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

10.4 Lubrication 437 lubricant (Clark et al. 1956). In such condition, there is Coefficient of friction 0.015 Boundary practically no wear. So, full-fluid lubrication is usually 0.010 lubrication desired but in practice, maintaining a continuous thick fluid 0.005 film becomes frequently impossible. Forming operations Partial fluid occur often under partial-fluid lubrication condition, in lubrication which the moving surfaces are partly separated by a liquid lubricant film and partly in direct contact or boundary Fluid lubrication lubrication condition, where the lubricant film becomes a few molecules thick, known as boundary lubricant. 0 hN/P Figure 10.21 shows the variations of COF in the regions of Normal running full-fluid, partial-fluid and boundary lubrication conditions Starting or with changes of the term ðgNÞ=P, where g is the viscosity of Stopping the lubricant, N is the speed of the moving surfaces in rev- olution per minute, and P is the applied load. This figure Fig. 10.21 Variations of COF in different lubrication regions with shows that with increasing the values of term ðgNÞ=P, which changes of the term ðgNÞ=P; where g is the viscosity of the lubricant, corresponds to an increase in viscosity and/or sliding speed N is the speed of the moving surfaces, and P is the applied load under a constant applied load P, the values of COF decrease sharply from high to low and then gradually increase after similar to that of pits and asperities on the moving surfaces; passing through a minimum. The full-fluid lubrication region even monomolecular layers of some compounds form begins at this minimum value and continues with further boundary lubricants that produce low friction. Although they increase in ðgNÞ=P, leading to gradual increase of COF. In form apparently continuous films separating the surfaces but hydrodynamic lubrication, sufficient sliding speed is owing to the thinness, they are of course quickly worn away required to maintain the film thickness greater than the and become discontinuous causing asperity contact. Since a surface roughness, but at a high speed the fluid is heated boundary lubricant is a very thin film of lubricant close to resulting in a decrease in viscosity, which in turn will cause a the breakdown level, it will produce a bright surface in the decrease of the film thickness and an increase in friction. finished product. Since increased speed lowers the viscosity and thereby increases friction, so a compromise between speed and vis- Boundary lubrication operates under the conditions of cosity is required. Finely divided lime or some other solids low values of ðgNÞ=P; corresponding to high applied load, may be added to fluid lubricants to increase their viscosity. low viscosity and slow speed of the moving surfaces. This However, the oil used as thick film usually breaks down due condition is characterized by high COF, as shown in to excessive loads or vibration or on starting and stopping of Fig. 10.21, and the viscosity of the lubricating oil no longer the working machine, which results in a direct contact of the is of primary importance, but rather the nature of rubbing moving surfaces leading to high friction and adhesion. surface is. In hydrodynamic bearing system, the condition of Boundary lubricants are generally obtained by addition of full-fluid lubrication exists, but it is somewhat rare in a relatively small amount of polar organic compounds, such working processes. However, it may occur by creating a as liquid or solid fatty acids, which are particularly effective, converging gap between the work-piece and the tools and by to lubricating oil. The polar groups, such as carboxyl, react high sliding speed, such as high-speed wire drawing and with the surface of work-piece to form a solid soapy rolling operations. For example, the viscous drag exerted on monomolecular layer, which is strongly bound to the surface a lubricant by a wire travelling at high speed may force of work-piece. The resulting film of the reaction products appreciable quantities of lubricant into a low-angle die, and offers an appreciable resistance to compression but low thus, thick fluid film is established. As the normal force resistance to shear, thereby greatly decreasing friction and increases or as the viscosity of the fluid and/or sliding speed wear. However, the shear strength of the film is pressure and decreases, the film thickness is reduced to about three to five temperature dependent. The COF decreases with increasing times the surface roughness. This produces some the thickness of boundary film up to 1000 Å, beyond which material-to-material contact resulting in an increase of COF the film becomes unstable leading to a rise in friction again. and wear rate. 10.4.3.2 Boundary Lubrication Boundary lubricants are generally thin organic films physi- cally absorbed or chemisorbed on the surface of work-piece. They are only a few molecules thick, whose dimension is

438 10 Fundamentals of Mechanical Working Evidence shows that when a boundary lubricant is solid it 10.4.3.4 Solid Lubricants performs the best, but with rise in temperature, friction and Any solid material of lower shear strength than the wear increase noticeably at a temperature near to the melting work-piece can in principal be used as a solid lubricant in point of the film because at the melting point there will be working processes. Neglecting ploughing and interlocking, disorientation of the long molecules leading to the break- the total sliding friction was shown to be equal to the down of the boundary film (Bowden and Tabor 1954 and shearing friction force as in (10.22), i.e. FT ¼ F ¼ Areal s; 1964). therefore, to obtain a low friction between two rubbing surfaces, the real area of contact, Areal, or the tangential shear Lubricating properties of long-chain solid organic com- stress s, or both, must be small. This implies that a material pounds are good, but they adhere less strongly to the surface of low shear strength and high hardness would be the most of work-piece than the reaction products, because long suitable one as lubricant, but such a combination is practi- hydrocarbon chains are oriented outwards in approximately cally impossible because materials of high hardness usually normal directions. The reaction of some materials, including have high shear strength. If, however, a thin film of a soft titanium alloys and stainless steel, with boundary lubricants metal is deposited on a base of hard material, the frictional does not occur easily, and these materials in the presence of shear strength will be the same as that of the soft metal, boundary lubricants are observed to be highly susceptible to while the hard base will largely offer the resistance to material pickup on tools. deformation. In such a case, the contact area Areal will not practically change, even under high applied loads, and the 10.4.3.3 Mixed-Film Lubrication friction will be low. With decreasing the film thickness, the Frequently, an excessive heat is generated due to friction friction will decrease, because the contact area Areal will under high pressure resulting in local rise of temperatures at decrease, whereas the frictional shear strength will remain the surface. One disadvantage of boundary lubricants is that the same. Further, with rise in temperature, the friction of organic compounds, usually used as boundary lubricants, very thin metallic film decreases steadily because of decrease decompose at temperatures of 250 °C or less. For this rea- in the shearing strength of the film, but there is a very little son, special additives are used to form more durable lubri- decrease in the hardness of the base metal because it has cating films that are capable of withstanding higher usually high melting point. But on reaching the melting temperatures. These additives are organic compounds hav- point of the metallic film, the friction begins to increase. ing certain active groups or radicals, such as chlorine, sul- Here, the wetting and spreading ability of the metal in the phur or phosphorus, which react with the surface of molten film over the surface of the hard base metal plays an work-piece to form inorganic films, such as chlorides, sul- important role in the changes that occur. phides or phosphides. But these additives are ineffective on chemically inert surfaces of metals like silver, titanium and Thin copper coatings have been used in forming opera- chromium. tions, including sinking of steel tubes. If a thin layer of copper with shear strength k = 150 MPa is deposited on a For example, chlorinated organic compounds used for work-piece made of steel (base metal) having uniaxial flow protection at higher temperatures react to form solid chlo- stress of r0 = 750 MPa, then the apparent value of coeffi- rides such as FeCl2 which decomposes at about 350 °C, but cient of friction l, depending to some extent on the strain the friction of a chloride film is usually higher than that of a hardening, should be about l ¼ ðkÞCu ðr0ÞSteel ¼ 150=750 ¼ true boundary film. Such compounds were first developed 0:2. Soft metal coatings like tin or lead have also been used for use in gears subjected to high pressures and are usually in some industrial processes. Cold tube drawing operation is referred to as extreme pressure (EP) additives, though their carried out with coatings of lead, which is not only very soft, action is primarily dependent on temperature. Sulphur but also very tenacious. If thick films of lead are used, compounds used as EP additives form sulphides such as FeS several heavy passes may be given to work-piece succes- which decomposes at about 700 °C. Though they are sively without relubrication. effective up to higher temperatures, the friction is somewhat higher still. Since the friction of EP additives is usually The most commonly used crystalline solid lubricants that higher than that of a true boundary film, so boundary can easily form sheared layers between the work-piece and lubricants and EP additives are often blended to produce tools are graphite and molybdenum disulphide (MoS2), used mixed-film lubrication for obtaining low friction over as either alone or as a colloidal suspension in oil or resin. Both large a temperature range as possible. Most deformation of them have lamellar structures that can undergo easy shear. processes are carried out predominantly with boundary Shearing aligns the lamellae parallel to the surface in the lubricant or mixed-film lubricant. direction of motion. The lamellae prevent contact between

10.4 Lubrication 439 the rubbing surfaces even under high loads, and thus, their mechanical forming operations, usually the following sim- lamellar structures account for their low shear strengths that plifying assumptions are made to obtain an easy solution of result in low friction coefficients. The best performance is the analysis: achieved with finer particles on relatively smooth surfaces at high speeds and large particles on relatively rough surfaces • The work-piece is assumed to be a plastic-rigid solid; i.e., at low speeds. On the surface of work-piece, they form elastic strain is neglected and only the plastic strain is continuous layers that offer an appreciable resistance to considered. This assumption is quite logical in the sense compression and exhibit low friction up to high tempera- that the extent of plastic strain involved in the forming tures. Graphite dispersed in tar or grease is useful in hot operations is quite large compared to the elastic strain. working of steel. • Because of large plastic strain involved in the forming Other useful solid lubricants include talc, polyethylene, operations, the strain is expressed in terms of true or polytetrafluoroethylene (Teflon), calcium fluoride, cerium natural strain, e, and stress in terms of true stress r. fluoride, boron nitride and tungsten disulphide. Solid poly- Constancy in volume is also maintained during plastic mers, solid waxes, solid soaps such as sodium- and calcium deformation in working processes. From (1.21), the stearate are widely used in cold working. Phosphate coating constant–volume relationship in terms of principal true are often applied to the work-piece to serve as a base for strain is expressed by retention of lubricant. e1 þ e2 þ e3 ¼ 0 ð10:29Þ 10.4.3.5 Melting Solids Lubrication is carried out by a thin film of liquid generated • Strain hardening is often neglected; i.e., the flow stress is by melting a solid lubricant in contact with a hot work-piece. assumed to be constant, which approaches the conditions For example, during extrusion at about 1000 °C, a pad of existing in the hot-working operations or for metals in a compacted glass powder is placed at the face of the die in highly cold-worked state. But in cases of cold working, front of the nose of a hot billet. Further, glass powder is used where strain hardening is present and flow stress varies to coat the billet surface in contact with the container for with strain, the above assumption will yield erroneous lubrication and thermal insulation. During extrusion, the hot results. When the metal is deformed from an initial strain billet progressively softens the glass pad and a molten film value of e1 to a final strain value of e2; a better approach of glass acting as a lubricant passes uniformly through the is to select the mean flow stress r0; given by the fol- die orifice, providing low friction, good surface finish and lowing (10.30), which can be used in the calculations of greatly improved die life (Sejournet and Delcroix 1955). forming load. Other melting solid lubricants include ice (Wallace 1960) 1 Ze2 and many inorganic compounds that can be applied at À suitable temperatures. For example, a blend of polyethylene r0 ¼ e2 e1 r0 de ð10:30Þ and wax produces good lubrication (Rogers and Rowe 1963) at a temperature of about 100–150 °C. e1 10.5 Mechanics of Working Process Equation (10.30) is explained in Fig. 10.22, which is The working processes are analytically studied to determine obviously better to select than to consider the flow stress at a mainly the forces required to produce a given deformation of mean value of true strain, i.e. at e ¼ ðe1 þ e2Þ=2. For the work-piece and also to predict reduction per pass or the example, to obtain the average flow stress r0, the expression maximum possible amount of deformation without causing for r from either (1.90a) or (1.90d) may be put in place of r0 the fracture to initiate. Such calculations are used to solve in (10.30). However, the following expression has been the problems of selecting or designing and modernization of the equipment to do a particular job. Calculations based on proposed (Shaw 1982) for the flow stress at large strains the theory of working process can also solve the problems of where e [ 1: intensification of deformation, development and improve- ment of plastic working techniques. Due to the quite com- r0 ¼ Kð1 À nÞ þ K n e ð10:31Þ plex nature of deformations and forces generally involved in where K and n are the same parameters as in (1.90a). • The work-piece is considered to be homogeneous and isotropic, practically in all analyses. • In most analyses, Von Mises’ yielding criterion is applied.

440 10 Fundamentals of Mechanical Working e2 surface of contact, which has been considered for the slab s0 de analysis. • All the stresses that act normal to the elemental slice e1 considered in this method are assumed to be principal stresses. • Frictional effects do not change the directions of principal stresses or produce internal distortion. s0 s0 In mechanical working operations, since compressive stresses play an important role, so they are considered as e1 e2 positive stresses and tensile stresses as negative, while the e reverse is true in the theory of plasticity, such as in any mechanical testing operation. Similarly, compressive strains Fig. 10.22 Diagrammatical illustration of mean flow stress will be considered as positive and tensile strains as negative in metalworking operations. The method of slab analysis has There are various methods of analysis, the names of been mainly considered in the text and applied to each and which are arranged below in order of increasing complexity every field of the working operations, wherever the defor- and ability to predict fine detail: mation stresses or loads have been determined. 10.5.2 Uniform-Deformation Energy Method 1. The slab method, Let us consider a simple case, where friction at the job–tool 2. Uniform-deformation energy method, interface and the influence of transverse stresses, i.e. plastic 3. Slip-line field theory, constraint, are neglected. Further, it is assumed that the 4. Upper- and lower-bound solutions, deformation is uniform or homogeneous; i.e., there is no 5. Finite element method. internal redundant deformation. That means we only con- sider the case of energy required for ideal work of plastic 10.5.1 Slab Method deformation. If the increment in length of the work-piece is dL, under an applied load of P, the increment of work will be dðW:D:Þ ¼ P dL ¼ r0 A dL The slab method of analysis is also known as free-body where r0 is the average flow stress, and A is the instanta- equilibrium approach. In this method, it is assumed that the neous cross-sectional area of the work-piece. The increment material deforms homogeneously in the deformation zone. of work per unit volume V will be Homogeneous deformation means plane sections remain plane during deformation; i.e., square elements of the defor- dðW :D:Þ ¼ r0A dL ¼ r0 de mation zone deform uniformly into rectangular elements. In V A L the 1920s, von Kármán, Hencky, Siebel and later on Sachs (Hoffman and Sachs 1953) developed the slab method, which If L1 is the initial length before deformation and L2 is the was the earliest approach of analysis based on the equilibrium length after deformation of the work-piece, the plastic of forces acting on an infinitely thin slab of material. The deformation energy per unit volume will be given by solution by this method is essentially the same to that used in the field of mechanics, where free-body equilibrium approach WZ:D: dðW:D:Þ ¼ W :D: Ze is considered. The slab method is helpful to estimate the role V V of friction on the deformation load. The important assump- UD ¼ ¼ r0 de tions in this method are as follows: 00 ð10:32Þ • The deformation is uniform or homogeneous; i.e., there is no internal redundant deformation. ZL2 dL ln L2 L L1 • Normal and tangential shear stresses are assumed to be ¼ r0 ¼ r0 constant within an infinitely small element of area on the L1 Now, let us take an example of drawing of a cylindrical wire, whose initial cross-sectional area A1 is drawn down to area A2, causing an increase in length from L1 to L2. From

10.5 Mechanics of Working Process 441 the constancy of volume, V, during plastic deformation, one they do not contribute directly to the changes in external can write, form of the worked product. The corresponding amount of extra or wasted work is called the redundant deformation or V ¼ A1 L1 ¼ A2 L2; or, L2 ¼ A1 ð10:33Þ work. Thus, we can say that the redundant deformation is L1 A2 involved in bending the metal fibres in one way and then back to the initial flow direction. Since the constraint for the And the reduction in cross-sectional area of the wire will metal to flow will be influenced by the friction at the job– be tool interface, the redundant deformation will depend upon the coefficient of friction. It must be noted that the redundant r ¼ A1 À A2 ¼ 1 À A2 ; or; A1 ¼ 1 1 r ð10:34Þ shearing increases the strain hardening, and the effect is A1 A1 A2 À additive, so that a number of light passes required to obtain a given overall reduction of area will harden the stock Now, the total work done under the applied load P is appreciably higher than one or two heavy passes required for the same reduction. W :D: ¼ P L2 ¼ V UD ¼ A2L2r0 ln L2 L1 Hence, the total deformation work per unit volume UT will be the summation of UD, UF and UR, which means [From (10.32) and ∵ V = A2L2]. Hence, from (10.33) and (10.34): ln A1 UT ¼ UD þ UF þ UR ð10:36Þ A2 Deformation load, P ¼ A2r0 Hence, the efficiency of a working process is expressed as 1 ð10:35aÞ À ¼ A2r0 ln r g ¼ ideal work ¼ UD ð10:37Þ 1 total work UT ) Deformation stress r ¼ P ¼ r0 ln A1 Depending on the particular working process, die A2 A2 geometry, frictional conditions and other process parameters, ð10:35bÞ values of efficiency vary widely; for example, typical values 1 of η are 75–95% for rolling and 30–60% for extrusion. The ¼ r0 ln À r less efficiency in extrusion is due to more expenditure of 1 energy for internal redundant deformation involved in extrusion, i.e. due to higher value of UR, in extrusion. Equations (10.35a) are the simple work formula for load and stress in an ideal homogeneous deformation without 10.5.3 Slip-Line Field Theory considering the contribution due to friction. Slip-line method is widely employed to determine stresses in Equation (10.32) gives the work per unit volume required a plane-strain deformation inside the body and on the contact for ideal plastic deformation UD, which is substantially less surface of the deformation zone. Slip-line field theory for than the total deformation work per unit volume in an actual plane strain can particularly be applied to calculate the working process. Because in deriving (10.32), we have overall forming loads when inhomogeneous plastic defor- neglected the work per unit volume required to overcome the mation is involved, as in the case of extrusion and drawing friction at the job–tool interface UF and the internal redun- operations, where slab method cannot estimate the working dant deformation or work per unit volume UR. Redundant loads correctly. Slip-line field analysis is based on the fol- deformation is the work required or energy expended for lowing assumptions: internal shear distortion in the work-piece caused by non-uniform plastic deformation which does not contribute • The material is considered to be isotropic and to the change in the external dimension or shape of the homogeneous. worked product. The amount of redundant deformation depends on friction and the geometry of the die or the • The material is considered as rigid-ideal plastic body; working process. Let us explain the redundant deformation i.e., elastic strain as well as strain hardening is neglected. taking an example of wire drawing, in which an element of the wire near the periphery moves initially towards the die in • At the interface between the dies or tools and the an axial direction, but on entering the die it is forced to move work-piece, the shear stresses are constant; usually, either with an inward radial velocity component due to constraint frictionless condition or sticking friction prevails. offered by the shape of the die. This is accomplished by shearing deformation within the wire. As the wire passes • Deformation under plane strain (two-dimensional strain) through the exit from the die, the element is sheared back condition is only considered, because an exact again, to advance once again in a direction parallel to the axis. Both these shear processes consume energy, though

442 10 Fundamentals of Mechanical Working three-dimensional solution of the problem often becomes r3 ¼ r1 þ r2 ¼ rm ð10:40cÞ too difficult. 2 • Effects of strain rate and temperature are ignored. For a non-strain-hardening material, k is constant every- Slip-line field theory is based on the fact that any general where, but rm may vary from point to point inside the metal. state of stress under plane strain (a triaxial state of stress) A complete stress solution can be found for the metal if it is possible to determine the magnitude of rm at each point condition consists of two components: (1) the hydrostatic inside the metal and the direction of k at that point. The slip component of stress or mean stress, rm, which does not affect yielding, and (2) the yield stress in pure shear, k. This lines, to be discussed subsequently, show the direction of k at any point, while the changes in the magnitude of rm can is shown in Fig. 10.23 and explained below, considering a be deduced from the angular rotation of the slip line between triaxial state of tensile principal stresses, r1, r2 and r3, in one point and another in the slip field. which r1 > r3 > r2. In order to calculate stress with the help of slip-line fields, From (1.85), since the principal stress in the direction ‘3’ let us now find the equations for stresses on a physical body in under plane strain condition is given by r3 ¼ ðr1 þ r2Þ=2; the xy coordinate system in terms of rm and k. Figure 10.24a so from (1.59), the hydrostatic component of stress or mean shows the state of stress on a physical body, which is repre- sented by Mohr’s circle given in Fig. 10.24b. Note that stress (considered as tensile) in plane strain is: according to convention, if a shear stress has a clockwise rm ¼ r1 þ r2 þ ðr1 þ r2Þ=2 ¼ r1 þ r2 ð10:38Þ sense of rotation about any point in the physical element, it is 3 2 ð10:39Þ considered to be positive and plotted above the horizontal axis From (1.86a) or (1.88b), we know that of the Mohr’s circle. From the Mohr’s circle diagram, the stresses can be expressed as r1 À r2 ¼ 2k; or; r1 À r2 ¼ k 2 rx ¼ rm þ k cos 2h ð10:41aÞ where k is the yield stress in pure shear. Therefore, from ry ¼ rm À k cos 2h ð10:41bÞ (10.38) and (10.39), r1 ¼ r1 þ r2 þ r1 À r2 ¼ r1 þ r2 þ r1 À r2 ¼ rm þk rz ¼ r3 ¼ rm ð10:41cÞ 2 2 2 sxy ¼ syx ¼ k sin 2h ð10:41dÞ ð10:40aÞ r2 ¼ r1 þ r2 À ðr1 À r2Þ ¼ r1 þ r2 À r1 À r2 ¼ rm À k where h is an arbitrary angle of inclination of r1 with the x- 2 2 2 axis on the physical element, as shown in Fig. 10.24c, and 2h is the angle between rx and r1 on Mohr’s circle. ð10:40bÞ sm sm sm = s1 + s2 = s3 10.5.3.1 Slip Lines kk 2 A slip line is the line, straight or curved, which is tangential at every point to the plane of maximum shear stress, i.e. a line of k = s1 s2 maximum shear stress, and is therefore oriented at 45° to the 2 axes of principal stresses. Hence, there are two orthogonal families of slip lines called a-lines and b-lines, as shown in sm sm Fig. 10.25. Along the tangents to the directions of slip lines, kk shear strain is a maximum and linear strain is zero. These slip lines have no relationship with the microscopic slip lines sm observed on the surface of plastically deformed crystals. For sm identification of slip lines, the convention is that an a-line is 45° to the right (i.e. lies at 45° in the clockwise direction) from Fig. 10.23 State of stress under plane strain condition the first principal direction ‘1’, along which the algebraically highest principal stress r1 acts, and a b-line departs to the left, i.e. in the counterclockwise direction from the direction of the algebraically highest principal stress r1 by the same angle. In other words, the direction of the algebraically largest principal stress (most tensile), i.e. r1, passes through the first and third quadrants formed by a right-handed aÀb coordinate system as illustrated in Fig. 10.25.

10.5 Mechanics of Working Process 443 Fig. 10.24 a State of stress on a (a) σy σx > σy physical body. b Mohr’s circle for τyx = τxy (a), where rm is shown as tensile. σx c Angular relationship between (b) the directions of principle stresses and x- and y-axes σy τ + τxy k τ max = k σ+ σ 2θ (c) y-axis O −τxy σ2 σ2 θ σ1 σx θ σ1 σ 1 x-axis σz =σ3 = σm σ2 If the angle of inclination of the tangent to an a-line s1 measured in an anticlockwise direction from the x-axis is denoted by /; then the slopes of slip lines of a- and y b -line b-families are, respectively f dy ¼ tan /; for aÀlines; kk dx and dy ¼ À cot /; for bÀlines: 45° a-line 45° s2 dx x If h is an arbitrary angle of inclination of the first prin- cipal direction ‘1’, (along which r1 acts) measured in an anticlockwise direction from the x-axis, it is evident from Fig. 10.25 that h ¼ p þ/ ð10:42Þ k 4 k Hence, from (10.42) and (10.41), we get q rx ¼ rm À k sin 2/ ð10:43aÞ ry ¼ rm þ k sin 2/ ð10:43bÞ Fig. 10.25 a- and b-lines tangential to the plane of maximum shear rz ¼ r2 ¼ rm ð10:43cÞ stress are shown for a curvilinear element ð10:43dÞ sxy ¼ syx ¼ k cos 2u Suppose the point P in a body stressed under plane strain condition is circumscribed symmetrically with a rectangle 10.5.3.2 Hencky’s Slip-Line Equations whose lengths of sides are @x and @y; as shown in Fig. 10.26. Hencky’s slip-line equations are developed from equations of Stresses acting under plane strain condition on the faces of force equilibrium in plane strain, which is derived below. the rectangular element are also shown in this figure. The rates of change of stress components with increase in x, when

444 10 Fundamentals of Mechanical Working  For a-lines, @rm À 2k @/ ¼ 0; y is kept constant, are @rx=@x; and @sxy @x: Similar @a @a expressions for direction y are @sxy @y and @ry @y: ð10:46aÞ Equilibrium of forces in the x-direction gives Or, @ ðrm À 2kuÞ ¼ 0 @a &  ' &  ' þ1 @rx 1 @sxy rx &2 @x @ x @y þ sxy þ 2 @y  @ y @x For b-lines, @rm þ 2k @/ ¼ 0;  ' &  @b @b ' or, ¼ rx À 1 @rx @x @y þ sxy À 1 @sxy @y @x @ ð10:46bÞ 2 @x 2 @y @b ðrm þ 2 k uÞ ¼ 0 ) @rx þ @sxy ¼ 0 ð10:44aÞ Thus, from (10.46), the variation of hydrostatic stress rm @x @y with change in direction of the slip lines is given by the Hencky’s slip-line equations as follows: Similarly, equilibrium of forces in the y-direction gives rm À / ¼ constant ¼ nðsayÞ; for an a-line ð10:47aÞ @ry þ @sxy ¼ 0 ð10:44bÞ 2k @y @x rm þ / ¼ constant ¼ gðsayÞ; for a b-line ð10:47bÞ Hence, from (10.43) and (10.44), we get 2k @ ðrm À k sin 2/Þ þ @ ðk cos 2uÞ ¼ 0; @x @y And; @ ðrm þ k sin 2/Þ þ @ ðk cos 2uÞ ¼ 0: 10.5.3.3 Stresses and Slip Lines @y @x at the Boundaries of a Plastic Body ) @rm À 2 k cos 2/ @/ À 2 k sin 2/ @/ ¼ 0 ð10:45aÞ (1) Free Surface @x @x @y The plastic zone sometimes extends to the free surface And; @rm þ2k cos 2/ @/ À 2k sin 2/ @/ ¼ 0 beyond the confines of the tool. There can be no tangential @y @y @x shear stress at a free surface. Thus, one of the principal planes is parallel to the free surface. Hence, slip lines must ð10:45bÞ meet a free surface at 45°. It is to be noted that there are two possible configurations, as shown in Fig. 10.27. In Fig. 10.25, if the coordinate axes, x and y, are chosen to coincide with the tangents to the slip lines, then / = 0, In Fig. 10.27a, r1 = 0, so from (10.40a): and therefore, cos 2u ¼ 1 and sin 2 u ¼ 0: Hence, (10.45) reduces to rm þ k ¼ 0; or; rm ¼ Àk: y 1 ∂σ y * From (10.40b), 2 ∂y r2 ¼ rm À k; ) r2 ¼ À2k: σy + ∂y In Fig. 10.27b, r2 = 0, so from (10.40b): τ xy + 1 ∂ τxy ∂y rm À k ¼ 0; or; rm ¼ k: 2 ∂y * From (10.40a), _ 1 ∂σ x τ xy + 1 ∂ τxy ∂x r1 ¼ rm þ k; ) r1 ¼ 2k: 2 ∂x 2 ∂x σx ∂x (2) Frictionless Interface P ( x, y) 1 ∂σ x ∂y σx + 2 ∂x ∂x τ xy _ 1 ∂ τxy ∂x ∂x 2 ∂x 1 ∂ τxy τ xy 2 ∂y ∂y σy 1 ∂σy ∂y There can be no resultant shear parallel to a smooth or 2 ∂y frictionless interface, so slip lines must meet a frictionless interface at 45°. x Fig. 10.26 Stresses acting on an elemental unit cube under plane strain condition

10.5 Mechanics of Working Process (a) s1 = 0 (b) s2 = 0 445 s1 Fig. 10.27 Slip lines meeting a s2 s1 free surface s2 s1 s2 b a b a (3) Axis of Symmetry 10.5.3.4 Simple State of Stress In case of a simple state of stress, one family of slip lines The centre-line of a symmetrical work-piece can have no (say, a-lines) consists of straight lines; curves orthogonal to resultant shear component along it, so the slip lines must the latter form the other family (b-family in this case). This make an angle of 45° with an axis of symmetry (centre-line). is shown in Fig. 10.29. (4) Perfectly Rough Interface Using (10.47a), along the a-lines, one can write If the friction is very high, the condition of sticking friction ðrmÞA À /A ¼ ðrmÞB À /B prevails. There is no interfacial movement; the material will 2k 2k yield beneath the interface, when the tangential shear stress reaches the value k, its yield stress in pure shear. Hence, ¼ ðrmÞC À /C ¼ constant ¼ n1 ðsayÞ sxy ¼ k is independent of the normal stress. Consequently, 2k one slip line meets the interface tangentially, i.e. at angle of 0° and the other normally, i.e. at angle of 90°, as shown in ) ðrmÞAÀðrmÞB ¼ /A À /B ð10:48Þ Fig. 10.28. 2k Hence, we can conclude Using (10.47b), along the b-lines, one can write • The slip lines, i.e. a- and b-lines, must meet a free surface ðrmÞA þ /A ¼ ðrmÞD þ /D ¼ ðrmÞG at 45°, because the stress normal to this surface is a 2k 2k 2k principal stress. þ /G ¼ ðrmÞJ þ /J ¼ constant ¼ g1 ðsayÞ • The slip lines must meet a frictionless interface at 45°. 2k • The slip lines must meet an axis of symmetry at 45°. • The slip lines meet rough surfaces having condition of C B sticking friction at 0° and 90°. p A τ xy = k F E D GH I α -lines J K L Fig. 10.28 Slip lines meeting a rough interface leading to the β -lines Fig. 10.29 Simple state of stress condition of sticking friction. p is the axial compressive stress; i.e., axial pressure, sx y, is the tangential shear stress, and k is the yield stress in pure shear, where sx y ¼ k

446 10 Fundamentals of Mechanical Working ðrmÞB þ /B ¼ ðrmÞE þ /E ¼ ðrmÞH 2k 2k 2k þ /H ¼ ðrmÞK þ /K ¼ constant ¼ g2 ðsayÞ β -family 2k ) g1 À g2 ¼ ðrmÞAÀðrmÞB þ ð/A À /BÞ ð10:49Þ 2k x Substituting (10.48) into (10.49) and noting that for a α -family straight a-line, /A ¼ /B; g1 À g2 ¼ 2ð/A À /BÞ ¼ 0; ) g1 ¼ g2: Similarly, it can be shown that g1 ¼ g2 ¼ g3 ¼ g4 ¼ Á Á Á ð10:50Þ Hence, the parameter η is constant over the whole field, Fig. 10.30 Uniform state of stress showing net of straight lines where the state of stress is simple and a-family is a family of straight lines. At the centre O, the stresses are discontinuous. Therefore, it is a singular point of a given stress field. Similarly, if the b-family is a family of straight lines, the parameter n is constant over the whole field, where the state In our example (Fig. 10.31), where the a-lines are of stress is simple, i.e. straight, the parameter η is constant, say η = η0. But n1 ¼ n2 ¼ n3 ¼ n4 ¼ Á Á Á ð10:51Þ rm þ/ ¼ g0 or; rm ¼ g0 À / 2k 2k ) rm ¼ 2kðg0 À /Þ ð10:52cÞ (1) Uniform State of Stress Hence, the normal stresses on radial and circumferential planes are linear function of /, the angle between the x-axis A special case of the simple state of stress is a uniform state and the a-line. of stress; in such regions, the slip-line network is formed by two orthogonal families of parallel straight lines, and 10.5.3.5 Hencky’s First Theorem therefore, the parameters n and η are constant. Let us con- Hencky’s first theorem states that if we pass from one slip sider any point, x, in the slip-line field (Fig. 10.30), where line to another of the same family along any slip line of the the state of stress is uniform. Then, other family, the hydrostatic stress, rm, and the angle, /, (the ðrmÞx À /x ¼ constant ¼ n ðsayÞ; and 2k O ðrmÞx 2k þ /x ¼ constant ¼ g ðsayÞ ) ðrmÞx ¼ k ðn þ g Þ ¼ constant ð10:52aÞ And, /x ¼ ðg À nÞ ¼ constant ð10:52bÞ σm σm 2 Since n and η are constant over the whole field, the hy- σm drostatic stress, rm; is also constant over the whole field, as shown by (10.52a). β -lines σm (2) Centred Fan-Type Field α -lines An important case of a simple state of stress is a slip-line field of the centred fan type, formed by a bunch of straight lines and concentric circular arcs, as shown in Fig. 10.31. Fig. 10.31 Centred fan-type field

10.5 Mechanics of Working Process 447 angle between the x-axis and the a-line) change by a con- ðrmÞA þ /A ¼ ðrmÞD þ /D stant amount. Referring to Fig. 10.32, along the a-lines, the 2k 2k theorem can be mathematically stated as follows: ¼ ðrmÞG þ /G ¼ constant ¼ g1 ðsayÞ 2k ðrmÞBÀðrmÞA¼ ðrmÞEÀðrmÞD¼ ðrmÞHÀðrmÞG ð10:53aÞ And; /B À /A ¼ /E À /D ¼ /H À /G ð10:53bÞ ðrmÞB þ /B ¼ ðrmÞE þ /E 2k 2k Similarly, along the b-lines, the theorem states mathe- ¼ ðrmÞH þ /H ¼ constant ¼ g2 ðsayÞ matically that 2k ðrmÞDÀðrmÞA¼ ðrmÞEÀðrmÞB¼ ðrmÞFÀðrmÞC ð10:54aÞ ðrmÞC þ /C ¼ ðrmÞF þ /F 2k 2k And; /D À /A ¼ /E À /B ¼ /F À /C ð10:54bÞ ¼ ðrmÞI þ /I ¼ constant ¼ g3 ðsayÞ 2k Proof ) ðrmÞA¼ kðn1 þ g1Þ; and /A ¼ g1 À n1 : 2 ðrmÞA À /A ¼ ðrmÞB À /B ðrmÞB¼ kðn1 þ g2Þ; and /B ¼ g2 À n1 : 2k 2k 2 ¼ ðrmÞC À /C ¼ constant ¼ n1 ðsayÞ g1 À n2 2k 2 ðrmÞD¼ kðn2 þ g1Þ; and /D ¼ : ðrmÞD À /D ¼ ðrmÞE À /E g2 À n2 2k 2k 2 ðrmÞE¼ kðn2 þ g2Þ; and /E ¼ : ðrmÞF ¼ 2k À /F ¼ constant ¼ n2 ðsayÞ ðrmÞH¼ kðn3 þ g2Þ; and /H ¼ g2 À n3 : 2 ðrmÞG À /G ¼ ðrmÞH À /H 2k 2k g1 À n3 ¼ ðrmÞI À /I ¼ ¼ n3 ðsayÞ ðrmÞG¼ kðn3 þ g1Þ; and /G ¼ 2 : 2k constant ) ðrmÞBÀðrmÞA¼ kðg2 À g1Þ; g2 À g1 and /B À /A ¼ 2 : ξ1 ðrmÞEÀðrmÞD¼ kðg2 À g1Þ; C g2 À g1 η and /E À /D ¼ 2 : 3 η2 ðrmÞHÀðrmÞG¼ kðg2 À g1Þ; η1 g2 À g1 B /H À /G ¼ 2 : A and ξ2 ) ðrmÞBÀðrmÞA¼ ðrmÞEÀðrmÞD ¼ ðrmÞHÀðrmÞG; F and /B À /A ¼ /E À /D ¼ /H À /G: E D Along the b-lines, (10.54) can also be similarly proved. GH ξ3 10.5.3.6 Numerical Method of Solution I A slip-line field consists of a- and b-lines intersecting at points called the nodal points. Any nodal point can be Fig. 10.32 Drawing to illustrate Hencky’s first theorem assigned the coordinates ð0; 0Þ: The coordinates of any other

448 10 Fundamentals of Mechanical Working point may be denoted by (m, n), m increasing along an a-line 10.5.3.7 Application of Slip-Line Field to Static and n increasing along a b-line, as shown in Fig. 10.33. By System Hencky’s first theorem, Let us consider plane-strain indentation with flat frictionless ðrmÞm;nÀðrmÞmÀ1;n¼ ðrmÞm;nÀ1ÀðrmÞmÀ1;nÀ1 platens for cases of various heights of work-piece with respect to the breadth of platen. Or; ðrmÞm;n¼ ðrmÞmÀ1;n þ ðrmÞm;nÀ1ÀðrmÞmÀ1;nÀ1 Case I: Strip Height Equal to Platen Breadth ð10:55aÞ Features are as follows: And, /m;n À /mÀ1;n ¼ /m;nÀ1 À /mÀ1;nÀ1 • The slip lines meet a frictionless interface at 45°. • The field is symmetrical about the centre-line. Or, /m;n ¼ /mÀ1;n þ /m;nÀ1 À /mÀ1;nÀ1 ð10:55bÞ • The centre-line of the strip is therefore an axis of sym- Again, metry and can have no resultant shear component along it, so the slip lines meet the centre-line also at 45°. dy=dx ¼ tan /; along an a-line; and It can be seen that in this particular case, the diagonals are dy=dx ¼ À cot /; along a b-line: the slip lines, so the uniform stress field shown in Fig. 10.34 fulfils the boundary conditions. It must be noted that the Replacing these differential equations by difference construction of a slip-line field involves the separation of rigid and plastic regions. For example, in Fig. 10.34, the equations, we get regions I and III are plastic and the regions II and IV are rigid. ym;n À ymÀ1;n ¼ À À Á tan/m;n þ /m  xm;n xmÀ1;n 2 Let us consider a point on the boundary slip-line BD. À1;n Since here the slip lines are at 45° to the horizontal, the ð10:56aÞ principal stresses r1 and r2 are horizontal and vertical. And, À ym;nÀ1 ¼ À À Á cot/m;n þ /m;nÀ1 Since here r1 = 0 so from (10.40a), rm þ k ¼ 0; or; ym;n À xm;n xm;nÀ1 2 rm ¼ Àk: ) From (10.40b), ð10:56bÞ r2 ¼ rm À k ¼ À2k: The above (10.56) is used to determine xm;n and ym;n: p ¼ 1: 2k ) The indentation pressure; p ¼ Àr2 ¼ 2k; or; p b (2,0) A 45° 45° B (1,0) I σ2 σ1 σ1 = 0 σ2 (0,0) (2,1) h II IV (1,1) (0,1) α -lines III C D (0,2) (1,2) (2,2) β -lines p Fig. 10.33 Coordinates of nodal points (intersecting points of a- and Fig. 10.34 Slip-line field for plane-strain indentation showing sepa- b-lines) ration of plastic regions, I and III, and rigid regions, II and IV, for a strip whose height, h, is equal to platen breadth, b

10.5 Mechanics of Working Process 449 Case II: Platen Breadth an Integral Multiple of Strip y-axis Height p b/2 With reference to Fig. 10.35, here also ðp=2 kÞ ¼ 1: Case III: Strip Height Greater than Platen Breadth x-axis With reference to Fig. 10.36, for different nodal points, the 45º 15º values of angle / are shown below: σ2 /0;0 ¼ À p ; /1;0 ¼ Àp ; (0,0) 4 6 pp /0;1 ¼ À 3 ; /1;1 ¼ À 4 : (0,1) (1,0) If p is the indentation pressure, then at the point of β -line (1,1) α -line indentation, r2 ¼ Àp: 45º 45º Now, from (10.40b), rm À k ¼ Àp; or; rm ¼ k À p; h/2 Or, rm ¼ 1 À p : p 2k 2 2k ) ðrmÞ0;0 ¼ 1 À p : 2k 2 2k Using (10.47a), along the a-line, ðrmÞ1;0 À /1;0 ¼ ðrmÞ0;0 À /0;0; Fig. 10.36 Slip-line field for plane-strain indentation for a strip whose 2k 2k height h is greater than the breadth b of platen ) ðrmÞ1;0 ¼ 1 À p þ p À p ) ðrmÞ1;1 ¼ 1 À p þ p À p þ p 2k 2 2k 4 6 2k 2 2k 12 6 4 ¼1À p þ p : ¼ 1À p þ p: 2 2k 12 2 2k 6 Using (10.47b), along the b-line, Now, from (10.43a), ðrmÞ1;1 þ /1;1 ¼ ðrmÞ1;0 þ /1;0; ðrxÞ1;0¼ ðrmÞ1;0 À k sin 2/1;0; 2k 2k Or, ðrxÞ1;0 ¼ 1 À p þ p À 1 sin  p 2À 2k 2 2k 12 2p 6 ¼1þ p 1 b = 3h þ sin À p p 2 12 2pffiffi 3 2k ¼1 þ p þ 3À p ; 2 12 4 2k h ) ðrxÞ1;0 ¼ 1:1948 À p ð10:57aÞ 2k 2k Similarly, ðrxÞ1;1 ¼ 1 Àp þ p À 1 sin À Á 2k ¼ 2k þ 16sin22 Â 2 /1;1 p 2 p À 1 þp p 262 4 2k Fig. 10.35 Slip-line field for plane-strain indentation for a strip whose ¼ 1 þ p þ 1À p ¼ 1þ pÀ p height h is one-third of the breadth b of platen 2 6 2 2k 6 2k

450 10 Fundamentals of Mechanical Working ) ðrxÞ1;1 ¼ 1:5236 À p ð10:57bÞ  2 k 2k Or, 0:8032  bp ¼ 1:1948  0:6124 b þ 1:3592 Along the a-line, we can write from (10.56a), 2k  0:1908 b; y1;1 À y0;1 ¼ À À Á tan/1;1 þ /0;1 ð10:57cÞ x1;1 x0;1 2 ) p ¼ 1:2339: 2k If b = the breadth of the platen, then the coordinates of the nodal points are Proceeding in this way, it can be shown that (i) x0;0 ¼ 0; and y0;0 ¼ À b ¼ À0:5 b: h 1.0 1.6064 2.4461 3.6613 5.4773 8.2540 2 b p 1.0 1.2339 1.4554 1.7239 2.0742 2.5074 2k (ii) x1;0 ¼ b À pbffiffi cos 60 Case IV: Single Punch Indentation of a Thick 2 2 (semi-infinite) Block (h/b ~ ∞) ¼ b À pbffiffi 1 ¼ 0:1464 b; and When the block is very thick, the zones of plastic deformation 2 2 2 do not extend completely across the block, and the problem becomes essentially that of single-sided indentation by one pffiffi punch. This may be considered with reference to Fig. 10.37, pbffiffi pbffiffi 3 ¼ À0:6124 b: which shows the slip-line field proposed by Hill (1950). y1;0 ¼ À 2 sin 60 ¼ À 2 2 (a) Construction of the Slip-Line Field: (iii) x0;1 ¼ Àx1;0 ¼ À0:1464 b; and y0;1 ¼ y1;0 ¼ À0:6124 b: Since the interface between the punch and the block is assumed to be frictionless, the slip lines must meet the (iv) x1;1 ¼ 0; hence, we get the value of y1;1 from (10.57c) interface at 45°. If all plastic deformation were restricted to a triangular region beneath the punch–block interface AB, it y1;1 þ 0:6124 b ¼ ð0:1464 bÞ tan 1  pÀ p would not be possible for the metal to move physically, 2 À 4 3 because it is fully constrained beneath and laterally by the surrounding rigid (elastic) material. Hence, it can only flow  upwards at the sides of the punch, which suggests that the ¼ ðÀ0:1464 bÞ tan 7p ; plastic zone must be extended along the free surface to AI and 24 BD. The slip lines must also meet this free surface at 45°. ) y1;1 ¼ ðÀ0:1464 bÞ Â 1:3032 À 0:6124 b ¼ À0:8032 b: As proposed by Hill (1950), the region beneath the indenter is divided into two parts. Metal from the zone OBF Again, the height h of the strip is: h ¼ 2  y1;1 ¼ flows along the slip lines within OFED to the free surface at 2  0:8032 b ¼ 1:6064 b; the right, while that from the zone OAG flows to the left. Each of the two plastic regions beneath the indenter consists Or; h ¼ 1:6064: of two uniform stress fields, sandwiching a centred fan-type b field. For example, the centred fan-type field BEF is sand- wiched between the uniform stress fields OBF and BDE. Z0 (b) Stress Determination from the Slip-Line Field Now; rx dy ¼0 To determine the stresses from the slip-line field, let us Àh2 consider a point D on the free surface BD. Since there can be no stress normal to this surface in this region, so at point D, Or; ðrxÞ1;0 y1;0 þ 1 n þ oÀ À Á ¼ 0; ðr1ÞD¼ 0: But from (10.40a), 2 ðrxÞ1;0 ðrxÞ1;1 y1;1 y1;0 ðr1ÞD¼ ðrmÞD þ k; Or; ðrxÞ1;0 y1;0 þ 1 &ðrxÞ1;0 þ ðr2xÞk1;1'Ày1;1 À Á ¼ 0; )ðrmÞD¼ Àk: 2k 2 2k y1;0  p2pkoÂÂððÀÀ00:8:6013224bbþÞ þ0:6211n214:1b9Þ 4¼8 À p Or, 1:1948 À 0; 2k þ 1:5236 À 2k

10.5 Mechanics of Working Process 451 Fig. 10.37 Slip-line field, p suggested by Hill, for indentation of a very thick block by a single σ1= 0 flat punch A OJ BK D I H 45º 45º σ2 45º σ2 Gy F σ1= 0 E α -line x β -line Since the slip line from D to E lies in the clockwise between the surface of the indenter and the indentation, pm, which is same as the Meyer hardness, is often estimated as direction from the direction of the algebraically highest three times the yield strength or flow stress r0 of the material principal stress r1, so it is an aÀline, and thus, OFED will according to (10.59). be an aÀline. Since slip lines must meet a frictionless 10.5.3.8 Application of Slip-Line Field to Steady interface at 45°, so for the OFED aÀline, at point O, /O ¼ Motion pp In many forming processes, the system is not static; rather, À; and at point D, /D ¼ þ : the metal flow rapidly attains a steady state which continues 4 4 throughout the operation. For these, it is essential to verify that the chosen slip-line field conforms to the requirements Along the OFED a-lines; one can write from (10.47a) of steady velocity—here lies the significance of velocity in slip-line field evaluation. Application of slip-line field to ðrmÞO À /O ¼ ðrmÞD À /D; or, ðrmÞO þ p ¼ Àk À p steady motion under frictionless condition has been dis- 2k 2k ; cussed with respect to extrusion in Chap. 13 and strip 2k 4 2k 4 drawing in Chap. 14, because both of the forming operations involve inhomogeneous deformation to a greater extent. The ) ðrmÞO ¼ À 1 À p ; or, ðrmÞO¼ Àkð1 þ pÞ: drawback of slip-line field is that it is valid only for 2k 2 2 plane-strain condition, and many forming operations like extrusion and wire drawing do not occur under plane-strain Again, from (10.40b), we get conditions. ðr2ÞO¼ ðrmÞOÀk;  p Or, ðr2ÞO¼ Àkð2 þ pÞ ¼ À2k 1þ : 2 If p is the punch pressure required to indent the block, then at the point of indentation, say at point O, ðr2ÞO¼ Àp: ) p ¼ À ðr2ÞO ¼ 1 þ p % 2:57 ð10:58Þ 10.5.4 Upper-Bound Technique 2k 2k 2 The slip-line field method has been unable to produce the Apcfficffiording to Von Mises’ yielding criterion, since k ¼ exact solution for deformation loads in many plastic forming r0 3 (1.63), operations. Johnson and his co-workers have developed alternative methods which can estimate the loads approxi- ) p ¼  þ p ¼ 2prffi0ffi  þ p ¼ 2:97r0 ’ 3r0 mately (Johnson and Kudo 1962; Avitzur 1968). These 2k 1 2 3 1 2 methods are ‘upper- and lower-bound techniques’, in which the solution by the upper-bound technique overestimates and ð10:59Þ that by the lower-bound method underestimates the required deformation load. The real load will lie between these upper This (10.59) shows that yield pressure required for the and lower bounds. It has been shown that values of load indentation of a semi-infinite (thick) block with a slim punch determined from kinematically admissible velocity field, is about three times the compressive stress required for the where a velocity vector diagram called the hodograph must yielding of a cylindrical block under frictionless condition. be satisfied, are upper bounds. In this, no attention is paid to Since the above plane strain indentation is analogous to a two-dimensional hardness test, so the mean pressure

452 10 Fundamentals of Mechanical Working satisfy the stress equilibrium conditions at any point in the 10.5.4.1 Derivation of Upper-Bound Equation deforming body. On the other hand, values of load deter- Let us now consider a simple derivation of the upper-bound mined from statically admissible stress field are lower equation (Johnson and Mellor 1973) along with a discussion of bounds, in which no attempt is made to satisfy the velocity the hodograph. For estimation of the external load required to conditions at any point in the deforming body. If the yielding deform a body, it is required to find the rate at which energy is criterion is not violated and the equilibrium equations and consumed by the internal flow field. To calculate internal the stress boundary conditions are satisfied by the assumed energy consumption on shear plane, consider an element of stress field, then it is statically admissible stress field. It may rigid material, ABCD of unit thickness, moving to the left with be noted that, of two kinematically possible solutions, the unit velocity v1, as shown in Fig. 10.38. When the undeformed one which leads to the smaller limiting load is more element ABCD lying right of the shear plane XÀX0 passes acceptable, while, of two statically possible solutions, the through the interface XÀX0; it is forced to change direction one which leads to the larger limiting load is more accept- and velocity. Hence, ABCD is deformed to a new shape A′B′C able. Generally, it is required to estimate loads that can ′D′, which is constrained to flow with a new velocity v2 at an perform certain forming operations rather than loads which angle a to the direction of v1. However, AD and A′D′ remain cannot. Since the upper-bound solution predicts the defor- parallel to the shear plane XÀX0: The velocity changes mation load greater than the exact value, it is more important caused by the deformation of the element are represented by a in forming operations than the lower-bound technique, and velocity vector diagram, called a hodograph, which is shown so this section will deal with the ‘upper-bound analysis’ in Fig. 10.39. The original unit velocity v1 and the changed only. Further, upper-bound solutions usually involve rather velocity v2 can be resolved into components perpendicular to simple graphical solutions which are much easier than the the shear plane XÀX0; say vp1 and vp2; and components solutions of slip-line field theory, whereas the lower bound parallel or tangential to XÀX0; say vt1 and vt2: Since the may involve considerable algebraic and trigonometric volume of material entering and leaving the interface XÀX0 manipulations, which may not be advantageous over the use per unit time must be equal, the normal components vp1 and of slip-line field. vp2 will be the same, i.e. vp1 ¼ vp2 ¼ vp (say), which is shown in the hodograph, in Fig. 10.39. The vector difference The upper-bound theorem was formulated by Prager and between v1 and v2 is called the velocity discontinuity along Hodge (1951). The upper-bound analysis is based on the the shear plane XÀX0, and in the hodograph, it has been assumption of an internal flow field that will produce the denoted by the vector v1Ã2; which must be parallel to the line shape change. The upper-bound theorem provides an esti- of shear, XÀX0: The velocity discontinuity along the surface mate of the external load required to deform a body that will XÀX0 produces shear within the deformed material, which in be equal to or greater than the correct load. According to turn gives rise to shear stress. The maximum resistance to this theorem, this external load can be calculated by shear that a material can develop is the shearing yield stress equating the rate of work due to external load with the rate represented by k. of internal energy consumption caused by the internal flow field. For consistency, the flow field can be checked with the X hodograph. The upper-bound analysis involves the follow- ing simplifying assumptions: • The material is considered to be isotropic, homogeneous v1 B and incompressible; i.e. the volume during plastic A deformation remains constant. Deformed α S • Strain hardening is neglected. v2 A' B' D • At the interface between the dies or tools and the C work-piece, either frictionless condition or sticking fric- tion prevails. C' Undeformed • The deformation occurs by rigid-body movements of triangular elements, where all of the material in a given D' element moves with the same velocity. X' • Usually, plane strain (two-dimensional strain) conditions are only considered, where deformation takes place by Fig. 10.38 Deformation of an element at shear plane, showing change shear on a few discrete planes and elsewhere the material in direction and velocity is rigid.

10.5 Mechanics of Working Process 453 X Fig. 10.40b 1shovwÃ12s¼thCaDt =fCroCm0; similar triangles v1vÃ12 ¼ CD=CC0; or since v1 ¼ 1: Therefore, dðW :D:Þ ¼ k S v1Ã2 ð10:60Þ dt vp = vp1 = vp2 For an upper-bound velocity field having more than one vt1 shear discontinuity, (10.60) can be written as vt2 dðW :D:Þ ¼ Xi k Si vià ¼ k Xi vià ð10:61Þ dt Si 1 v1 = 1 α 1 v*12 Since the work hardening is neglected, hence k is constant v2 and taken outside the summation sign in (10.61). Most of the deformation fields for upper-bound solutions have a number of rigid blocks of polygons, where all of the material inside a given polygon moves with the same velocity. X' 10.5.4.2 Upper-Bound Solution for Indentation Fig. 10.39 Hodograph for Fig. 10.38 of a Semi-infinite Slab The internal energy expended; i.e., the work done Let us consider plane-strain frictionless indentation of a semi- infinite slab with a flat indenter. Because of symmetry, let us required for changing the shape of the element from ABCD consider only the flow field to the right of the centre point O in to A′B′C′D′ will now be considered (see Fig. 10.40). If XÀX0 Fig. 10.41, which shows an upper-bound field for plane-strain frictionless indentation and the corresponding hodograph for is a straight line and the length of the line crossing the the right-hand part of the field. In Fig. 10.41a, the whole zone discontinuity surface XÀX0 is denoted by S, then the internal OACDBO flows plastically, whereas the metal outside the energy expended in deforming the element is the work done, triangles, i.e. below OACD, is rigid. As the indenter moves W:D: ¼ ½k Sð1ފ CC0; where k is the maximum shear stress in downward with the assumed velocity v0 = 1, all particles in plane-strain deformation and the factor of unity comes from the region ‘1’ (i.e. in the triangle OAB) under the indenter will move vertically downward with the velocity of the indenter, the assumption of unit thickness of the element. So, the rate v0 = 1, but are constrained to slide parallel to the rigid of internal energy consumption along the shear discontinu- boundary OA on which shear occurs. Hence, a horizontal ity, i.e. the rate at which the work is performed, will be velocity component represented by vOB (Fig. 10.41b) is pro- duced. As the particles cross the boundary BA, their velocities dðW :D:Þ=dt ¼i.e½.ðtkS¼ÞCDCC0Š;=atn; dwdhðeWre:Dt :Þis=dtht e¼ti½mðkeSÞfCoCr 0DŠCDCto: cross XÀX0; are changed again by shear parallel to AB and they are com- Comparison of the triangle C0CD with the hodograph in pelled to move parallel to AC. The velocity triangle for this region BAC is thus represented by the original velocity v1 ¼ vOà A to the left of BA, the change in velocity vÃAB parallel to AB and the absolute velocity parallel to AC, which is thus equal to B' (a) (b) A' AB C' C' D' α α v2 DC DC α v*12 v1 Fig. 10.40 a Superimposition of a distorted element on the undeformed one at interface XÀX0: b Similar triangles between the distortion of the element and the velocities

454 (a) 10 Fundamentals of Mechanical Working Fig. 10.41 a An upper-bound (b) field for plane-strain frictionless indentation. b Hodograph for the v3 = v*CD v*CB right-hand part of the field shown in (a) p v0 = 1 60º v2 = v*AC b/2 60º OB D v1 = v*OA v*AB v0 1 3 60º 2 60º vOB 60º C A v2 ¼ vAÃC: Further, when the particles cross the boundary BC, 10.5.4.3 Upper-Bound Solution for Compression their velocities are changed by shear parallel to CB, and hence, Let us consider plane-strain frictionless compression of a a shear velocity vCà B is added. In the triangle BCD, the final work-piece with smooth platens. If b is the breadth of the absolute velocity parallel to CD is consequently given by compression platen and h is the height of the work-piece, v3 ¼ vCà D: then let us consider the case where b/h = 2.5. An upper-bound field for plane-strain frictionless compression Since all the triangles in the hodograph are equilateral and the corresponding hodograph are shown in Fig. 10.42. The flat compression platens move with velocity, v0 = 1. having 60° angles, so from the hodograph, the velocity Materials shear along the discontinuity lines AO, BO, CO and DO, whose lengths are discontinuities, AO ¼ BO ¼ CO ¼ DO ¼ b=ð2 cos hÞ; vÃOA ¼ vAÃB ¼ vAÃC ¼ vCà B ¼ vCà D ¼ v0 ¼ 1 ¼ p2ffiffi : sin 60 sin 60 3 Again, the lengths of discontinuity lines are: where h is the angle made by the above discontinuity lines OA ¼ AB ¼ AC ¼ CB ¼ CD ¼ b=2; where b is the with the horizontal flat surfaces of the compression plates. breadth of the flat indenter. The velocity discontinuities along these lines are Hence, the rate at which internal energy is consumed on the right-hand part of the field is vÃAO ¼ vÃBO ¼ vCà O ¼ vDà O ¼ v0=sin h: dðW:D:Þ ¼ k X Svà ¼ ÂÀ Á vOà A Á þ À Á vÃABÁ Hence, the rate of internal energy consumption along the k OA AB shear discontinuity is þdtÀACÁ vAÃCÁ þ À Á vCà BÁ þ À Á vÃCD Áà dðW :D:Þ X và ÂÀ vAà O Á À vÃBO Á CB CD dt k AO þ BO ! ¼ k S ¼ Á Á þ 4 k b v0 ¼ 5k b p2ffiffi ¼ 5pkffibffi ð10:62Þ ¼ À vCà O Á À vDÃ!OÁà sin 2h CO Á þ DO Á ¼ 23 3 b  v0  If p is the pressure required to cause indentation, then the 4k 2 cos h sin h ð10:64Þ rate at which external work is performed on the right-hand part of the field is If p is the pressure required for compression, then the rate at which external work is performed by the movement of dðW :D:Þ ¼ p b ðv0Þ ¼ p b ð1Þ ¼ pb ð10:63Þ compression platens is dt 2 2 2 Equating (10.62) with (10.63) for dW/dt, we obtain dðW :D:Þ ¼ 2p b v0 ð10:65Þ dt pb ¼ 5pkffibffi ; or, p ¼ p5ffiffi ¼ 2:89: where the factor of 2 comes from the fact that the work-piece 23 2k 3 is being compressed by the pressure p from both sides with

10.5 Mechanics of Working Process 455 Fig. 10.42 a An upper-bound (a) (b) v3 = v0 v*OB field for plane-strain compression. v*OC b Hodograph corresponding to (a) p v*OD v*OA v0 = 1 A C v4 v2 h4 2 θ1 θ D O B v0 = 1 3 b p v1 = v0 the compression platens. Equating (10.64) with (10.65) for Substituting the value of sin 2h in (10.66), we get dðW:D:Þ=dt; we obtain n2 þ ðb=hÞ2 4 k b v0 p ¼ 1 ¼ 2nðb=hÞ ð10:69Þ sin 2h 2k sin 2h 2p b v0 ¼ Since b/h = 2.5, when the value of n = 1, we get from ) p ¼ 1 ð10:66Þ (10.69) þ ð2:5Þ2 2k sin 2h 2ð2:5Þ p ¼ 1 ¼ 7:25 ¼ 1:45 ð10:70Þ 2k 5 Since h 1 As expected, (10.70) and (10.67) yield the same value of b 2:5 tan h ¼ ¼ ; p=2 k: 2 tan h 2=2:5 5 Substituting the value of n = 2, 3, 4, 5 in (10.69), we get, þ tan2 þ ð1=6:25Þ 7:25 ) sin 2h ¼ 1 h ¼ 1 ¼ : respectively, the following values of p : 2k 4 þ ð2:5Þ2 From (10.66), we get, When n = 2, p ¼ 2  2  2:5 ¼ 10:25 ¼ 1:025: 2k 10 p ¼ 7:25 ¼ 1:45 ð10:67Þ When n = 3, p ¼ 9 þ ð2:5Þ2 ¼ 15:25 ¼ 1:067: 2k 5 2k 2  3  2:5 15 Let the breadth of the compression platen b be divided When n = 4, p ¼ 16 þ ð2:5Þ2 ¼ 22:25 ¼ 1:1125: into n divisions (see Fig. 10.43) so that we obtain a better 2k 2  4  2:5 20 solution for the pressure p, when values of b/h are large. In such case, according to (10.64), we can write When n = 5, p ¼ 25 þ ð2:5Þ2 ¼ 31:25 ¼ 1:25: 2k 2  5  2:5 25 dðW :D:Þ Â À ÁbvÃAOÁà  v0 ! dt ¼ k 4n AO cos h sin h 4 k b v0 ¼ 4n k  sin 2h 2n ¼ ð10:68Þ p Since the end result of (10.68) is identical to that of v0 = 1 AC (10.64), so (10.66) is also applicable in the above case, θ where the platen breadth b has been divided into n divisions. O h Hence, tan h ¼ h=2 ¼ nh ; D v0 = 1 B b=ð2nÞ b b ) sin 2h ¼ 2 tan h h ¼ ð2nhÞ=b ¼ 2nhb p þ tan2 þ ½ðnhÞ=bŠ2 b2 þ n2h2 1 1 ¼ 2nðb=hÞ : Fig. 10.43 An upper-bound field for plane-strain compression when ðb=hÞ2 þ n2 the breadth of the compression platen b is divided into n divisions for large values of b/h

456 10 Fundamentals of Mechanical Working If there is a friction, the lowest value among the solutions working operations and the initiation of defects, without using experimental data, can be made quickly and reliably must have an odd number of triangles, because the middle using FEM method. For the concept of FEM, the reader is referred to literatures (Segerlind 1976; Zienkiewicz 1977; triangle does not slide. Hence, we consider the value of p/2k, Heubner and Thornton 1982). where n = 3. So, the answer will be p ¼ 1:067: 2k 10.5.5 Finite Element Method 10.6 Deformation-Zone Geometry Finite element analysis is a very powerful method to The geometry of deformation zone of a conical converging determine the distributions of stress, strain and displacement die strongly affects the redundant deformation, the frictional in plane stress, plane strain and axisymmetric conditions for work and thereby the working load. The deformation-zone both steady-state and non-steady-state deformation prob- geometry also affects the properties of the product, such as lems, which are too complex to analyse by strictly analytical its homogeneity, porosity, residual stress patterns and methods. Since about 1960, finite element method cracking tendency. The deformation-zone geometry is (FEM) has been applied extensively to solve complex defined by the ratio of the mean thickness or height, h, or the problems of stress and strain involving equations of alge- mean diameter, Dm, of the work-piece to the contact length, braic ones. To provide good accuracy with FEM method, the Lc, between the die and the work-piece in the deformation deformation zone in a body is divided into a network of very zone. This ratio is designated by the parameter, D, which, for small elements, interconnected at a finite number of nodal the simple case of parallel sided (non-converging) dies, is: points. Next, a set of simultaneous force–displacement equations are developed and need to be solved, for which a D ¼ h or; D ¼ Dm ð10:71Þ digital computer is essential. From the solution of these Lc Lc equations, complete distributions of stress, strain and dis- placement are calculated. As applied to problems in metal- For different deformation processes, the expression for working plasticity, the method relates the stresses acting on the faces of an element to the incremental displacement the contact length, Lc, will vary, but the mean thickness or produced. A limitation to calculations is that linear rela- height, h, or diameter, Dm, of the work-piece will be given tionships such as Hooke’s law for elastic deformation and the Levy–Mises equations for plastic deformation need to be by the same expression: assumed. Nonlinear strain-hardening and anisotropic prop- erties of the job material in only some optimized forms as h ¼ h1 þ h2 or; Dm ¼ D1 þ D2 ð10:72Þ well as frictional conditions at the die–job interfaces can be 2 2 incorporated in this technique. To apply this technique, inputs, such as the stress–strain behaviour of the job material where h1 = thickness or height of ingoing stock, D1 = diam- as a function of temperature and strain rate, heat transfer and eter of ingoing stock, and h2 = thickness or height of outgoing fictional characteristics of the job and the die or tool, are product, D2 = diameter of outgoing product. For different required. deformation processes, where material flows through a con- verging channel, the parameter, D, in terms of the reduction in Early applications of FEM to problems of plasticity dealt area r of the work-piece has been shown below: with elastic–plastic solutions. Since in these problems, the use of very small strain increments with elastic calculation (a) Plane-Strain Extrusion or Drawing of Strip: made at each increment is required, a very large amount of computer storage is needed. FEM was practically applied to The contact length, Lc, is given by metalworking analysis by Kobayashi (Lee and Kobayashi 1971; Kobayashi and Shah 1978) using a matrix method. In ðh1 À h2Þ=2 ¼ sin a; or; Lc ¼ h1 À h2 ð10:73aÞ this method, rigid plastic behaviour is assumed ignoring Lc 2 sin a elastic strains in comparison with larger plastic strains, and therefore, the use of relatively large increments of strain can where a = the approach semi-angle of the conical die in be made, which reduces the requirements of computer con- radians. If w is the width of the strip, which remains constant siderably. Determination of the temperature distribution and maintains the plane strain condition, then the reduction throughout the work-piece, investigation of the influence of in area, r, of the strip is die geometry on defect formation and die fill and prediction of microstructural changes in the job material during hot r ¼ wðh1 À h2Þ ; or; h2 ¼ 1 À r ð10:74Þ wh1 h1

10.6 Deformation-Zone Geometry 457 With the help of (10.72), (10.73a) and (10.74), the D (d) Flat Rolling: parameter in terms of r will be Roll gap is like a converging channel. The reduction in area, D ¼ h ¼ h1 þ h2 sin a ¼ 1 þ ðh2=h1Þ sin a ¼ 2 À r sin a r, is given by (10.74), because the width, w, of the stock Lc h1 À h2 1 À ðh2=h1Þ r remains constant in most cases and thus maintains the plane strain condition in rolling. The contact length, Lc, in rolling ð10:75Þ is nearly equal to the projected length of the arc of contact, L, known as the length of the deformation zone. L is given (b) Axisymmetric Extrusion or Drawing of Rod or Wire: by (12.11b) in Chap. 12, which is given below in terms of r, using (10.74) As D1 or D2 is the diameter of rod or wire, so the reduction in area, r, is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi L ¼ Rðh1 À h2Þ ¼ Rh1½1 À ðh2=h1ފ ¼ Rh1r  À ÁÃ pffiffiffiffiffiffiffiffiffiffi p DÀ 12 À ÁD22 4; D2 1Àr ð10:81Þ r¼ pD21 4 or; ¼ ð10:76Þ D1 Considering Lc = L, and with the help of (10.72), (10.74) and (10.81), the D parameter in terms of r will be For the round section, the contact length, Lc, is given by D ¼ h ¼ h ¼ hp1 þffiffiffiffiffihffiffi2ffiffi ¼ À1pþffiffiðffiffihffiffi2ffiffi=ffiÁh1Þ ¼ À p2ffiffiffiffiÀffiÁrpffiffiffiffi ðD1 À D2Þ=2 ¼ sin a; or; Lc ¼ D1 À D2 ð10:73bÞ Lc Lrffiffiffiffi2ffiffi Rh1r 2 Rh1r h1 2 Rr h1 Lc 2 sin a ¼ 2 À r h1 With the help of (10.72), (10.73b) and (10.76), the D 2 Rr parameter in terms of r will be: ð10:82Þ D ¼ Dm ¼ D1 þ D2 sin a ¼ 1 þ ðD2=D1Þ sin a It has been shown (Backofen 1972) that for the friction- Lc D1 À D2 1 À ðD2=D1Þ less plane-strain indentation of a rigid-ideal plastic material, À pffi1ffiffiffiÀffiffiffiffiffirffiÁ2 the greater the value of D, the higher is the die pressure ¼ 1 þ pffiffiffiffiffiffiffiffiffiffi sin a ¼ 1 þ sin a required for yielding. Since the contact area between the p1ffiffiffiffiÀffiffiffiffirffiffi work-piece and the die increases with decreasing D, so the 1À 1Àr r lower the value of D, the greater is the effect of friction at the interface between the work-piece and the tool. Internal ð10:77Þ cracks, such as centre burst or chevron cracks in drawn or extruded rods, develop as a result of ‘secondary tensile (c) Tube Drawing Over a Moving Cylindrical Mandrel: stresses’ which typically occur with greater values of D and because of low friction at the tool–work-piece interface. As h1 or h2 refers to the thickness of wall of the tube, the contact length, Lc, is given by 10.7 Anisotropy of Mechanical Properties h1 À h2 ¼ sin a; or; Lc ¼ h1 À h2 ð10:78Þ It is frequently observed that the mechanical properties of Lc sin a worked or wrought product are different for different ori- entations of the test specimen. The variation of properties If Dm is the mean diameter of the tube, which remains with orientation of the body is called anisotropy. For constant maintaining the plane strain condition, then the example, high anisotropy of strength properties is exhibited reduction in area, r, of the strip is by a single crystal. Generally, there are two types of ani- sotropy—(a) crystallographic texture and (b) mechanical r ¼ pDmðh1 À h2Þ ; or, h2 ¼ 1 À r ð10:79Þ fibering. pDmh1 h1 When a material undergoes severe deformation, such as With the help of (10.72), (10.78) and (10.79), the D in rolling or wire drawing, a preferred orientation or crys- parameter in terms of r will be tallographic texture may develop, in which certain crystal- lographic planes or directions may align themselves in a D ¼ h ¼ h1 þ h2 sin a ¼ 1 þ ðh2=h1Þ sin a ¼ Lc ah À r2i½1 À ðh2=h1ފ 2Àr 2ðh1 À h2Þ ð10:80Þ sin a ¼ sin 1 2r r 2

458 10 Fundamentals of Mechanical Working preferred manner with respect to the direction of greatest which is normal to both the longitudinal (fibre) and deformation. A strong crystallographic texture may lead to short-transverse directions. It has been observed that anisotropy in mechanical properties. This may result in an vacuum-melted steels of a given composition generally show uneven response of the material during forming processes. less anisotropy and result in higher values of transverse The development of anisotropy in polycrystalline material reduction of area than conventionally killed, air-melted due to texture affects mostly the yield strength and to a lesser steels of the same composition because the latter have higher extent the tensile strength. The strength properties in the non-metallic inclusion contents and these inclusions are one longitudinal (main) direction of working will be higher or of the sources that are responsible for higher anisotropy and lower than those in the transverse direction depending on the lower transverse ductility. However, grain flow can be used type of texture. The effect of texture on anisotropy is the to make a product with superior performance if the maxi- maximum in formed sheets which have been severely mum tensile stress in the component during service acts in worked. A practical example of texture induced by rolling the direction of grain flow. The resistance to initiation and operation is the formation of ‘ears’ or non-uniform defor- propagation of crack is the maximum in the direction normal mation in cups deep-drawn from rolled stocks. Recrystal- to the grain flow because grain boundaries and elongated lization reduces the strain energy incorporated due to prior inclusions act to reduce the sharpness of advancing crack deformation and changes the initial deformation texture to and this blunting process decreases the stress concentration recrystallization texture or annealing texture, but a different ahead of the crack tip. On the other hand, the crack propa- type of anisotropy still prevails due to the changes in the gation is enhanced by the grain flow if the direction of crystallographic orientation caused by recrystallization or applied tensile stress is normal to the fibre. annealing. Transverse properties of greater values are required in such Mechanical fibering, on the other hand, is produced when applications, like pressure vessels and gun tubes, which are structural discontinuities such as inclusions, cavities, segre- subjected to high internal pressures because the highest gation and second-phase constituents are oriented parallel to principal stress acts in the circumferential direction, which the longitudinal (main) direction of mechanical working, will be normal to the longitudinal working direction of these which is the direction of maximum strain. Inclusions or tubes or cylinders. Figure 10.44 (Wells and Mehl 1949) second-phase particles which are initially spheroidal in shows schematically the variation of the maximum and shape will be deformed into an ellipsoidal shape parallel to minimum values of reduction of area with the angle between the longitudinal working direction, if they are softer and the longitudinal forging direction and the tensile specimen more ductile than the matrix. If they are harder and stronger axis for steel. Both the maximum and minimum values of than the matrix, they will remain mainly undeformed, while reduction of area for different orientations of specimen are brittle inclusions or particles after deformation will be bro- shown in this figure, because highly scattered data are usually ken into fragments which will be aligned in the longitudinal obtained in measurements of transverse reduction of area. working direction. The preferred alignment of second-phase Figure 10.45 (Wells and Mehl 1949) shows schematically the particles, inclusions, cavities during cold working and hot- variation of the longitudinal and transverse reduction of area working and the preferred fragmentation of grains during with forging ratio, which is defined as the ratio of cold working give rise to the fibre structure or flow lines, cross-sectional area of the initial work-piece before forging to which can be viewed at low magnification after etching the that of the finished forged product. To obtain an optimum worked product. The existence of a fibre structure is the balance between the longitudinal and transverse ductility, it is characteristic of wrought products, especially of all forgings, often necessary to maintain a forging ratio of 2 to 3:1, which and is not considered to be a defect of wrought product. means the limiting amount of deformation ranges from 50 to Cast, machined or powder metallurgy products will never roughly 70% reduction in cross-sectional area. show flow lines or fibre structure. The occurrence and severity of fibering depends on many factors, such as the 10.8 Solved Problems amount of working or reduction given to the work-piece, composition and the extent of different structural disconti- 10.8.1. A cylindrical work-piece with a height of 30 mm, nuities (inclusions, segregation, etc.). An important conse- initially at a temperature of 600 °C, is compressed under quence of mechanical fibering is the development of frictionless condition for 15 s with flat dies, preheated to a directional properties. Generally, the tensile ductility mea- temperature of 250 °C, under an average effective true stress of sured by reduction of area, fatigue properties and impact 250 MPa resulting in an effective true strain of 1.0. Calculate properties will be the highest in the longitudinal or working the final temperature of the cylindrical work-piece in °C, (fibre) direction, the lowest in the direction of minimum assuming 95% of the deformation work is converted into heat. dimension of the product, known as the short-transverse The properties of the material of the cylinder are as follows: direction, and intermediate in the long-transverse direction,

10.8 Solved Problems 459 Longitudinal Transverse Longitudinal Max Min Reduction of area, % Reduction of area, % 0 20 40 60 80 Angle, degree Transverse Fig. 10.44 Schematic variation of the maximum and minimum values of reduction of area with the angle between the longitudinal forging direction and the tensile specimen axis for steel (Wells and Mehl 1949) Density = 2880 kg m−3, specific heat = 1000 J kg−1 1:1 3:1 5:1 7:1 K−1, heat transfer coefficient between the work-piece mate- Forging ration rial and the dies = 500 W m−2 K−1 and thermal conductivity = 78 W m−1 K−1. Fig. 10.45 Schematic variation of the longitudinal and transverse reduction of area with forging ratio (Wells and Mehl 1949) Solution Given that for the work-piece, the density is q = 2880 kg Hence, our assumption is quite reasonable, and (10.12) is m−3, the specific heat is Csp.heat = 1000 J kg−1 K−1; the heat applicable. Given that the initial temperature of the cylin- transfer coefficient between the work-piece material and the dies is htr.coeff. = 500 W m−2 K−1; and the thermal conduc- drical work-piece is T0 ¼ ð600 þ 273Þ K ¼ 873 K, the initial tivity is kth.cond. = 78 W m−1 K−1 Semiheight of the cylin- temperature of the dies is Td = (250 + 273) K = 523 K, and drical work-piece is L = 0.03 m/2 = 0.015 m. the deformation time is tn = 15 s. By applying (10.12), we obtain the temperature, Tn, of the work-piece as follows: Since the work-piece is subjected to an average effective  true stress, rm ¼ 250 Â 106 N m2; resulting in an effective htr:coeff : tn true strain, e ¼ 1:0; in which the fraction of the deformation Tn ¼ Td þ ðT0 À TdÞ exp À L q Csp:heat work converted into heat is b = 0.95, so the rise in tem- 500 Â 15  Â 2880 Â perature for a frictionless (ideal) plastic deformation process ¼ 523 þ ð873 À 523Þ exp À 0:015 1000 K according to (10.3) is: ¼ ð523 þ 350 Â 0:8406Þ K ¼ 817:2 K ¼ 544:2 C: ÀÁ brme 0:95 Â 250 Â 106 Â 1:0 DTD ¼ q Csp:heat ¼ 2880 Â 1000 K Hence, addition of ΔTD with Tn will give the final tem- perature, T, of the work-piece after a deformation time of ¼ 82:5 K ¼ 82:5 C: tn = 15 s: Equation (10.12) can be used to determine the tempera- T ¼ DTD þ Tn ¼ 82:5 C þ 544:2 C ¼ 626:7 C: ture of the work-piece due to heat transfer between the work-piece and the dies, if Newtonian cooling is assumed, in 10.8.2. Two rectangular specimens of the same material are which ðhtr:coeff: LÞ=kth:cond: 0:1; Let us check the validity of given identical reduction of 20% in height in plane strain our assumption: compression tests using flat dies in identical frictional condi- tions where the width of the specimen remains constant. The htr:coeff: L ¼ 500 Â 0:015 ¼ 0:096\\0:1: respective initial length and height of one specimen are 100 kth:cond: 78 and 50 mm that requires an average deformation pressure of

460 10 Fundamentals of Mechanical Working 200 MPa for reduction and those of another specimen are 200 Since the deformation has not been carried out under and 50 mm that requires an average deformation pressure of frictionless condition, so the coefficient of friction at the 400 MPa for reduction. Estimate the coefficient of friction, l, specimen–die interface will be: at the specimen–die interface. l ¼ 0:35: Solution After 20% reduction in height, the height of the first speci- 10.8.3. In the ring-compression test used for measuring men is: Coulomb’s coefficient of friction l, and interface friction factor m, the ring specimen with outer diameter of 150 mm h1 ¼ ð1 À 0:2Þ Â 50 mm ¼ 40 mm: is reduced in height by 40%. Find the values of l and m for the above upset ring test using the calibration curves given, From the constancy in volume, the length of the first respectively, in Figs. 10.19 and 10.20: specimen after reduction will be L1 ¼ 100  50 ¼ 125 mm: (a) If the outer diameter of the deformed ring is 179.4 mm. 40 (b) If the outer diameter of the deformed ring is 185.94 mm. Similarly, after 20% reduction in height, the height of the Solution second specimen is: If the initial outer diameter of the ring is OD1 = 150 mm, h2 ¼ ð1 À 0:2Þ Â 50 ¼ 40 mm: then the initial inner diameter the above ring will be ID1 ¼ ð3  150Þ=6 mm ¼ 75 mm, and its initial height will be Hence, the length of the second specimen after reduction H1 ¼ ð2  150Þ=6 mm ¼ 50 mm, since the calibration will be curves in Figs. 10.19 and 10.20 are based on the dimensions OD:ID:H in the ratio 6:3:2. Given that the height reduction, L2 ¼ 200  50 ¼ 250 mm: i.e. DH ¼ ðH1 À H2Þ=H1 ¼ 0:4; or the height of the 40 deformed ring is: Since the average deformation pressure for the first H2 ¼ ð1 À 0:4ÞH1 ¼ 30 mm: specimen, p1 ¼ 200 MPa, and that for the second specimen, p2 ¼ 400 MPa, from (10.28), one can write p2 rr0000 ¼ ½expfðl L2Þ=h2g À 1Š=fðl L2Þ=h2g (a) Given that the final outer diameter of the deformed p1 ½expfðl L1Þ=h1g À 1Š=fðl L1Þ=h1g ¼ exp½ðl L2Þ=h2Š À 1  h2 L1 ring is OD2 = 179.4 mm. From the principle of constancy in exp½ðl L1Þ=h1Š À 1 h1 L2 volume during plastic deformation, we can calculate the final inner diameter ‘ID2’ of the compressed ring: where r00 is the plane-strain deformation resistance of the Volume ¼ p À1502 À 752Á 50 ¼ p À179:42 À ID22Á 30: material for both specimens. 4 4 ) 400 ¼ exp½ð250 lÞ=40Š À 1  40  125 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 200 exp½ð125 lÞ=40Š À 1 40  250 ) ID2 ¼ 179:42 À 28125 mm ¼ 63:7 mm: Or, expð6:25 lÞ À 1 ¼ 400  40  250 ¼ 4; The percentage change in the inside diameter of the expð3:125 lÞ À 1 200  40  125 deformed ring is: Or, expð6:25 lÞ À 1 ¼ 4 expð3:125 lÞ À 4;  DID% ¼ ID1 À ID2  100 ¼ 75 À 63:7  100 ¼ 15%: ID1 75 Or, Àe3:125 lÁ2À4Àe3:125 lÁ þ 3 ¼ 0; Since there is a decrease in the internal diameter of the ring after deformation, so the percentage change in the inside qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi diameter gives a positive value. ) e3:125 l ¼ ÀðÀ4Þ Æ ðÀ4Þ2Àð4  1  3Þ ¼ 3 or 1: At DH% ¼ 40%; and DID% ¼ 15%; the value of Cou- lomb’s coefficient of friction from Fig. 10.19 is found to be 2Â1 in between the curve for l = 0.12 and that for l = 0.15; ) l ¼ ln 3 or ln 1 ¼ 0:35 or 0: 3:125 3:125

10.8 Solved Problems 461 hence, we can take an average value, which will be true stresses required are 180 MPa for Al and 360 MPa for l = 0.135. Ti. Compare the temperature rise caused by only ideal plastic deformation between Al and Ti, assuming 95% of the And the value of interface friction factor from Fig. 10.20 deformation work is converted into heat. Given that the is found to be m = 0.5. density of Al is 2700 kg m−3 and that of Ti is 4500 kg m−3 and the specific heat of Al is 900 J kg−1 K−1 and that of Ti (b) Given that the final outer diameter of the deformed ring is is 520 J kg−1 K−1. OD2 = 185.94 mm. From the principle of constancy in volume during plastic deformation: 10.Ex.4. In a ring-compression test, a specimen of 15 mm in height with outside diameter (OD) 45 mm and inside Volume ¼ p À1502 À 752Á 50 ¼ p À185:942 À ID22Á 30: diameter (ID) 22.5 mm is reduced in height by 50%. In 4 4 which of the following two cases the interface friction factor will be higher? Justify your answer mathematically consid- pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ering identical reduction in height: ) ID2 ¼ 185:942 À 28125 mm ¼ 80:3 mm: (a) OD after deformation = 57 mm, The percentage change in the inside diameter of the (b) OD after deformation = 61 mm. deformed ring is: 10.Ex.5. A material has been predeformed by hot working  up to a true strain e ¼ 0:4: It is required to further hot work DID% ¼ ID1 À ID2 Â 100 ¼ 75 À 80:3 Â 100 ¼ À7%: the material from the true strain of 0.4–1.9. Determine the mean flow stress in that strain range if the flow stress r0 ID1 75 follows the relation: Since there is an increase in the internal diameter of the (a) r0ðMPaÞ ¼ 100ð1 þ eÞ: ring after deformation, so the percentage change in the inside (b) r0ðMPaÞ ¼ 100 þ 317 e0:54: diameter gives a negative value. (c) Comment on the answers obtained from (a) and (b) with respect to the flow stress obtained at a mean value of true At D H % ¼ 40%; and DID% ¼ À7%; the value of Cou- strain. lomb’s coefficient of friction from Fig. 10.19 is found to be in between the curve for l = 0.04 and that for l = 0.03; hence, 10.Ex.6. A round bar of 50 mm diameter is reduced to 40 we can take an average value, which will be l = 0.035. and 30 mm in two steps using proper dies. If the flow stress r0 follows the relation: r0ðMPaÞ ¼ 600 e0:2; determine the And the value of interface friction factor from Fig. 10.20 following: is found to be m = 0.15. (a) Mean flow stress. Exercise (b) Work done per unit volume for ideal plastic deformation. 10.Ex.1. If a circular disc having a uniaxial flow stress of 10.Ex.7. Indicate the correct or most appropriate answer 500 MPa is deformed using solid lubricant having a uniaxial from the following multiple choices: flow stress of 86.61 MPa, what will be the maximum coef- ficient of friction according to Von Mises’ yielding criterion? (a) Full-fluid film lubrication occurs in wire drawing if the sliding velocity of the wire and fluid viscosity 10.Ex.2. If a 35-mm-thin flat plate initially at a temperature of 800 °C is hot worked for 1 min with dies, preheated to a (A) both increase; temperature of 300 °C, calculate the final temperature of the (B) increases and decreases, respectively; plate in °C, due to only heat transfer between the thin plate (C) decreases and increases, respectively; and the dies (without consideration of temperature rise due (D) both decrease. to plastic deformation and friction). The properties of the material of the plate are as follows: (b) Classification of metal forming processes into hot and cold working is based on the following parameter: Density = 3000 kg m−3, specific heat = 1000 J kg−1 K−1, heat transfer coefficient between the plate material and (A) equicohesive temperature; the dies = 400 W m−2 K−1 and thermal conductivity (B) recrystallization temperature; = 90 W m−1 K−1. (C) transformation temperature; (D) solidus temperature. 10.Ex.3. Plastic deformation is carried out quickly at room temperature on aluminium (Al) and titanium (Ti) to produce an effective true strain of 0.9, for which the average effective

462 10 Fundamentals of Mechanical Working (c) When a solder wire is bent back and forth at room Earles, S.W.E., Powell, D.G.: Proc. IME. 181, 171–179 (1966/67) temperature, it does not strain-harden, because of Ettles, C.M.M.C.: J. Tribol. ASME 108, 98–104 (1986) Fukui, S., Ohi, T., Kudo, H., Takita, I., Seino, J.: Some aspects of (A) immobilization of the dislocations during the bending process; friction in metal-strip drawing. Int. J. Mech. Sci. 4, 297–314 (1962) Goddard, J., Wilman, H.: Wear 5, 114–135 (1962) (B) preferred orientation of the grains; Green, A.P.: Friction between unlubricated metals. Proc. Roy. Soc. (C) preferential growth of grains in the direction of London, A 228, 191–204 (1955) deformation; Grzesik, W., Nieslony, P.: Wear 256, 108–117 (2004) (D) the recrystallization temperature being below the room Harris, J.N.: Mechanical Working of Metals Theory and Practice temperature. pp. 72, 76. Pergamon Press, Oxford (1983) Heubner, K.H., Thornton, E.A.: The Finite Element Method for Answer to Exercise Problems Engineers. Wiley, New York (1982) 10.Ex.1. 0.1. Hill, R.: The Mathematical Theory of Plasticity p. 255. Oxford 10.Ex.2. 616.5 °C. 10.Ex.3. 63.33 K for Al and 131.43 K for Ti. University Press Inc., New York (1950) 10.Ex.4. (a) Change in ID after reduction is +35.5%; Hirst, S., Ursell, D.H.: Some limiting factors in extrusion. Metal Treat. (b) change in ID after reduction is −16%. Interface friction factor for (a) is higher, since % change in ID of deformed Drop Forging 25, 409 (1958) ring is algebraically higher for (a) than for (b). Hoffman, O., Sachs, G.: Introduction to the Theory of Plasticity for 10.Ex.5. (a) 215 MPa. (b) 435.28 MPa. (c) For (a), both are same, because the stress–strain relation is linear. For (b), Engineers. McGraw-Hill Book Company, New York (1953) flow stress at a mean value of true strain is 441.85 MPa, Johnson, R.H.: Superplast. Met. 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Forging 11 Chapter Objectives • Classification of forging processes and different types of forging operations. • Forging equipments, describing gravity drop hammer, power drop or steam hammer, mechanical forging press and hydraulic forging press. • Open-die forging, coefficient of spread and its relation with bite ratio, ‘spread law’ with definitions of spread ratio and squeeze ratio. • Closed-die forging or impression-die forging with discussion on some die design factors, such as flash and flash land ratio, draft angles, corner and fillet radii and location of parting line. • Material loss due to scale formation, discard, croppings, slug waste, during forging. • Plane strain forging of uniformly thick rectangular plate: distributions of pressure and longitudinal stress, average pressure and total load under conditions of Coulomb sliding friction, sticking friction and mixed sticking–sliding friction. • Plane strain forging of strip with inclined dies: die pressure and strip thickness at neutral plane and its location. • Forging of flat circular disk: distributions of pressure and longitudinal stress, average pressure and total load under conditions of Coulomb sliding friction, sticking friction and mixed sticking–sliding friction. • Forging of circular disk by conical pointed dies: die pressure and condition for complete elimination of barrelling of the work-piece. • Forging defects and fibre structure. • Problems and solutions. 11.1 Classification of Forging Processes impact blows are imparted to the work-piece by power-driven hammers. Instead of impact blows, forging can Forging is probably the oldest method of forming processes also be performed by the application of a slow-speed and was known even during prehistoric days when the squeezing (compressive) force by means of hydraulic or desired shapes were made manually from a hot work-piece electrically powered mechanical devices. by using hand-held tools and hammers. Example is the smith forging operation of ancient days, which would involve the Forging is the deformation of material between two dies manual application of impact force for deformation by the for obtaining a desired configuration by hammering or blacksmith by means of a hammer. Nowadays, the repeated pressing that involves, respectively, rapid or slow applica- tion of compressive stress. So depending on the rate of load application, there are two major classes of equipments, © Springer Nature Singapore Pte Ltd. 2018 465 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_11

466 11 Forging which are competent to manufacture components ranging Most forging operations are carried out hot, although from a small size like a bolt to a massive size, such as a certain materials may be cold forged. The choice of tem- turbine rotor or an entire aeroplane wing. Hence, forging perature is decided by such factors as—(a) ease and operations can be divided into two major classes depending cheapness of deformation; (b) required mechanical proper- on equipments used. ties; and (c) surface finish. The last two factors are vital. Over 90% of the forging processes are hot. The reason for (1) Impact forging, or hammer forging, or drop forging: It limited application of cold forging process is the higher is the process that uses a forging hammer or drop consumption of power, the excessive wear of dies and the hammer, which delivers rapid impact blows to the smaller amount of possible deformation. One example of surface of the material, and deformation occurs over a cold forging is the coining operation, where finer details are very short period. To achieve the desired size and shape embossed in cold condition on the surface of a metal piece. of the product, usually repeated blows are required. There are two broad categories of forging processes, both of With impact forging, the pressure is at the maximum which are carried out in hot condition as well as in cold intensity when the hammer touches the material and it condition: decreases rapidly in intensity as the energy of the blow is absorbed in deforming the material. Therefore, (1) Open-die forging: In this process, the work-piece is impact forging results in deformation primarily in the deformed between two flat dies or dies of very simple surface layers of the work-pieces, which is a disad- shape that do not restrict the flow of material during the vantage for this equipment. Hence, the impact forging compression of the part. The process is used is not applied to a large work-piece because it will (i) mostly result in an inhomogeneous structure and a (a) for simple-shaped large objects, or non-uniform distribution of mechanical properties from (b) when the number of parts produced is small; the outside layers to the centre. For example, impact and forging of a large cast ingot at high temperature will (ii) often to preshape the work-piece for closed-die show a hot-worked structure at the outside surfaces but forging. the interior will be still as cast. The application of impact forging is therefore limited to comparatively (2) Closed-die forging, or impression-die forging: In small work-pieces. closed-die forging, the deformation of work-piece is carried out to obtain the desired configuration by (2) Press forging: It is the process based on forging press, squeezing the workpiece between carefully machined which instead of repeated blows, subjects the material matching two die blocks or two die halves carrying the to a slow-speed squeeze force causing material to yield impressions of the desired final shape. The closed and results in a deeper penetration of deformed zone. In cavity formed by the dies restricts the flow of material this process, the operation is completed in one stroke during the compression process causing the work-piece and the alignment of both halves of the die is less to deform under high pressure and thus produces pre- problematic than in drop forging. Hence, parts pro- cision forging of desired geometric shape with dimen- duced by this process have greater dimensional accu- sions closer to those of the desired final product. This racy than drop forging, although a press is usually close dimensional tolerance saves the cost of machin- expensive than a hammer. In press forging, the pressure ing the forged product. This process is normally used to gradually increases as the material is being deformed forge smaller parts. and the pressure reaches to the maximum value just prior to the release of the pressure. Since the effect of 11.2 Types of Forging Operations the applied stress penetrates totally up to the centre of the work-piece in press forging, the final product shows (1) Upsetting Operation a complete homogeneous structure and uniform mechanical properties throughout the cross-section. It is an example of the simplest forging operation in which a Since a high quality object with close dimensional work-piece (usually cylindrical in shape) is compressed tolerance is produced by a much slower process using between two flat dies in order to reduce its height with a much larger and costlier equipment, press forged view to increasing its transversal dimensions. The components are more expensive than impact forged articles.

11.2 Types of Forging Operations 467 compression test is a small-scale prototype of this process. Step GRIPPING DIE With the advancement of dies, the material attempts to flow (1) outwards but there is less flow at the end surfaces of the work-piece in contact with the dies because of interfacial HEADING TOOL friction forces than at the mid-height plane. The uninter- rupted outward flow of material at the mid-height plane leads Step to lateral expansion of the central portion of the upset (2) cylinder, resulting in a barrelled profile, as shown in Fig. 2.13. Forging machines for upsetting operations are Step known as ‘upsetters’ or ‘headers’, which are usually hori- (3) zontal mechanical presses. These are used for forging of symmetrical shapes from round bar stocks, such as bolts, rivets and gear blanks. Upsetting is commonly used to per- form a localized forging operation, i.e. when a portion of the work-piece needs to be forged, for example the forging of bolt head at one end of a rod. The formation of a bolt head by hot forging process is shown schematically in Fig. 11.1. For localized upsetting operations, one end of the work-piece is clamped or gripped with a die and longitudinal compression of the other free end of the work-piece is car- ried out with a movable ram openly or within a die cavity. Localized upsetting operations may be of two types, as shown in Fig. 11.2. • Open upsetting and Fig. 11.1 Diagram illustrating the formation of a bolt head by a hot • Closed upsetting. forging process As upsetting involves longitudinal compression of the edging dies but the horizontal (longitudinal) flow of the work-piece, so to prevent buckling of the unsupported material is prevented by the dies. So by this operation, length, l, to be forged, the following rules are observed. cross-sectional area of the work-piece in its longitudinal direction is increased at certain places and reduced at others • In an open upsetting, l 3 d; where d is the diameter of by distribution of material. the work-piece. (3) Fullering Operation • If l [ 3 d; a closed upsetting operation should be per- formed with an inner diameter of the die, D 1:5 d Fullering operation is mostly used as an earlier step to pre- shape the work-piece for closed-die forging operation. This • If during closed upsetting, l extends beyond the die operation involves the reduction in the cross-sectional area cavity by an amount l1; then l1 d: of a portion of the work-piece that may undergo slight elongation. The material flows in the outward directions and (2) Edging Operation away from the centre of the fullering dies to both sides, as shown in Fig. 11.4. In fullering operation, the material is Edging operation is shown in Fig. 11.3. It is often applied to distributed away from an area in contrast to the gathering of preshape the work-piece for closed-die forging operation. In material into a localized area that occurs in edging operation. edging operation, the end of work-piece material is inserted Example of fullering operation is the forging of a connecting into the longitudinal gap of dies having concave or semi- rod for an internal combustion engine. circular deforming faces and deformed there. This operation is used to gather material at the end of the work-pieces by shaping their edges. In this operation, the material flows freely in the lateral directions to fill up the cavity of the

468 11 Forging Fig. 11.2 Upsetting operation Die Ram l Job gripped d Job d tightly D Clamp l Ram Head forged Head forged (a) Closed upsetting. (b) Open upsetting. l1 D ≤ 1.5 d d (c) Condition on unsupported length. Force Die Flow of Die material (a) Before edging (b) After edging Fig. 11.3 Edging operation (4) Cogging or Drawing Down or Drawing Out Force Operation Fig. 11.4 Fullering operation Cogging operation is usually an open-die forging operation, in which flat or slightly convex shaped dies are used to die. After each reduction, the work-piece is moved forward reduce the cross-sectional area of a large work-piece with so that the unreduced segment of the work piece can be simultaneous increase in the length, without causing any deformed by the dies. The thickness of the work-piece is change in the shape of the cross-section. Cogging operation reduced from one end to another in a stepwise sequence, and is shown in Fig. 11.5. The component is deformed in a series the length of deformation zone at any point of time will be a of steps because the work-piece is larger than the size of the small fraction of the work-piece. The completion of the

11.2 Types of Forging Operations 469 1. 2. DIE 3. BITE DIE DIE FORCE FORCE WORK WORK WORK FORCE WORK IS MOVED FORCE DIE FORWARD DIE Fig. 11.5 Cogging operation DIE forging work-piece may require several passes. The length tapering their ends. Impact forging equipment is used for of deformation zone at any instant is called the bite. The rotary swaging operation, also known as radial forging, thickness of the work-piece can be reduced to a greater where repeated blows are obtained on the work-piece by the extent if the width of the bite is decreased. The main reciprocating radial movement of usually two or four con- advantage of cogging operation is that smaller machinery cave dies, as shown in Fig. 11.6. requiring less power can be employed for deformation of a large work-piece. Formation of a desired component may (6) Roll Forging Operation require a series of forging processes, in which cogging operation often may be just one forging process. Sometimes This operation is performed with a pair of rolls having cogging operation may directly form products such as metal semicircular matching grooves over half of the circumfer- fences. ence of each of the rolls that are held by two parallel shafts, as shown in Fig. 11.7. This operation is used to reduce the (5) Swaging Operation diameters of rods and tubes as well as for tapering their ends. The heated work-piece is placed at the maximum gap Swaging is carried out to reduce the diameter of the between the grooves of the two grooved rolls. After a half work-piece, having circular cross-sectional area, with con- current increase in length with concave dies that will result Roll in a product of smaller diameter. This operation is generally used to reduce the diameters of rods and tubes as well as for Forged shape Workpiece Roll Fig. 11.6 Principle of rotary swaging Fig. 11.7 Principle of roll forging

470 11 Forging Punch geometry and strength of the work-piece, the temperature of forging operation, the desired dimensional accuracy and Coined mechanical properties of the forged product, the amount or workpiece number of parts to be produced, time of production, cost of the machine and manufacturing process, etc. Die Classification of forging equipments is based on the Fig. 11.8 Coining operation principle of their operations (Altan et al. 1973; Tool and Manufacturing Engineers Handbook 1984). Depending on revolution of the rolls, the work-piece is rolled out. The the methods of transfer of energy or force from the machine work-piece is then put in another pair of grooved rolls to the work-piece, forging equipments are broadly classified having the smaller groove, and the forging operation is into the following two categories: continued until the desired dimension is achieved. In roll forging, matching grooves on rolls may have various shapes. • Drop forging hammer and Basically the final product, such as tapered leaf springs and • Forging press. knives, may also be made by this process using specially designed rolls. This process can also be used as a prelimi- In both forging equipments, the lower face of a movable nary forming operation prior to other forging and forming ram carries an upper die containing one part of the impres- operations, such as in producing crank shafts and various sion that shapes the forging and a lower die containing the components used in automobile industries. remainder of the impression is keyed into an anvil cap, which is tightly fixed in place on a stationary anvil. The work-piece rests on the lower die and is deformed by the upper die during the downstroke of the ram, as shown in Figs. 11.9, 11.10, 11.11 and 11.12. Thus, both equipments deliver the energy or force to the work-piece through the forging die, although they differ in many respects. (7) Coining Operation 11.3.1 Drop Forging Hammer Coining is usually a closed-die forging operation performed The term drop forging is used for such forging processes with hammers in a cold state with or without the formation where a hammer or ram along with the upper die drops in a of flash. Figure 11.8 illustrates the flashless closed-die linear path from a certain height towards the lower die coining process. In this process, a flat and thin work-piece placed on the anvil. When the two dies meet, the falling can made to vary in the thickness because of lateral con- weight strikes the work-piece resting on the lower die, and straints. As the name indicates, this process is largely applied the kinetic energy of the hammer is rapidly transferred to the to make coins but can also be used to produce similar other work-piece. This supplies the load to forge and form the articles that require a well-defined impression of the die part. Although a great amount of energy is absorbed by the surface. In order to produce fine details of a coin or other work-piece on impact, substantial amount of energy is articles, the forming pressure required can be as large as five transferred to the ground and to the machine. The weight of to six times the flow stress of the metal. Moreover, use of ram is used to rate (Lyman 1970; Halter 1983) the forging lubricants cannot be allowed in this operation, because hammers. However, since the deformation of material occurs reproduction of fine details of the die surface is prevented at the expense of kinetic energy of the falling weight, the due to the entrapment of lubricants in die cavities. Since this forging hammers or drop hammers are classified as energy- process gives the desired dimensional accuracy (sizing) and restricted machines. During a deformation stroke when the an improved surface finish of the products, so to achieve die faces contact each other, the plastic deformation of the these, coining may also be used in association with other work-piece or the elastic deformation of the dies or tools and forging processes. the machine continues until the total kinetic energy is con- sumed. Hence, it is more appropriate to rate these machines 11.3 Forging Equipments in terms of energy delivered. Different types of machines are available to perform a Drop forging hammer is the least expensive and the most forging operation, but selection of a specific forging machine versatile type of forging equipment to carry out a forming depends on many factors. These factors are the size, operation and good for mass production of complex shapes. Hammers are primarily used for hot forging; for coining;

11.3 Forging Equipments 471 Roll Down Cylinder Up Piston Board Ram Ram Upper moving die Job Job Lower fixed die Anvil Anvil (b) Power Drop Hammer (a) Board Drop Hammer Fig. 11.9 Schematic diagrams of drop forging hammer (a) (b) (c) Motion of Direction of Motion of Direction of Drive shaft ram and shaft rotation ram and shaft rotation crank upper die upper die Connecting Drive shaft rod crank Drive shaft crank Linear Connecting guide rod Connecting Job rod Upper RAM (moving) Job Upper RAM Upper RAM die (moving) die Linear Job Lower guide die Lower fixed die Linear die Lower guide fixed die BASE BASE BASE Fig. 11.10 Crank press. a Ram and upper die are at the top position motion of the ram and upper die begins to deform the job. c Ram and and move towards the job when force is applied to the ram through the upper die are at their bottom positions resulting in the closure of the connecting rod. b Continuous pressure exerted by the downward mould that completes the forging operation

472 11 Forging Friction disk Flywheel and, to a limited extent, for sheet metal forming of parts manufactured in small quantities—for example, in the Drive shaft air-craft/air-frame industries. A hammer can strike the sur- (reversible) face of material between 60 and 190 impact blows per minute depending on the size and capacity of the machine. Screw The weight of a ram or hammer may vary from few hundred to several thousand kilograms and that of the anvil is com- Ram monly 15–20 times or may be as high as 30 times the weight of the hammer because a heavy base is required to absorb the Dies Work piece tremendous impact blow of the hammer. The forging ham- mer produces a high forging load most economically and Anvil contacts with the job under pressure for the shortest dura- tion, which reduces the heat transfer from the hot work-piece Fig. 11.11 Screw press to the colder dies. The contact time generally ranges from 1 to 10 ms. However, the dimensional accuracy of the (a) hammer-forged product is inferior to that obtainable in presses. Drop forging hammer can function only under the Fluid influence of gravity or in addition to the gravity, its power can be increased by additional sources of force. Depending on the above, the following two basic types of forging hammers are used: • Gravity drop hammer and • Power drop hammer or steam hammer. Cylinder (b) Cylinder (c) Cylinder Fluid Fluid Fluid out Fluid in Fluid in Fluid in Fluid out Fluid out Piston Piston Piston Linear guide Upper RAM RAM Job (moving) Upper Upper RAM die die die Linear Linear Lower guide Lower guide Lower fixed die die die Job BASE BASE BASE Fig. 11.12 Hydraulic press. a A higher pressure of the fluid below the than the fluid below it causes the piston along with the ram and the piston than the fluid above it causes the piston along with the ram and upper die to move downwards. c The fluid pressure delivered through the upper die to rise. b A higher pressure of the fluid above the piston the apparatus closes the mould and forms the part

11.3 Forging Equipments 473 11.3.1.1 Gravity Drop Hammer forging hammer to drop under the force of gravity to pro- The term gravity drop forging implies that the force of duce the blow energy. Other gravity drop hammers include gravity is the only force employed by the hammer for its chain drop hammer, where the ram is connected to a chain, drop from the required height and thus to acquire the energy and air-, steam- or oil-lift drop hammer, where the ram is to forge the work-piece. The speed ranges of gravity drop connected to a piston, which is lifted by applying upward hammer vary from 3.6 to 4.8 m/s (Semiatin 2005). Board force through air, steam or oil. drop hammer is one kind of drop forging machine dependent only on gravity, where a hardwood board is attached to the 11.3.1.2 Power Drop Hammer or Steam Hammer ram. This equipment is shown in Fig. 11.9a. There are two This equipment is shown in Fig. 11.9b. The principle behind friction rolls that grip the board. When the rolls rotate the the power drop hammer is similar to that of the board hammer friction forces between the board and rolls can raise the except that in the former, the hammer is powered most board, ram and upper die attached to the ram. Initially before commonly by compressed air, steam or hydraulic pressure proceeding for a blow to the surface of a material, the and thus, the power hammer provides greater forging hammer or ram assembly is raised to the required height. capacity than the board hammer. In addition to the gravita- Then the rolls are pulled apart and the board is released that tional influence, the pressure of compressed air, steam or causes the forging hammer to drop under the gravitational hydraulic oil accelerates the ram on the downstroke and thus force to produce the blow energy. Ram and upper die strike increases the energy of blow in power hammer. Steam, the work-piece resting on the lower die and the kinetic hydraulic oil or air also serves to raise the ram on the energy of the hammer is rapidly transferred to the upstroke. In this machine, the upper end of the ram is attached work-piece. During a working blow, the kinetic energy to a piston that moves up and down within a cylinder. There transforms into deformation energy, which can develop are one slide valve at the upper end and another slide valve at considerable force required to forge and form the the lower end of the cylinder, both of which are controlled by part. However, immediately after striking the surface of the operator with lever. Steam, hydraulic oil or air is admitted work-piece, the board is again raised by the rolls for the next to the cylinder through the upper slide valve for downstroke blow and in this manner, repeated blows are continued as of the ram and through the lower slide valve for upward long as the desired size and shape of the product are movement of the ram. The power drop hammer can operate achieved. with either single stroke or repeated blows of the ram and die on the work-piece. In a power drop hammer, the total energy When the hammer is at rest prior to its drop, the potential supplied by the blow is given by energy is equal to product of the weight of the hammer and the height of fall, which is again equal to the kinetic energy UPower Drop Hammer ¼ 1 mv2 þ pAH ¼ ðmg þ pAÞH ð11:2Þ of the hammer at the start of deformation of the work-piece. 2 Hence, the total energy supplied by the blow is equal to the kinetic energy or the potential energy and is given by where UGravity Drop Hammer ¼ 1 mv2 ¼ mgH ð11:1aÞ m total mass of the piston, ram and upper die, 2 usually called mass of ram; p air, hydraulic oil or steam pressure acting on where the ram piston on down stroke; A cross-sectional area of the ram piston; m total mass of the board, ram and upper die, usually v; g; and H have the usual meanings, as mentioned with called mass of ram; (11.1a). v velocity of the ram at the start of deformation of the Power drop hammers are rated by the weight of the work-piece; striking mass excluding the upper die, which may range from 450 to 31,750 kg (Semiatin 2005). A steel anvil block g acceleration due to gravity; weighing 453,600 kg or more is required for a hammer rated H fall height of the ram. at 22,700 kg. The capacity of available hammers ranges from about 5 to 310 kN, and the weights of forgings pro- Belt drop hammer is another kind of drop forging duced by this equipment may vary from a few kg to several machine whose function is similar to that of the board drop tonnes. The speed ranges of power drop hammers vary from hammer, except that a belt is used instead of a board. In this 3 to 9 m/s (Semiatin 2005). The advantage of the power machine, two rolls grip the belt that is attached to the ram at hammer over the board hammer is that the blow energy in one end and to a wall at another end. The rolls raise the ram the power hammer can be regulated by varying the air or and upper die through the belt to the required height, where the rolls hold the hammer but make the belt slack. When the upper roller moves away, the belt is released that causes the

474 11 Forging steam pressure, whereas in the board hammer the falling of hammers varies from 0.8 to 0.9 for soft blows, i.e. for weight and height of drop are fixed. Power hammer is pre- small load and large displacement, and from 0.2 to 0.5 for ferred to board hammer for closed-die forging operation. hard blows, i.e. for high load and small displacement (Semiatin 1988). In power drop hammer, the transmission of kinetic energy of the ram in the anvil block and foundation is as much as 11.3.2 Forging Press 15–25% and may rise up to 80% in finishing blows, where the actual deformation per stroke is comparatively slight. In this, the capability of the machine to perform the forging Due to the transmitted energy, a large stress is imposed on operation depends on the length of the press stroke or the the anvil block and may even break the anvil. The trans- available maximum load capacity. Presses are rated on the mitted energy also develops damaging shocks and vibrations basis of the load developed at the end of stroke. The press in the foundation and surrounding grounds. To avoid or deforms the work-piece with a slow-speed squeeze force and minimize this, it is necessary to use shock-absorbing mate- produces components in a single closing of die, while rials, such as iron felt or timber, in the anvil block founda- component is produced by repeated impact blows on the tions, which adds substantially to the cost of the foundation. work-piece in drop forging. Hence, a much better dimen- To minimize the loss of substantial energy in the anvil block sional accuracy of the forged product can be achieved in and foundation, counter blow hammer machine is used, in press forging than in drop forging. The press also provides which two opposed rams move from top and bottom sides. deeper forging penetration resulting in more homogeneous Steam, hot air or cold air is used to accelerate the upper ram properties of the forged product. However, initial cost of a downward while a steel band (for smaller capacities) or a press is much higher than that of a hammer and large pro- hydraulic coupling system (for larger capacities) is used to duction runs are required to justify the expensive press. accelerate the lower ram upward at the same time. The lower ram including the die assembly is about 10% heavier than Presses vary in size and in the extent of force that they can the upper ram. Both the rams strike the work-piece at the produce. Depending on the methods of force delivery system same time so that practically, all of the energy is absorbed by employed to deform the work-piece, the following two basic the job and very little energy is lost through vibration in the types of forging presses are available. foundation and environment. Therefore, the foundation required for a counter blow hammer will be smaller than that • Mechanical press and for an anvil hammer of comparable capacity. Horizontal • Hydraulic press. counter blow hammers are also available in which two These press machines are not only used for forging opposed die-carrying rams move horizontally by compressed operations but they are also used in metal extrusion and air. The speed ranges of counter blow hammers vary from sheet metal forming and in the production of plastic parts. 4.5 to 9 m/s (Semiatin 2005). When both rams in coun- The diagram shows that the force applied by the press to the terblow hammers have approximately the same weight, the work-piece acts vertically downwards, but in the presses total energy supplied by the blow is equal to double of that used for extrusion, the force is applied horizontally, i.e. in a by the gravity drop hammer and hence, given by: path normal to the direction of gravitational force, though the working principles for both types of press are the same. UCounter ¼ 2   ¼ m  vT 2 mvT2 Selection of a press type primarily depends on the type of the 1 mv2 ¼ 4 manufacturing process to be performed. For example, the Blow Hammer 2 2 general requirements of a press for forging, extrusion, impact extrusion or sheet metal forming are all different. ð11:1bÞ Other important factors in selecting a press include the press capacity and the length of press stroke over which the force where is delivered. Again, the required press capacity is decided by the size of the work-piece and the type of operation and the m the mass of ram; stroke length is related to the type of operation. v velocity of one ram; and vT the actual velocity of the blow of the two rams ¼ 2v: 11.3.2.1 Mechanical Press Mechanical presses include a wide range of different It must be noted that during a working stroke, the total machine types. Mechanical presses are stroke-restricted nominal energy, say UT ; of a hammer is not fully utilized for deformation, because a small amount of energy is lost to the environment in the form of vibration and noise. The ratio of useful energy available for deformation, say UA; to UT is called the blow efficiency, g; of hammers, i.e. g ¼ UA=UT : g

11.3 Forging Equipments 475 machines since their capability to perform the forging flywheel is converted into linear motion by multiple threads operation is determined by the length of the press stroke and the available load at various locations of the stroke. The on the spindle and thus, the screw pushes the ram in a linear energy in a mechanical press is obtained from the rotational energy of a motor. Mechanical presses are usually not used path. Screw presses are largely applied in Europe for both for extrusion because it often requires a more uniform force over a larger distance. Apart from forging, mechanical cold and hot closed-die forging. These presses are classified presses are commonly used in sheet metal forming and may often be used for impact extrusion, where the requirement is as energy-restricted machines because their capability to a rapid and quickly repeatable application of force over a shorter distance. perform the forging operation is determined by the energy Most mechanical presses function by using an eccentric available in the flywheel of the press. Screw presses are crank that transforms the rotary motion into reciprocating linear motion of the press slide to perform the pressing similar to hydraulic presses in that they can produce a force action. This type of equipment is called crank press, which is shown in Fig. 11.10. A drive shaft is attached with a crank of uniform magnitude over a large stroke length and are link used in the press, which rotates with the drive shaft and linked to a connecting rod by a rotary joint. This connecting relatively slow-speed machine requiring a longer contact rod is further connected to a ram by a rotary joint. During the rotational motion of drive shaft crank, the connecting rod time with the work-piece. Screw presses have load capacities swings back and forth that causes the ram operating in a slider joint to travel in a linear path in both directions. ranging from 1.3 to 280 MN (150–31,500 tonnes) and the During the downward movement of the ram, the upper die attached to it compresses the work-piece resting on the lower pressing speeds varying from 0.5 to 1.2 m/s (Semiatin die and this compressive force forms the desired part. 2005). The force in the crank press varies in both magnitude and speed of application throughout the length of the press stroke. During the stroke of a mechanical press, the total energy This press is the most suitable for forgings of low profile, because the stroke length of ram in this press is shorter than supplied is given by that in a hydraulic press or hammer. Less bulky dies can be used, and die life is longer with a press than with a hammer UMechanical ¼ 1 I Àx21 Àx22 Á ¼ 1 I  p 2 Àn12Àn22Á ð11:3Þ because the press squeezes the work-piece slowly unlike the 2 2 30 rapid impact blow of the hammer. These presses can give up Press to 50 strokes per minute. They are slower than the forging hammers but are generally faster than hydraulic presses or where screw presses (actually, the screw press may also be classified as a mechanical press). Mechanical presses have load I moment of inertia of the flywheel; capacities ranging from about 2.2 to 142.3 MN (250– x1 initial angular velocity, in rad s−1; 16,000 tonnes) and the pressing speeds varying from 0.06 to x2 angular velocity after deformation, in rad s−1; 1.5 m/s (Semiatin 2005). Next to the forging hammer, the most extensively used machine for closed-die forging oper- n1 initial speed of flywheel, in rpm; ation is the mechanical press. The production rate of a mechanical press is similar to that of a hammer, but both n2 speed of flywheel after deformation, in rpm. preliminary and finishing steps of a forging operation may not be possible to carry out in the same mechanical press Depending on the mode of transforming the rotational because each blow of the same press exerts equal force. motion of a motor to the linear motion of the ram, other types of mechanical presses are also available, such as rack Mechanical presses may also be screw-driven, called and pinion press, eccentric press and knuckle joint press. In screw presses where the rotational energy of a motor is used the rack and pinion press, a pinion, which is a rotating round to turn a large screw. In a screw press, the ram is attached by gear, attached to the drive shaft transfers the rotational a rotational joint to a spindle, which is actually a large screw. energy of motor through a rack (which may be considered as A reversible drive shaft by using a friction disk produces a a round gear of infinite radius) to provide force for the rotary motion to a flywheel, which is connected to the desired linear motion of the ram. The eccentric press uses a spindle, as shown in Fig. 11.11. The rotational motion of the motor-driven eccentric round shaft that may completely rotate within a connecting rod attached to the ram. With the rotation of motor, the overall centre of the shaft changes causing the shaft to change position and thus provides motion to the connecting rod that moves the ram linearly in a slider joint. The knuckle joint press uses a powerful linkage design, through which a completely rotating drive shaft crank transforms the rotational energy of a motor to a single dimension translational energy. 11.3.2.2 Hydraulic Press Hydraulic presses are the most powerful class of presses. They are load-restricted machines because their capability to perform the forging operation depends on the available maximum load capacity of the press. Hydraulic presses use hydraulic pressure or fluid pressure and a piston to generate the deformation load. The load supplied for deformation will be equal to the product of the fluid pressure and the

476 11 Forging cross-sectional area of piston head. Water, certain types of problem. Thus, the disadvantages of a hydraulic press emulsion, or mineral oil may be used as the working fluid. include its slow speed with consequent cooling of the job The fluid pressure can be increased or decreased by means of and shorter die life, its cost and vastness. pumps and valves to move the piston downwards or upwards in a chamber or cylinder. The functioning of the hydraulic 11.4 Open-Die Forging press depends on difference in the fluid pressure between above and below the piston. For the piston to rise, the fluid Open-die forging is employed widely for cogging operation pressure below the piston must be higher than that above the where the cross-sectional area of usually large work-piece of piston. So during the upstroke of piston, the pump inserts the relatively simple shape is reduced between flat dies or dies of fluid through the bottom channel into the cylinder below the very simple shape by slow squeezing action in a large piston to increase the pressure under the piston and simul- hydraulic press or by repeated blows in a power hammer. In taneously takes the fluid out of the cylinder through the top open-die forging, shape of forged product is manipulated channel to decrease the pressure above the piston. In the next manually. Figure 11.13 shows the cogging operation in step, the downward movement of the piston is required to open-die forging, where the work-piece moves gradually to deform the work-piece, which necessitates a higher pressure the right after deformation, and the shaded area on the of the fluid above the piston than the fluid below it. So work-piece indicates the region of the material ready for during the downstroke of piston, the pump takes the fluid out forging in contact with the upper die. The thickness of the of the cylinder through the bottom channel to decrease the work-piece is reduced from one end to another in a stepwise pressure under the piston and simultaneously inserts the fluid sequence because the tool is usually shorter in length than through the top channel into the cylinder to increase the the work-piece and the length of deformation zone at any pressure above the piston. The upper die is attached below point of time will be a small fraction of the work-piece. The the ram, which forms the lower part of piston. During the completion of the forging work may require several passes. downstroke of piston, the ram through the upper die com- The reduction in thickness of the work-piece is accompanied presses the work-piece that rests on the lower die placed by elongation in the longitudinal direction and spreading in above a base or an anvil and this compressive force forms the lateral directions. Relative amounts of elongation and the desired part. The hydraulic press is shown in Fig. 11.12. spread will depend upon the ratio of the length of defor- mation zone at any instant to the predeformed initial width This press can control and even vary ram velocity during of the work-piece, called the bite ratio b=w1: Since the the stroke by changing the fluid pressure. An important deformation involved is large, so the true strains are used to characteristic of the hydraulic press is that it can produce the express the spread and elongation, which may be defined maximum press load at any point over the entire stroke (Tomlinson and Stringer 1959) as follows: length of the ram. This character makes the press perfectly suitable for forging operations where extrusion flow, i.e. Coefficient of spread, flow parallel to the direction of die motion takes place. Such a press is very commonly used for extrusion operations, but SCoeff: ¼ width increase ¼ lnðw2=w1Þ ð11:4Þ a horizontal hydraulic press is often employed. Hydraulic thickness reduction lnðh1=h2Þ presses have load capacities ranging from about 2.2 to 623 MN (250–70,000 tonnes) and the pressing speeds b w2 varying from 0.03 to 0.8 m/s (Semiatin 2005). The largest w1 h2 forgings are always produced by large hydraulic presses. The above-mentioned important characteristic, flexibility of h1 operation and greater capacity of a hydraulic press are its advantages over a mechanical press. The hydraulic press Die operates at a constant speed, but it is a relatively slow-speed b machine, which involves a longer contact time with the job. This may result in heat loss from the job during hot working Fig. 11.13 Cogging operation in open-die forging. Contact between and die deterioration causing the die life to decrease. On the job and upper die will occur at the place shown by shaded area other hand, a hydraulic press produces forgings of close dimensional tolerance resulting from its slow squeezing action. However, the initial cost of a hydraulic press is higher than that of a mechanical press of equal capacity. Factors for converting between the capacity of presses and hammers are available (Lyman 1970). Some hydraulic presses may be as large as buildings causing accommodation

11.4 Open-Die Forging 477 where w1 and w2 are, respectively, the predeformed initial Often (11.4) is expressed in terms of the ‘spread law’, as and post-forged final widths of the work-piece; h1 and h2 are, shown by the following (11.10). Let us define spread ratio respectively, its predeformed initial and post-reduced final and squeeze ratio as follows: thicknesses. Spread ratio ¼ w2=w1 ¼ b; ðsayÞ; During compressive deformation, there will be barrelling and squeeze ratio ¼ h2=h1 ¼ c; ðsayÞ: of the work-piece, which makes the measurement of the final width w2 and thus the width natural strain difficult and Equation (11.4) can be written in terms of b and c as unreliable, but the increase in length and reduction in follows: thickness can be measured precisely. Therefore, change in the magnitude of width can be obtained in terms of length SCoeff: lnðh1=h2Þ ¼ lnðw2=w1Þ; increase and thickness reduction from the constancy of 1SCoeff: volume relationship, which is given by Or; ln c ¼ ln b h1w1l1 ¼ h2w2l2; or; h2w2l2 ¼ 1 ð11:5Þ ) b ¼ 1SCoeff: ð11:10Þ h1w1l1 c where l1 and l2 are, respectively, the preforged initial and Ship propeller shafts, pressure vessels, gun tubes and post-deformed final lengths of the work-piece. Taking nat- rings are the examples of components that are made in open-die forging. Since large sections are frequently used in ural logarithm to both the sides of equality sign in (11.5), we open-die forging, care must be taken so that a homogeneous deformation zone penetrates up to the centre of the get work-piece. To minimize inhomogeneous deformation, a minimum bite ratio of 1/3 is recommended. Further for a      ð11:6Þ given geometry of tooling, a critical deformation may pro- ln h2 þ ln w2 þ ln l2 ¼ 0; duce surface laps (forging defect) at the step that separates the deformed from the undeformed segment of the h1   w1   l1   work-piece, because only that portion of the surface under Or; ln w2 ¼ ln h1 À ln l2 the bite is being forged at any instant. To avoid formation of laps, Wistreich and Shutt (1959) recommended that the w1 h2 l1 reciprocal of squeeze ratio, i.e. h1=h2 must not exceed 1.3. Hence from (11.4) and (11.6), we can write the coefficient of spread in terms of length increase and thickness reduction as follows: SCoeff: ¼ lnðh1=h2Þ À lnðl2=l1Þ ¼ 1 À lnðl2=l1Þ ð11:7Þ lnðh1=h2Þ lnðh1=h2Þ From (11.7), we can define the coefficient of elongation as 11.5 Closed-Die or Impression-Die Forging follows: Carefully machined matching two die blocks or two die Coefficient of elongation ¼ 1 À SCoeff: halves carrying the impressions of the desired final shape are used in closed-die forging for production of precision forg- ¼ length increase ings with dimensions closer to those of the desired final thickness reduction product. Mass or large-lot production of parts is generally lnðl2=l1Þ required to justify the expensive dies. When the two die halves ¼ lnðh1=h2Þ ð11:8Þ are assembled, it will form one or several internal cavities, called die impressions and forging may be performed in either Clearly, if SCoeff: ¼ 1; then there would be no elongation single- or multiple-impression dies. The single-impression at all and the reduction in thickness would all appear as dies are suitable for forgings of simple shapes, while multiple- spread, while if SCoeff: ¼ 0; there would be no spread at all impression dies containing preliminary impression in addition and all of the reduction in thickness would appear as only to the finish impression are used for forgings of irregular or complex shape; sometimes the preliminary and finish elongation. impression are made in two or more separate dies. The process is called closed-die forging because closing of the dies com- From the examination of Tomlinson and Stringer (1959), pletes the deformation. The work-piece after forging will acquire the geometric dimensions of the die cavity provided it was found that the value of SCoeff: depended mainly on the the cavity is completely filled during the process. bite ratio b=w1 and the relation between coefficient of spread and bite ratio in cogging operation was given by   b 2 0:36 b 0:054 SCoeff: ¼ 0:14 þ À w1 ð11:9Þ w1

478 11 Forging In closed-die forging, the work-piece is usually preshaped of flash may be a small fraction of the total weight for forgings by placing first into the fullering impression and then into of simple shapes but may exceed the weight of the actual edging impression, so that material is distributed in the forgings for those of complex shapes. The formation of a very correct places for subsequent forging. Now if required, the wide flash is not desired because it increases the forging stock undergoes some other processing operation, e.g. pressure. Therefore to limit the width of flash, a ridge, known bending is required for a connecting rod. The preformed as a flash gutter, is usually provided as shown in Fig. 11.15. stock is then placed in the cavity of the semifinished Actually, flash consists of two parts: the flash at the land and impression, called the blocking impression, where the stock that in the gutter. The portion of the flash adjacent to the part is is rough-forged to close to the final shape. Blocking opera- the flash land, which is generally constructed as two parallel tion usually causes the greatest change in the shape of the surfaces, and the flash portion outside the land is gutter. forging and thus reduces the wear of the finishing impres- However, in the final step of a closed-die forging, a trimming sion. Finally, the forging is transferred to the finishing die is used to remove the flash formed in the forged product impression, where the final shape and dimensions are obtained from the die of finishing impression. Figure 11.16 imparted to the forged product. The blocking cavity and the shows the steps involved in a closed-die forging operation. finishing cavity are usually machined into the same die Because of the flash, instead of the name closed-die forging, a block, in which fullering and edging impressions are often better name to describe the process would be impression-die placed on the edges. For complex final shapes of the forged forging. As an example, the forging sequence for a connecting products, more than one preshaping or blocking operation is rod with impression die has been shown in Fig. 11.17. necessary to achieve a steady flow of material that will cause a gradual change in shape and size. It is possible to carry out flashless closed-die forging operation by constraining the entire work-piece within the It is very important to fill the die cavity completely, for closed-die cavity in such a way that no material can flow out which excess material is required in the work-piece. Thus, the of the cavity to form the flash during the compression of the volume of the initial work-piece is somewhat greater than that part. But then the filling of the die cavity requires too close of the die cavity because it is very difficult to put just the right control of the raw stock, and that is usually more expensive. amount of material in the right places during closed-die The volume of the initial stock in flashless closed die must operation. When the dies come together for the finishing step, be equal to or slightly greater that of the finished part. The the excess material flows out of the finishing die cavity as a die cavity will not be filled-up completely if the raw material thin ribbon of material called flash or fin. Actually, a flash is is less than the required amount while too much amount of waste material appearing at the parting line around the material will cause a dangerous build up of forging pressure perimeter of the forged product and increases forging load by because of restriction of material flow within the closed five to ten time, as seen from Fig. 11.14 that shows a plot of cavity. Too much pressure within the die cavity can damage forging load versus forging stroke (die advance). The weight the die and machinery. In view of the above, the closed-die forging with a flash is more widely employed in spite of the Die cavity completely filled Upper die Flash gutter Forging lood Flash begins Forging completeto form Forging Dies contact workpiece Flash Forging storke Lower die Fig. 11.14 Typical curve of forging load versus stroke (die advance) Fig. 11.15 Flash and flash gutter shown in the sectional view of for closed-die forging closed-die forging

11.5 Closed-Die or Impression-Die Forging 479 Trimming die Billet Preshaped Rough-forge Finishing die Fig. 11.16 Steps involved in a closed-die forging operation Fig. 11.17 Sequence of impression-die forging for a connecting rod Initial forging stock Fullering Edging Bending Blocking Finishing loss of material in the trimmed off flash. Depending on the aluminium alloys, magnesium alloys, copper alloys, car- size of the forging, flash material losses may vary between 5 bon and low-alloy steels, martensitic stainless steels, and 15%, which may increase up to 30% for tall, narrow maraging steels, austenitic stainless steels, nickel alloys, forgings and for blades (Sabroff et al. 1964). semi-austenitic PH stainless steels, titanium alloys, iron-base superalloys, cobalt-base superalloys, niobium alloys, tanta- Materials selected for die must have certain properties, lum alloys, molybdenum alloys, nickel-base superalloys, such as good hardenability, resistance to wear, toughness, tungsten alloys. resistance to plastic deformation, resistance to thermal and mechanical fatigue (Semiatin 1988). Die materials include Chilling of the workpiece by the colder dies is a problem hot work tool steels (AISI H series) and some alloy steels, when closed-die forging operations are carried out by con- such as AISI 4300 or 4100 series. Some dies are cast but ventional hot forging processes. To get rid of this problem, more are forged, sunk and then heat-treated to develop closed-die forging operations are carried out by either hardness lower than the maximum value but with sufficient isothermal forging process or hot-die forging process in toughness in order to achieve shock resistance property. which die temperatures are appreciably higher than those used in conventional hot forging processes. These processing In closed-die forging, the basic requirements of a forging techniques are primarily applied to manufacture air-frame stock material are: structures and jet-engine components made of titanium and nickel-base alloys, which are difficult to forge. Superalloys • Strength or flow stress of the material must be low so that and refractory metals are used as die materials when dies are die pressure does not exceed capabilities of practical die heated above 650 °C. In isothermal forging process, the dies materials. are maintained at the same temperature as the forging stock. Hence, the die chill is eliminated completely and the stock is • Forgeability of the material must allow the required maintained at a constant temperature. In this, forging can be amount of deformation without failure. With increase in performed at extremely slow strain rates, thus providing the temperature, the forgeability generally increases, but at advantage of low-strain-rate sensitivity of flow stress of the same time, grain growth also takes place, which certain alloys. This process can produce net shape forgings decreases forgeability in some alloy systems. In other that can readily be used without machining, or near-net shape alloys, forgeability is largely influenced by the charac- parts that require minimal machining. The disadvantage is teristics of second phase constituents. that the choice of die materials and lubricants are limited due to the use of hot dies. In hot-die forging process, the die In order of increasing forging difficulty, forging materials may be arranged as follows:

480 11 Forging temperatures are lower than those in isothermal forging, but Forging pressure higher than those in conventional hot forging. Typical die temperatures are 110–225 °C lower than the stock tempera- Parting line Web ture. Compared to conventional hot forging, this process PL L obviously reduces the die chill and produces near-net shape parts. Therefore, both isothermal and hot-die forging pro- Rib cesses are referred to as near-net shape forging processes. The advantage of hot-die forging process over isothermal forging process is that the lowering of die temperature allows wider selection of die materials. However, it is to be noted that there are two basic types of material flow: • Upsetting, where flow is perpendicular to the direction of Fig. 11.18 Webs perpendicular to the direction of forging pressure die motion, and and ribs parallel • Extrusion, where flow is parallel to the direction of die motion. It is obvious that upsetting flow is relatively easy, while extrusion flow is much more difficult. However, both types of flow of material can occur Fig. 11.19 Metal flow in webs and ribs simultaneously. Ideally, finishing step should have upsetting flow towards the die cavity without additional shear at the than 50%, and flash loss is three times greater, in the forging die–stock interface because friction, forging load and die of a rib than in the forging of a web. Even to fill a rib wear will be minimized during this type of flow. becomes more difficult if it is displaced from the centre of the forging, as shown in Fig. 11.19. In general, the rib height As far as the size and weight of parts involved in the should not exceed eight times the rib width (Sabroff et al. impression-die forging are concerned, components produced 1964). The degree of difficulty in impression-die forging is by this forging are of limited size, the weights of which observed to be more for shapes with one dimension sub- range from a few decagrams to several tons. Products that stantially bigger than the other two, which form roughly are successfully forged in closed dies weigh as much as 70% of impression-die forgings. 25,400 kg, although more than 70% of the closed-die forged products weigh 0.9 kg or less. The success of a closed-die forging operation depends considerably on the design of the die because die is the one of Within the overall size and weight limitations, the basic the most critical components of the closed-die forging opera- factor that determines the difficulty of forging a given con- tion. The aims of die design are to achieve easy flow of figuration is the surface-to-volume ratio of the part. If a part material, reduction of forging load and power requirement, has the lowest surface-to-volume ratio, such as Spherical and reduction of die wear and breakage, elimination of forging block-like shapes, it is the easiest to forge because the slow defects, etc. Some die design factors for impression-die forging rate of cooling of the part does not cause the flow stress of the that involves controlling of flash land ratio, providing proper material and thereby the forging load to increase rapidly. The draft angles and adequate enough corner and fillet radii, and more a part differs from these simple shapes—with thin, long selecting location of good parting line are discussed below. sections, holes and recesses, ribs and projections—the higher its surface-to-volume ratio and its forging difficulty. 11.5.1 Flash Thin section of the part that is parallel to the direction of The formation of flash is an important part in the production forging pressure is called a rib and that perpendicular to the by impression-die forging. The process of flash formation direction of forging pressure is called a web. These are with single-impression dies is illustrated in Fig. 11.20. shown in Fig. 11.18. Rib and web, both are more difficult to forge than thicker sections, because the material in them cools rapidly, building up high resistance to deformation. However, a vertical rib is even more difficult to form than a horizontal web since a web is formed by upsetting flow of material under the direct application of compressive force, whereas a rib is formed by extrusion flow of material when the upsetting flow of material is hindered by the formation of a thin flash. As a result, die life is usually decreased by more

11.5 Closed-Die or Impression-Die Forging 481 (a) (b) HAMMER HAMMER top die top die Flash gutter bottom die ANVIL bottom die ANVIL (d) (e) (c) HAMMER top die bottom die ANVIL Fig. 11.20 Schematically illustrating the process of flash formation impression. c Formation and thinning of flash between the two die faces with single-impression dies. a A work-piece is placed in a stationary with further advancement of the top half die. d Forged part joined with bottom half die. b With approach of the top half die, the work-piece is flash. e Forged part after trimming of flash compressed until its enlarged sides touch the side walls of the die In this, the work-piece of a simple shape, say with a rect- enough stress is built up in the flash to force the material into angular cross-section, is placed in a stationary bottom half the more intricate details of the cavities, which successfully die. As the top half die approaches, the rectangular completes the process. Thus, flash serves mainly two purposes: cross-section is compressed until its enlarged sides touch the side walls of the die impression. At this point, excess (1) Formation of flash ensures that excess material is present material begins to flow laterally and form the flash between in the work-piece for complete filling of the die cavity. the two die faces and this flash is gradually thinned with Further, flash provides a passage for escape of the excess further advancement of the top half die. Thus, the dies are material from the die cavity and assists to avoid cracking only closed by virtue of flash formation, leading to of die that might happen due to build up of high die three-dimensional control of the flow of material. In this pressure resulting from excess material in the die cavity. sense, the flash becomes effectively part of the die. (2) Before the flash opens up into a gutter, it must proceed At the stage of initiation of flash formation, the whole cavity through a narrow passage, called flash land. During the will not be completely filled for intricate shapes. Thin flash flow of material through the flash land, the material formed with further die approach cools gradually, and a high experiences friction and this friction resists further flow

482 11 Forging of material out of the die cavity. In addition, the thin flash flash land ratio depends on the material being forged, the cools rapidly and the drop of temperature increases its weight of the forging, the complexity of the forged part flow resistance. We can recall from Chap. 2 that flow and the forging equipment used. The flash land ratio resistance during compression of a thin section is more varies from 2:1 to 5:1 (Semiatin 1988). Lower ratios are than that of a thick section. Thus, friction, drop of tem- used in presses, whereas higher ratios are used in ham- perature and thinness of flash—all factors increase mers. It has been seen from experiments that die-filling resistance to deformation of material in the flash area, capacity increases with increasing flash land ratio up to which in turn causes to build up a very high pressure a value of about five above which the improvement is inside the bulk of the work-piece. This high pressure very small, whereas extremely high die pressure causes forces the material to flow into the die impressions severe die wear resulting in a shorter die life. hitherto unfilled so that, at the end of the stroke, all • The flash represents a loss of material. To minimize this impressions including the most intricate details in the die loss, a narrowing flash has been used. Since the tapered cavity are completely filled with the work-piece material. designs of flash provide considerably more constraint to metal flow than do parallel designs of flash (Sabroff et al. The thinner the flash the greater is the cooling rate, and 1964) and thus generate extremely high pressures, which the higher is the flow resistance and the die pressure in the aid very efficiently in filling the die. Although the tapered flash area. Similarly, the wider the flash land the more is the flash designs reduce flash-metal losses, the requirement frictional resistance to deformation and the higher will be the of greater forging pressure imposes greater stresses on die pressure in the flash area. Although thinner flash and forging dies. A tapered flash design is generally used wider flash land can ensure complete filling of the die cavity when the material savings justify the use of larger forging by increasing the forging pressure even under unfavourable equipment required for applying greater forging pressure. conditions, such forced filling of the die is undesirable Hence, to a certain extent, the choice of such flash design because of short die life and high power requirements. becomes a matter of economics. Hence, the pressure within the forging die cavity and form- filling capacity are often controlled by varying the land Finally, the designer has to consider that the volume of width to thickness ratio of the flash, called the ‘flash land raw stock must be equal to the volume of finished part plus ratio’. Dies must be designed in such a way that minimum the volume of flash. flash is used to do the job, for which the following guide lines are provided. 11.5.2 Draft • The flash thickness is usually maintained at 3% of the The amount of taper provided on all internal and external side maximum thickness of forging. surfaces of the forging and on projections to facilitate its easy removal from the dies is called draft. The corresponding taper • An enough large flash gutter (a cavity in the die halves for on the internal and external side walls of the die impression is the excess material), which must be thicker than the flash, also called draft. Almost all forgings require draft and the is usually provided so that the gutter does not become larger the draft angle, the easier will be the removal of forged pressurized or fill up with excess material. In such cases, products from the dies. Easier to forge materials, such as the flash material is allowed to flow into the gutter and the aluminium and magnesium alloys, generally require less draft flash is reduced in thickness only over a part of its width, angles than difficult to forge materials, such as steel, nickel i.e. the formation of a wide flash is prevented. This and titanium alloys. However, there is an effort to minimize reduces the pressure exerted on the die faces by pre- draft, because in most cases, it represents consumption of venting unnecessary increase of forging load. However, extra material without contributing anything to the mechani- there are commonly used four gutter designs, which are cal utility of the forged product. Forging designs with zero parallel, conventional, tapered open and tapered closed draft angles require dies with special knockouts. For example, (Semiatin 1988). Choice of gutter design generally hydraulic and mechanical presses, where knockout arrange- depends on the properties of the forging material, the type ments can readily be used, can produce forgings with zero of forging equipment used, the overall pressures exerted draft angles. On the other hand, forgings with zero draft angles in the die cavity and the forging temperature. are seldom produced on drop hammers because there is a risk of die breakage if knockout equipment is used in drop hammer • In designing the flash, its width and thickness are so forging (Sabroff et al. 1964). adjusted that the extrusion flow through the narrow opening of flash becomes more difficult than the filling of During cooling of a forging part within the die, the exte- all recesses in the die cavity; but at the same time, rior surfaces of the part shrink away from the walls of die excessively high forging pressure must not be created so as to cause wear and breakage of the die. The value of

11.5 Closed-Die or Impression-Die Forging 483 Inside Outside During filling of the die cavity, the material of work-piece draft draft flows and changes directions that will depend upon the geometry of the forged product. If the parts have fillet turns Fig. 11.21 Draft angles showing outside draft angle is smaller than or corners of larger radii, the material will follow smoothly the inside one and completely the contours of the die around the fillets or corners without creating any vacancies or forging defects. cavities, while the interior surfaces of the part shrink onto On the other hand, sharp fillets or corners may not allow the bosses in the die. Therefore, interior draft angles will usually material to follow completely the path of fillets or corners, be larger than exterior draft angles, as shown in Fig. 11.21. leading to the formation of voids or/and forging defects like The values of draft angle may vary from 3° to 15° that depend laps or cold shuts (Sabroff et al. 1964), as shown in on many factors. The deeper the die cavity, the greater the Fig. 11.22. Good design for successful impression-die draft angle required to ensure smooth release of the forging. forging operation must provide fillet and corner radii as The exterior draft angles normally vary from 3° to 7° and the large as possible to allow for easy flow of material, to reduce interior draft angles range from 5° to 10° (Semiatin 1988). die wear and to avoid fracture at the corners. It is to be noted that in a forging sequence all concave radii on the perform 11.5.3 Radii forging should be greater than the radii on the final forged part. Small radii at the fillets and corners are undesirable for many reasons. They may cause For forgings of stainless steels, titanium and high- temperature alloys, the absolute minimum fillet radius of • Difficulty in the flow of material during forging and the parts is generally 6.35 mm (0.25 inch). If the fillets are at the formation of vacancies or/and forging defects, like laps base of tall ribs and bosses, their radii have to be increased or cold shuts; because the die cavity at the base of tall section experiences greater difficulty of filling. Except for very small forgings, • Decrease of die life by increasing die wear; the absolute minimums for comer radii are usually about half • Development of stress concentration within the die cavity the minimums for fillet radii (Sabroff et al. 1964). Thus, the minimum comer radius of bosses and other edges is usually and fracture of material near the corners. 3.175 mm (0.125 inch) for the stainless steels, titanium and high-temperature alloys, while for aluminium forgings of comparable shape and size, a smaller comer radius of 1.5875 mm (0.0625 inch), which is one-half of the above value, would be used. For the end of a rib, it is preferred to have a radius that will make the end a full semicircle, i.e. the radius at the end of a rib must be equal to half of the end-thickness of that rib. (a) (b) (c) Forging defect Fig. 11.22 Sharp edges leading to forging defects. Flow of metal is shown by arrow symbols as the metal is gradually compressed from a to b and b to c

484 (b) 11 Forging FORGING PLANE (a) PARTING LINE PARTING LINE AND FORGING PLANE (c) COUNTER LOCK (d) (e) Fig. 11.23 Straight and broken parting lines 11.5.4 Parting Line Some general rules for location of parting line are as follows: The parting line is the line along the surface of a forging where the dies meet as well as the flash is formed. The plane (1) The location of parting line in most forgings is gener- that includes the principal die face and is normal to the ally at the largest cross-section of the forging part direction of die motion is called the parting plane or the (Semiatin 1988), because it is easier to spread metal by forging plane. The parting line lying on the forging plane upsetting flow than extrusion flow into deep die forms a straight parting line. However, the parting line does impressions, which requires greater force. not always remain on the forging plane. If any part of the parting line is inclined to the forging plane, a broken parting (2) For simple, symmetrical, shallow sections, a straight line is formed. Straight and broken parting lines are shown in parting line is usually selected that lies on the forging Fig. 11.23. Sometimes the shape of forging part is such that plane, i.e. situated at the centre (mid-height) of the sec- it is more economical or even essential to make a broken tion, as shown in Fig. 11.24. To make dies simpler and parting line. For closed-die forging of a certain shape, the cheaper, keep the parting line in one plane, if possible. first step in the die design is to establish the shape and location of the parting line (Sabroff et al. 1964). Depending (3) When it is more economical or even essential to make a on the geometry of the final component, the parting line may broken parting line, the inclined part of the broken be straight or broken, by which other design factors such as parting line produces horizontal components of the die design and construction, grain flow, and trimming pro- forging force, which attempt to shift the die halves cedure are influenced. The locations of the parting line vary, sideways, as shown in Fig. 11.23b, c. These horizontal depending on the type of forging equipment employed and, forces are automatically balanced out in symmetrical to some extent, on the forging metal. parts, such as shown in Fig. 11.23b, but in forging of asymmetrical sections requiring a broken parting line, the side thrust must be minimized or eliminated by adopting any of the following means: Fig. 11.24 Straight parting line in symmetrical section

11.5 Closed-Die or Impression-Die Forging 485 • Side thrusts trying to displace the die halves must be wall. If the parting line is located at any point below the restrained by expensive counterlocks built into the die top of the wall but above the centre of the bottom web blocks, as shown in Fig. 11.23c. the grain flow may be disrupted leading to defects in the forging (Semiatin 1988). In the forging of channel • Figure 11.23d shows that the position of the part in the section, Fig. 11.26c shows the position of parting line at die has been tilted to balance out forces. The parting line the bottom of the web, which is the most economical as tilting can sometimes eliminate the need for counter- all of the impression is in one die and is also less locks, but at the expense of a very complex parting line detrimental than the parting line shown in Fig. 11.26a, and increased die-making costs. b, but the preferred location of parting line is at the ends of the ribs as shown in Fig. 11.26d, because it results in • Often the most economical solution to balance the side smooth and good grain flow patterns. forces without tilting the parting line can be achieved by placing two identical forgings in a mirror image position Figure 11.27 illustrates a variety of simple shapes as shown in Fig. 11.23e. showing bad and good parting line locations. The reasons for the good locations of parting line are (Sabroff et al. 1964): (4) During making of rib and web type forgings with deep ribs, the preferred location of parting line is generally Case 1: The preferred choice avoids deep die impressions in near the top of the rib formed in the upper die as shown either die that might otherwise increase the forging difficulty in Fig. 11.25, where restriction of lateral flow aids and develop die wear and breakage. For this reason, the filling of the narrow rib. Further, this assists to avoid longest dimension of a ribbed component is generally placed the formation of folds (forging defect) caused by in the plane of forging rather than normal to the plane. material ‘sucking through’ into the flash land if the Case 2: The preferred choice is the central location of the placement of parting line is too low. The use of parting line, which requires less material to match drafts at excessive die lubricant can also produce this type of edges. Thus, reduction of draft saves material. defect. Case 3: The preferred choice avoids side thrust which would cause the dies to shift sideways. (5) Since metal flows partly towards the parting line during Case 4: The preferred location of the parting line produces forging, improper positioning of the parting line causes the most desirable uninterrupted grain flow pattern. to rupture the grain flow patterns of the material (Semiatin 1988) and may also create unfilled section 11.5.5 Design Steps (Fig. 11.26). The plane formed at the break point of the grain flow patterns is the weakest plane, which will The design of a part to be produced by closed-die forging easily fracture during service under a stress acting requires the following forecast: normal to it. Hence, the parting line of the forging dies must be so located that there is the minimum disruption • Volume and weight of the stock to be forged. to the grain flow lines. To create such a situation, for • Number of preform forging steps and their arrangements. example, in a forging having a vertical wall adjacent to a bottom web section, a parting line on the outer side of the wall should be located either at the top of the wall or adjacent to the web section and near the bottom of the Fig. 11.25 a Preferred parting (a) (b) line position in rib and web type forgings. b Restriction of lateral flow aids rib filling Parting line

486 11 Forging (a) (b) Metal Grain structure May have flow lines is ruptured at unfilled section the parting line Parting line (c) (d) Metal flow lines Most economical as all of the Metal impression is in one die flow lines Parting at the ends of the ribs results in good grain structure Fig. 11.26 Parting line location and its influence on the grain flow patterns that cause forging defects. c The most economical parting line patterns in the forging of channel section. a, b Show improper that is less detrimental than (a) and (b). d The preferred location of positioning of the parting line resulting in rupture of the grain flow parting line resulting in smooth and good grain flow patterns Parting line Parting line Bad Good Bad Good Avoid deep die impressions Centre location reduces draft Case 1 Case 2 Parting line Parting line Bad Good Bad Good Keep parting line in one plane if possible high parting line on dished part Case 3 gives uninterrupted grain flow Case 4 Fig. 11.27 Simple shapes showing bad and good parting line locations with specific reasons

11.5 Closed-Die or Impression-Die Forging 487 • Sizes of flash in preforming and finishing dies. of the ingot. Further, material is lost due to croppings, which • Requirements of load and energy for each forging step. involve the trimming scrap formed at the end of forging oper- ation. If holes are required in forgings, there will be waste of First and third of the above requirements have been dis- material when punching holes in forgings. This waste is known cussed by Altan and Henning (1972). The most difficult as as slug. Croppings include both trimming scrap as well as slug, well as critical step in forging design is the design of preform and this is unavoidable loss or waste during forging operation. forging. Minimum flash loss, complete die fill, and Considering the above-mentioned all losses, the initial weight defect-free forging are ensured by correct preform design. or volume of work-piece is determined from the following Some general considerations regarding forging design have formula on the basis of the weight or volume of the forging: been dealt in literatures (Lange 1958, Altan et al. 1983). The reader is also referred to Akgerman et al. (1973), where In case of forging from an ingot, some guidelines used in designing preforming dies have been discussed. WIN ¼ WFORG: þ WH:DISC: þ WB:DISC: þ WSC: þ WCROP: ð11:11aÞ In conclusion, we can summarize the steps to be followed while planning to design the forging die. These steps are: Or, VIN ¼ VFORG: þ VH:DISC: þ VB:DISC: þ VSC: þ VCROP: ð11:11bÞ (1) First step in die design is to decide the final shape of the forging with due consideration to machining and In case of forging from a rolled stock, shrinkage allowances, draft angles, fillet and corner radii, any web or rib dimensions and the position of the WIN ¼ WFORG: þ WSC: þ WCROP: ð11:12aÞ parting line. In hot forging, shrinkage allowance must be included in the dimensions of die to compensate for Or, VIN ¼ VFORG: þ VSC: þ VCROP: ð11:12bÞ the contraction that occurs on cooling of the product. where WIN or VIN is, respectively, the initial weight or vol- (2) The next step in die design is to calculate for the design ume of the work-piece that may be an ingot or a rolled stock; of flash and gutter. WFORG: or VFORG: is, respectively, the weight or volume (3) Before actually starting the die design, decide the se- of the forging; quence of steps before reaching the final finishing impression and design the product with its accurate WH:DISC: or VH:DISC: is, respectively, the weight or volume dimensioning for each of these technological steps, of the ingot head discard; which are fullering, edging, some other processing operations if required, blocking and finally finishing. WB:DISC: or VB:DISC: is, respectively, the weight or volume of the ingot bottom discard; (4) Next step is to select the die material and die block sizes. WSC: or VSC: is, respectively, the weight or volume of the scale loss; (5) The next job is to locate the positions of the impres- sions for each of the above technological steps on the WCROP: or VCROP: is, respectively, the weight or volume die block. The normal practice is to keep the finishing of the croppings that include trimming scrap and slug, if any. impressions more or less in the middle of the die block. The weight or volume of ingot head discard is usually (6) The last step is to arrange for locking of the dies, if considered as 20% (ranges from 14 to 30%) of the weight or necessary. volume of the ingot, and that for ingot bottom discard is usually taken as 5% (ranges from 5 to 10%). Scale loss is 11.6 Material Loss During Forging generally taken as 2% (varies from 2 to 3%) of the weight or (Kamenshchikov et al. 1964) volume of the ingot or rolled stock for each full heating and generally considered as 1.75% (varies from 1.5 to 2%) for The initial work-piece material in forging may be an ingot or a each subsequent extra heating. The weight of the trimming rolled stock of various cross-sections and lengths. Generally, scrap, i.e. the croppings excluding slug depends upon the light forgings are made from rolled stocks, while ingots are complexity of the forged product and the processing method considered for heavy forgings. Since most of the forging employed; for forgings of simple shape, such as shafts or operations are carried out at high temperature, so obviously, discs, it ranges from 5 to 8 or 10%, but it increases with the there will be loss of material due to formation of scale during complexity of the forgings and for certain intricate forged heating. Apart from scale loss, there are various forms of products it may reach 30% of the forged product. The volume material losses. If forgings are made from ingots, there will be of the croppings excluding slug from each end of cylindrical loss of material due to discard from the head as well as bottom forging sections of diameter ‘D’ has been estimated to be 0.21 Â D3, in case of press forging, and 0.23 Â D3, in case of hammer forging, whereas that from each end of rectan- gular forging sections of width ‘W’ and height ‘H’ has been