred in AZ31B-O is 134987481.2N/m2. The increased amount of radial stresses in magnesium alloys with in the range r is 65-95mm, in AZ31B-O is 56923386.04N/m2 and in AZ61A-F is 83487673.9N/m2. So among theorder of increase mount in hoop stresses of magnesium alloys as AZ31B-O < AZ61A– F. Thehoop stresses are in the magnesium alloys are high at r is 95mm, low at r is 65mm.So the hoopstresses are directly proportional to the radial distance from job axis were obtained. The hoopstresses are depends on the viscosity, fluid pressure and process parameters. The higher values of 50
    • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July-December (2012), © IAEMEhoop stresses are gives the minimizing the drawing time and higher in forming limits. These hoopstresses are used to get better results of formability of magnesium alloys.ACKNOWLEDGEMENTOne of the authors (Dr.R.Uday Kumar) thanks the management, director and principal ofMahatma Gandhi Institute of Technology for encouraging and granting permission to carry outthis work.REFERENCES[1] J.M. Alexander, “An appraisal of the theory of deep drawing”, Met. Rev. 5 (19) (1960) 349–409.[2]D.F. Eary, E.A. Reed, “Techniques of Press-working Sheet Metal”, prentice-Hall, New Jersey,1974, pp. 100–172.[3]W. Panknin, W. “Mulhauser, Principles of the hydro form process,” Mittleilungen derforschungrges Blechvererbeitung 24 (1957) 269–277.[4] B. Larsen, “Hydromechanic forming of sheet metal”, Sheet Met.Ind. (Feb. 1977) 162–166.[5] K. Nakamura, “Sheet metal forming with hydraulic counter pressure” in Japan, Ann. CIRP 36(1) (1987) 191–194.[6] S. Thiruvarudchelvan, “A novel pressure augmented hydraulic-deep-drawing process for highdraw ratios,” J. Mater. Proc.Technol. 54 (1995) 355–361.[7] K. Lange, “Handbook of Metal forming,” McGraw-Hill, New York, 1985, pp. 20.21–20.24.[8] S. Yossifon, J. Tirosh, “on the permissible fluid-pressure path in hydroforming deep drawingprocesses—analysis of failures and experiments,” Trans. ASME J. Eng. Ind. 110 (1988) 146–152.[9] S. Thiruvarudchelvan, W. Lewis, “A note on hydro forming with constant fluid pressure,” J.Mater. Process. Technol. 88 (1999) 51–56.[10] K. Oberlander, “The hydromechanical deep drawing of sheet metals-II,” Blech Rohre Profile4 (1982) 161–164.[11] D.Y. Yang, J.B. Kim, D.W. Lee, “Investigations into the manufacturing of very long cups byhydromechnical deep drawing and ironing with controlled radial pressure,” Ann. CIRP 44 (1995)255–258.[12] S.H. Zhang, J. Danckert, “Development of hydro-mechanical deep drawing”, J. Mater.Process. Technol. 83 (1998) 14–25. 51