mechanical-properties-and-working-of-metals-and-alloys_compress

690 14 Drawing: Flat Strip, Round Bar and Tube 14.Ex.3. Calculate the smallest diameter up to which a of 400 rpm. Assuming the maximum possible reduction in a non-hardening wire with an initial diameter of 4 mm could single pass under frictionless homogeneous ideal deforma- be homogeneously drawn without friction and back tension tion condition without any back tension and no slippage in consecutive three passes. Write a formula for the between the wire and the drawing block, calculate the obtainable smallest diameter of a wire in terms of its initial diameter of the drawing block. diameter after consecutive passes for n times. 14.Ex.9. The flow stress of a non-strain-hardening wire with 14.Ex.4. A stainless steel tube having plane-strain defor- initial diameter of 4 mm is 600 MPa. The wire is subjected mation resistance of 700 MPa is subjected to close-pass to the maximum ideal reduction under frictionless condition. drawing through a conical die with a semi-angle of 20°. If the wire is homogeneously drawn without any back ten- Close-pass drawing is performed separately using a conical sion at a speed of 300 m=min; what will be the overall power fixed plug with a semi-angle of 10° and a cylindrical man- requirement for the deformation occurring in a single pass drel. In both cases, the cross-sectional area is homoge- assuming the efficiency of the power unit to be 85%? neously reduced by 30% under plane strain condition with the application of a pulling force of 220 KN. Assume that 14.Ex.10. If the interface friction factor is 0.1 and the the coefficient of friction is 0.08 at the tube–die as well as at drawing stress during sound flow shows a minimum for a the tube–support interface. Compute the maximum area of conical converging die with a total included angle of 24 the drawn product in both cases and indicate which one will degree, calculate the percentage reduction during the draw- produce larger area. ing operation under the above condition. 14.Ex.5. Compare the load required for close-pass cylin- 14.Ex.11. An electric motor with a power of 22.5 kW is drical fixed plug drawing with that for close-pass cylindrical required for frictionless drawing of a round steel bar at a mandrel drawing of a copper tube through a conical die speed of 1:5 m=s using a draw bench. If the diameter of the having a semi-angle of 12° under plane-strain condition. The drawn rod after a single pass is 12 mm and average flow wall thickness of the tube before drawing is 3 mm and that stress of steel is 300 MPa, calculate the initial diameter of after drawing is 2 mm, while the mean diameter of the tube the bar that has been drawn homogeneously under friction- remains unaltered. Assume that the coefficient of friction is less ideal condition without any back tension assuming 0.08 at the tube–die as well as at the tube–support interface efficiency of the motor to be 100%. and neglect the redundant deformation, if any. 14.Ex.12. A metal wire of 12.7 mm diameter is drawn 14.Ex.6. An aluminium tube is close-pass drawn homoge- homogeneously through a conical converging die with a neously with cylindrical fixed plug through a conical die total die angle of 12° to produce a wire of 10.2 mm diam- with a semi-angle of 20° under plane-strain condition to eter. The coefficient of friction at the job–die interface is 0.1. produce 30% reduction in area without any change in the If a back tensile force of 6334 N is applied during drawing diameter. If the coefficient of friction at all interfaces is 0.06 operation and the metal obeys the Ludwik stress–strain and 0.12 obtained by using two different lubricants, what relation with yield stress of 207 MPa, strength coefficient of will be the percentage contribution of friction to the drawing 301 MPa and strain-hardening exponent of 0.54, determine stress for both the lubricants? the maximum possible reduction for the above strain-hardened wire. 14.Ex.7. A 30 mm wide by 5-mm-thick copper strip is given 40% reduction of area by plane-strain drawing without any 14.Ex.13. What is the maximum possible reduction in a back tension through wedge-shaped dies with an included single close pass with a non-hardened tube on a fixed par- total angle of 16°. A cylindrical rod of the same initial allel plug through a conical die with an included total angle cross-sectional area is given an equal reduction in area by of 30°, where the coefficient of friction is 0.1 at the tube–die drawing through a conical die having the same included as well as the tube–plug interface? Assume that plane strain angle. Assume that the average plane strain flow stress of condition exists in the actual deformation zone, and neglect copper is 280 MPa and the coefficient of friction at the job– redundant deformation, if any. die interface is 0.1. Neglecting the redundant deformation, determine the loads required for strip drawing as well as for 14.Ex.14. A 100-mm-wide metal strip of 4 mm thickness is rod drawing and compare the drawing loads. homogeneously drawn without any back tension under plane-strain condition at a speed of 1:5 m=s through a 14.Ex.8. If a non-strain-hardening wire is entering a die at a wedge-shaped dies with total included angle of 24° to reduce speed of 2.25 m=s and drawing block is rotating at a speed its thickness to 3 mm. The average uniaxial flow stress of the

14.10 Solved Problems 691 strip metal is 440 MPa. The coefficient of friction at the (A) Impact extrusion; (B) Rolling; strip–die interface is 0.09, and the density of the strip metal (C) Forging; (D) Cold drawing. is 8800 kg=m3: Determine the total work expended in drawing an 800 kg coil of the metal strip and the time (i) The type of defects observed in rod and wire drawing is: required to draw the coil. (A) Alligatoring; (B) Flash cracking; 14.Ex.15. Indicate the correct or most appropriate answer (C) Chevron cracking; (D) Earing. from the following multiple choices: (j) Which of the following will give the highest optimum (a) Which of the following will give the lowest optimum semicone die angle? semicone die angle? (A) Strip drawing with friction factor m ¼ 0:6; (A) Aluminium with 40% reduction; (B) Rod drawing with m ¼ 0:6; (B) Copper with 40% reduction; (C) Strip drawing with friction factor m ¼ 0:3; (C) Aluminium with 20% reduction; (D) Rod drawing with m ¼ 0:3: (D) Copper with 20% reduction. Answer to Exercise Problems (b) The aim of patenting heat treatment, applied during wire 14.Ex.1. 300 MPa; 154.39 MPa. drawing operation of mostly high-carbon steel, is to cause the formation of 14.Ex.2. 201.1 kN. (A) martensite; 14.Ex.3. init0i.a8l9dimamme; ter/SÀpmffieaffiffixlffilffipeffiffisÁtn. diameter after nth (B) coarse pearlite with proeutectoid phase; passes = (C) fine pearlite with proeutectoid phase; (D) fine pearlite with perhaps some upper bainite, without 14.Ex.4. 551.6 mm2; 881.2 mm2; larger area in mandrel the formation of proeutectoid phase. drawing. (c) The critical cone angle of the die for dead-zone formation increases if friction and % reduction 14.Ex.5. 51% greater in plug drawing than in mandrel (A) both increase; drawing. (B) respectively increases and decreases; (C) respectively decreases and increases; 14.Ex.6. 25.45% and 47.87%. (D) both decrease. 14.Ex.7. 18.47 kN and 16 kN; 15.4% higher load required (d) A wire drawn in the sound flow region from diameter of 3 mm to diameter of 2.59 mm, with an interface friction for strip drawing. factor of 0.2, will require a minimum stress, if the semicone die angle is 14.Ex.8. 292 mm. (A) 08°; (B) 10°; (C) 12°; (D) 14°. 14.Ex.9. 16.3 kW. (e) The smallest diameter up to which a wire of 4.1 mm 14.Ex.10. 44.28%. diameter could theoretically (under frictionless condition) be drawn in consecutive two passes, is very close to 14.Ex.11. 14.9 mm. 14.Ex.12. 63.6%. 14.Ex.13. 46.17%. 14.Ex.14. 17.8 MJ; 202 s. 14.Ex.15. (a) (D) Copper with 20% reduction. (b) (D) fine pearlite with perhaps some upper bainite, without the for- mation of proeutectoid phase. (c) (C) respectively decreases and increases. (d) (C) 12°. (e) (B) 1.5 mm. (f) (C) tubes. (g) (C) decrease. (h) (D) Cold drawing. (i) (C) Chevron cracking. (j) (A) Strip–drawing with friction factor m ¼ 0:6: (A) 1.4 mm; (B) 1.5 mm; (C) 1.6 mm; (D) 2.5 mm. References (f) Sinking is a process to produce Avitzur, B.: Metal Forming: Processes and Analysis, TMH edn, (A) billets; (B) plates and sheets; (C) tubes; (D) slabs. pp. 159–162, 172. McGraw-Hill, Inc., New York (1977), Tata McGraw-Hill publishing Company Limited, New Delhi (1968) (g) If the angle of conical converging die is increased during shaving mode of metal flow, the wire drawing stress will Bland, D.R., Ford, H.: The calculation of roll force and torque in cold strip rolling with tensions. Proc. Inst. Mech. Eng. London 159, 144– (A) remain constant; (B) increase; 153 (1948) (C) decrease; (D) first decrease and then increase. Caddell, R.M., Atkins, A.G.: Trans. ASME J. Eng. Ind. 90, 411–419 (h) The manufacturing process for hypodermic needle is: (1968)

692 14 Drawing: Flat Strip, Round Bar and Tube Dove, A.B. (ed.): Steel Wire Handbook, vol. 4. The Wire Association Pernis, R., Kasala, J.: The influence of the die and floating plug International (1980) geometry on the drawing process of tubing. Int. J. Adv. Manuf. Technol. 65, 1081–1089 (2013) Gokyü, I., Ōkubo, T.: Studies on the roller die (in English). Tetsu To Hagane Overseas 4(1), 45–52 (1964) Pomp, A.: The Manufacture and Properties of Steel Wire (Stahldraht), Translated from the German by Bernhoeft, C.P. The Wire Industry Hosford, W.F., Caddell, R.M.: Metal Forming, pp. 162–163. Limited, London (1964) Prentice-Hall Inc, Englewood Cliffs, NJ (1983) Rowe, G.W.: Principles of Industrial Metalworking Processes, p 135. Hosford, W.F., Caddell, R.M.: Metal Forming Mechanics and Metal- Edward Arnold (Publishers) Ltd., UK, and CBS Publishers and lurgy, 4th edn, p. 169. Cambridge University Press, New York Distributors, India (1977) (2011) Sachs, G., Baldwin, W.M.: Stress analysis of tube sinking. Trans. Kalpakjian, S., Schimid, S.R.: Manufacturing Processes for Engineer- ASME 68, 655–662 (1946) ing Materials, 5th edn, p. 321. Licensees of Pearson Education in South Asia, Dorling Kindersley (India) Pvt. Ltd., New Delhi Sachs, G., Lubahn, J.D., Tracy, D.P.: Drawing of thin-walled tubing (2011) with a moving mandrel through a single stationary die. J. Appl. Mech. Trans. ASME 11, 199–210 (1944) McGannon, H.E. (ed.): The Making, Shaping and Treating of Steel, 8th edn. United States Steel Corporation, Pittsburgh, PA, pp. 782–783, Wistreich, J.G.: Proc. Inst. Mech. Eng. 169, 654 (1955) 787, 796–798 (1964) Wright, R.N.: Wire Technology 4, 57–61 (1976)

Deep Drawing 15 Chapter Objectives • Deep drawing: definition and fundamentals. Stresses and deformation in a deep-drawn cup. • Deep drawing load and its components and their variations with punch movement. Derivation of mathematical expression for load and its comparison with Sachs relation. • Formability: properties of work metal for optimal formability. Strain distribution in a forming operation determined by strain-hardening exponent, strain rate sensitivity and plastic strain ratio. Allowable maximum deformation level represented graphi- cally as forming limit diagram. • Deep drawability and its relation with plastic strain ratio. Conditions for optimal drawability. Measurement of drawability by drawing ratio using Swift cup test. Limiting draw ratio to express deformation limit in deep drawing. • Effects of process variables that include drawing ratio and redrawing operations, profile radii of die and punch, punch-to-die clearance, drawing speed, friction and lubrication, restraint of metal flow, such as use of a blank holder. Effects of material parameters that include sheet thickness and anisotropy. • Evaluation of formability of sheet metal using different type tests, such as Marciniak biaxial stretching test, Swift cup test, Ericksen and Olsen cup tests, Fukui conical cup test, hole expansion test and forming limit diagram using hemispherical punch method. • Defects in deep-drawn components. • Problems and solutions. 15.1 Fundamentals of Deep Drawing applications of this process was the production of artillery shells and cartridge cases. When the ratio of depth of the One of the most important and widely used drawing pro- product to its diameter (or the smallest dimension of its cesses is the production of closed bottom cylindrical or opening) is greater than 1, the process is known as deep rectangular containers from thin metal sheets. This process is drawing, whereas when the ratio is less than 1, it is con- sometimes called shell drawing, because one of the earliest sidered as shallow drawing. The text will refer to deep drawing. Deep drawing process is capable to form bathtubs, © Springer Nature Singapore Pte Ltd. 2018 693 A. Bhaduri, Mechanical Properties and Working of Metals and Alloys, Springer Series in Materials Science 264, https://doi.org/10.1007/978-981-10-7209-3_15

694 15 Deep Drawing cooking pots, beverage cans, sinks, pressure vessels, auto- Blank mobile body panels and parts. Metals that can be deep drawn include plain carbon steel, stainless steel, aluminium and its Cup alloys, copper and its alloy and titanium. Fig. 15.2 Diameter reduction of a circular blank to the diameter of a In the simplest form, the deep drawing operation is car- cylindrical cup top circumference by shrink forming during deep ried out usually at a cold working temperature by placing a drawing operation flat, thin, circular blank of sheet metal of appropriate size over a shaped die and pressing the blank into the die with a Original length of periphery Finallength of punch. Any attempt to perform deep drawing at a hot of blank cup top working temperature may result in necking and failure of the circumference work material. The deep drawing operation is shown Decrease in length by puckering schematically in Fig. 15.1. When the punch advances, the blank is drawn inwards and over the die profile. The cir- Decrease in length by thickening cumference of the original blank must decrease to form the top circle of the cup, called shrink forming, which involves Fig. 15.3 Schematic diagram to show the decrease in length of the the reduction of blank diameter to the diameter of a cylin- blank during deep drawing, either by puckering or by thickening drical cup top circumference, as shown in Fig. 15.2. Since the volume of material must remain constant, the decrease in failure during service when subjected to corrosive environ- the length of the periphery must be compensated in two ment. So, the regions with large amounts of neither shrink ways—by either buckling or wrinkling as the blank is thin, leading to problem like buckling nor stretch associated with or by thickening (increase in another dimension) of metal, as thinning are desirable during drawing operation, and in shown in Fig. 15.3. Relieving of circumferential compres- general, it is the best if the thickness and the surface area sion by thickening of metal is preferred to buckling or remain constant. To avoid wrinkling, tearing or unwanted wrinkling in deep drawing process. To suppress buckling or change in thickness during drawing of a part, the control of wrinkling, a pressure ring or blank holder is used to com- metal flow is usually accomplished through the use of some press the blank against the upper surface of the die during form of pressure pad or ring. forming operation. The force of blank holder causes an increase in either thickness or radial length. Once wrinkling Deep drawing may be performed on a single-action press begins, the blank holder is raised from the surface of the with only one movement that is available, or on a blank metal so that further wrinkles can form easily. Gen- double-action press offering two or more independent erally, a double-action press is used to apply hold-down motions. Simpler operations can be performed with force and punch force. single-action presses, where springs or air pressure is often applied to hold the blank between the pressure ring and Many deep-drawn products contain regions of both upper die. The drawing of complex parts is usually per- shrink and stretch forming. In stretch forming, the circum- formed on double-action press where hold-down force ference and diameter both increase, with a corresponding applied to the pressure ring can be varied as required. Either reduction in thickness. This thin section may cause tearing of mechanically or hydraulically operated presses can be used the parts during forming operation or subsequent premature for deep drawing operations, but the hydraulic presses are preferred because the rate of the punch travel can be better Punch controlled and in some hydraulic system, a more nearly uniform pressure can be applied during the entire stroke. Pressure ring The flow of metal in a drawing operation is generally not homogeneous, because of poor rolling and many other Metal process and metallurgical variables. To ensure the final dimensions and uniformity of the product, excess of starting Blank Cup Die Fig. 15.1 Deep drawing of a circular blank to form a cylindrical cup

15.1 Fundamentals of Deep Drawing 695 Fig. 15.4 Stripping of the drawn (a) (b) part from the punch by using Die trimming recess Trimming recess Deep drawing with die Action of trimming recess at having trimming recess end of deep drawing operation material in the blank and trimming of undesired extra portion cylindrical cup-shaped article from flat sheet metals using a in the finished drawn part may be required in most cases. simple circular-shaped die. Such trimming operation can be performed either manually or by using a separate trimming die. Obviously, the manu- 15.1.1 Stresses and Deformation facturing cost is increased due to such trimming operation in a Deep-Drawn Cup because it not only converts some of the starting material to scrap but also involves an additional operation. Trimming Different types of deformation of the metal are involved in operation may often be excluded through better process and the deep drawing of a cylindrical cup. Stresses and defor- metallurgical control and the use of complicated shaped mation developed in a pie-shaped segment of a cup drawn blanks. In most deep drawing operations, the container from a circular blank are illustrated in Fig. 15.5. The three formed has a solid bottom with a flange at the top and this distinct zones in the deep-drawn cup are flange, cup wall and retained flange is trimmed later in the processing. In some punch region. cases, a straight wall cup shape without any flange is formed by fully drawing the blank into the female die cavity and When a rounded punch is used to form the base of the then ejecting through the die throat. The stripping of the cup, the metal at the centre of the blank under the nose of the drawn part from the punch can be accomplished by punch is folded around the punch profile and in doing so, it machining a minor depression or recess into the underside of the drawing die, as shown in Fig. 15.4a. During the upward Flange stroke of the punch, the drawn cup is released from the punch pressure and tends to spring back. At the end of the Cup wall drawing operation, the die having trimming recess prevents the drawn part from travelling along with the punch during Punch region its upward movement, as illustrated in Fig. 15.4b. 0 The success of deep drawing operation depends on the design aspects of this process, which involves many vari- Fig. 15.5 Stresses and deformation in a segment from a deep-drawn ables such as the force applied by the pressure ring, the cup profile radius of the die and the punch, the type of lubricant, press speed, the clearance between the punch and the die opening and the drawing ratio. Further to achieve the opti- mum drawing conditions, it is required to have knowledge about the strain distribution in the fully drawn product as well as the way in which the strains are generated in different phases of the drawing process. The die used in drawing operation can have any shape from a simple circular to the complex assemblies required for bodies of motor cars, for which several drawing opera- tions may be required. The text will illustrate the principles of deep drawing by considering the formation of a

696 15 Deep Drawing is thinned down. The metal in the region of rounded punch is thickness caused by free thickening, the metal in these areas subjected to stretching; i.e., the biaxial tensile stress acts on will be compressed or ironed and reduced in thickness under the metal in this region. The thinning of the blank occurs the radial compressive stress developed by the pressure of most seriously from stretching over the head of the rounded the punch and die. The degree of ironing usually increases punch and particularly between the punch head and die. In towards the top of the cup. The above clearance may be order to minimize the thinning as far as possible, there must uniform or it may decrease from top to bottom of the die in be high friction on the punch but low friction everywhere order to distribute the load more evenly over the metal in the else. It is found that if the punch is slightly roughened and cup wall. the lubrication of the punch area is minimized, the drawa- bility will increase. However, the die opening must be The punch applies load to the bottom of the cup and smooth and well lubricated with a suitable drawing com- subsequently, the applied load is transferred to the sidewall pound. When a flat-headed punch is used to form the base of of the cup. A narrow ring of material in the cup wall between the cup, then the bottom of the cup does not undergo any the punch head and die has been neither bent over the die or deformation and remains at the original thickness. the punch nor drawn radially is only subjected to plane strain tensile straining and thinning throughout the drawing oper- The metal in the outer part of the blank is drawn radially ation. Mostly this region fails by necking followed by inward towards the opening of the die. The continuous tearing at a stress which is approximately equapl ffiffito UTS decrease of outer circumference of the original blank to that multiplied by the plane strain factor of 2 3 : This of the finished cup due to the inward flow of material approximate maximum stress at fracture is (Dieter 1988): induces a compressive hoop stress. So the flange of the cup is subjected to a tensile stress in the radial direction and a Maximum stress at fracture; rmax ¼ Pmax ¼ p2ffiffi Su compressive stress in the circumferential or hoop direction pDpt 3 of the circular blank. When the compressive hoop stress exceeds a limit, it may result in plastic wrinkling of the cup ð15:1Þ flange, which cannot be ironed out afterwards. If the wrin- kled or buckled metal is drawn into the die during drawing where operation, it will increase the strain in the region of the punch head to the point at which tearing of the work-piece Pmax maximum deep drawing load at fracture; would occur soon after the start of the draw. Since a blank Dp diameter of the punch; holder is used to suppress wrinkling, the compressive hoop t wall thickness of the cup; stress causes the metal in the flange to thicken. The more the Su ultimate tensile strength (UTS) of the work material. metal flows inwards, the greater will be the increase in the thickness. Ultimately, the flange becomes so thick that it When a flat-headed punch is used to form the base of the comes in contact with the blank holder of constant clearance cup, the bottom of the cup remains at the original state type and tries to push the blank holder upwards. Since the without undergoing any deformation, and the metal over the blank holder is kept firmly and cannot move upwards, the punch radius is stretch formed with no sliding of metal over metal in the flange region will be squeezed by the blank the punch. On the other hand, the regions in the flange and in holder. This kind of deformation is called ironing. The the cup wall pass successively through three distinct stages: extent of ironing increases with the rate of inward flow. (a) inward radial drawing of the flange towards the die Ironing imposes large loads on the blank and the punch. The throat, (b) die throat stretching over the die radius, together main factor that decides the maximum possible reduction in with sliding and (c) die profile stretch forming, accompanied deep drawing operation is the ironing load. by sliding along the die surface. However, there is a degree of ironing which increases towards the top of the cup. The However, as the metal is drawn over the die radius, first cup drawn finally will therefore be in a state of inhomoge- bending and then straightening and sliding occurs and neous deformation, in which the undeformed cup-base will simultaneously a tensile stress acts on the metal. The plastic remain in its original metallurgical condition while the bending under tensile stress makes the blank thin and thus, degree of deformation or working will increase towards the the thick portion in the flange region produced by the cir- top of the cup. Hence, the mechanical properties of an end cumferential shrinking is reduced. The metal in the cup wall product will be non-uniform. Differentially annealed blanks is stretch formed between the punch and the die, together have been used with an attempt to produce more uniform with sliding along the die surface. The cup wall metal is mechanical properties in the final product. This process subjected to a state of stress that consists of biaxial tensile involves the flame annealing of the edges of a rotated stress combined with radial compressive stress. If the cold-worked circular sheet blanks so that the rim is annealed clearance between the die and the punch is less than the and the centre remains in the cold-worked condition. This assists to produce a final part of uniform properties.

15.2 Deep-Drawing Load 697 15.2 Deep-Drawing Load • Produce a cup wall thinner than the bottom of the cup; • Produce a uniform cup wall; The deep drawing load, i.e. the load applied on the punch for • Produce a tapered cup wall, if required, such as cartridge production of a cup, is the summation of the following three components: cases; • Rectify the natural thickening of wall that takes place 1. The load required for ideal deformation of the material; 2. The load necessary to overcome friction; towards the top edge of a drawn cup. 3. The load necessary for ironing, if present. The drawing load exhibits therefore two peaks, one early Figure 15.6 illustrates the variation of total drawing load as well as the above three components of it with the and the other late in the drawing cycle. The locations and movement of the punch. If the flow stress of material and the wall thickness of cup are assumed to be constant, the ideal relative heights of these peaks depend upon many factors, load of deformation is a function only of natural logarithm of drawing ratio. The drawing ratio is the ratio of the original such as lubrication conditions, blank holder pressure or diameter of the blank to the internal diameter of the cup, measured by the punch diameter. This component being clearance and drawing ratio. One measurement (Swift 1952) independent of punch travel would appear as a horizontal straight line in Fig. 15.6. The blank holder force contributes of the work required to form cup revealed that 70% of the in a major way to the friction force and hence, the magnitude of friction force depends upon the surface area of the total work was spent for the radial drawing of the metal, 13% material under the blank holder. This force component rises to a peak value at an early stage and then drop with increase for overcoming friction and 17% for bending and in the punch travel, because the surface area of the material in contact with the die and the blank holder continually restraightening of the blank around the radius of the die. The decreases as the drawing proceeds. The ironing load occurs late in the process after the side wall thickness of the cup has last factor, i.e. the load required for bending and unbending exceeded the clearance between the punch and the die and increases with the advancement of drawing operation. around the die radius, was not considered in the above Ironing is basically the reduction of the wall thickness of a cup, usually without causing any change in the diameter. analysis of the variation of total drawing load against the Because of constancy of volume, the height of an ironed cup will obviously be greater than that of the cup which has not punch movement. From an analysis of the forces in equi- been ironed and hence, the cup height will increase with the degree of ironing. Objective of ironing is to librium during the deep drawing of cup, an approximate equation for the total punch (drawing) load as a function of the original diameter of the sheet blank at any stage in the drawing cycle was developed by Sachs (1930), Sachs and Van Horn (1940) and the Sachs relation is as follows:    Dp ! lp ln D0 l 2HB D0 exp P¼ pDp tð1:1r0Þ þ þ B Dp 2 ð15:2Þ where P total punch (drawing) load; r0 average flow stress of work material; Dp diameter of the punch; D0 original diameter of the sheet blank; t wall thickness of the cup; Fig. 15.6 Deep drawing load Drawing load versus punch movement diagram Total load Ideal load Ironing load Friction load Punch Movement

698 15 Deep Drawing HB blank holder force; • The work material is isotropic. l interfacial coefficient of friction; • There is no change in the thickness of work material, B force required to bend and unbend the sheet blank. though varying amount of thickening and thinning of the In (15.2), the first term represents the ideal deformation work piece is unavoidable in deep drawing operation. load required for the formation of the cup, and the second Hence, the wall thickness of the deep-drawn cup is the term is the friction load under the blank holder. The friction same as the original thickness, t0, of the sheet blank. at the die radius is considered by the exponential term out- • There is no ironing of the sheet blank, and so, the force side the bracket, and the force needed to bend and required for ironing is not considered. restraighten the sheet blank around the die radius is • Coulomb’s law of sliding friction holds good, and the accounted by the quantity B. coefficient of friction, l, is constant at the surfaces of the material in contact with the die, blank holder and punch. 15.2.1 Derivation of Mathematical Expression • The force required to bend and unbend the sheet blank is not considered. Derivation of mathematical expression for the ideal defor- Let the radii of the sheet blank, the die, and the punch are, mation and friction load is very difficult task because each component is again divided into several subcomponents. respectively, R0, Rd and Rp. Since thickening and thinning of The loads required for radial drawing, for stretching over the the work piece have been nÀeglected,Á so the clearance die radius, for drawing the material through the throat of the between the die and the punch Rd À Rp , which is the cup die and finally for stretching over the punch radius have to be considered to derive the mathematical expression for deep wall thickness, is equal to the original thickness of the sheet drawing load. A more accurate and complete mathematical treatment for deep drawing has been presented by Chung blank, t0. The radius of the die corner, Rcd, that of the punch, and Swift (1951), and Alexander (1960). Rcp , and a clearance of ‘c’ maintained between the blank holder and the punch are shown in Fig. 15.7. Derivation of mathematical expression for the drawing load correlating the initial and final dimensions of the job Let us first consider the segment of the sheet blank will be considered in the text in a simplified way with some between the blank holder and the top surface of die. Fig- assumptions and compared with Sachs relation given by (15.2). Figure 15.7 shows that a flat circular blank is being ure 15.8 shows stresses acting on an element in this area. deep drawn through a simple circular-shaped die with a flat-headed punch into a flat-bottomed cylindrical cup. The The segment of the sheet blank between the blank holder and following assumptions are considered in our analysis: the die is subjected to a pure radial drawing, where the principal stresses are the radial drawing stress rr ¼ r1, the circumferential stress rh ¼ r2 and the normal stress on the work piece applied by the blank holder rB ¼ r3. If the blank holder force is HB, the stress exerted by blank holder on the blank is: rH ¼ À HB Á ð15:3Þ p R20 À R2d P Punch HB HB Blank c σθ holder Rcp G Flat Rp E circular blank Job axis H σr + d σr C Die t0 rF dr σr D R0 t0 Rcd dθ t0 Rd AB r σθ r=0 r=0 Fig. 15.7 Deep drawing of a flat circular blank to form a flat-bottomed Fig. 15.8 Stresses acting on an element of the sheet blank between the cylindrical cup blank holder and the top surface of die during deep drawing