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

386 9 Fracture be in the direction of crack propagation. The ‘river patterns’ planes. Dimples and tear ridges are often observed around appear due to the movement of the cleavage crack across the the periphery of the facets on the quasi-cleavage fracture high-angle grain boundary along a number of parallel but surface. The size of the quasi-cleavage facet usually matches slightly offset cleavage planes, which form a series of par- with that of prior austenite grain unlike cleavage facet. Most allel plateaus and connecting ledges. The cleavage steps may important is that river lines of quasi-cleavage fracture are also be formed by the intersection of the cleavage crack with ridges and not steps as in cleavage fracture. On two-halves screw dislocations when the cleavage crack crosses a of a fracture surface, there is a matching of ridges in low-angle twist boundary. quasi-cleavage in contrast to matching of steps in cleavage, as shown in Fig. 9.11. Information achieved from cleavage facets may not be so useful as far as failure analysis is concerned. However, the 9.5.3.4 Intergranular Fracture phase responsible for fracture of an alloy may be detected by The microscopic fracture mechanism that perhaps can be observing the shape of the cleavage facet and comparing it most easily identified is the intergranular or intercrystalline with the morphology of various phases in that alloy. Fur- fracture which involves the propagation of crack along the thermore, in materials that experience a transition of grain boundary. The three-dimensional shape of the grains is microscopic fracture mechanism from MVC to cleavage often revealed by the morphology of the fracture surface. fracture, the presence of the cleavage mechanism may be The intergranular fracture exhibits a much smoother surface related to a general set of external variables. For example, in generally without any cleavage steps. This fracture usually most mild steel alloys (in which the above transition of shows little necking and often is associated with little total fracture mechanism occurs), the observation of cleavage elongation of the specimen. The intergranular fracture thus indicates that those materials were exposed to some com- results in a brittle fracture. The appearance of the fracture binations of a set of external variables like high strain rate, surface predicts that the energy absorbed in an intergranular low temperature and/or a high triaxial tensile stress field as brittle fracture is much lower than that in a transgranular produced in the presence of a notch. brittle cleavage fracture. The intergranular fracture is caused by nucleation and coalescence of microvoids at inclusions or 9.5.3.3 Quasi-cleavage Fracture second-phase particles located along the grain boundaries. Quasi-cleavage fracture is mainly observed in the fracture of Mostly, brittle fractures occur in a transgranular manner, but quenched and tempered steel at low temperature and differs there are a variety of situations narrated below, which can from cleavage fracture in some aspects. The quasi-cleavage result in the occurrence of the brittle intergranular fractures: facets on the fracture surface do not exhibit truly cleavage Fig. 9.11 Schematically (a) (b) showing matching of steps in cleavage and that of ridges in quasi-cleavage, on two-halves of a fracture surface Cleavage step Quasi-cleavage ridge

9.5 Characteristic Features of Fracture Process 387 (1) When a brittle phase precipitates or a film of brittle area of crack requires an expenditure of energy to overcome constituent exists at the grain boundaries, as found in the binding force of atoms; i.e., an increase in surface energy molybdenum alloys containing oxygen, nitrogen, car- is required. Since the propagation of a crack must be bon or sensitized austenitic stainless steel. accompanied by a reduction in system energy, the increase in surface energy (as it increases the system energy) acts to (2) When the impurity elements segregate at the grain resist crack extension. This may be overcome if the addi- boundaries causing to lower the surface energy suffi- tional surface energy is supplied by the decrease in elastic ciently. This leads to decohesion between adjacent stored strain energy. The elastic stored strain energy is grains; for example, the embrittlement is produced by released as the crack advances, and thus, it acts as a driving segregation of embrittling elements like Sb, P, Sn or force for crack propagation. This release occurs because the As, at the grain boundaries during the temper embrit- elastic strain cannot be continuous across the cracked region tlement of alloy steels, addition of antimony to copper and no elastic energy can be stored there. The criterion for and oxygen to iron. the propagation of a pre-existing crack under an applied constant tensile stress postulated by Griffith is: (3) When decohesion between adjacent grains occurs in association with an aggressive atmosphere like hydro- • A pre-existing crack will propagate if the decrease in gen gas and liquid metals, resulting respectively, in elastic stored strain energy is at least equal to the hydrogen embrittlement and liquid metal embrittlement. additional surface energy necessary to form the new crack surface. (4) When chemical dissolution along grain boundaries occurs leading to stress corrosion cracking. The magnitude of the applied tensile stress required to just spread a certain sized crack into a brittle fracture can be (5) When a material possesses less than five numbers of determined by using the above Griffith criterion. Consider an independent slip systems, grain-boundary separation elliptical shaped sharp crack in an infinitely wide thin plate may occur since continuity between adjacent grains of thickness t, where the crack runs from the front to the cannot be maintained during plastic deformation. back face, i.e. along the thickness direction of the plate as shown in Fig. 9.12, and the plate is considered to be thin for (6) When there are grain-boundary cracking and cavity treating the problem under plane stress condition. The effect formations at high temperatures associated with of an internal crack of certain size on the fracture behaviour stress-rupture conditions. is the same as that of a surface crack of length equal to half of the length of the internal crack. Assume that the length of (7) During low-cycle fatigue performed at high temperature the internal crack is 2a while that of the surface crack is a; in air, there is a transition from transgranular to brittle both of which are oriented normal to the applied tensile intergranular fracture with decrease in frequency of stress. When the tensile stress ra is applied to the plate in its cycling because of gain-boundary oxidation. longitudinal direction, the crack tends to grow in the width direction of the plate. Since two surfaces are created when 9.6 Griffith Theory of Brittle Fracture the crack propagates, the total surface energy term Us is the product of total surface area, which is 2 2a t for internal A. Griffith (1920) was the first person to explain the dis- crack or 2 a t for surface crack and the specific surface crepancy between the theoretical cohesive strength and the energy cs; whose unit is energy/unit area. So, Us is given by observed fracture strength of a brittle material. The theory proposed by Griffith is applicable only to a perfectly brittle Us ¼ 4at cs ðfor internal crackÞ ð9:22Þ elastic solid, such as glass. According to him, a brittle or 2at cs ðfor surface crackÞ material contains a lot of fine and sharp cracks that can produce such stress concentration that the local stress level at The elastic strain energy per unit volume of the material is the tip of the crack is raised to reach the theoretical cohesive given by the area under the linear elastic stress–strain curve, strength, while the applied tensile stress remains at a much i.e. rae=2; where ra and e are, respectively, the applied lower level than the theoretical value. It has been mentioned longitudinal stress and the resulting longitudinal strain in the earlier in Sect. 9.2 that there are several ways by which elastic range. Substituting for in terems of the Young’s cracks or microcracks are created in a material. So, the modulus, E; the elastic strain energy per unit volume, ðUEÞV; critical step usually is the determination of stress required to for a thin plate, i.e. under plane stress condition, becomes propagate a pre-existing crack or microcrack in a brittle material to cause its complete fracture and the criterion proposed by Griffith was based on this. When a pre-existing crack grows to increase its dimension, the increase in surface

388 9 Fracture Stress, σa Stress, σa πa 2a πa 2 a 2 Crack Centre line a t Stress, σa Stress, σa Fig. 9.12 Illustration of Griffith criterion, showing that a plate of Fig. 9.13 Model for approximate calculation of release of strain thickness t contains an internal crack of length 2a and a surface crack of energy length a; both of which are oriented normal to the applied tensile stress ra  V ¼ 4  1  a  pa  t ðfor internal crackÞ 2  ra2 1 ðUEÞV¼ 2E ð9:23aÞ or 2  2  a  pa  t ðfor surface crackÞ The surrounding regions of the internal as well as the ¼ p2a2t ðfor internal crackÞ or pa2t ðfor surface crackÞ surface crack are free from stresses where there are no lines of force as shown in Fig. 9.3, while the remaining regions ð9:23bÞ continue to experience the applied stress ra: Suppose the stress-free (also strain-free) regions above and below the When the crack is formed, the released total elastic strain internal as well as the surface crack are roughly triangular in shape (Knott 1973) with a width of each triangle equal energy UE is obtained by multiplying ðUEÞV from (9.23a) to a; as illustrated for an internal crack of semilength a in with V from (9.23b) as follows: Fig. 9.13. In agreement with Griffith accurate method of calculation for the elastic strain energy release, the height UE ¼ À ra2  p2a2t ðfor internal crackÞ of each stress-free triangular region is considered to be pa; 2E as shown in Fig. 9.13. Since the number of strain-free triangular regions in internal crack is four and that in or À r2a  pa2t ðfor surface crackÞ surface crack is two, so the volume V of this strain-free 2E zone will be ¼ À pra2a2t ðfor internal crackÞ E or À pra2a2t ðfor surface crackÞ ð9:24Þ 2E

9.6 Griffith Theory of Brittle Fracture 389 A negative sign is used in (9.24) to indicate that total Fig. 9.14, the critical crack sizes for applied tensile stresses ra1 and ra2 are, respectively, a1à and a2Ã; where a1Ã\\a2Ã; since elastic strain energy UE is decreased when the crack grows. ra1 [ ra2 : Hence, crack size of aÃ1 will not propagate at an Hence, the total change in the potential energy, DU; due to applied tensile stress of ra2 due to increase in DU with crack the formation of the crack is given by extension, whereas crack size of a2à will propagate sponta- neously at an applied tensile stress of ra1 due to decrease in DU ¼ U À U0 ¼ Us þ UE DU with crack extension. ¼ 4atcs À pra2a2t ðfor internal crackÞ ð9:25Þ According to Griffith theory, if the increase in Us is bal- E anced by the decrease in UE; the crack will grow under a constant applied tensile stress ra; which then becomes frac- or 2atcs À pra2a2t ðfor surface crackÞ ture strength rf of the material. Hence, the critical condition 2E where for fracture is achieved when an incremental increase in the U potential energy of the body with crack; crack size causes no change in the total potential energy DU U0 potential energy of the body without crack. of the system, i.e. when dðDUÞ=da ¼ 0; or dU=da ¼ 0; since The surface energy Us (crack retarding force) varies lin- early (9.22), and the elastic strain energy UE (crack driving U0 does not vary with the crack length. Applying this con- force) varies quadratically (9.24) with the crack size dition to (9.25), and remembering that when fracture occurs, a. Hence, at a given stress level, spontaneous propagation of the applied stress ra ¼ rf (the fracture strength), we get cracks below a critical size will not occur because Us dominates UE leading to an increase in DU (9.25) with crack pa2tr2f ! extension. On the other hand, cracks above a critical size E propagate spontaneously as UE [ Us and DU deceases. The d 4acst À ¼ 0 ðfor internal crackÞ; or, above can be illustrated with reference to Fig. 9.14 that da shows a plot of DU as a function of a; for two different ! applied stresses ðraÞ: DU passes through a maximum at a pa2trf2 critical value of a equal to aÃ; beyond which the crack d 2acst À 2E ¼ 0 ðfor surface crackÞ: propagates under an applied constant tensile stress because da there is a reduction in the system energy DU: From Fig. 9.14, it can be seen that the greater the value of ra; the Or; 4cst À 2patrf2 ¼ 0 ðfor internal crackÞ smaller will be the critical value of a; i.e. aÃ: The critical E value aà can be found by setting dðDUÞ=da ¼ 0: In or; 2cst À patr2f ¼ 0 ðfor surface crackÞ: E Since t ¼6 0; ) 2cs ¼ parf2 ðfor internal and surface cracksÞ ð9:26Þ E Equation (9.26) represents the equilibrium condition. σa1 > σ a2 From (9.26), we get the fracture strength, rf ; for a thin plate, i.e. under plane stress condition (biaxial stress condition) as ΔU2* ΔU 2Ecs rffiffiffiffiffiffiffiffiffi pa 2Ecs ΔU1* σ a2 rf2 ¼ ; or, rf ¼ pa ð9:27Þ σ a1 a* Equation (9.27) is applicable to the internal as well as the 1 a surface crack in plane stress condition. The equation indi- a* cates that the fracture strength in a brittle material is inver- 2 sely proportional to the square root of the size of the microcrack. Thus, the fracture strength will become double Fig. 9.14 Total change in the potential energy, DU; as a function of if the crack length is reduced by a factor of four. the crack semilength, a; for two applied tensile stresses, ra1 and ra2 ; If we consider a plate which is thick in comparison with where ra1 [ ra2 : the crack length, then the plate is subjected to plane strain condition, i.e. a condition of triaxial state of stress where the strain is suppressed in one direction, say e3 ¼ 0: Under plane strain condition, we know from (1.39a) that e1 ¼ ½ð1 À m2Þr1 À mð1 þ mÞr2Š=E: Since at the free surface of the crack, lateral contraction can occur without any restraint from the empty crack, so r2 ¼ 0 (see Fig. 9.7). Hence, in

390 9 Fracture plane strain, the tensile strain e1 ¼ e; associated with an The application of surface active agents or chemicals may applied tensile stress r1 ¼ ra; is e ¼ ½rað1 À m2ފ=E; where reduce the surface energy term cs and thus the fracture m ¼ Poisson’s ratio. Therefore, the total elastic strain energy strength of a material. Hence, the fracture of a brittle solid is UE released under plane strain is as follows: sensitive to its surface condition, and this sensitiveness is UE ¼ Àrae  p2a2t called the Joffe effect (Joffe 1928). The influence of surface 2 energy on the fracture strength can be used for the benefit of mankind. For example, rock drilling will become easier if ¼ À r2að1 À m2Þ Â p2a2t ðfor internal crackÞ 2E surface active agents are used. For ice, its bending fracture strength in air is about 1 MPa, but this strength decreases to Or, UE ¼ Àrae  pa2t 2 0.5 MPa when methyl chloride is sprayed on ice to reduce its surface energy. The surface energy of copper is 1.8 J/m2 ¼ À ra2ð1 À m2Þ Â pa2t ðfor surface crackÞ: which is reduced to 1 J/m2 by addition of 0.5% antimony to 2E copper. ) UE ¼ À pra2a2tð1 À m2Þ ðfor internal crackÞ E ð9:28Þ 9.6.1 Applicability of Griffith Theory or À pr2aa2tð1 À m2Þ ðfor surface crackÞ 2E The fracture strength of a perfectly brittle material like glass In analogy with the earlier derivation for plane stress is satisfactorily predicted by the Griffith equation. But the condition, the fracture strength for a brittle material under Griffith relation is not applicable to metals because when the plane strain condition using (9.28) is given by observed fracture strength of metal like zinc crystal is sub- sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi stituted in the Griffith equation it results in a critical crack rf ¼ 2Ecs ð9:29Þ size of several millimetres, which may exceed the smaller pað1 À m2Þ dimension of the specimen. Equation (9.29) is applicable to the internal as well as the Equation (9.16) shows that the crack-tip radius qt is surface crack in plane strain condition. Since Poisson’s ratio included in the equation for fracture strength derived from m is 0.25 for a perfectly isotropic elastic material and close to 0.33 for most materials, the fracture strength for a given the stress concentration point of view, while (9.27) and brittle material subjected to plane stress condition given by (9.27) does not appear to differ greatly from that under plane (9.29) derived from the Griffith theory do not involve qt strain condition given by (9.29). term, although the crack is assumed to be very sharp having It is to be noted that in a given material, surface cracks of a very small crack-tip radius. To evaluate qt in the Griffith length a are more effective than internal cracks of length 2a: relation, (9.27) is compared with the following (9.30), which If grips holding a brittle material do not extend up to the surface, then surface cracks are ineffective due to the is obtained by rearranging (9.16) as follows: absence of stresses on the surface and the material shows a higher strength. In a brittle material having different sizes of rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cracks, the longest crack oriented normal to the stress axis is 2Ecs pqt % 2Ecs qt the most effective and therefore propagates first to produce rf ¼ pa 8r0 pa 3r0 ð9:30Þ fracture, as the applied tensile stress is increased. If one of the broken pieces from the first test is tested for the second When qt ¼ ð8r0Þ=p % 3r0; (9.30) reduces to the Griffith time, a higher strength is usually observed because of the equation for plane-stress fracture strength given by (9.27). elimination of the most effective crack in the first test. Even When the crack-tip radius becomes infinitely small, i.e. after the Griffith condition is satisfied, a crack may not propagate if the stress concentration at the crack tip is not approaches zero, it will not be reasonable to expect the sufficient to break the interatomic bonds because the inter- atomic bonds at the crack tip must be broken for the prop- fracture strength of the material to approach zero value. So, agation of the crack and the rupture of bonds would be when the value of qt decreases below ð8r0Þ=p % 3r0; the possible only if the stress concentration at the crack tip is value of fracture strength rf is believed to be not less than equal to the theoretical cohesive strength of the material. that given by the Griffith equation, which is expected to give the minimum value for the plane-stress fracture strength of a brittle material containing the sharpest possible fine crack. Hence, it can be concluded that in plane stress condition, the fracture strength for a brittle material is given by the Griffith (9.27) when qt ð8r0Þ=p or aproximately 3r0; while that is given by the (9.16) derived from the stress concentration point of view when qt [ ð8r0Þ=p or aproximately [ 3r0:

9.6 Griffith Theory of Brittle Fracture 391 9.6.2 Modification of Griffith Theory Equations (9.32a) and (9.32b) are simplified forms of Orowan equation for fracture strength. Now, observing the Even when metals fail in a completely brittle manner, some similarity between (9.30) and (9.32a) and equating them for amount of plastic deformation precedes this brittle fracture. This has been confirmed from studies of fracture surfaces by the fracture strength rf , a relation between the crack-tip X-ray diffraction (Klier 1951; Chang 1955; Felbeck and radius qt; and the plastic deformation energy cp; can be Orowan 1955) and by metallographic studies of fracture. developed as follows: Even a small amount of plastic deformation occurring at the tip of crack prior to fracture is expected to blunt the crack tip rffiffiffiffiffiffi ¼ rfficffiffipffi; or qt / cp ð9:33Þ and increase the radius qt; which makes the fracture strength qt of a brittle metal greater than that of a perfectly brittle elastic material. Thus, Griffith equation for the fracture strength is 3r0 cs not valid for brittle metals. One way to get the fracture strength of a brittle metal is to use (9.16) derived from the Equation (9.33) shows that the higher the plastic defor- stress concentration point of view. The other way to deter- mation energy cp; i.e. the higher the amount of plastic mine the fracture strength of a brittle metal is to use the deformation ahead of the crack tip prior to fracture, the more following (9.31), suggested by Orowan (1950). is the blunting of the crack tip resulting in a higher crack-tip radius qt: The limitation of Orowan equation is that it is difficult to measure the plastic deformation energy cp: 9.6.2.1 Orowan Relation 9.6.2.2 Irwin Approach Considering the plastic deformation in the fracture process In the equatioÀns for fÁracture, Griffith used the term cs and of brittle metals, Orowan suggested to modify the Griffith Orowan used cs þ cp ; both of which require an increase in relation and include the energy of plastic deformation in energy; i.e., both are energy sink terms. Instead, Irwin (9.27) and (9.29) derived from the Griffith theory, respec- (1949) and Irwin et al. (1958) considered to use the energy tively, for plane stress and strain conditions. The Griffith source term for the fracture strength of a brittle material equations in plane stress and strain conditions modified by which undergoes a small amount of plastic deformation prior Orowan are given below, respectively, by (9.31a) and to its fracture. The energy source term is the release of the (9.31b). elastic strain energy per unit crack surface-area increment, whose unit is the energy per unit area, i.e. J/m2 or, N/m. This rf ¼ sffiffiffiffiffiffiÀffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÁffiffi ¼ sffi2ffiffiEffiffifficffiffisffiffiffiffiffiffi1ffiffiffiþffiffiffiffifficffiffipffiffiffiffiffi ð9:31aÞ is called elastic strain energy release rate and denoted by G. 2E cs þ cp pa cs ð9:31bÞ This elastic strain energy release rate G is also referred to as pa the crack driving force or the crack extension force and easy to measure unlike the term cp: Irwin showed that sffiffiffiffiffiffiÀffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÁffiffi sffipffiffiaffiffiffið2ffi1ffiffiEffiffiÀfficffiffisffiffimffiffi2ffiffiÞffiffiffiffiffiffi1ffiffiffiþffiffiffiffifficfficffipffisffiffiffiffi 2E cs þ cp In plane stress condition, pað1 À m2Þ rf ¼ ¼ G ¼ par2a E ð9:34aÞ And in plane strain condition, where cp ¼ plastic deformation energy, whose unit is G ¼ par2að1 À m2Þ ð9:34bÞ energy/unit area. The estimation of cp shows a variation E from about 102–103 J/m2, whereas cs varies from about 1– 2 J/m2. Since cp ) cs; i:e: cp=cs ) 1; neglecting 1 from The above equations can be explained in the following way. According to definition, G can be expressed as G ¼ both equations of (9.31), we get ÀdUE=dA; where dUE is the change in elastic strain energy which is preceded by a negative sign showing a decrease in In plane stress condition, energy and dA is the increment in crack surface area, in which rf ¼ sffi2ffiffiEffiffifficffiffisffiffiffiffiffifficffiffipffiffiffiffiffi ¼ rffiffiffiffiffiffiffiffiffi ð9:32aÞ dA ¼ 2t da for internal crack or dA ¼ t da for surface crack, pa cs 2Ecp t ¼ thickness of specimen having through-thickness crack pa and da ¼ increment in crack length in the width direction of specimen subjected to loading normal to the crack plane in In plane strain condition, the longitudinal direction. Considering the expression for UE rf ¼ sffiffiffiffiffiffi2ffiffiffiEffiffifficffiffisffiffiffiffiffiffiffiffiffiffiffifficffiffipffiffiffiffiffi ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð9:32bÞ given by (9.24), G in plane stress condition is given by pað1 À m2Þ cs 2Ecp pað1 À m2Þ

392 9 Fracture G ¼ À dUE ¼ À dUE ðfor internal crackÞ energy. The necessary additional surface energy is supplied dA 2tda by the release of elastic strain energy, ÀdUE; and the work done by the external body force, Pdd (Irwin and Kies 1954). or, À dUE ðfor surface crackÞ The work done Pdd consumes some energy and is a positive tda energy term, whereas dUE is an energy source term and so it is preceded by a negative sign. Hence, the elastic strain ¼ pra22at ¼ par2a energy release rate for a body of thickness t is expressed by 2tE E (9.37) ðin plane stress; for surface and internal cracksÞ: Similarly, considering the expression for UE given by G ¼ P dd À dUE ð9:37Þ (9.28), G in plane strain condition can be obtained, which is tda tda same as that given by (9.34b). When this crack driving force G reaches a critical value Figure 9.15a shows that an axial tensile load P is applied in the elastic range through pins to a test specimen of Gc, the crack will extend in an unstable manner leading to thickness t having a single notch at its edge. The sharpest possible crack is introduced in the specimen by fatigue fracture of the material and ra becomes rf : So, (9.34) cycling at the tip of the notch produced by machining. The initial crack length a in the specimen is taken as the sum- changes to rffiffiffiffiffiffiffiffi mation of the depth of machined notch and the length of fatigue crack. When the specimen is elastically loaded, the In plane stress condition; rf ¼ EGc ð9:35aÞ displacement of this crack (also known as the load dis- pa placement), d; is measured with a clip gage placed at the mouth of the crack as a function of the load P; and load sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P versus displacement d curve is determined for this crack length a. The amount of stored elastic strain energy UE is the And in plane strain condition; rf ¼ EGc area under this linear load versus displacement curve and pað1 À m2Þ given by ð9:35bÞ Note that a directly measurable term, Gc, is included in the Irwin’s fracture stress equation, whereas Orowan’s fracture stress equation includes the term cp; which is diffi- cult to measure. Equations (9.35) are among the most important relations P2 M in the field of fracture mechanics. The resistance to unstable UE ¼ 1 Pd ¼ 1 ð9:38Þ 2 2 propagation of crack leading to fracture of a material having pre-existing cracks or discontinuities is called the fracture where toughness of that material. Hence, the critical value of elastic strain energy release rate G that makes the crack to propagate M ¼ P=d body stiffness for crack length a: as a fracture is called the fracture toughness, Gc; and is When the existing crack grows by an amount da so that considered as a material parameter. By comparing (9.35a) the crack length becomes a0; where a0 ¼ a þ da; the deter- with (9.31a) or (9.35b) with (9.31b), we obtain mination of the critical elastic strain energy release rate Gc ÀÁ ð9:36Þ can be carried out under the following conditions: Gc ¼ 2 cs þ cp 9.7 Elastic Strain Energy Release Rate (1) Fixed grip condition: In this case, the test specimen is In this section, we will consider the significance of elastic rigidly gripped so that an increment in crack growth by strain energy release rate G and the experimental measure- ment of its critical value Gc using single-edge sharply not- an amount da causes the load to drop from P1 to say ched test specimen. Let us consider that a tensile load, P; P2; with consequent decrease in the body stiffness from acting on a precracked body of thickness t produces a load M1 to say M2; but there is no displacement of load, i.e. displacement equal to dd and extends the crack by an d1 ¼ d2 ðsayÞ: In such case, the ratio of load to body amount da: The extension of crack creates new surface stiffness will remain constant, since from Fig. 9.15b requiring additional energy but decreases the elastic strain d1 ¼ d2 ¼ P1 ¼ P2 ð9:39Þ M1 M2

9.7 Elastic Strain Energy Release Rate 393 P1 P1′ = P1 Fig. 9.15 a A test specimen of (a) P (b) M1 P2 M2 thickness t with notch at its edge dδ P da for determination of crack extension force G: b Load–crack displacement behaviour of cracked specimen for case where crack length increases by an amount da: OP2 corresponds to fixed grip case, while 0P01 corresponds to fixed load condition a t δ 1 δ 2 (Fixed load) 0 dδ δ P δ1 = δ2 (Fixed grip) Since from (9.39) P ¼ constant, P01 ¼ constant, so d1 ¼ P1=M1 and d2 ¼ P10 =M2 ¼ M P1=M2: Now, the crack driving force for specimen of thickness t under constant-load condition, i.e. for P ¼  @ð1=MÞ @P 1 P @a constant, can be obtained by substituting UE from (9.38) ) @a M þ ¼ 0; into (9.37) and considering d ¼ P=M; as follows: 1 @P @ð1=MÞ ð9:40Þ M @a @a Or, ¼ ÀP  Hence, the crack driving force, Gd for specimen of GP ¼ P @d À @ 1 thickness t under fixed grip condition, i.e. for d ¼ constant, t@a t@a Pd  @d 2 1 @a 1 @ P can be obtained by substituting @d=@a ¼ 0 into (9.37) and ¼ 2t P ¼ 2t P @a M ð9:43Þ considering (9.38) for UE as follows: 1 P2 @ð1=MÞ 2t @a  @ P2! ¼ @UE ¼À1 @a M! d Gd ¼ À t@a d 2t Equations (9.42) and (9.43) show that Gd ¼ GP: There- fore, the elastic strain energy release rate is the same for both ¼À1 2P @P @ð1=MÞ ð9:41Þ conditions of fixed grip and constant load; i.e., G is inde- 2t M @a @a pendent of the type of load application. Further, it is þ P2 observed from (9.42) and (9.43) that the elastic strain energy release rate for a body of unit thickness is a function of load By substituting the result of (9.40) into (9.41), we get the and the slope of the compliance versus crack length curve. When the crack grows in an unstable manner leading to following (9.42) for Gd for specimen with thickness = t: fracture of the material, the critical elastic strain energy release rate or the fracture toughness Gc for a body having a 1 @ð1=MÞ @ð1=MÞ ! thickness = t and that for a body of unit thickness are 2t @a @a respectively, given by Gd ¼ À À2P2 þ P2 1 @ð1=MÞ ð9:42Þ 2t @a ¼ P2 where 1=M ¼ compliance of the cracked body ¼ d=P: The Gc ¼ Pf2 @ð1=MÞ ð9:44aÞ compliance increases with increase of crack length and vice 2t @a versa. (2) Constant-load condition: In this case, an increment in Gc ¼ Pf2 @ð1=MÞ ð9:44bÞ 2 @a crack growth by an amount da results in an increase in d by an amount dd; say from d1 to d2; with consequent where Pf ¼ load at which fracture of the material takes drop in the body stiffness from M1 to M2 (say), as shown place. in Fig. 9.15b. Since the load is kept fixed, i.e. P1 ¼

394 9 Fracture Equations (9.44) are applicable to brittle materials cap- plate having a through-thickness internal sharp crack is able of plastic deformation, but the degree of plasticity at the loaded with uniaxial tensile stress ra; applied normal to the crack tip prior to the fracture must be very small. plane of the crack, the elastic stress distribution at the crack tip in terms of the notation shown in Fig. 9.18 is given by For cracked bodies containing cracks of increasing the following (9.45): lengths, with a1\\a2\\a3\\a4; and so on, the load–dis- placement plot can be made experimentally in a manner rffiffiffiffi  1 þ sin h sin 3h ! similar to that described above. For fixed grip case, i.e. for a cos h 22 fixed displacement of load d; with increasing the crack length ry ¼ ra 2r 2 ð9:45aÞ the loads will drop that will make P1 [ P2 [ P3 [ P4; and so on, as shown in Fig. 9.16a. Similarly, for constant-load rffiffiffiffi  ! case where P1 ¼ constant, with increasing the crack length a h 1 À sin h sin 3h the load displacements d will increase, which will make rx ¼ ra cos 22 ð9:45bÞ d1\\d2\\d3\\d4; and so on, as shown in Fig. 9.16b. The 2r 2 slope of the linear plot for load versus displacement is the stiffness of the body, and this stiffness decreases with rffiffiffiffi ! increasing the crack length for fixed grip as well as fixed load a h h 3h conditions, which means M1 [ M2 [ M3 [ M4; and so on sxy ¼ ra sin cos cos ð9:45cÞ for both cases. Now, measure these slopes of these linear plots for various crack sizes and plot the reciprocal of stiff- 2r 2 2 2 ness, i.e. compliance versus crack length curve. Since spec- imen compliance 1=M ¼ d=P; the variation of the values of where 1=M as a function of the crack length a takes the form given in Fig. 9.16c. Once the compliance as a function of crack ra applied stress calculated on the basis of the gross length has been established for a given specimen configura- cross-sectional area of a specimen without taking into tion, the critical elastic strain energy release rate or the account the effect of crack, i.e. fracture toughness Gc of the specimen can be determined from (9.44) by noting the load at fracture Pf; provided the ra ¼ ðwidth  applied load specimen ¼ P ; degree of plasticity at the tip of crack is kept to a minimum. thicknessÞ of w t 9.8 Stress Intensity Factor ry normal stress acting in the y-direction; rx normal stress acting in the x-direction; Fracture mechanics is concerned with the study of the sxy shear stress lying on a plane normal to the x-direction fracture of flawed components that may be based on a stress analysis of the elastic stress distribution near a crack tip and acting in the y-direction; located in a linear elastic body using the concepts of elastic theory. Although fracture mechanics can mainly be divided a half of the length of an internal crack; into three parts—(1) linear elastic fracture mechanics or r arbitrary radial distance from the crack tip; and LEFM, (2) elastic–plastic fracture mechanics or EPFM h arbitrary angle of orientation ahead of the crack. (3) gross yielding fracture mechanics or GYFM, discussions in the text will be limited to LEFM and consideration of the The crack-tip stresses given by (9.45) are valid if rest two divisions is beyond the scope of the text. Generally, LEFM approach works well for high-strength materials, i.e. a [ r [ qt; where qt is the crack-tip radius. For any distance for materials whose yield strength [ E=150: where E is the directly ahead of the crack tip, i.e. at an orientation angle of modulus of elasticity, but LEFM is less universally appli- cable for low-strength structural materials. h ¼ 0; (9.45) are reduced to Irwin modified the Airy stress function approach used by rffiffiffiffi Westergaard (1939) and published solutions (Irwin 1958) for a stress distribution at the crack tip for three major modes of ry ¼ rx ¼ ra 2r and sxy ¼ 0 loading, as shown in Fig. 9.17. If an infinitely wide thin From (9.45), Irwin pointed out that pthffieffi stress fields in the vicinity of a crack depend on ðra  aÞ: He defined this stress field parameter as the stress intensity factor K: For a sharp elastic through-thickness small central crack of length 2a in an infinitely wide plate loaded with uniaxial tension, K is given by pffiffiffiffiffi ð9:46Þ K ¼ ra pa Note tphaffiffitffiffiffitffihffiffieffiffiffiffidffiffiffiiffimffiffiffiffieffiffinffiffisffiffiiffioffiffiffinffiffiffiffioffi f K is pthffieffiffiffiunit of stress multi- plied by the unit of length i.e. Pa m; or N mÀ3=2: Using K defined by (9.46), the crack-tip stresses from (9.45) can be rewritten in terms of K as

9.8 Stress Intensity Factor 395 Fig. 9.16 Method of (a) establishing compliance versus crack length relation for a given Crack length, a1 < a2 < a3 < a4 specimen configuration. Effect of P1 crack length on load–crack displacement behaviour: a fixed P2 (c) grip case and b fixed load case. c Compliance as a function of a1 M1 crack length P M2 P3 a2 M3 P4 1 a3 M a4 M4 0 δ 1 = Constant a1 a2 a3 a4 a δ Compliance–crack length curve Fixed grip (b) Crack length, a1 < a2 < a3 < a4 a3 a4 a1 a2 P1 P M1 M2 0 M3 δ M4 δ1 δ2 δ3 δ4 Fixed load y x Fig. 9.17 Different types of y y crack surface displacement, x x z associated with three major modes of loading z z Mode I Mode II Mode III

396 9 Fracture Stress, σ a (i) Y ¼ 1; for a sharp elastic through-thickness small central crack of length 2a in an infinitely wide plate σy loaded with uniaxial tension. (ii) Y ¼ 1:1; for a sharp elastic through-thickness small τ xy surface crack with depth a from surface in an infi- y σx σx nitelyrwffiffiiffidffiffiffieffiffiffiffipffiffilffiaffiffitffiffieffi loaded with uniaxial tension. w pa; 2a τ yx (iii) Y ¼ pa tan w for a centrally located w r σz through-thickness crack of length 2a in a plate of θx σy ρt width w loaded with uniaxial tension, where w ¼ 6a: t (iv) Y ¼ 2 ; for an embedded circular flaw of radius a or a p semicircular surface flaw of radius a in an infinitely z wide plate loaded with uniaxial tension. Fig. 9.18 Distribution of elastic stresses in the vicinity of the tip of a crack (v) Y ¼ ð1:12Þ22 ; if semicircular surface flaw of radius p a lies along two free surfaces in an infinitely wide  plate loaded with uniaxial tension. ry ¼ pKffiffiffiffiffiffiffi cos h 1þ sin h 3h ð9:47aÞ (vi) The empirical expression rprffioffiffipffiffiffioffiffisffiffieffiffid by Feddersen 2pr 2 sin ð9:47bÞ sec pa; for a partial 22 ð9:47cÞ w (1967) reveals that Y ¼ h  pKffiffiffiffiffiffiffi 2 1 À sin h sin 3h through-thickness crack in a plate of width w loaded rx ¼ 2pr cos 22 with uniaxial tension, where a is the depth of pene-  tration of the crack through the component wall sxy ¼ pKffiffiffiffiffiffiffi h h 3h thickness t and width w ¼ 2t: 2pr sin cos cos 22 2 It is clear from (9.47) that these local stresses at the root of At this point, it is worthy to mention that information provided by the stress intensity factor K is more than that by the crack could rise to excessively high levels as r approaches the stress concentration factor, Kt: Because Kt contains only the geometrical variables related to crack, i.e. the crack zero, but this high rise does not happen because of the onset length and the crack-tip radius, whereas K includes both the stress level and the crack geometry having distinct appear- of plastic deformation at the crack tip, as discussed in ance of the crack length with assumption of very sharp crack-tip radius. Sect. 9.4. The stress distribution around any crack or flaw can be conveniently described by the stress intensity factor K: If the values of K are identical for different cracks or flaws of various geometry, then the magnitudes of the stress field around each of the cracks or flaws are exactly alike. Irwin showed that K is a function of applied stress and crack length, 9.8.1 Different Crack Surface Displacements where the functionality depends on the manner of load Figure 9.17 shows that there are different types of crack application and configurations of cracked components. Many surface displacement, which are associated with three major such functions for various configurations of cracked com- modes of loading. They are as follows: ponents and different types of loading have been determined (1) Mode I or crack opening mode or tensile mode, where a tensile stress is applied in a direction perpendicular to with the theory of elasticity and can be obtained from the the surfaces of the crack so that the crack faces move directly apart. This type of loading is experienced by the literatures of fracture mechanics (Paris and Sih 1965; Tada majority of engineering cracked components and that’s why it is the usual mode for fracture toughness testing. et al. 1973; Sih 1973; Murakapmffiiffi 1987). However, there may be situations where K / ð1= aÞ or K is independent of a, (2) Mode II or sliding or forward shear or in-plane shear mode, where a shear stress is applied in a direction per- and these have also been reported in these literatures. pendicular to the leading edge of the crack but in the surface of the crack, and as a result, the crack faces slide Generally, the stress intensity factor is expressed by over one another in a direction normal to the leading edge pffiffiffiffiffi ð9:48Þ K ¼ Yra pa where Y is a parameter that depends on the types of loading and the geometry of crack and specimen configurations. For example, the different values of Y are given below (Irwin et al. 1967; Tada et al. 1973; Dieter 1988; Hertzberg 1989).

9.8 Stress Intensity Factor 397 of the crack. This type of loading is observed less fre- applied load and the length of crack in the component, but quently and its importance in the field of engineering is the unique value of the stress intensity factor required to little, although a mixed Mode I–II loading may be found. cause fracture is called the critical stress intensity factor or (3) Mode III or tearing or parallel shear or antiplane shear fracture toughness Kc: Therefore, from the above analogy, it mode, where a shear stress is applied in a direction may be said that stress is to strength as the stress intensity parallel to the leading edge of the crack so that the crack factor is to fracture toughness. A critical value of the stress faces move relative to one another in a direction parallel intensity factor determined for the Mode I type of loading to the leading edge of the crack. This type of loading under a plane strain state of stress is designated as KIc and involves pure shear and is observed in a notched round known as plane-strain fracture toughness. The plane-strain bar loaded in torsion. fracture toughness KIc is a measure of the inherent resistance of the material to brittle fracture, i.e. unstable rapid fracture 9.8.2 Relationship Between Energy Release Rate in the presence of a crack-like defect. It is a basic material and Stress Intensity Factor property just like the yield strength of material. In dealing with fracture mechanics, the stress intensity factor From Chap. 1, we already know that the tensile beha- K is preferred to the elastic strain energy release rate G viour of a material depends on the temperature and strain rate of testing. Since the values of plane-strain fracture because analytical determination of K is more flexible, toughness KIc drop often sharply with increase in the although G has a more direct physical significance to the strength of a material (Hertzberg 1989) and the strength of a material usually increases as the deformation temperature is process of fracture. We can relate these two parameters as decreased or the strain rate is increased, it is expected that the values of plane-strain fracture toughness KIc for a given follows: material may decrease with decreasing test temperature and increasing strain rate. This occurs particularly for structural From (9.34a) and (9.46), steel (Hertzberg 1989). Further, KIc for a given alloy strongly depends on a number of metallurgical variables, pffiffiffiffiffiffi pffiffiffiffiffi ð9:49aÞ such as melting practice, heat treatment, texture, inclusions, EG ¼ ra pa ¼ K; impurities. Experimental determination of KIc is performed by tests usually conducted at room temperature at low strain ) In plane stress condition, K2 ¼ GE rates of approximately 10À 5 sÀ1: Studies are now being conducted to establish an acceptable standard for dynamic From (9.34b) and (9.46), KIc level, called KId; at strain rates of about 10 s−1, which corresponds to impact loading conditions. rffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi EG ra pa 1 À m2 ¼ ¼ K; GE ð9:49bÞ À m2Þ ) In plane strain condition; K2 ¼ ð1 9.8.3 Fracture Toughness 9.9 Plastic Zone at Crack Tip Once we know the stress intensity factor K for a particular Whenever the stresses given by (9.47) exceed the yield configuration of cracked component under a definite type of strength of the material, the material will yield and a zone of loading, the maximum value of K required to cause fracture plasticity will be developed at the crack tip. This is illus- of this component is then possible to determine. The fracture trated in Fig. 9.19. To obtain an estimate of the size of this toughness of the material is defined as the critical value of plastic zone, let us consider the elastic stresses that exist at stress intensity factor, which makes the crack propagate to any distance directly ahead of the crack tip. The elastic stress fracture. The fracture toughness is denoted by Kc in the in the loading direction, i.e. in the y-direction at an orien- literature. We may say the relation between stress and tation angle of h ¼ 0, is given by strength is analogous to that between the stress intensity factor and fracture toughness, which is explained below. ry ¼ pKffiffiffiffiffiffiffi ð9:50Þ Levels of stress experienced by a member vary with the 2pr values of the applied load and the cross-sectional area of the member, but the stress required for permanent plastic Since the elastic stress ry increases with decreasing r; as deformation or for fracture has a unique value for a given shown in (9.50), the elastic stress ry given by (9.50) will material and is, respectively, called the yield strength or exceed the yield strength r0 at some distance r from the fracture strength. Similarly, the magnitudes of the stress crack tip. Let us assume that at a distance r\\rp from the intensity factor K at the crack tip depend on the values of the crack tip, ry [ r0; and at the elastic–plastic boundary, i.e. at

398 9 Fracture rpc ¼ 1 Kc2 ð9:53aÞ 2p r02 ð9:53bÞ K σy ÀÁ 1 KI2c σ0 σy = rpc P:Strain¼ 6p r02 2πr Modified stress distribution elastic + plastic 9.9.1 Effective Stress Intensity Factor r As a result of crack-tip plasticity, the stiffness of body becomes lower compared to the strictly elastic situation. rp Hence, the presence of the plastic zone makes the material behave as if the crack were longer than its actual size. Irwin Fig. 9.19 Size estimation of plastic zone at the crack tip. ‘Effective’ (1958) proposed that the effective crack length is the actual crack length is shown to be initial crack length plus the radius of the initial crack length a plus the radius of the plastic zone. plastic zone, rp Hence, the effective crack length will increase the applied stress intensity factor K; which may be termed as effective stress intensity factor and denoted by Keff: Now, on substi- tuting Keff for K into (9.51), the effective crack length, aeff ; in plane stress condition will be given by aeff ¼ a þ rp ¼ aþ 1 Ke2ff ð9:54Þ 2p r02 r ¼ rp; the elastic stress is truncated at the value of yield strength making ry ¼ r0; as shown in Fig. 9.19. As a first Similarly, using (9.52), the effective crack length, approximation, if the distance rp is assumed to be the size of ð9:55Þ the plastic zone, then ðaeff ÞP:Strain; in plane strain condition is given by ÀÁ ÀÁ 1 Ke2ff P:Strain r0 ¼ pffiKffiffiffiffiffiffiffiffi ; or; ¼ 1 K2 ð9:51Þ ðaeff ÞP:Strain¼ a þ rp P:Strain¼ aþ 6p r02 2prp 2p r02 rp Since the plastic-zone size is a function of the effective However, the diameter of plastic zone must be larger than stress intensity factor and the effective stress intensity factor rp given by (9.51). The shaded region in Fig. 9.19, where ry [ r0; represents the load-carrying capability of the is a function of the effective crack length that involves the material that must be compensated for by increasing the size of the plastic zone. More detailed analysis shows that the plastic-zone size, the value of the effective stress intensity diameter of plastic zone in plane stress condition is 2rp: Hence, the radius, rp; of plastic zone in plane stress con- factor must be determined by an iterative process. For a dition is given by (9.51). sharp elastic through-thickness small central crack of length Since the triaxial stress field in plane strain condition restricts the extent of plastic deformation, the size of plastic 2a in an infinitely wide plate loaded with uniaxial tension, zone is suppressed and becomes smaller than that given by where the stress intensity factor K is defined by (9.46), (9.51). Estimation (McClintock and Irwin 1965) shows that the radius of plastic zone in plane strain condition is iteration is not necessary and the effective stress intensity factor may be determined directly in plane stress condition using (9.54) and in plane strain condition using (9.55), as shown below. In plane stress, the effective stress intensity factor is: ÀÁ 1 K2 ð9:52Þ Keff ¼ pffiffiffiffiffiffiffiffiffi ¼ rasffipffiffiffiffiffiffiaffiffiffiþffiffiffiffiffi2ffi1ffiffipffiffiffiKffirffiffieffi202ffifffifffiffiffiffi; rp P:Strain¼ 6p r20 ra paeff When a material fractures under plane stress or plane Or; Ke2ff ¼ ra2 p12aþrra0r22a2K#re202¼ff ;ra2pa Or; \" À strainÀconÁdition, the respective critical radius of plastic zone rpc or rpc P:Strain at fracture can be obtained on substituting the Ke2ff 1 fracture toughness Kc or the plane-strain fracture toughness KIc for K, respectively, into (9.51) or (9.52), as given below:

9.9 Plastic Zone at Crack Tip 399 pffiffiffiffiffi causing plane stress condition to predominate and the 1rarpaa2#1=2 K maximum value of fracture toughness is exhibited by the ) Keff ¼ \" ¼ \" 2#1=2 ð9:56Þ material. It is to be noted that if the thickness of specimen is 1 1  further decreased below the value of t1; as shown in À À 1 ra Fig. 9.20, the fracture toughness would gradually decrease instead of increase because less volume of material would be 2 r0 2 r0 available in a thinner specimen for absorption of energy in the process of plastic deformation. Conversely, when the Similarly, in plane strain condition, the effective stress thickness of specimen is increased above t1, the fracture intensity factor is: toughness starts to drop sharply due to increase of plastic constraint. When the specimen becomes sufficiently thick, pffiffiffiffiffi say thickness t ¼ t2; the plastic constraint reaches the max- ra pa K imum level leading to the development of an absolute plane ðKeff ÞP:Strain¼ \" 1 ra2 #1=2 ¼ \" 1 ra2#1=2 strain state of stress with a high degree of triaxiality at the 1 1 crack tip and the fracture toughness drops to a value that À 6 r0 À 6 r0 may be one-third or less than that of the plane stress value. This plane-strain fracture toughness KIc does not drop fur- ð9:57Þ ther with increasing thickness of specimen, i.e. reaches the lowest limit of saturation with respect to the thickness The above (9.56) and (9.57) show that the effective stress increase of specimen and thus provides the conservative intensity factor is always greater than the applied K; but the values of toughness of engineering materials in any appli- difference between them increases with the ratio of the cation. It means that the value of KIc determined in the applied stress ra to the yield strength r0 of the material. laboratory for a given material with a specimen having the Under low applied stress conditions or for high-yield-strength minimum thickness of t2; as shown in Fig. 9.20, must be the materials, this difference will be very small; i.e., the effective same for an engineering component of the same material stress intensity factor will be a little higher than the applied K; with a thickness greater than t2: So, the basic difference and this occurs when the ratio of plastic-zone size to crack between Kc and KIc lies in the fact that Kc is dependent on length is very small. So, when the size of the plastic zone is both specimen thickness and metallurgical variables, while very small relative to the actual crack length, the plasticity KIc is independent of specimen thickness and depends only correction to the stress intensity factor is generally neglected on metallurgical variables, such as inclusions, impurities, in practice. On the other hand, under high applied stress melting practice, heat treatment, texture. The fracture conditions or for low-yield-strength materials, the effective stress intensity factor will be considerably greater than the applied level of stress intensity factor. This happens when the ratio of the size of plastic zone to the actual crack length becomes appreciably large, and in such cases, the applied stress intensity factor requires the plastic-zone correction. 9.10 Fracture Toughness: Plane Stress Versus Plane Strain The state of stress operating at the crack tip decides the size Kc of the plastic zone, as discussed in the previous section. As shown by (9.51) and (9.52), the radius of the plastic zone in KIc thin specimen subjected to plane stress condition is three times larger than that in thick specimen experiencing plane t2 t1 strain condition. Because when the thickness of specimen is 1 large in a direction parallel to the crack front, a large t through-thickness stress is induced that limits the plastic deformation in that direction. Since the fracture toughness of Fig. 9.20 Variation in fracture toughness, Kc; with thickness, t, of the material increases with the extent of plasticity prior to specimen fracture, i.e. with the size of the plastic zone at the crack tip and since the plastic-zone size increases with the decrease in the thickness of specimen, it follows that the fracture toughness Kc increases with the decrease in the thickness t of specimen, as shown in Fig. 9.20. Suppose when the thick- ness of specimen is t ¼ t1; the intensity of the constraining stresses acting at the crack tip reaches the minimum level

400 9 Fracture 100 Mixed-mode fracture Mixed mode Fracture stress % Flat fracture Kc Plane strain Plane stress Plane-strain fracture KIc Thickness, t 0 Fig. 9.21 Effect of specimen thickness, t, on fracture stress 0.1 1.0 rp /t toughness also changes with external variables like tem- Fig. 9.22 Effect of ratio of plastic-zone size to specimen thickness on perature and strain rate, and KIc is usually observed to fracture toughness and macroscopic appearance of fracture surface. decrease with decrease in temperature and increase in strain Plane-stress condition is associated with maximum toughness and slant rate. Since KIc is a basic material property just like the yield fracture, whereas plane-strain condition is associated with minimum strength of material, so the best way for comparison of toughness and flat fracture different materials with varying thickness with respect to their inherent fracture toughness levels is to compare their prevailing at the crack tip. When the state of plane stress KIc values. Once KIc is determined for a particular material, associated with the maximum toughness exists at rp ! t; it is possible to compute the fracture strength of that material fracture often occurs on those planes having maximum for a given flaw size. Figure 9.21 shows the effect of the resolved shear stress, i.e. along planes oriented at an angle of specimen thickness t on the measured fracture stress. 45° to the axis of applied stress, and thus, slant fracture appears on the fracture surface. When ðt=10Þ rp\\t; the Since the effects of the state of stress on fracture toughness state of stress is neither a full plane stress nor an absolute are influenced by the ratio of the plastic-zone size or radius rp plane strain condition, and this mixed state of stress having to the specimen thickness t, it is useful to consider the tran- partially plane stress and partially plane strain conditions sition in the state of stress in terms of rp=t; where rp is cal- associated with the intermediate toughness produces a mixed culated arbitrarily with the relation for plane stress condition mode of fracture on the fracture surface, which shows a flat as given by (9.51). Figure 9.22 shows a variation of fracture fracture with 45° shear lips. The relative proportion of flat toughness Kc with rp=t; instead of only 1=t; as shown in and slant fracture depends on the fracture toughness of the Fig. 9.20. It is observed from experience that a state of plane material, and as the toughness decreases, the proportion of sÀtressÁ exists and fracture toughness Kc is high when flat fracture increases. When the plane strain condition rp=t ! 1, and on the other side, a condition of plane strain associated with the minimum toughness is developed at will be present at the crack tip resÀultingÁ in the lowest value of rp\\ðt=10Þ; fracture tends to occur in a plane containing the fracture toughness, i.e. KIc; if rp=t \\ð1=10Þ (Hertzberg maximum net section stress, i.e. in a plane oriented at an À1989).ÁSince at any given level of stress intensity factor rp / angle of 90° to the axis of applied stress, and thus, flat 1=r02 [see (9.51)], so the increase of the yield strength r0 by fracture appears on the fracture surface. a factor of two using some treatment will reduce the plastic-zone size rp by a factor of four, which in turn will 9.11 Plane-Strain Fracture Toughness ðKIcÞ cause the sample thickness t required for the development of Testing a plane-strain condition to reduce by the same factor of four, assuming that KIc has not been changed by that treatment. An important material property in fracture prevention is the Similarly, the sample thickness required to achieve a state of plane-strain fracture toughness, ðKIcÞ. This property charac- plane stress will depend on the yield strength of the material. terizes the resistance of a material to fracture under a neutral Hence, it is clear that a state of plane strain can exist even in a environment in the presence of a sharp crack under the very thin piece of a high-yield-strength material, whereas an maximum plastic constraint in the tensile mode of loading, absolute plane-stress condition may never develop in a very such that a triaxial state of tensile stress producing a plane thick piece of a low-yield-strength material. strain condition is developed near the crack front and the crack-tip plastic-zone size is small compared to the crack Again, the transition in the macroscopic mode of fracture from slant to flat or flat to slant depends on the state of stress

9.11 Plane-Strain Fracture Toughness … 401 size, specimen thickness and ligament ahead of the crack. condition the effective stress intensity factor would be a little The ðKIcÞ value represents the lowest limiting value of frac- higher than the applied K; and the plastic-zone correction to ture toughness, and this value may be used to get a relation the stress intensity factor would be unnecessary. between the fracture strength and flaw size for a component subjected to the above-mentioned severe tensile constraint in The plane-strain fracture toughness ðKIcÞ is determined in service. The value of KIc determined for a particular thickness the laboratory by using test methods standardized by the of specimen of a given material may or may not be valid ASTM Standard (ASTM E399 2009). This standard is rec- because a valid KIc value is obtained only when a specimen ommended for precise details. A summary of the most thicker than the previous one does not produce a toughness important features of the test is given in this section. The test value lower than the previous one for any given material. procedure described in this section can only be applied to materials with limited ductility, since the method of analysis From the examination of the fracture toughness of various is based on linear elastic fracture mechanics. High-strength high-strength materials, Brown and Srawley (1966) found steel, aluminium and titanium alloy are typical examples of empirically that both specimen thickness t and crack length such materials. a must be greater than a certain minimum value to achieve a plane strain condition and a valid KIc measurement. This Various types of specimen have been proposed to deter- critical minimum value is given by the following (9.58a) and mine KIc: The types of specimen recommended by the (9.58b): ASTM for the plane-strain fracture toughness testing are as follows: Thickness of specimen; t ! 2:5KIc2 ð9:58aÞ r0 • A single-edge notched plain-sided compact tension specimen, which is pin loaded with uniaxial tension.  2 2:5 KIc • A single-edge notched plain-sided bend specimen loaded And crack length; a ! r0 ð9:58bÞ by three-point bending. where r0 ¼ 0.2% offset yield strength of the material in the • A single-edge notched disk-shaped compact specimen environment and at the temperature and strain rate used in loaded in tension. the test. • A single-edge notched arc-shaped specimen loaded in tension. The form of (9.53b) is the same as that of (9.58a) or (9.58b), because the ratio ðKIc=r0Þ2 is common to all of • A single-edge notched arc-shaped specimen loaded in these equations. This suggests that the size of plastic zone is bending. related to the minimum specimen thickness and crack length. 9.11.1 Specimen Size, Configurations, Substitution for ðKIc=r0Þ2 from (9.53b) into (9.58a) and and Preparation (9.58b) gives the condition for the development of a state of plane sÀtraiÁn in terms of critical plane-strain–plastic-zone If a is the crack length (total length of machined notch plus radius rpc P:Strain at fracture, which is shown by the fol- fatigue crack), t is the thickness, and w is the width of the lowing (9.59a), (9.59b) and (9.59c): test specimen, then the proportions of the specimen are such that a is nominally between 0:45w and 0:55w; and for bend ÀÁ specimen, 1 w=t 4, whereas for tension specimen, t and a ! 2:5 Â 6p rpc P:Strain; 2 w=t 4: It is recommended that ÀÁ h ÀÁ i • The width w is nominally two times the thickness t; i.e. Or, t and a ! 47:124 rpc P:Strain % 50 rpc P:Strain w=t ¼ 2: ð9:59aÞ • Similarly, the crack length a should be nominally equal to one-half the width w; i.e. a ¼ w=2: ) Thickness of specimen; t ! 50 ÀÁ P:Strain ; ÀÁ rpc From the recommended dimensional relationship, it is ð9:59bÞ clear that the specimen ligament size ðw À aÞ ¼ a (the crack Or, rpc P:Strain 2% of t ð9:59cÞ size). Hence, from (9.58b), it can be seen that the specimen ligament size ðw À aÞ must not be less than 2:5ðKIc=r0Þ2 for And crack length; a ! 50 ÀÁ P:Strain; a valid KIc measurement. ÀÁ rpc The above first two types of specimens, i.e. a single-edge Or; rpc P:Strain 2% of a notched plain-sided compact tension specimen and a Equation (9.59c) shows that for the development of plane strain condition at the crack tip, the critical radius of the plastic zone at fracture must not exceed 2% of the crack length. As mentioned earlier in Sect. 9.9.1, under such

402 9 Fracture single-edge notched plain-sided bend specimen, are the most P a=t common specimen configurations and shown, respectively, t w w = 2t in Figs. 9.23 and 9.24. According to ASTM Standard, side grooving for the above compact tension and bend specimens a s = 8t is allowed as an alternative to plain-sided specimens. The schematic diagram of cross-section of side groove configu- s ration is shown in Fig. 9.25. If tN is the minimum thickness of specimen between the roots of the side grooves, the total 2.1 w 2.1 w thickness reduction will be in such a way that tN ! 0:75t: It (Minimum) (Minimum) is recommended to make thickness reduction 10% per side so that tN ¼ 0:8t: The side groove can have any included Fig. 9.24 A single-edge notched bend specimen for KIc testing, angle less than 90° and a root radius of 0:5 Æ 0:2 mm: The loaded by three-point bending location of the root of side groove should be along the specimen centreline. Side grooves are made to increase the Radius 0.5 ± 0.2 mm level of constraint in comparison with recommended plain-sided specimen. A more uniform state of stress is ≤ 90° developed along the crack front, and the development of tN shear lip is inhibited due to the increased constraint. As a result, a side-grooved specimen is expected to give a lower tN ≥ 0.75t value of KIc than a plain-sided specimen, particularly when the specimens are thin or exhibit Type I load–displacement t curve as shown in Fig. 9.26. For materials where structural Fig. 9.25 Schematic diagram of cross-section of side-grooved situations are such that plasticity is more highly constrained specimen by the geometry of crack front such as the case for a corner or surface crack or by structural details such as notches, A Pmax A Pmax A keyways, radii, it may be better to represent fracture toughness by the value of KIc obtained from a side-grooved specimen. For materials with structural situations where no constraint is imposed on surface plasticity and shear lip development such as through crack in the region of uniform thickness, it may be better to represent fracture toughness by the value of KIc obtained from a plain-sided specimen. The initial selection of a thickness of specimen, from which a valid KIc value will be determined, is often based on an estimated value of KIc of the material and may be found P P5 = PQ PQ Pmax = PQ 1.25 w P5 P5 Diameter 0.25 w 0.275 w a t Load, P 0.275 w w a=t w = 2t Type I Type II Type III Diameter 0.25w OOO P Displacement, δ Fig. 9.23 A single-edge notched compact tension specimen for KIc Fig. 9.26 Three major types of typical load–crack displacement testing, loaded with uniaxial tension curves, obtained during KIc testing

9.11 Plane-Strain Fracture Toughness … 403 from (9.58a). An alternative way is to use the ratio of yield • The length of the fatigue crack for the straight strength r0 to elastic modulus E of the material for the initial through-thickness notch must be at least 2.5% of the selection of thickness of a specimen which is provided in the width w or 1.3 mm, whichever is larger. following Table 9.1, based on ASTM Standard E 399. 9.11.2 Test, Interpretation of Result When it has been established that the minimum values of and Calculation of ðKIcÞ t recommended in the preceding Table 9.1 are substantially higher than 2:5ðKIc=r0Þ2; then a correspondingly lower During the test, when the load P is applied to the specimen, values of t can be used. On the other hand, if the form of the the relative displacement across the open end of the notch at available material makes it impossible to obtain a specimen the specimen edge proportional to the displacement of crack, thickness t greater than 2:5ðKIc=r0Þ2; then it is not possible d; is measured with a clip gage placed at the mouth of the to carry out a valid KIc test. notch. The testing machine used to determine KIc must be able to provide for a continuous autographic record of load After the initial selection and preparation of a specimen, a P versus crack displacement d curve. The typical load–crack straight through-thickness slot terminating in V-notch with displacement curves, i.e. P versus d curves obtained during an included angle less than 90° and root radius of 0.08 mm KIc testing of different materials, are of three major types, as or less is machined on the surface of the specimen in such a shown in Fig. 9.26. Type I load–displacement curve shows way so that the width of notch will be minimum 1.6 mm but the increase of load with displacement till the point of less than one-tenth of the width of specimen. There are also maximum load Pmax required for fracture and represents the other kinds of starter notches, like slot ending in drilled hole behaviour of a relatively less brittle material in which the and chevron notch. The crack starter notch must be normal crack propagates at a relatively slow rate without any indi- to the specimen surface and to the intended direction of the cation of the onset of unstable fracture. Type II P versus d crack propagation. Afterwards, a sharpest crack is produced curve exhibits a sharp drop of load at certain point of dis- at the root of notch by subjecting the specimen to low-cycle placement followed by a rise of load with further displace- fatigue in high-strain mode – typically of the order of 1000 ment till the point of maximum load Pmax required for fatigue cycles with a strain of around 0.03. For side-grooved fracture. The sharp drop of load occurs due to sudden specimens, it is recommended that side-grooving operation unstable rapid propagation of crack, while the rise of load follows the fatigue precracking to produce nearly straight arises from a slow rate of crack propagation. This type of fatigue precrack fronts. The initial crack length a in the curve is obtained for materials of intermediate brittle nature. specimen is taken as the summation of the depth of Type III P versus d curve exhibits a sharp load drop at the machined notch and the length of fatigue crack. Cycling for point of maximum load Pmax where the crack propagates in fatigue precracking is continued until the following an unstable rapid manner to cause complete fracture of requirements are satisfied. material. This type of curve is characteristic of the most brittle ‘elastic material’. • The initial crack length a; i.e. total length of machined notch plus fatigue crack, is nominally equal to thickness In order to establish that a valid KIc value has been t and is between 0:45w and 0:55w: determined, it is necessary first to calculate a conditional value of fracture toughness denoted by KQ; which requires a Table 9.1 Guideline for selection of a specimen size from the ratio of graphical construction on the load–displacement test record. yield strength to elastic modulus for a valid KIc test (ASTM E399 2009) As per ASTM Standard E 399-09, the following procedures are adopted for this graphical construction and the evaluation Yield strength ¼ r0 Minimum recommended of KQ: Elastic modulus E specimen thickness t (mm) • Draw a secant line OP5; as shown in Fig. 9.26, from the 0.0050 to less than 0.0057 76 origin of the curve on the load P versus displacement d test record with a slope that is 5% less than the tangent 0.0057 to less than 0.0062 64 OA to the initial part of the curve, i.e. ðP=dÞ5¼ 0:95ðP=dÞ0; where ðP=dÞ5¼ slope of secant line OP5; 0.0062 to less than 0.0065 51 and ðP=dÞ0¼ slope of tangent OA: For compact tension and three-point bend specimens, a 5% reduction in slope 0.0065 to less than 0.0068 44 is roughly equal to a 2% increment in the effective crack 0.0068 to less than 0.0071 38 0.0071 to less than 0.0075 32 0.0075 to less than 0.0080 25 0.0080 to less than 0.0085 19 0.0085 to less than 0.0100 13 0.0100 or greater 6.4

404 9 Fracture length of the specimen, which reflects a minimal crack For plane-sided three-point bend specimen: extension and plastic-zone correction (Brown and KQ ¼ PQs a ð9:62aÞ Srawley 1966). P5 is defined as the load where the secant tw3=2 fw ð9:62bÞ line OP5 intersects with the original curve on the test record. For side-grooved three-point bend specimen: • To calculate KQ; it is required to determine a load PQ that is defined as follows: KQ ¼ pffitffiPtffiNffiffiQws3=2 fa w (i) If the load at every point on the record which precedes where: P5 is less than P5; then PQ ¼ P5; as seen in Type I curve of Fig. 9.26. rffiffiffi a  a &2:15 3:93 a   2'!+ a 1 2:7 a (ii) If, however, there is a highest load preceding P5 which a *3 w 1:99 À À À þ exceeds it, then this highest recorded load is PQ; as w w w w w seen in Types II and III curves of Fig. 9.26. f ¼  þ 2a1 a 3=2 1 2 À ww ð9:63Þ • Calculate the ratio Pmax=PQ; where Pmax is the maximum For the above equations from (9.60) to (9.63), definitions recorded load that the specimen is able to withstand. If of different terms are: ðPmax=PQÞ [ 1:10; then the test performed is not a valid KIc test because KQ may not bear any relation with KIc: In PQ load in N, such case, a thicker specimen with dimensions at least t specimen thickness in m, 1.5 times the dimensions of the previous specimen, for tN minimum specimen thickness between the roots of the which ðPmax=PQÞ [ 1:10; is prepared for further testing, or the user is referred to Test Method E1820 (ASTM) on side grooves (Fig. 9.25) in m, elastic–plastic fracture toughness. If ðPmax=PQÞ 1:10; w specimen depth (Fig. 9.24) or dimension in width then proceed to compute KQ according to the following (9.60) and (9.61) for a compact tension specimen and direction (Fig. 9.23) in m, (9.62) and (9.63) for a three-point bend specimen. The a crack length in m, measured after fracture and crack length a used in the following equations is mea- s span (Fig. 9.24) in m sured after the fracture of specimen. The following equations, adopted from ASTM Standard (ASTM E399 • Next assuming KQ ¼ KIc, calculate the factor 2009), are established on the basis of elastic stress 2:5ðKIc=r0Þ2: As per (9.58), if the quantity 2:5ðKIc=r0Þ2 analysis of specimens. is less than or equal to both the thickness t and crack length a of specimen or the specimen ligament size, w À pffiffiffiffi a; then KQ ¼ KIc; and the test is valid. Otherwise, the test KQ is expressed in units of Pa m in the following (9.60) is not a valid KIc test and a thicker and/or more deeply and (9.62): cracked specimen must be used for further testing in For plane-sided compact tension specimen: order to determine a valid KIc value. The thickness of KQ ¼ pPQffiffiffi fa ð9:60aÞ new specimen can be estimated by using the computed tw w value of KQ through (9.58a). If the fracture toughness KIc For side-grooved compact tension specimen: of a material is properly determined, it must be inde- pendent of crack geometry, specimen configuration or fa loading system. w KQ ¼ pffiffiPffiffiffiQpffiffiffi ð9:60bÞ ttN w where: 9.11.3 Kc from KIc 2 a & þ  a À  a 2 þ 14:72wa 3 À5:6wa 4 '5737 Sometimes, the materials are available in gages thinner that 664 2 þ 0:886 4:64 13:32 those required for valid KIc measurement. Consideration of a w w w KIc values to determine the maximum allowable flaw size f w ¼  a 3=2 a or the maximum allowable operating stress ra would be 1 À w ð9:61Þ

9.11 Plane-Strain Fracture Toughness … 405 inappropriate for such thin materials, where plane strain crack that can easily be detected and repaired is allowed by the designer, the maximum allowable stress, i.e. the oppeffiffirffiffi-ffi conditions would not prevail. For such cases, it is required to ating stress ra, will be fixed and must be less than KIc= pa according to (9.66). know Kc values. Irwin (Irwin et al. 1967) has shown an empirical relation to obtain an estimate of Kc from the KIc value determined in the region near plane strain, as given below: Example II utuuvffi42ffiffiffi1ffiffiffiþffiffiffiffi1ffiffiffi:ffi4ffiffi(ffiffiffiffi1ffiffiffiffiffiffiffiKffiffiffiIfficffiffiffiffiffiffi2ffi)ffiffiffiffi2ffiffi53ffiffiffi If a certain aluminium alloy is selected by the designer for t r0 Kc ¼ KIc ð9:64Þ application in the wing skin of an aircraft, then KIc is fixed. In addition, the operating stress ra has been kept at a high level to maximize the payload capacity. After fixing KIc and 9.12 Design Philosophy with Fracture ðrKa;Ict=hpe pffiffimffiraaxÞ2imaucmcoradllionwg atbole(9fl.a6w6).sizUesuaalmlyu, stthbee less than Toughness allowable flaw size a is kept to a very low level, which is often undetectable with NDT (non-destructive testing) techniques. The fracture condition of a component in service will be Note that this above limit of the allowable flaw size must controlled by the interaction of the plane-strain fracture also be applied to the diameter of any rivet hole present in toughness KIc of the component with the operating stress ra the component, which is explained by one case history, as and size of crack a present in the component. Once a certain follows, related to the above example of the wing skin of an material is selected for application in service, then the KIc aircraft. In this case, a certain aluminium alloy was used in value is fixed. Now, during service, if the operating stress ra reaches the value of the fracture propagation stress rf for a the wing skin of an aircraft, where a fatigue crack propa- certain flaw size a; the applied stress intensity factor K is gated from a rivet hole in one of the wing plates and exceeded the maximum allowable flaw size defined by increased to KIc value of the material. For example, the (9.66), resulting in fracture. In fact, the diameter of the rivet condition of fracture for an infinitely wide cracked plate can hole was bigger than the maximum allowable flaw size that be obtained from (9.46) and given by could be endured by the selected material under the applied pffiffiffiffiffi stress. If the diameter of the rivet hole would be fixed KIc ¼ rf pa ð9:65Þ beforehand as the maximum allowable flaw size, the fracture Mostly, all design against fracture is based on maintain- could have been prevented by reducing the operating stres- ing the operating stress ra at a lower level than the fracture ses and/or by selecting a material tougher than this alu- propagation stress rf ; assuming that the component contains flaws of a certain size. Hence, to have ra\\rf for prevention minium alloy for the wing plates. of fracture, the applied K\\KIc; and this is expressed as Example III pffiffiffiffiffi ð9:66Þ KIc [ ra pa The development of a ‘leak-before-break’ condition in a pressure vessel may sometimes prevent the complete fracture For designing against the fracture of a component, the as explained below. How this condition is developed is above relation may be applied in one of the various ways. shown in Fig. 9.27. Suppose that a semi-elliptical surface Once any two variables out of the three variables in (9.66), crack is located at the inner surface of the pressure vessel. i.e. fracture toughness KIc; operating stress ra; and flaw size The crack is of external length 2a and projects from inner a; are decided by the designer, the third factor is automati- surface into the wall of the vessel with the distance b; as cally fixed by the (9.66). But the designer must first decide shown in Fig. 9.27. The major plane of the crack is oriented the most important factor required for design of the com- perpendicular to the direction of the circumferential hoop ponent so that it can safely operate. The design philosophy is stress produced by the internal pressure. It must be noted that illustrated with some examples as follows: under the above condition, the growth rate of a semi-elliptical surface crack in a direction parallel to the Example I minor axis of the elliptical crack is likely to be more rapid. So, the crack grows till it assumes a semicircular configu- If a material that must endure the attacks of a liquid metal ration. Afterwards, the crack continues to expand in a cir- environment, such as in some nuclear reactor, is required to cular manner till it breaks the outer surface of the wall, and be selected, the designer must give emphasis on the corro- thus, a through-thickness crack is created, through which the sion resistance property of the material. Once a suitable fluid of the pressure vessel is allowed to escape. At the point corrosion-resistant material is selected, the value of KIc is of breaking through the remaining unbroken ligaments fixed. Further, if the presence of a relatively large stable

406 9 Fracture 2b=2t t b 2a Fig. 9.27 Diagram showing growth of semi-elliptical surface crack located at the inner surface of a pressure vessel to semicircular configuration. At leak condition ðb ¼ tÞ; remaining unbroken ligaments (shaded zones) of the pressure vessel break open to create through-thickness crack (shaded zones in Fig. 9.27), if the through-thickness crack is (a) The sharpest possible crack would be one where the assumed to remain in a semicircular configuration with a crack-tip radius qt ¼ r0; and thus, from given data, qt ¼ 0:3 Â 10À9 m: Since qt ¼ r0; i.e. qt\\ð8r0Þ=p; so the Griffith radius b; then the vessel wall thickness t ¼ b: Hence, the equation (9.27) will be applicable for the determination of stress intensity factor K from (9.46) can be written as the facture strength. The fracture strength of the material pffiffiffiffiffi pffiffiffiffi ð9:67Þ obtained from (9.27) is: K ¼ ra pb ¼ ra pt Now, as long as the applied K\\KIc; complete fracture of rffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the pressure vessel will not occur even though fluid leakage 2Ecs ¼ 2 Â ð100 Â 109Þ Â 1:5 has started. When a crack of length equal to at least twice the rf ¼ pa pð3 Â 10À6Þ N=m2 vessel wall thickness can be allowed, i.e. is stable under the existing stresses, then the ‘leak-before-break’ condition will ¼ 178:4 Â 106 Pa generally prevail. ¼ 178:4 MPa. The theoretical cohesive strength of the material obtained from (9.9) is: rffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ecs ¼ ð100 Â 109Þ1:5 9.13 Solved Problems rc ¼ r0 0:3 Â 10À9 N=m2 9.13.1. A brittle material has an internal crack of length 6 ¼ 22:36 Â 109Pa lm, oriented normal to the applied stress. Its equilibrium atomic spacing is 0.3 nm, specific surface energy is ¼ 22:36 GPa: 1:5 J=m2; and modulus of elasticity is 100 GPa. Assume plane stress condition. Hence, (a) For the sharpest possible crack, determine its fracture E ¼ 100 Â 109 ¼ 560:54: and strength and theoretical cohesive strength. Compare its rf 178:4 Â 106 fracture strength with its theoretical cohesive strength in terms of its elastic modulus. E ¼ 100 Â 109 ¼ 4:47; (b) When the crack-tip radius is 1 nm, determine the fracture rc 22:36 Â 109 strength of the same material. ) rf ¼ E ; whereas; rc ¼ E : Solution 560:54 4:47 Given that the internal crack length is 2a ¼ 6 Â 10À6 m; i.e. (b) When the crack-tip radius is qt ¼ 1 Â 10À9 m; (9.16) a ¼ 3 Â 10À6 m; the equilibrium atomic spacing is r0 ¼ derived from the stress concentration point of view will be 0:3 Â 10À9 m; the specific surface energy is cs ¼ 1:5 J=m2; the modulus of elasticity is E ¼ 100 Â 109 N=m2: applicable for the determination of the facture strength, because ð8r0Þ=p ¼ ð8 Â 0:3 Â 10À9Þ=p m ¼ 0:764 Â 10À9 m; and so qt [ ð8r0Þ=p:

9.13 Solved Problems 407 The fracture strength of the material obtained from Yield sstrffiffieffiffinffiffigffiffiffitffihffiffi;ffiffiðffiffirffiffiffi0ffiffiÞffiffiSffiecond (9.16) is: ¼ 1 À ÁK2 s2pffiffiffiffirffiffipffiffiffiffiSffiffiefficffiffioffinffiffidffiffiffiffiffi rf ¼ rffiffiffiffiffiffiffiffiffiffi ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N=m2 À pÁa Ecsqt ð100 Â 109Þ Â 1:5 Â ð1 Â 10À9Þ 2p rp Second 4r0a ¼ ra 4 Â ð0:3 Â 10À9Þ Â ð3 Â 10À6Þ ¼ 204 Â 106 Pa ¼ 204 MPa: À 106Á sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 375 ¼ Â Â 2 Â 0:009 N/m2 ð3:5 Â 10À3Þ 9.13.2. Two infinitely wide plates of steel are heat treated to produce different yield strengths. Each of the plates has a ¼ 425:21 Â 106 N/m2 central through-thickness flaw of length 18 mm and is sub- jected to a tensile stress of 375 MPa normal to the crack ¼ 425:21 MPa: plane. The plastic-zone diameters ahead of the crack tips are found to be 0.5 mm in one plate and 7 mm in another plate. (b) The effective stress intensity factors in plane stress Assuming plane stress condition, determine the following condition for the first and second plates can be obtained from for both plates: (9.56), and the respective values are given below: (a) Yield strength of the material. ra pffiffiffiffiffi (b) Effective stress intensity level at the crack tip. & pa (c) Percentage increase in effective stress intensity level with ðKeff ÞFirst ¼ \" 1 ra '2#1=2 respect to applied stress intensity level. Comment on the 1 plasticity correction from these derived values. À 2 ðr0ÞFirst À Á pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 375 Â 106 Â pð0:009Þ pffiffiffiffi ¼ \"  375 Â 106 2#1=2 Pa m 1 1591 Â 106 Solution À 1 2 Equation (9.51) shows that the radius of plastic zone in plane ¼ 63:95 Â 106 pffiffiffiffi Pa m K2 pffiffiffiffi stress condition is rp ¼ 1 r02 ; where r0 ¼ yield strength of ¼ 63:95 MPa m 2p the material and K ¼ stress intensity factor, which for a sharp pffiffiffiffiffi ra pa elastic through-thickness central crack of length 2a in an ðKeff ÞSecond ¼ \" & '2#1=2 1 infinitely wide plate lpoaffidffiffiffieffid with uniaxial tension is expressed À 1 ra by (9.48) as K ¼ ra pa; where a ¼ 0:009 m, (given), and 2 ðr0ÞSecond À Á pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ra ¼ applied stress ¼ 375 Â 106 N=m2; (given). 375 Â 106 Â pð0:009Þ pffiffiffiffi (a) For first plate, where ÀÁ 0:25 Â 10À3 m, ¼ \"  375 Â 106 2#1=2 Pa m rp First¼ 1 425:21 Â 106 À 1 2 Yield sstrffiffieffiffinffiffigffiffiffitffihffiffi;ffiffiffiffiðffiffirffiffi0ÞFirst ¼ 1 À KÁ2 ¼ 80:66 Â 106 pffiffiffiffi s2pffiffiffiffirffiffipffiffiffiffiFffiffiiffirffisffitffiffiffiffi Pa m pffiffiffiffi ¼ 80:66 MPa m: ¼ ra ÀpaÁ (c) Since each of the plates is subjected to the same value of 2p rp First stress intensity level, so the applied stress intensity level for sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi each of the plates according to (9.48) is: ¼ À Â 106Á Â 2 Â 0:009 10À3Þ N/m2 ¼ pffiffiffiffiffi 375 ð0:25 Â Kapplied ¼ ra pa À ¼ 1591 Â 106 N/m2 375 Â 106Á Â pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi pð0:009Þ Pa m pffiffiffiffi ¼ 1591 MPa: ¼ 63 Â 106 Pa m ÀÁ pffiffiffiffi rp Second¼ 10À3 ¼ 63 MPa m: For second plate, where 3:5 Â m,

408 9 Fracture Therefore, percentage increases in effective stress inten- rffiffiffiffiffiffiffiffi sity level with respect to applied stress intensity level for the EGc first and second plates are, respectively: rf ¼ sffiffipffiffiaffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðKeff ÞFirstÀKapplied  100 ¼ ð202  109Þ Â ð40  103Þ N=m2 Kapplied pð0:01Þ ¼ 63:95 À 63  100 ¼ 1:5%: ¼ 507:14  106 Pa 63 ¼ 507:14 MPa: ðKeff ÞSecond À Kapplied  100 Kapplied (ii) For surface crack length of a ¼ 0:04 m; the fracture strength in plane stress condition according to (9.35a) is: ¼ 80:66 À 63  100 ¼ 28%: 63 rffiffiffiffiffiffiffiffi EGc Comments rf ¼ sffiffipffiffiaffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The effective stress intensity factor is only 1.5% higher than ¼ ð202  109Þ Â ð40  103Þ N/m2 the applied K for the first plate having high yield strength pð0:04Þ and showing a very small ratio of plastic-zone size to crack length. On the other hand, for the second plate having low ¼ 253:57  106 Pa yield strength and showing an appreciably large ratio of plastic-zone size to crack length, the effective stress intensity ¼ 253:57 MPa: factor is 28% greater than the applied K; and the increase is considerably high. So, for the first plate, the plasticity cor- (b) It is plane strain condition and given that the critical rection to the stress intensity factor may be neglected, but for elastic strain energy release rate is Gc ¼ 20  103 J=m2: For the second plate, the applied stress intensity factor requires centre crack length of 2a ¼ 0:02 m; i.e. for a ¼ 0:01 m; the the plastic-zone correction. fracture strength in plane strain condition according to 9.13.3. The critical elastic strain energy release rate Gc of a (9.35b) is: thin mild steel plate in plane stress condition is 40 kJ=m2, and the shear modulus of steel is 76 GPa. Considering sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Poisson’s ratio of steel to be 0.33, find the fracture strengths for the following through-thickness sharp linear cracks ori- rf ¼ EGc ented normal to the stress axis: pað1 À m2Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð202  109Þ Â ð20  103Þ N=m2 pð0:01Þð1 À 0:332Þ ¼ 379:88  106 Pa ¼ 379:88 MPa: (a) (i) 20 mm length remaining in the centre of the plate and Exercise (ii) 40 mm length remaining at the surface of the plate. 9.Ex.1. A sheet of glass has a number of surface cracks (b) When Gc is reduced to 20 kJ=m2 by increasing the plate varying in length from 2 to 4 lm; oriented normal to the thickness to cause plane strain condition, what will be the stress axis. If the Young’s modulus of the glass is 70 GPa fracture strength for a 20-mm-long through-thickness sharp and its fracture strength under plane stress condition is linear central crack oriented normal to the stress axis? 1=700th of the Young’s modulus, estimate its Solution Given that the shear modulus is G ¼ 76  109 Pa; and the (a) Specific surface energy. Poisson’s ratio is m ¼ 0:33: From (5.18), we get the modulus (b) Fracture strength under plane strain condition, if Pois- of elasticity: son’s ratio is 0.25. E ¼ 2Gð1 þ mÞ 9.Ex.2. If a glass sample having an internal crack of length À 109Á 5 lm, oriented normal to the stress axis, is scratched with a ¼ 2  76   ð1 þ 0:33Þ Pa diamond tool to produce a surface crack of same length having the same orientation, will the fracture strength under ¼ 202  109 N=m2: plane stress condition change? If it changes, by how much would it increase or decrease? The Young’s modulus of the (a) It is plane stress condition and given that the critical glass is 70 GPa, and the specific surface energy is 1 J=m2: elastic strain energy release rate is Gc ¼ 40  103 J=m2: (i) For centre crack length of 2a ¼ 0:02 m; i.e. for a ¼ 9.Ex.3. A glass sample having an internal crack of length 0:01 m; the fracture strength in plane stress condition 4 lm is fractured under plane stress condition by applying a according to (9.35a) is:

9.13 Solved Problems 409 tensile stress along an axis inclined at 50° to the crack surface. (a) Which steel sheet will you select under the above con- If Young’s modulus of the glass is 70 GPa, determine the dition? Justify it mathematically. value of the applied stress that causes the fracture of the glass, (b) If the maximum flaw size tolerated by the accepted steel would be allowed for the rejected steel, then what value of (a) if the specific surface energy is 1 J=m2: design stress would be indicated by that rejected steel? How (b) if the specific surface energy is reduced to 0:5 J=m2; due much increase or decrease would it be than the design stress to application of some surface active agents. fixed for the accepted steel? 9.Ex.4. It is found that a crack of 4 mm length, oriented 9.Ex.9. A 3-m-wide thin metal plate has a residual axial normal to the stress axis, is present inside a thin steel sample tensile stress of the order of 250 MPa. The critical crack under plane stress condition. If the brittle fracture strength of extension force of this metal in plane stress condition is the steel at low temperature is 200 MPa, its modulus of known to be 5 kJ=m2: If the centre of the sheet contains a elasticity is 200 GPa and the specific surface energy is linear sharp crack oriented normal to the tensile stress axis, 1:5 J=m2; calculate the plastic work done by the crack as it calculate the critical size of the crack which would rapidly propagates. grow. Shear modulus of the metal ¼ 75 GPa, and Poisson’s ratio ¼ 0:33: 9.Ex.5. For a single-edge notched wide steel specimen of 1 mm thick, the values of specimen compliance ð1=MÞ are 9.Ex.10. An infinitely wide plate of a material has a central found to be 1:435 Â 10À5 mm NÀ1 and 1:585 Â through-thickness flaw of length 15 mm and is subjected to a 10À5 mm NÀ1 for crack sizes of 3 and 6 mm at a critical tensile stress of 300 MPa normal to the crack plane. The yield strength of the material is 400 MPa. Assuming plane load value of 11 kN. Assume that the crack size dependence stress condition, calculate the following at the tip of the of compliance is linear within the range of specified crack crack: sizes, and compute the following: (a) Stress intensity factor. (a) Critical elastic strain energy release rate Gc of the steel (b) Plastic-zone size. specimen. (c) Effective stress intensity factor and comment upon the (b) Fracture stress of the steel specimen in plane stress requirement of plastic-zone correction factor. condition, if the crack length would be 12 mm. Assume that Young’s modulus of the steel is 207 GPa. 9.Ex.11. For plane-strain fracture toughness ðKIcÞ testing, a wide plate is to be made from a mapteffiffirffiffiial whose plane-strain 9.Ex.6. The stress intensity factor for a centrally located fracture toughness value is 40 MPa m and 0.2% offset yield through-thickness crack of length 2a in a 4340 steel plate strength is 575 MPa. loapdeffiffidffiffiffipwffiffiffiffiiffitffihffiffiffiffiffiffiffiffiuffiffiffinffiffiiffiaffiffiffixffiffiiffiaffiffilffi tension is given by K ¼ (a) If the thickness of the plate of the material is 6 mm, will ra pa ð6=pÞ tanðp=6Þ: If the applied stress ra ¼ this result in a valid KIc measurement? If not, estimate plane 200 MPpa ffiffiaffiffind the plane-strain fracture toughness KIc ¼ stress fracture toughness value ðKcÞ for the 6-mm-thick plate 46 MPa m; calculate the maximum permissible value of a; from the given value of KIc: which does not allow the crack to propagate as a brittle (b) For the given value of KIc; what would be the minimum fracture. thickness of the plate for a valid KIc measurement? 9.Ex.7. The stress intensity factor for a partial 9.Ex.12. Indicate the correct or most appropriate answer through-thickness crack in a plate of Ti–6pAffilffi–ffiffiVffipaffilffiffilffioffiffiffiyffiffiffilffiffioffiffiaffiffidffiffieffiffiffid from the following multiple choices: with uniaxial tension is given by K ¼ ra pa secðpa=2tÞ; where a is the depth of penetration of the crack through the (a) The appearance of intergranular fracture suggests that the component wall thickness t. If a ¼ 6 mm, t ¼ 18pmffiffiffimffi , and following mechanism is responsible for the fracture: the plane-strain fracture toughness KIc ¼ 57 MPa m; what will be the maximum permissible value of the applied stress (A) Ductile fracture; (B) Brittle cleavage fracture; ra; so that the crack does not propagate as a brittle fracture? (C) Fatigue failure; (D) High-temperature creep failure. 9.Ex.8. The available flaw detection procedure requires that (b) If a uniaxial tensile load is applied in the longitudinal direction of an infinitely wide plate, the theoretical stress the critical flaw size must be greater than 2 mm. A design concentration factor for a through-thickness circular hole in the plate will be stress level is fixed at two-thirds of the tensile strength level (A) 1; (B) 2; (C) 3; (D) 4; (E) none of the above. of the material. Two large steel sheets with the tensile strength values of 1400 and 1700 MPa are available. The values of MthePiarppffimlffiaffiffinaen-dstr4a0inMfrPaacptumffirffiffieffi: toughness are, respec- tively, 70

410 9 Fracture (c) For a perfectly brittle material, if the crack length is Beachem, C.D.: Metall. Trans. 6A, 377–383 (1975) increased by a factor of four, the fracture stress will change Birkle, A.J., Wei, R.P., Pellissier, G.E.: Trans. ASM 59, 981 (1966) by a factor of Brown, W.F. Jr., Srawley, J.E.: Plane Strain Crack Toughness Testing (A) 4; (B) 2; (C) ½; (D) ¼. of High Strength Metallic Materials. ASTM STP 410 (1966) Chang, L.C.: J. Mech. Phys. Solids 3, 212–217 (1955) (d) The fracture surface morphology of a ductile fracture Dieter, G.E.: Mechanical Metallurgy, 3rd edn, p. 352. McGraw-Hill shows: Book Company (UK) Limited, London (1988) (A) Dimples; (B) Cleavage; (C) Striations; (D) Veins. Feddersen, C.: ASTM STP 410, 77 (1967) Felbeck, D.K., Orowan, E.: Welding J. 34, 570s–757s (1955) (e) The fracture surface morphology of a brittle fracture Griffith, A.A.: Philos. Trans. R. Soc. London 221A, 163 (1920) (This shows: article has been republished with additional commentary in 1968. (A) Dimples; (B) Cleavage; (C) Veins; (D) Striations. Trans. ASM 61, 871) Hertzberg, R.W.: Deformation and Fracture Mechanics of Engineering Answer to Exercise Problems Materials, 3rd edn. Wiley, New York, pp. 254, 281–286, 297, 304– 305, 307 (1989) 9.Ex.1. (a) 0.9 J m−2; (b) 103.2 MPa. Hull, D.: Acta Metall. 9, 191 (1961) Hull, D.: In: Drucker, D.C., Gilman, J.J. (eds.) Fracture in Solids, 9.Ex.2. Decrease by 39.1 MPa. pp. 417–453. Interscience Publishers Inc., New York (1963) Inglis, C.E.: Trans. Inst. Nav. Archit., London 55(pt. I), 219–230 (1913) 9.Ex.3. (a) 194.86 MPa; (b) 137.8 MPa. Irwin, G.R.: Fracturing of Metals, p. 147. ASM, Cleveland, Ohio (1949) 9.Ex.4. 626.8 J m−2. Irwin, G.R.: Handbuch der Physik, vol. VI, p. 551. Springer, Berlin 9.Ex.5. (a) 30.25 kN m−1; (b) 407.55 MPa. (1958) Irwin, G.R., Kies, J.A.: Weld. J. Res. Suppl. 33, 193s (1954) 9.Ex.6. 15.27 mm. Irwin, G.R., Kies, J.A., Smith, H.L.: Proc. ASTM 58, 640–660 (1958) Irwin, G.R., Krafft, J.M., Paris, P.C., Wells, A.A.: Basic Aspects of 9.Ex.7. 386 MPa. Crack Growth and Fracture. NRL Report 6598, Naval Research Laboratory, Washington, D.C., 21 Nov 1967, pp. 9–10, 38 (1967) 9.Ex.8. (a) Weaker steel with strength of 1400 MPa is to be Joffe, A.F.: The Physics of Crystals. McGraw-Hill Book Company, New York (1928) selected, since the maximum flaw size tolerated by it is Klier, E.P.: Trans. ASM 43, 935–957 (1951) Knott, J.F.: Linear elastic fracture mechanics. In: Fundamentals of 3.58 mm that meets the minimum flaw size requirement Fracture Mechanics. Butterworth & Co (Publishers) Ltd., London, p. 101 (1973) (greater than 2 mm), whereas the maximum flaw size tol- McClintock, F.A., Irwin, G.R.: ASTM STP 381, 84 (1965) Murakami, Y. (ed.): Stress Intensity Factors Handbook. Pergamon erated by stronger steel with strength of 1700 MPa is Oxford (1987) Orowan, E.: Trans. Inst. Eng. Schipbuild, Scot. 89, 165 (1945) 0.79 mm; (b) 533.4 MPa, 399.93 MPa decrease. Orowan, E.: Fatigue and Fracture of Metals, p. 139. MIT Press, Cambridge, MA (1950) 9.Ex.9. 10.16 mm. pffiffiffiffi pffiffiffiffi Paris, P.C., Sih, G.C.M.: In: Srawley, J.E., Brown, W.F. (eds.) Fracture 9.Ex.10. (a) 46 MPa m; (b): 2.1 mm; (c) 64 MPa m: Toughness Testing, ASTM STP No. 381, Philadelphia, PA., p. 30 (1965) Hence, correction is essential, since stress intensity factor Passoja, D.R., Hill, D.C.: Metall. Trans. 5, 1851 (1974) Peterson, R.E.: Stress-Concentration Design Factors. Wiley, New York increases by Nmoo;re55th:3anM3P9a%p.mffiffiffiffi; (b) 12.1 mm. (1974) 9.Ex.11. (a) Reed-Hill, R.E.: Fracture. In: Physical Metallurgy Principles, 2nd edn., Litton Educational Publishing, Inc., New York, p. 753 (1973) 9.Ex.12. (a) (D) High-temperature creep failure. (b) (C) 3. Sih, G.C.M.: Handbook of Stress Intensity Factors. Lehigh University, Bethlehem, PA (1973) (c) (C) 1/2. (d) (A) Dimples. (e) (B) Cleavage. Tada, H., Paris, P.C., Irwin, G.R.: The Stress Analysis of Cracks Handbook. Del Research, Hellertown, PA (1973) References Van Stone, R.H., Cox, T.B., Low, J.R. Jr., Psioda, J.A.: Int. Met. Rev. 30(4), 157 (1985) ASTM E399 (editorially corrected in 2010): Standard Test Method for Westergaard, H.M.: Trans. ASME J. Appl. Mech. 61, 49 (1939) Linear-Elastic Plane-Strain Fracture Toughness KIc of Metallic Materials. Designation: E399-09. ASTM International, PA (2009). doi:https://doi.org/10.1520/E0399-09E02 Beachem, C.D.: Trans. ASME J. Basic Eng. Ser. D 87, 299 (1965)

Part II Mechanical Working of Metals and Alloys

Fundamentals of Mechanical Working 10 Chapter Objectives • Classification of mechanical forming processes and their main objectives. • Hot working and cold working, and their comparison. Cold-work-anneal cycle and temperature limits for hot working. Warm working, its purpose and advantages. • Temperature change during working of deforming metal, depending on its ideal plastic deformation, friction at its interface with tools or dies, and heat transfer between them. • Effects of strain rate in working processes. Effect of varying pressure and strain rate on allowable hot working temperature range. • Friction: Coulomb’s law of sliding friction and factors affecting Coulomb’s coeffi- cient of sliding friction (COF). • Shear friction factor and sticking friction. Maximum value of COF under sticking condition according to Von Mises’ and Tresca yielding criteria. Difference and advantages of shear friction factor model over Coulomb’s model of friction. • Evaluations of friction factor and COF by ring-compression test. • Adverse as well as beneficial effects of friction on mechanical working. • Material pickup on tools. Functions and characteristics of a lubricant. • Lubrication mechanism: hydrodynamic or full-fluid or thick-film lubrication, boundary lubrication, mixed-film lubrication, solid lubricants and melting solids. • Mechanics of working process: slab method, uniform-deformation energy method, slip-line field theory, upper- and lower-bound solutions and finite element method. • Slip-line field theory: slip lines, Hencky’s slip-line equations, stresses and slip lines at the boundaries of a plastic body, simple state of stress, Hencky’s first theorem, numerical method of solution, application of slip-line field to static system and steady motion. • Upper-bound technique: derivation of its equation, its solutions for indentation of a semi-infinite slab and for compression. • Deformation-zone geometry in terms of reduction in area of work-piece for different deformation processes. • Anisotropy of mechanical properties: crystallographic texture and mechanical fibering. • Problems and solutions. © Springer Nature Singapore Pte Ltd. 2018 413 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_10

414 10 Fundamentals of Mechanical Working 10.1 Classification of Mechanical Forming and extrusion, whereas all forming processes including the Processes above three can be carried out by cold working. Another method of classification depends on the flow patterns during Metals are used only after the desired or useful shapes such deformation. The whole range of working processes is cov- as tubes, rods, sheets are given to them by different manu- ered by two kinds of flow patterns. During the deformation facturing processes, such as cycle, the flow pattern will either change continuously or remain unaltered. Examples of the former are forging, • Casting; extrusion, deep drawing and stretch forming. The flow pattern • Compacting of metal powder; in rolling is static and that in wire drawing is quasi-static, and • Joining processes, such as welding; so both processes fall into the latter category. However, this • Machining; and classification method is not extensively used. • Mechanical working or forming. The most important and normally adopted method of Out of the above manufacturing processes, we are con- classification is based on the stress generated in the work cerned with only mechanical working or forming operations. metal during deformation. In all working processes, the Mechanical working or forming is the creation of desired initial stress applied to a work-piece by tools or dies used for shapes by plastic deformation in which the displacement of the deformation is either uniaxial or biaxial. But during material from one location to another occurs, but the mass of deformation, the work-piece is subjected to a triaxial stress all solids and practically the volume of non-porous solids field. The reason for this is that as soon as the deformation remain constant. Plastic working processes which reduce an starts the flow of metal produces friction between the ingot or billet to a semifinished product of simple shape, work-piece and the tools or dies, and thus, further stresses such as sheet, plate and bar, are called ‘primary mechanical are induced to give a triaxial state of stress. Anyone of the working processes’ or ‘processing operation’. Forming following three kinds of stress systems is originally applied methods, such as wire drawing and tube drawing, and most to the work-piece. These are as follows: sheet metal forming operations like deep drawing, which work on a part to produce a final finished shape, are called • Uniaxial compressive stress, ‘secondary mechanical working processes’ or ‘fabrication’. • Uniaxial tensile stress, • Biaxial tensile stresses. 10.1.1 Aims of Mechanical Working Depending on the above-mentioned applied stress sys- tem, working processes are divided into the following three Main objectives of working processes are as follows: categories: • To obtain desired shape and size of the product metal. 1. Direct compression processes, where uniaxial compres- sive stress is applied directly to the work-piece. The Other essential purpose of working is to improve and/or reaction of the work-piece with the dies or tools induces obtain the mechanical properties (strength, toughness, etc.) two more compressive stresses that act on two mutually of a product required for a specific application through perpendicular planes as shown in Fig. 10.1. This category • Redistribution of the microstructural constituents in the Applied compressive stress Induced parent metal and/or compressive stress • Refinement of grain structures in the parent metal and/or • Imparting strain hardening to the parent metal. 10.1.2 Different Forming Processes Induced Induced compressive compressive Forming processes can be classified into different categories stress in a number of ways. One method of classification is based on stress the temperature of working, i.e. whether the process has been carried out by cold or by hot working. But this method of Induced classification is not suitable because hot working can be used compressive mainly for three forming processes which are forging, rolling stress Applied compressive stress Fig. 10.1 Direct compression process

10.1 Classification of Mechanical Forming Processes 415 Hammer Applied compressive stress Induced compressive stress Metal Dies Induced compressive Induced stress compressive stress Induced compressive stress Anvil Applied compressive Forging stress Fig. 10.2 Forging process, showing direct compression Applied compressive stress Induced Fig. 10.3 Rolling, showing compressive direct compression stress Induced Induced compressive compressive stress stress Rolling Induced Applied compressive compressive stress stress Fig. 10.4 Extrusion process, Induced showing direct compression Induced compressive stress Ram Applied Applied Extrusion compressive compressive stress stress Induced Induced compressive stress includes forging, rolling and extrusion processes where Figs. 10.6 and 10.7. Tube drawing and flat-strip drawing any element in the deformation zone is subjected to a are also included in indirect compression systems of triaxial state of compressive stress—one applied and deformation. the other two induced, as shown in Figs. 10.2, 10.3 3. Tension processes, where biaxial tensile stresses are and 10.4. applied directly to the work-piece. The applied stresses 2. Indirect compression processes, where uniaxial tensile will induce a compressive stress that acts on a mutually stress is applied directly to the work-piece. The reaction perpendicular plane as shown in Fig. 10.8. The example of of the work-piece with the dies or tools induces two more this category is stretch forming. Stretch forming is essen- compressive stresses that act on two mutually perpen- tially applied for the production of shapes in sheet metal. In dicular planes as shown in Fig. 10.5. Wire drawing and this process, a metal sheet blank is placed over a form deep drawing fall into this category as shown in block and pulled by tensile forces applied to both ends of

416 10 Fundamentals of Mechanical Working Applied tensile stress the sheet in such a way that the material is wrapped around Induced the profile of the form block by plastic deformation to compressive obtain the final shape. It is a cold working process. Out of stress all working processes, the least used process is stretch forming, which will not be considered further. Induced compressive stress This last method of classification based on the stress system has close similarity with the first classification Fig. 10.5 Indirect compression process method based on the working temperature. It may be noted that the direct compression processes can be carried out by cold or by hot working, while indirect compression pro- cesses can only be carried out by cold working. This is one of the reasons for which this last classification method has received its importance and wide acceptance. Fig. 10.6 Wire drawing, Reaction of Reaction of showing indirect compression job with die job with die Back Tensile pull Applied Applied tension Conical die tension tension Reaction of Reaction of job with die job with die Wire drawing Fig. 10.7 Deep drawing, Punch force Reaction of showing indirect compression job with tool Applied Applied tension tension Reaction of job with tool Deep drawing Fig. 10.8 Stretch forming Applied tension Induced compression process, showing biaxial tension Metal Tensile pull Former Applied Applied tension tension Stretch Forming Tensile pull Applied tension Tensile pull Induced compression

10.1 Classification of Mechanical Forming Processes 417 Deformation force Lubrication simple manner that the plastic deformation of a material Heat flow system above its recrystallization temperature is hot working, whereas that below its recrystallization temperature is cold Tools Product working. The temperature at which a heavily cold-worked Friction material completely recrystallizes in 1 h is called its re- Lubrication crystallization temperature, TR, which is roughly estimated Undeformed Deformation system from its melting temperature, TM, measured in Kelvin and Metal Zone approximately given by TR (in K) ’ (0.4–0.6) Â TM (in K). Hence, the higher the melting point of metal, the higher is Friction the recrystallization temperature, provided other factors affecting TR remains the same. Since lead and tin are Heat flow Tools low-melting metals and they recrystallize rapidly at room temperature after large deformation, so the room temperature Deformation force working of these metals constitutes hot working. On the Fig. 10.9 A simplified view of total deformation processing system other hand, working of tungsten (a high-melting metal) at 1100 °C, which is in hot-working range for steel, constitutes A simplified view of total deformation processing system is cold working, because TR for tungsten is above 1100 °C. provided in Fig. 10.9. This diagram shows that the work-piece has been divided into three segments—undeformed metal, Dividing line between the hot and cold working opera- deformation zone and product. The work-piece is in contact tions is explained below and illustrated in Fig. 10.10, which with non-deforming dies or tools, which may undergo elastic is a schematic diagram of hardness versus working tem- deformation. Heat flows from the hot work-piece to the tools. perature showing the effect of strain rate. When a material is Upon loading, the friction force acts along the work-piece– plastically deformed, its hardness tends to increase because tool interface which is generally lubricated. Lubrication not of strain hardening, but as the deformation temperature only reduces the friction but also prolongs the tool life by increases the rate of strain hardening decreases, and at some minimizing the wear of tools, controls surface finish of the critical temperature, recrystallization takes place, which product, cools the work-piece and/or the tools and thermally makes the material softer. Thus, during plastic deformation insulates the work-piece and the tools. of a material at some critical temperature, two opposing effects act simultaneously on the material — a hardening 10.2 Temperature and Strain Rate effect due to plastic deformation and a softening effect due to recrystallization, and the former effect decreases, while the On the basis of the working temperature, forming operations latter effect increases with increase in temperature. For a are commonly divided into hot working and cold working. given rate of deformation, there must be some temperature at Hot working is defined as the plastic deformation carried out which the rate of hardening is exactly equal to that of soft- under such conditions of temperature and strain rate that ening. Deformation of the material above this temperature is recrystallization occurs as the work-piece is deformed and called hot working, while that below this temperature is the strain hardening caused by the working operation is called cold working. When these two opposing effects just relieved instantaneously, whereas cold working is the plastic balance, the material can be plastically deformed continu- deformation under such conditions of temperature and strain ously without causing the deformation load to increase. If rate that recrystallization does not take place during the the strain rate is increased, the deformation temperature will working operation and the strain hardening caused by have to be increased to such an extent that the increase in the deformation is not relieved. Since the effect of the increase in hardening rate is balanced by the increase in the softening temperature on the strength and ductility properties of a rate. Thus, hot working temperature in case of rapid rate of material is opposite to that due to increasing strain rate and working, such as hammer forging, is higher than that for both temperature and strain rate can vary during the working slow rate of working like press forging, as shown Fig. 10.10. operations, so both parameters are always included in the definitions of hot or cold working to consider their joint 10.2.1 Cold-Work-Anneal Cycle effects. From the above definitions, it can be stated in a Crystalline materials in the unstrained condition consist of equiaxed grains and exhibit isotropic mechanical properties; i.e., the same properties are exhibited in all directions. When a single-phase structure having one kind of grains is cold worked, the randomly oriented equiaxed grains will be

418 10 Fundamentals of Mechanical Working Rapid rate of working Fig. 10.10 Schematic illustration of the effect of Slow rate of working temperature on hardness working with variation in the rate of working Hardness Unworked Cold Hot Cold Hot worked worked worked worked Working temperature deformed and preferably oriented in the direction of flow of of cold working. This sequence of repeated cold working material and produce elongated grains that will result in and annealing is frequently called the cold-work-anneal anisotropic or directional mechanical properties; i.e., prop- cycle. The changes in mechanical properties and erties will vary with the direction of testing. At very high microstructure involved in this cycle are schematically amount of cold working, the grains will be so elongated that illustrated in Fig. 10.11 (Smith 1969), in which the three the structure appears fibrous. Further, cold working produces stages of annealing, viz. recovery, recrystallization and an increase in the dislocation density; for example, a heavily grain growth, are shown to occur progressively with cold-worked metal contains about 1010 number of disloca- increase of annealing temperature, assuming specific con- tions per mm2, while in fully annealed condition it contains stant annealing time at each temperature. It is to be noted about 104–106 dislocations per mm2. The duplex structure that the driving forces for both recovery and recrystallization consisting of a soft and ductile phase and a hard and brittle are the release of stored strain energy, which resulted from phase behaves in a similar manner on cold working except dislocation interaction leading to a higher state of internal that the grains of hard and brittle phase will tend to break stress during the process of cold work, whereas the driving into fragments. These fragmented grains will appear as force for grain growth is the decrease in surface energy since stringers, which will be preferably oriented in the direction grain-boundary area per unit volume decreases with an of flow of material, i.e. in the longitudinal direction. The increase in grain size. mechanical properties of a duplex structure will tend to exhibit more anisotropy than those of a single-phase struc- The recrystallization annealing treatment, also known as ture. As a result of cold working, the ductility of a material process annealing or process and recrystallization drops to a very low value, whereas the hardness, UTS and annealing treatment, consists in heating a cold-worked yield strength all increase to a maximum value, which is material above its recrystallization temperature, TR, holding practically found to be about 2.5–3 times the values of those for proper time (generally 1 h for 1 in. thickness or diam- in the annealed (softest) condition. The notched-bar Charpy eter of the work-piece) and then cooling by any desired or Izod impact toughness rises with cold working up to a rate. It is important to mention here that a minimum per- maximum and then gradually falls (Harris 1983). centage of prior cold work, known as the critical amount of cold work, of the order of around 5–7% for most materials, If a material is continuously subjected to excessive cold is necessary for recrystallization to occur on heating. The working, it will break before it acquires the desired shape purpose of the recrystallization annealing treatment is to and dimension. So, to achieve the required shape and size of soften the cold-worked material and restore the ductility so the final product without any crack, cold working operations that the work-piece can be further cold worked without are normally interrupted and carried out repeatedly in several formation of any crack. This heat treatment does not cause steps, with introduction of intermittent recrystallization any phase change but is associated with the microstructural annealing treatments applied for several times between steps change that involves the formation of new strain-free

10.2 Temperature and Strain Rate 419 Hardness Recovery Grain growth Strength Ductility Recrystallization Grain size -- Ductility Amount of cold work New Strength -- Hardness grains Old grains (Specified constant time at each temperature) Temperature Original Cold - worked Cold - worked Initial More Complete partial Complete structure and recovered recrystal- recrystal- recrystal- grain grain lization lization lization growth growth Fig. 10.11 Schematic representation of the cold-work-anneal cycle illustrating the effects on mechanical properties and microstructure (Smith 1969) randomly oriented equiaxed grains by replacement of the mechanical properties of the material. Further, if the grains previous strain-hardened preferably oriented elongated after cold working and annealing are too coarse, the surface grains with a high density of crystal imperfections produced finish of the metal on machining will be rough and an ‘or- by prior cold working operation. The end result of this ange peel’ effect in which the surface appearance of the part treatment is a restoration of ductile microstructure with a resembles to that of an orange will be exhibited after low dislocation density of the order of 104–106 dislocations pressing. The final temperature in the annealing furnace is an per mm2, which is again capable of undergoing significant important factor in industry. This temperature must be as cold deformation. low as possible, while ensuring complete recrystallization in adequate time. In industry, the annealing temperature of the Although the requirement for intermediate annealing metal is considered as roughly three fourths of the melting increases the cost of cold forming, especially for reactive point measured in Kelvin, for example 450 °C for alu- materials which need inert environment or vacuum for minium and 800 °C for copper (Harris 1983). annealing, at the same time suitable adjustment of the cold-work-anneal cycle can produce a part with any desired It is to be noted that the faster the nucleation rate and the degree of strain hardening, which cannot be achieved by hot slower the growth rate of grains, the finer is the recrystallized working operations. If the finished product is desired to be in grain size dR, and hence, the fineness of recrystallized grain the softened condition, the final operation must be recrys- size dR depends on the following factors. tallization annealing that will follow the last cold working step required to provide the desired size and shape. In this (1) As the degree of prior deformation increases, nucleation case, the aim will be to obtain fine recrystallized grains in is favoured and the recrystallized grain size dR becomes order to have good toughness; otherwise, the toughness will finer. Because increasing degree of prior deformation decrease if the finished product is heated to the grain growth will result in an increase in the number of high stress- or stage. Therefore, the recrystallized grain size, dR, obtained high energy-points acting as the nucleation sites, from by annealing treatment is very important in determining the which greater number of grains will be recrystallized.

420 10 Fundamentals of Mechanical Working (2) As the time at any temperature above the recrystal- procedure of cold working followed by annealing is not lization temperature increases, grain growth is favoured advisable because the recrystallization process progresses and dR increases. relatively fast and is quite sensitive to slight temperature variations in the furnace and thus difficult to control. (3) As the annealing temperature above the recrystalliza- tion temperature increases, grain growth is favoured and 10.2.2 Temperature Limits for Hot Working dR increases. In practice, the equipment used to carry out the deformation (4) The faster the rate of heating to the annealing temper- process decides the strain rate to be imparted to the ature, the finer is the recrystallized grain size dR because work-piece, but for given equipment the strain rates become the shorter time of heating will lead to the formation of fixed, whereas the deformation temperature can be changed more nuclei and restrict the growth of grains resulting in or regulated as per desire. Since the higher the deformation a fine-grained product. temperature, the lower the flow stress at a given strain rate and the higher is the maximum amount of deformation (5) The greater the amount and the finer the distribution of possible under a given load; therefore, it is advantageous to insoluble impurities, the finer is the recrystallized grain deform the work-piece at as high a temperature as possible. size dR because insoluble impurities not only act as Hence, the upper limit for hot working should be the solidus centres of heterogeneous nucleation and increase the temperature of the work-piece. Generally, the maximum nucleation rate, but also pin the grain boundaries and temperature for hot working is limited to 50 °C below the act as barriers to the growth of grains. temperature at which either melting of the lowest melting constituent in the material or excessive oxidation at the grain (6) As the rate of cooling from the annealing temperature boundary takes place. Incipient melting of a lower melting decreases, grain growth is favoured and dR increases. constituent, often present only in minute amounts, that is segregated at grain boundaries forms a grain-boundary film (7) The finer the initial grain size of the material, the finer of the lower melting constituent. When the material is is the recrystallized grain size dR, provided other factors stressed or deformed, only a very small amount of this remain the same because more grain-boundary areas per intergranular film causes the material to break apart by unit volume in an initially fine-grained material act as separation along grain boundaries rather than to deform and centres of heterogeneous nucleation and increase the results in a scrapped product. Such a condition is called hot nucleation rate resulting in a fine-grained product. shortness or burning. Since brittleness developed by hot shortness or burning hinders hot-working operations, the However, if it is desired to obtain a final product having upper limit for hot working is generally 50 °C below the strength more than that in the fully annealed condition, then melting point. Sometimes, the upper temperature limit for the final operation must be a cold working step following the hot working is maintained below the temperature at which last recrystallization annealing treatment. The degree of cold the material undergoes allotropic or other phase transfor- reduction required to impart to the work-piece depends on the mation producing allotrope or phase of higher flow stress extent of the desired strength — the higher the percentage of than the parent one. cold reduction, the higher is the strength. However, the cold-worked material must be finally subjected to stress relief Since hot working is the deformation followed by annealing or stress-relieving treatment to remove residual instantaneous recrystallization, the lower temperature limit stresses, which may otherwise cause stress corrosion cracking for hot working of a material is the lowest temperature at or produce distortion in the cold-worked member. The stress which the rate of recrystallization is rapid enough to remove relief annealing or stress relieving treatment consists in instantly the effects of deformation on structure and prop- heating a cold-worked material below its recrystallization erties of the material. Since time is required for recrystal- temperature, TR, holding long enough to reduce residual lization and the effects of working are instantaneous, so in stresses by recovery process and then cooling slowly enough practice the indication of deformation remains at the end of to minimize the development of new residual stresses. Since hot working processes unless the processes are slow enough the mechanical properties of the material are essentially to permit full recrystallization. However, the lower temper- unchanged during this treatment, a cold-worked material after ature limit for hot working, TL, is related to the recrystal- stress-relieving treatment will maintain the strengthening lization temperature, TR, of the individual material, and produced by cold working without the harmful effects of neither TR nor TL is a fixed temperature in the sense of a residual stresses present in the as-deformed condition. The melting temperature; rather, both of them depend in the above procedure of recrystallization annealing followed by similar manner mainly on the following variables. The final cold working and subsequent stress relieving treatment is successfully applied to develop a certain combination of strength and ductility in the final product, which might be possible to achieve if a fully cold-worked material is partially softened by recrystallization annealing. But this latter

10.2 Temperature and Strain Rate 421 influence of individual factor on TR and TL has been dis- finishing working temperature is lowered to the point, which cussed below, assuming that other factors do not change. is usually just above the minimum hot working temperature so that during cooling from this finishing temperature there (1) The greater the amount of deformation, the more the will be negligible grain growth. In the last pass, a large stored strain energy and the less is the thermal energy or amount of deformation is imparted to the work-piece so that temperature required to cause recrystallization, and a product with recrystallized fine grain sizes, which is usu- thus, the more will be the decrease in TR and TL. ally desired, can be achieved. (2) Increasing the duration at temperature of hot working 10.2.3 Hot Working Versus Cold Working decreases TR and TL, because recrystallization is a time- and temperature-dependent phenomenon. However, When the hot working operation is compared with the cold increasing hot working temperature by 10 °C is working operation, it can be seen from the following dis- approximately equivalent to doubling the hot working cussion that each of these operations has some advantages as duration, and hence, the temperature is a more impor- well as disadvantages. tant factor than the time. Advantages of hot working over cold working (3) The higher the rate of working, i.e. the strain rate, the more is the increase in TR and TL. It is because of the Hot working (HW) Cold working (CW) necessity of increasing softening rate by recrystalliza- tion so as to balance the increased rate of strain hard- (1) Energy or power or load (1) Energy or power or load ening caused by rapid rate of working. required for deformation is less, required for deformation is because flow stress decreases higher, because flow stress is (4) The more rapid the cooling rate from the temperature of with increasing temperature and higher at lower temperatures hot working, the more will be the increase in TR and TL. remains essentially constant and increases with deformation When the material is cooled rapidly, less time is during deformation due to rapid due to strain hardening that is available for recrystallization to occur, and so it is elimination of strain hardening not relieved in CW. Due to required to increase the recrystallization temperature. by the process of requirement of higher recrystallization. Due to deformation load, expensive and (5) The finer the initial grain size of the material, the higher requirement of less deformation more powerful equipments are the ratio of grain-boundary area to volume and the more load, cheaper and less powerful required for deformation in CW is the strain hardening for the same degree of defor- equipments are required to carry mation. This causes an increase in the stored strain out the deformation process energy and the decrease in the thermal energy or tem- perature required to cause recrystallization, which (2) Much larger deformation (2) Due to strain hardening, results in the decrease of TR and TL. without cracking is possible to subsequent deformation achieve, since recrystallization becomes increasingly difficult, (6) As compositional purity of the material increases, the does not allow the deformation and so total possible energy required to overcome the rigidity of the distorted resistance to increase with strain deformation is less. Excessive lattice decreases, which causes to decrease TR and TL. On room temperature working the other hand, solid solution alloying additions always causes cracking due to raise TR and TL, due to increase in the rigidity of the lattice. decreased toughness. To achieve large deformation, Hot working is generally carried out at temperatures intermittent recrystallization above 0:6 Tm and at strain rates that vary from 0.5 to annealing is essential 500 s−1. In hot working, usually large strain of the order of 200–400% is imparted to the work-piece. Several steps or (3) Coarse columnar grains of (3) Elongated distorted grains passes are mostly used to carry out hot working operations. casting and distorted grains with high density of To convert a cast ingot into a wrought product, commonly produced by deformation, both dislocations are produced by the first step is to employ hot working processes, such as are transformed into strain-free CW. Chemical heterogeneity is forging, rolling or extrusion at a temperature near the upper smaller equiaxed recrystallized not reduced due to lower rate of temperature limit for hot working. During the intermediate grains in HW. Further, high diffusivity, and cavities are not passes, the working temperature is gradually decreased but temperature of HW reduces eliminated at temperatures of kept well above the lower temperature limit for hot working. chemical inhomogeneities of CW This processing at high temperatures is not only economical cast ingot due to rapid rate of due to decrease in the flow stress but also makes larger diffusion and eliminates deformation possible, which is desirable at the initial blowholes and porosities by breakdown of the ingot. Of course, there will be some grain welding these cavities growth subsequent to the recrystallization at these high temperatures, which is not desired. In the last pass, the (4) Mechanical properties are (4) Elongated grains result in a isotropic due to formation of marked anisotropy in the equiaxed grains mechanical properties; i.e., properties vary with the testing direction (continued)

422 10 Fundamentals of Mechanical Working Hot working (HW) Cold working (CW) Hot working (HW) Cold working (CW) (5) Ductility of the product is (5) Ductility and toughness of non-uniformity in mechanical (7) No discontinuous yield point usually high. HW improves the product are usually low. To properties over the cross- is observed in cold-worked toughness over the cast state make the product ductile, the section of the product low-carbon steels due to refinement of grains and end operation after the CW elimination of cavities, the operation is over must be (7) A discontinuous yield point presence of which might cause recrystallization annealing is exhibited by hot-worked stress concentrations. low-carbon steels Toughness is usually better in a hot-worked metal which has The above comparison clearly indicates that if the fine recrystallized grains objective is to change the shape of a material as rapidly and economically as possible, without giving importance to Advantages of cold working over hot working surface finish, dimensional accuracy and final mechanical properties, hot working must be used. On the contrary, cold Hot working (HW) Cold working (CW) working is the most suitable if it is desired to have products with high-strength and hardness, good surface finish, (1) Good surface finish is (1) Surface finish is of a high dimensional accuracy of a high order. difficult to achieve due to oxide order; i.e., CW usually produces and scale formation; i.e., HW a clean and bright surface usually produces an oxidized discoloured surface 10.2.4 Warm Working (2) Dimensional tolerances for (2) Dimensional accuracy of the Cold working of most metals is carried out usually at room products are greater, because product is of a high order temperature. Sometimes, a compromise is made between hot allowance must be provided for and cold or room temperature working by working metals at expansion during heating and an intermediate temperature, i.e. by working metals in contraction during cooling warmed condition. This is called warm working, which is defined as the plastic deformation carried out at temperatures (3) Involvement of high (3) As the working temperature below the recrystallization temperature of the material but temperature usually causes is usually low, no such surface above (3/10)th of the recrystallization temperature on an surface reaction to occur reaction or oxidation and absolute scale. The temperature in warm working should be between the material and the material loss or surface low enough to provide sufficient strain hardening for atmosphere. HW ordinarily decarburization of steel takes improvement in mechanical properties, but high enough to carried out in air results in place prevent excessive deformation loads and the possibility of oxidation that causes a fracture on working. The purpose of warm working is to substantial amount of material combine the advantages of both cold or room temperature loss. In case of steel, surface working and hot working into a single operation. The decarburization also takes place advantages of warm working are as follows: and the removal of this decarburized layer often requires 1. The extent of possible deformation without fracture in extensive surface finishing each step of working is more in warm working than in room temperature working, and so the number of work- (4) Severe embrittlement of (4) Reactive metals are not ing steps required to get the desired shape of the material reactive metals like Mo, Ti and embrittled by oxygen during will be less in the former than in the latter. W occurs in the presence of CW, but during annealing of oxygen, and use of inert reactive metals in the 2. Deformation load required in warm working is reduced atmosphere or protection from cold-work-anneal cycle, compared to that in room temperature working due to the air by a suitable barrier is embrittlement will take place, decrease in the flow stress of material with increase in the required for their HW and so annealing must be working temperature. performed in vacuum or inert atmosphere 3. Energy consumption is less in warm working than that in room temperature working because the intermittent (5) Products are generally soft (5) Products have high strength annealing is not required in the former. and have low strength and hardness. Any desired degree of strength and ductility 4. Warm working offers improved dimensional control in can be achieved by suitably comparison with hot working. adjusting the cold-work-anneal cycle 5. Quality of surfaces is better in warm working than that in hot working. (6) As deformation and cooling (6) Cold-worked and annealed rate at the surface are greater product shows more uniformity than those at the interior, so the in the microstructure and recrystallized grains at the mechanical properties over the surface are finer than those at cross-section the interior. This difference in the microstructure results in the (continued)

10.2 Temperature and Strain Rate 423 Warm working at moderate temperatures may be applied The heat absorbed per unit volume, ðHDÞV ; by the to wire drawing and sheet metal forming operations to take work-piece is given by advantages of yield strength reduction and ductility increase without facing the expenditure and associated problems of ðHDÞV ¼ q Csp:heatDTD ð10:2Þ high-temperature heating. Warm working is particularly suitable to hard heat-resistant alloys that are difficult to cold where q = the density of the work-piece, Csp:heat = the work and that contain phases which would soften or melt at specific heat of the work-piece, and DTD = the rise in tem- temperatures of hot working. However, warm working can perature due to plastic deformation. be used for medium carbon and nickel–chrome steel between 400 and 700 °C, but it is not usually applied to Since bðUDÞV ¼ ðHDÞV ; so from (10.1b) and (10.2) we age-hardening alloys. obtain the rise in temperature, DTD; for a frictionless (ideal) plastic deformation process under adiabatic condition, in which no heat is transferred to the surroundings, as follows: 10.2.5 Temperature Change During Working DTD ¼ brme ð10:3Þ qCsp:heat During mechanical working, the temperature of the 10.2.5.2 Friction at the Work-Piece–Die Interface deforming material depends on According to Coulomb’s law of friction, friction force F is given by the product of the friction coefficient and the nor- • The ideal plastic deformation of the work-piece. mal force at the work-piece/die interface, that is • The friction at the interface of the work-piece and the F ¼ lpA ð10:4Þ tool or die. • The initial temperatures of the work-piece and the tools where or dies and the heat transfer between them. 10.2.5.1 Ideal Plastic Deformation l the coefficient of friction at the interface of the work-piece and the die, The heat generated due to plastic deformation causes the p the stress normal to the interface of the work-piece and initial temperature of the work-piece to rise during working. the die, and The strain energy per unit volume, ðUDÞV ; consumed in the plastic deformation is equal to the area under the flow curve A the surface area at the interface of the work-piece and of effective stress versus strain up to the effective strain e; the die. imparted to the work-piece, as shown below. Work done ðW:D:FÞ by the friction force F is equal to Ze ð10:1aÞ W:D:F ¼ F Â displacement ð10:5Þ ðUDÞV ¼ r de ¼ rme ¼ F Â ðv DtÞ ¼ l p Av Dt 0 where e = the effective strain up to which the material is where v = the velocity at the interface of the work-piece and plastically deformed, and rm = the average value of effective the die, and Dt = the time interval of consideration. stress, r; over the strain range of 0–e. This work done ðW:D:FÞ is converted to heat that is The effective stress and strain are considered to describe absorbed by the work-piece, and the temperature of the the flow curve because they are independent of the state of work-piece increases. The heat absorbed, HF; by the stress. However, only a small fraction of this strain energy work-piece is given by UD is stored in the material as the energy associated with the defect structure, mainly dislocations and vacancies, which HF ¼ V q Csp:heat DTF ð10:6Þ may vary from about 5% at low strains to 1 or 2% at high strains. The rest energy is converted to heat that causes to where V = the volume of the work-piece subjected to the rise the initial temperature of the material, which can be increase in temperature, and DTF = the rise in temperature obtained from (10.1a) as due to friction. bðUDÞV ¼ b rm e ð10:1bÞ Since W:D:F ¼ HF; so from (10.6) and (10.5) we obtain the rise in temperature, DTF; due to friction: where b = the fraction of strain energy released as heat DTF ¼ l p Av Dt ð10:7Þ %0.98. V q Csp:heat

424 10 Fundamentals of Mechanical Working 10.2.5.3 Heat Transfer Between the Work-Piece The rate of heat lost by the plate or cylinder and the Tools or Dies ¼ ÀV q Csp:heat dT ð10:8Þ The temperature usually becomes the maximum at the dt interface of the work-piece and the die, where the heat is generated by the friction, and thereafter, it drops down The rate of heat transfer to the tools or dies ð10:9Þ towards the interior of the work-piece and into the die. ¼ htr:coeff: AðT À TdÞ Assume that the work-piece is initially at a uniform tem- perature of T0 and cooled during deformation between two where die surfaces, whose initial uniform temperature was Td. If the work-piece is thick, the internal temperature distribution V the volume of the plate or cylinder, within it will vary with time, as shown in Fig. 10.12a. If the A the surface area of the plate or cylinder that is work-piece has a thin section and/or a high thermal con- exposed to the tools or dies, ductivity, kth:cond:; then the temperature gradients within the T the instantaneous temperature of the plate or work-piece can be neglected, as shown in Fig. 10.12b, and cylinder, the temperature of the work-piece may be considered to be a q the density of the work-piece (plate or cylinder), function of time only. For simplicity, let us consider the Csp:heat the specific heat of the work-piece (plate or cylinder). deforming material to be a thin flat plate or cylinder having high thermal conductivity kth:cond:: For the following analysis Now, equating (10.8) with (10.9), we get to be valid, Newtonian cooling is assumed, in which ðhtr:coeff: LÞ=kth:cond: 0:1; where ÀV q Csp:heat dT ¼ htr:coeff: AðT À TdÞ; dt htr:coeff: the heat transfer coefficient between the deforming On rearranging, T dT ¼ À htr:coeff: A dt ð10:10Þ L material and the tools or dies, À Td V q Csp:heat the ratio of the volume V; of the work-piece to the kth:cond: surface area A; in contact with the tools or dies Initially, at time t = 0, the temperature of the work-piece = V/A and T = T0 and after a deformation time of t = tn, assume that the the thermal conductivity of the deforming temperature of the work-piece at that instant is T = Tn. So, material. on integrating (10.10) over the time interval from t = 0 to t = tn, we obtain the temperature, Tn, of the work-piece as As the uniform initial temperature of the plate or cylinder, follows: T0, is higher than that of the dies, Td, so the heat will be lost from the plate or cylinder and transferred to the tools or dies. ZTn Ztn dT htr:coeff: A T À Td ¼ À V q Csp:heat dt; (a) (b) T0   0 Tn À Td htr:coeff: A T0 Time t = 0 T0 Or; ln T0 À Td ¼ À V qCsp:heat tn; Time t = 0 htr:coeff: A tn  V q Csp:heat ) Tn ¼ Td þ ðT0 À TdÞ exp À ð10:11Þ Since L = V/A, as defined earlier, now substituting A/ V = 1/L, in (10.11), we obtain Time t = ∞  Td htr:coeff: tn Time t = ∞ Tn ¼ Td þ ðT0 À TdÞ exp À L q Csp:heat ð10:12Þ Td x=0 x=L Equation (10.12) is the basic relation for the variation of the average temperature of the work-piece (assumed to be a (Temperature) T thin one), which is being cooled during deformation between two die surfaces. However, the parameter L in (10.12) x depends on the geometry of the work-piece and its surface (Thickness of area in contact with the tools or dies. work-piece from its centre) For example, in case of a thin flat plate or cylinder cooled between two flat dies or tools, the two flat surface areas of Fig. 10.12 Transient temperature distribution during cooling of a a thick the plate or cylinder are exposed to the tools or dies. So, the plate and b a thin plate

10.2 Temperature and Strain Rate 425 parameter L, which is the ratio of the volume to those flat • To make the deformation possible for many metals that areas of the plate or cylinder, is given by are difficult to form like titanium and tungsten alloys. L ¼ V ¼ ðflat surface areaÞ Â thickness of plate, or, length of cylinder It is seen that increasing strain rate and decreasing tem- A 2  ðflat surface areaÞ perature increase the flow stress of the work-piece and therefore the deformation load, and so the loads required to ¼ thickness of plate, or, length of cylinder ; carry out deformation can be minimized by choosing their 2 values. The smaller the deformation load, the less will be the power required and the smaller and cheaper will be the i.e. L is the semithickness of the plate or the semilength of equipment required for deformation. the cylinder. In open-die forging where a cylindrical work-piece is Similarly, when the curved surface of a thin cylinder is compressed between two flat dies, the true strain rate, e_; is cooled during deformation between concave dies or tools, usually defined as the curved surface area of the cylinder is exposed to the tools or dies. So, the parameter, L; which is the ratio of the volume e_ ¼ de ¼ dh=h ¼ dh=dt ¼ v to that curved surface area of the cylinder will be given by pðradiusÞ2Âlength dt dt hh ð10:14Þ 2 p radius  length L ¼ V ¼ ¼ radius of cylinder ; A 2 where i.e. L is the semiradius of the cylinder. h the instantaneous height of the work-piece, Hence, adding the temperature rise due to plastic defor- v the deformation velocity and dh the reduction in height of the work-piece in the time mation (10.3) and friction (10.7) with the temperature of the work-piece due to heat transfer between the work-piece and interval dt. the tools or dies (10.12), the final average temperature, T, of the work-piece at a time t = tn is given by Since in many working processes converging dies are T ¼ DTD þ DTF þ Tn ¼ b rm e þ l p Av Dt used to deform the work-piece where between the die gap, qCsp:heat V qC!sp:heat htr:coeff: tn ð10:13Þ h varies with distance x, along the longitudinal axis of the L q Csp:heat work-piece, it is appropriate to define a distance average þ Td þ ðT0 À TdÞ exp À strain rate e_x; as follows: 1 ZL L e_x ¼ e_ dx ð10:15Þ 10.2.6 Strain-Rate Effects 0 The principal effects of increasing the strain rate or defor- where L = the length of deformation zone, i.e. contact length mation velocity in the working processes are as follows: between the work-piece and the tools. • To increase the flow stress of the work-piece and there- Evaluation of average strain rate in terms of time is more fore the deformation load. acceptable. If the travelling time of the work-piece through the die is tn, the time average strain rate e_t; is given by • To decrease the ductility of the work-piece. • To increase the temperature of the work-piece because of 1 Ztn tn adiabatic heating. e_t ¼ e_ dt ð10:16Þ • To improve lubrication at the tool–job interface, so long 0 as the lubricant film can be maintained. • To cause heating of the fluid lubricant resulting in a Typical values of deformation velocity and strain rate decrease in viscosity which in turn will cause a decrease encountered in different testing and forming operations are in the film thickness due to squeezing out of the lubricant from the work-piece–tool interface and consequent listed in Table 10.1. It must be noted from Table 10.1 that increase in the interfacial friction. • To decrease the coefficient of sliding friction in the the crosshead velocity of the standard tensile testing machine regime of high-speed sliding (Williams and Griffen 1964; is significantly lower than the deformation velocity of most Earles and Powell 1966/67; Kadhim and Earles 1966/67; commercial forming equipment. These deformation veloci- Ettles 1986), as discussed in Sect. 10.3.1.1. ties in the working operations can produce high strain rates, if deformation is concentrated into a narrow zone. For example, a strain rate of more than 105 s−1 can be attained during drawing of a thin wire at a velocity of 40 m s−1.

426 10 Fundamentals of Mechanical Working Table 10.1 Typical values of Operation Velocity (m s−1) Strain rate (s−1) velocity and strain rate for Tension test 6 Â 10−7–6 Â 10−3 10−5–10−1 different testing and forming Superplastic forming 6 Â 10−7–6 Â 10−4 Less than 10−2 operations (Dieter 1988) Mechanical press 0.06–1.5 0.5–500 Hydraulic forging press 0.06–0.30 Hydraulic extrusion press 3 Â 10−3–3.0 102–104 Charpy impact test 3–6 102–104 Forging hammer and rolling 3–10 5–3 Â 102 Rod drawing 0.15–1.5 102–4 Â 105 Wire drawing 10–40 104–108 High energy rate forming 30–200 From Table 10.1, it can be seen that the forming opera- glass blowing operation can be used to form these alloys, tions at the two extremes of the strain-rate spectrum are high and these forming operations are very cheap. The flow stress energy rate forming (HERF) and superplastic forming. required for superplastic forming operation is usually quite In HERF, as the name suggests, the delivery of deformation low, of the order of 5–30 MPa. This advantage is useful energy occurs at a much rapid rate than in conventional during mechanical working operations of superalloys that forming operations. HERF processes are also known as are difficult to work and for embossing finer details in sev- high-velocity forming processes, because forging, extrusion, eral applications. sheet forming, etc., are carried out by these processes with forming velocities as high as 200 m s−1, in contrast to the 10.2.7 Choice of Allowable Hot Working maximum forming velocity of 40 m s−1, as obtainable in the Temperature Range wire drawing. The energy of these processes is used to produce high particle velocities unlike the conventional In discussing the choice of allowable hot working temper- practice. For many materials, the ductility, measured by ature range, it is the initial preheat temperature of the engineering strain to fracture, increases with strain rate work-piece that is important. The heat of plastic deformation beyond the usual working range until the strain rate reaches causes this initial temperature to rise during working, if the a critical value at which there is a sharp drop of ductility. strain rate is high approaching adiabatic deformation con- HERF processes have been discussed in Chap. 16. dition. On the other hand, the temperature of the work-piece may drop due to loss of heat to the surroundings, if the strain On the other hand, superplastic forming must be per- rate is low. If the final temperature of the work-piece is too formed at temperatures above 0:4 Á Tm; (where Tm is the high, a coarse-grain-sized product having poor strength and melting point of the superplastic material in Kelvin) and toughness will be obtained or excessively high final tem- generally at strain rates below 0.01 s−1. This is the limiting perature may result in hot shortness or burning. If the final strain rate above which no superplastic forming can be temperature is too low, the result will be a cold-worked performed and below which superplastic forming can be product. Hence, the allowable hot working temperature performed over a range of strain rate. Generally, the material range can be explained diagrammatically with the help of which undergoes superplastic forming must have a fine grain Fig. 10.13, in which ‘work-piece temperature’ as abscissa size of the order of 1–3 lm which must not be allowed to and ‘amount of deformation’ as ordinate have been consid- increase by recrystallization, although forming temperatures ered and the effects of other variables controlling working are sufficient for recrystallization. Mainly for this reason, the range, viz. applied pressure and strain rate, have been two-phase structure, mostly of eutectic or eutectoid com- included in the diagram. positions, is required for superplastic forming, although some single-phase metals with very fine grain size also show For a given working pressure and temperature, there will be superplastic behaviour (Johnson 1970). An important char- an upper limit of deformation that the work-piece can undergo. acteristic of a superplastic material is that it has a high value This deformation limit is assumed to base on the flow resis- of strain-rate sensitivity m (0.3 < m < 1.0), due to which the tance of the work-piece and not on its ductility. Under the same material exhibits a pronounced resistance to necking and a applied pressure, if the temperature of the work-piece increases high elongation up to 20Â or more. The behaviour of the maximum amount of possible deformation will increase superplastic material is called superplasticity, which has due to lowering of the flow stress with increasing temperature. been discussed in Sect. 1.11.2 of Chap. 1. In Fig. 10.13, a given applied pressure has been indicated by a line AB, below which the deformation is allowed but above Some examples of superplastic alloys are Al–33% Cu, Zn–22% Al, Fe–25% Cr–6.5% Ni. Processes analogous to

10.2 Temperature and Strain Rate 427 A given applied B pressure p3 > p2 > p1 ... D ε3 > ε2 > ε1 Increasing Zone of pressure (p) no deformation Amount of deformation Zone of Deformation, percentno deformation. Hot . ε1 short Isothermal. ε2 A given strain rate Cold short ε3 Strain p3 Adiabatic rate. Deformation p2 (ε ) A zone p1 Solidus Solidus Possible temperature deformation Workpiece temperature C Fig. 10.13 Schematically showing maximum possible hot working Workpiece temperature range for a given applied pressure and strain rate Fig. 10.14 Schematic effect of varying pressure and strain rate on the allowable hot working range which the deformation is not allowed under that applied applied pressure. Thus, when the strain rate becomes the pressure. Apart from pressure, the temperature range allowed most rapid leading to adiabatic condition, the maximum for possible deformation depends also on the strain rate. If amount of possible deformation at any work-piece tem- deformation is carried out very slowly and the amount of perature for a given applied pressure will be the least. deformation is very less so that there is no rise in the temper- ature of the work-piece, obviously the solidus temperature of Hence, we can conclude that increasing strain rate will the work-piece is the upper limit on the temperature scale, decrease the deformation range of a material, whereas although this limiting temperature will decrease somewhat due increasing applied pressure will have the opposite effect. to the risk of incipient melting or hot shortness. At a given Again, since the temperature of deformation depends on the strain rate of deformation, the greater the amount of deforma- strain rate of deformation, so both of them must be con- tion, the greater is the temperature rise of the work-piece, and trolled jointly in most forming operations. hence, the work-piece temperature will have to be lowered to a greater extent to maintain its final temperature below the hot 10.3 Friction shortness temperature. In Fig. 10.13, a given strain rate has been represented by a line CD having a negative slope, below It is well established that whenever two solid surfaces in which the deformation is allowed but above which the defor- contact move relative to each other, a force is developed at mation is not allowed at that strain rate and thus the line CD will the contact surface to oppose this movement. This resisting limit the upper temperature of the hot working range. Fig- force is commonly called the friction force. The friction ure 10.14 shows schematically the effects of varying pressure plays an important role in the mechanical working processes, and strain rate on the allowable hot working temperature range, where the surfaces of work-piece and the deforming tools are after the work of Hirst and Ursell (1958). This diagram shows: in contact with each other during deformation. The friction stress, often referred to as frictional resistance, is measured • As the applied pressure increases, the maximum amount in force units per unit surface area of contact and denoted by of possible deformation or the allowable working range s, that is directed opposite to the relative motion between the at a given work-piece temperature and strain rate will two solids. Since the surface area of contact is a boundary of increase, because higher the applied pressure, the more is the deformed material, the frictional stress or resistance that the margin above the flow resistance of the material. is parallel to the contacting surfaces will be the tangential shear stress at the boundary of the material. The phe- • The higher the strain rate, the greater will be the retention nomenon of friction has been much studied (Bowden and of heat in the work-piece, and hence, the work-piece Tabor 1954 and 1964; Bishop 1958; Rooyan and Backofen temperature will have to be lowered to a greater extent to 1960; Pinkus and Sternlicht 1961; Fukui et al. 1962), but the maintain its final temperature below the hot shortness knowledge about friction is yet very little for the formulation temperature. As the strain rate increases, the flow stress of of the exact functional relationship between the friction the material will increase, which will cause the maximum stress and other variables, upon which the friction depends. amount of possible deformation or the allowable working range to decrease at a given work-piece temperature and

428 10 Fundamentals of Mechanical Working These variables include nature of materials, shape of the apparent area of contact, A, between a pair of sliding sur- specimen, deformation velocity, temperature and humidity faces and the magnitude of sliding velocity. Coulomb con- or atmospheric conditions. The understanding of friction ducted experiments on dry unlubricated clean surfaces. phenomenon requires the study of tribology science, an interdisciplinary field involving surface chemistry, mechan- The elements of friction theory (Bowden and Tabor 1954 ics and material science. Basic studies of friction have been and 1964) are based on the observation that all apparently mostly carried out under light loads without causing plastic flat and smooth material surfaces are in reality not perfectly deformation of the work-piece, such as study of friction smooth when considered on an atomic scale. Real surfaces during movement of a shaft in a bearing. Since the contain many irregularities in the form of peaks and valleys, mechanics of friction is an extremely complicated phe- which are large when viewed on a microscopic scale. The nomenon, our discussion on friction will be restricted to a magnitude of these irregularities may be of the order of very elementary level. Of the numerous developed mathe- thousands of angstrom units for highly polished surface matical descriptions of friction, the most important ones are finishes and increases with the degree of surface roughness. considered in the text. When two solid surfaces are placed in contact or one slides over another under a light load, a real contact between these 10.3.1 Coulomb’s Law of Sliding Friction surfaces takes place only at a few relatively isolated peak spots, which are called asperities, as shown in Fig. 10.15. The fundamental laws of friction, declared first by Amonton The summation of the contact areas at all of the asperities is in 1699 and later by Coulomb in 1781, are that during equal to the real area of contact Areal. At the beginning of sliding the tangential frictional force F is directly propor- contact, since Areal tends to be very small under light loads, tional to the applied normal load P and is independent of the so the local pressure at the asperities will be sufficiently high to cause plastic deformation of the ductile materials, such as metals, at the contact regions, as shown in Fig. 10.16 Fig. 10.15 Surface irregularities Areal = a1 + a2 + + an Asperities showing asperities and their contact Fig. 10.16 Real contact and a1 a2 adhesion during sliding of solids. P Elastic deformation Plastic deformation occurs at the asperity junctions (Bowden 1957) F = Areal τ Plastic flow Plastic flow

10.3 Friction 429 (Bowden 1957). As the load normal to the contacting sur- l ¼ F ¼ Pðs=pÞ ¼ s ð10:20Þ faces is increased, the extent of these asperity contacts will PPp increase and fresh contacts will be created. In this model, since the contact surfaces of the materials are dry, high where l is called the Coulomb’s coefficient of sliding friction pressure and plastic deformation at the asperities will cause and the other terms are the same as described in (10.17) and cold welding of asperity junctions. The strength of this (10.18). The coefficient of friction (COF), l, is a dimen- welded bond depends on several factors, such as materials in sionless scalar quantity and considered as constant for a contact, temperature, cleanliness of the contacting surfaces, given pair of materials under constant temperature and sur- i.e. the presence of a thin oxide film, layers of moisture or face conditions. COF is also said to be independent of gas molecules. Under increasing load, the number and areas deformation velocity, applied normal load and area of con- of the welded junctions will increase and these welded tact. Equation (10.20) shows that l is directly proportional junction areas, Areal, will carry the entire load between the to s and inversely proportional to p. If two materials of two surfaces. It must be noted that in most cases, the real different hardness slide over one another, the values of s and area of contact Areal is only a small fraction of the apparent p of the softer material have to be considered in (10.20). It area of contact A between a pair of sliding surfaces and can has been seen that for the same value of s, the coefficient of be given by friction will be lower for the harder material because of its higher yield pressure p (Green 1955), which being a normal Areal ¼ P or; P ¼ Areal p ð10:17Þ stress is equivalent to the hardness of a material for a plas- p tically deformed asperity. According to Von Mises’ yielding criterion, generally applied to most of the working processes, where the value of l varies from 0 to 0.577, although this value varies from 0 to 0.5 according to Tresca yielding criterion, P the applied load normal to the contacting surfaces; both of which have been analysed in Sect. 10.3.2. p the yield pressure, i.e. the yield stress in compression In practice, the sliding friction may involve additional of the softer material normal to the contacting effects such as ploughing out softer work-piece by the surfaces that causes the plastic deformation of the asperities of the harder tools and interlocking of the surface softer material and irregularities. Thus, the total sliding friction force FT during Areal the real area of contact, which is the sum of the area sliding will be equal to the sum of the three components as of welded junction formed. follows: The sliding of one surface above the other will be pos- FT ¼ F þ FP þ FI ð10:21Þ sible only after the welded asperity junctions begin to shear. The shearing force F required to tear apart all welded where junctions is F ¼ Areal s ð10:18Þ FP is the frictional force due to ploughing, FI is the frictional force due to interlocking of the surface where irregularities, and F the shearing friction force necessary to tear apart all F is the shearing friction force necessary to tear apart all welded junctions that acts in a direction parallel to the contacting surfaces; welded junctions. Areal the real area of contact, which is independent of the Since the sliding surfaces are usually well ground and size of the sliding surfaces; and relatively smooth, the frictional resistance caused by surface irregularities is practically negligible. The frictional resis- s the tangential shear stress at the contacting surfaces tance due to ploughing depends on the difference in hardness for shearing through the asperities. of the two sliding surfaces as well as on their smoothness. When a hard material, such as tools or dies, slides over a soft On substituting for Areal from (10.17) in (10.18), we one like work-piece, the asperities of the hard material (tools obtain or dies) will penetrate the soft material (work-piece) to an appreciable depth below the surface and displace a volume F ¼ P s ¼ P s ð10:19Þ of material from work-piece proportional to the total length p p of sliding and the cross-sectional area of the asperities. This is known as ploughing. The ploughing force (Goddard and Equation (10.19) indicates that F, like Areal, is directly Wilman 1962) is related to the flow properties of the proportional to P. Since the coefficient of friction, l, is work-piece and the size and shape of the asperities. defined by the ratio F to P, hence using (10.19), we obtain

430 10 Fundamentals of Mechanical Working Ploughing is particularly strong if the surface of the hard more typical in case of a ground finish roll surface. For cold material is rough and has asperities of pyramidal shape. But drawing and deep drawing of steel, copper and brass using when dies or tools are smooth and the shape of the asperities hardened polished dies and efficient lubricants, the typical is quite rounded, the friction contribution from ploughing is values of l vary from 0.05 to 0.15. negligible. When a soft material slides over a hard one, the ploughing does not take place, but small fragments of the The value of l for hot working is usually higher than that soft material adhere to the hard one. for room temperature cold working, because oxidation takes place easily at elevated temperature and roughens the sur- In most cases, interlocking and ploughing are neglected, faces of the work-piece and the tools. It is better to say that and thus, the total friction force will be due to the shearing the value of l will depend more on the frictional charac- friction only, which will be given by teristics of the oxide film formed at elevated temperature than on the temperature of working. For example, the rolling FT ¼ F ¼ Areal s ð10:22Þ of steel at a temperature between 375 and 900 °C shows a value of l = 0.4, but on rolling the same steel above the 10.3.1.1 Friction in Forming Operations temperature of 1095 °C, a value of l = 0.2 is found. This The friction forces developed between the forming tools and drop in the value of l is due to a change in the frictional the work-piece during plastic deformation increase the characteristics of the oxide film formed at different resistance to plastic deformation offered by the work-piece, temperatures. which in turn will increase the normal pressure required to deform the work-piece. The friction between the tools and It has been observed in the regime of high-speed sliding the work-piece gives rise to tangential shear stresses along that the coefficient of sliding friction (COF) decreases with the contact surfaces. The relationship between the tangential the increase of the sliding speed (Williams and Griffen 1964; shear stress s, the applied pressure normal to the interface Earles and Powell 1966/67; Kadhim and Earles 1966/67; between the work-piece and the tools, p, and the coefficient Ettles 1986) and the statement that COF is independent of of friction l is generally expressed by Coulomb’s law of sliding velocity, as in the Coulomb’s law of friction, is found sliding friction as expressed by (10.20), which is repeated to be invalid. Some researchers (Kragelsky et al. 1982) say below for convenience. that the reason of decreasing COF with increasing sliding speed is the softening of the asperities over the frictional s¼l ð10:20Þ interface. Increasing deformation velocity produces an p improved lubrication at the work-piece–tool interface, so long as the lubricant film can be maintained, and this pres- It is assumed in practically all theories of mechanical ence of lubricant will reduce the interface shear stress, which working that in the presence of lubricant between the in turn will decrease the COF. In high-speed forming oper- work-piece and the tools, the (10.20) will be applicable. In ations, extremely high sliding speed results in lubrication (10.20), the values of s and p of the work-piece have to be breakdown between the work-piece and the tools, but at the considered, because the tools are harder than the work-piece, same time, it may generate sufficient temperature to melt the and so the latter undergoes plastic deformation, although the contact surface of the work-piece. This will produce a thin former may elastically deform. molten layer of material that will act as a lubricant at the work-piece–tool interface. This lubricant reduces the inter- It was observed by several authors (Chen et al. 2002; face shear stress, and the COF drops to a low value that will Grzesik and Nieslony 2004; Bonnet et al. 2008; Rech et al. depend on the thickness and viscosity of the liquid layer. 2009; Klinkova et al. 2011; Spijker et al. 2012) that the value of l depends upon the work-piece material, the Several studies (Rooyan and Backofen 1960; Pearsall and material used for the dies or tools, the surface roughness of Backofen 1963; Chen et al. 2002; Yasuhisa 2003; Chowd- the work-piece and the dies or tools and the efficiency of the hury et al. 2011; Al-Samarai et al. 2012) have shown that the lubricant, the normal load, the sliding speed, the temperature friction coefficient varies with the normal pressure and it of deformation and the relative humidity. For example, the usually decreases with the increase of applied normal load, value of l in working operations may be very low of the which will be explained in the next section. order of 0.01–0.05 under conditions of slow speed and excellent lubrication with highly polished tool surfaces. The It has also been found (Chowdhury et al. 2012) that the cold rolling of mild steel with flood lubrication may have a value of friction coefficient decreases with the increase of value of l = 0.05. For cold rolling of most metals, typical relative humidity. The increase of relative humidity may value is l = 0.10, with polished rolls, whereas l = 0.15 is moisten the contact surface that will cause some lubricating effect, and hence, the friction force will be reduced.

10.3 Friction 431 10.3.2 Shear Friction Factor lubrication may be difficult. Sticking may sometimes occur over the whole contact area of the work-piece, as in hot The foregoing Coulomb’s model of friction is valid for light rolling, or sticking often occurs over the contact area where normal load when the real area of contact Areal ( A, the deformation pressure is high. For example, in a hot-forged apparent area of contact. In such case, the COF is inde- work-piece, sticking friction may exist in the interior where pendent of applied normal load, which is known as Amon- deformation pressure is high, while sliding occurs in the ton’s law and is the basis on which Coulomb’s law stands. If periphery experiencing low deformation pressure. The the normal load P is gradually increased, Areal increases and interface friction factor m is also considered as constant for a approaches the apparent area of contact, A. Once the value of given pair of materials under constant temperature and sur- Areal becomes equal to that of A, the frictional shear force face conditions and is said to be independent of deformation F will not increase further even if the applied normal load velocity. P is increased because the frictional shear stress s, which is equal to F/A, cannot exceed the yield stress in pure shear, k, Von Mises’ yielding criterion is generally applied to most of the deformed work-piece material. When the maximum of the workinpg ffipffi rocesses, and according to this criterion, value of frictional shear stress smax ¼ k; the material at the since k ¼ r0 3; [see (1.63)], (10.24a) becomes interface of the work-piece will stick to the tool and sub- surface yielding of the work-piece will take place. s ¼ m pr0ffiffi ð10:24bÞ 3 When the normal load is high, the asperities undergo extensive plastic deformation causing the plastic zones Thus, according to Von Mises’ yielding criterion, the below each asperity to overlap. Then, due to this plastic deformation and the accompanying plastic constraint, the maxpimffiffium shear stress that a material can withstand is bond strength at the work-piece–tool interface is more than r0 3: Since the maximum frictional shear stress smax ¼ k; the shear strength of the material in a subsurface zone, say and the interface normal pressure p developed in most s0. Now, considering Areal % A, and thus, F ¼ A s0; the coefficient of friction, l, for subsurface shear can be obtained working processes is at least equal to the uniaxial yield from (10.20) as follows: stress, r0, of the material, i.e. pmin = r0, so the maximum value of Coulomb’s coefficient of sliding friction lmax under sticking condition according to Von Mises’ yielding criterion l ¼ A s0 will be P ð10:23Þ pffiffi smax k r0 3 ¼ p1ffiffi ¼ 0:557 lmax ¼ pmin ¼ r0 ¼ r0 3 ð10:25Þ Since A and s0 are nearly constant, (10.23) shows that as According to Tresca yielding criterion, the value of lmax is the normal load P increases the coefficient of friction l decreases, if bulk plastic deformation is taking place. lmax ¼ smax ¼ k ¼ r0=2 ¼ 1 ¼ 0:5 ð10:26Þ pmin r0 r0 2 However, at high contact pressure levels, the linear rela- tionship between s and p, as described by (10.20), may not be This value of lmax is convenient to remember and is often valid and the COF becomes meaningless when the value of quoted. In this context, it is to be noted that p can exceed r0 (l  p) exceeds k, which is the maximum value of s. Hence, to and reach a multiple of r0, in many forming processes. avoid this limitation of Coulomb’s model, some investigators However, lmax under plane strain condition, where pmin ¼ prefer to consider the following model of shear friction factor. r00 (plane-strain flow stress), give the same value of 0.5 In this model, it is assumed that the interface shear stress s is according to both yielding criteria of Von Mises and Tresca some constant fraction m of the yield stress in pure shear, k, of the deformed work-piece material, where m is called the in- as shown by the following (10.27): terface friction factor. Mathematically, it is expressed as s ¼ mk ð10:24aÞ ðlmax ÞP:Strain ¼ smax ¼ k ¼ k ¼ 0:5 ð10:27Þ pmin r00 2k The interface friction factor m is a dimensionless scalar quantity. The limits for m are 0 m 1. When m = 0, it If the frictional shear stress s is plotted as a function of is a condition of perfect sliding, i.e. a frictionless condition. the normal contact pressure p, the curve obtained will consist When m = 1, it results in the sticking friction, where no of theoretically two straight lines, as shown in Fig. 10.17. In relative motion between the work-piece and the tools takes this model, s at low pressure is proportional to p showing a place; i.e., movement along the interface is arrested, and constant coefficient of sliding friction l, given by Coulomb’s subsequent deformation occurs by subsurface shearing. model. However, s equals the yield stress in pure shear, k, Sticking friction often takes place in hot working, where and becomes constant at high interface pressure, p.

432 10 Fundamentals of Mechanical Working τ limitation of the above is the possibility of non-availability of information on r0 at the desired e_ and T: Of course, l can τ = k = constant be determined without knowing r0 by carrying out two sets k of plane strain compression test (Takahashi and Alexander 1961–62) on flat rectangular plate. If two specimens of the μ = τ = constant same material are given same reduction under identical p frictional condition to produce two values of the ratio of the deformation-zone length, L, to the deformation-zone height, h; and if the average deformation pressure, pC:F:; for each value of L=h is measured, then l can be evaluated by solving simultaneously the two sets of the following (10.28), which has been derived in Chap. 11 (see 11.30). pC:F: ¼ exp½ðl LÞ=hŠ À 1 ð10:28Þ r00 ðl LÞ=h p where r00 ¼ plane strain deformation resistance, i.e. plane Fig. 10.17 Hypothetical variation of frictional shear stress with strain flow stress in compression ¼ 2 k; where k is the yield normal contact pressure stress in pure shear. The shear friction factor model basically differs from and The measurement of Coulomb’s coefficient of friction, l; has certain advantages over the Coulomb’s model of fric- tion. These differences and advantages are as follows: in rolling has been discussed in Chap. 12. • The interface frictional shear stress s is independent of 10.3.3.1 Ring-Compression Test normal contact pressure p in shear friction factor model, To evaluate friction factor ‘m’, or Coulomb’s friction coef- whereas s depends on p in Coulomb’s model of friction. ficient ‘µ’, the ring-compression test suggested by Kunogi (1954) and Kudo (1955) and developed by Male and • The friction factor m is independent of p, but the Cou- Cockcroft (1964–1965) is widely used particularly for bulk lomb’s COF, l / 1=p; when the maximum frictional deformation processes such as forging. In this test, a thin flat shear stress, smax ¼ k; because k is not affected by p. ring of specific dimensions is upset in the axial direction between flat dies. The resulting changes in dimensions of the • The friction factor model provides easier computational ring depend on the extent of reduction in the thickness analysis of working processes by introducing mathe- direction and the frictional conditions existing at the inter- matical simplification with the use of m, although Cou- faces between the ring and the dies. The changes in lomb’s model with the use of l has been mostly applied dimensions of the ring after compression under high and low in the analysis of working processes. The use of m is interfacial frictional conditions in comparison with the particularly suitable in hot working processes associated original ring have been shown schematically in Fig. 10.18. If with large deformations, such as extrusion and forging the interfacial friction were zero, the ring would deform in operations, whereas l is used in cold working processes, the same manner as a solid disk; i.e., the inner diameter (ID) such as wire drawing and cold rolling operations. would expand radially outward by the same percentage as the outer diameter (OD) as the height of the ring is reduced. 10.3.3 Measurement of Friction With increasing friction during compression, the rate of increase of inner diameter decreases. For compression under The Coulomb’s coefficient of friction l, at any desired strain low interfacial friction, the inner diameter increases but is rate e_ and temperature T, can be determined from the ana- smaller than the zero-friction case. If the friction exceeds a lytical relation developed between the average deformation critical value depending on a certain reduction in height, pressure with Coulomb’s friction, pC:F:, and the flow stress, frictional resistance to outward radial flow becomes so r0, of the material. For this, it is required to know the value high that some of the ring material flows radially inward of r0 of the test specimen and to measure pC:F: of the test towards the centre, and so there must be a no-slip location carried out at the above same e_ and T. For example, in the between the inner and outer diameters. Hence, if the inter- compression of a flat circular disk of uniform height, if pC:F: facial friction is higher, the inner diameter actually decreases is measured and r0 is known at the desired e_ and T; then l during compression, while the outer diameter will increase can be calculated with (11.90), derived in Chap. 11. The as usual.

10.3 Friction 433 OD1 Coefficient 0.40 of friction, µ = 0.577 ID1 original ring 80 0.30 70 ID2 60 0.20 high friction 50Decrease in internal diameter of ring, % 0.15 0.12 ID2 40 low friction Fig. 10.18 Ring–compression test. Original ring with outer diameter 30 0.10 of OD1 and inner diameter of ID1 is at the top. The inner diameter ID2 0.09 of the compressed ring is lower than ID1 under high interface friction, shown in the middle, and higher than ID1 under low interface friction, 20 0.08 shown in the bottom 0.07 The changes of inner diameter as a function of the 10 0.06 reduction percentage in height of the ring have been pre- 0.055 sented as calibration curves for a variety of interfacial fric- tional conditions. Adopting the initial dimensions of the ring 0 0.05 in the following ratios of measures, outer diameter:inner diameter:height = 6:3:2, Male and Cockcroft (1964–1965) -10 0.04 established the calibration curves by experimental method to evaluate Coulomb’s friction coefficient ‘µ’ at the interfaces -20 between the ring and the tools, as shown in Fig. 10.19 and 0.03 Lee and Altan (1972) presented the calibration curves for ring of 6:3:2, obtained by applying the upper-bound method, -30 to evaluate friction factor ‘m’ at the interfaces between the ring and the tools, as shown in Fig. 10.20. With the help of -40 these calibration curves, the values of ‘µ’ and ‘m’ at the 0.02 interface can be determined by measuring the inner diameter and the change in the height of the compressed ring, as 0 explained below: -50 (1) Measure the initial outer diameter ‘OD1’, inner diameter 0 10 20 30 40 50 60 70 ‘ID1’, height ‘H1’ of the above thin ring and compress Reduction in heoght, % the ring to reduce its height to any desired level. Fig. 10.19 Friction calibration curves in terms of Coulomb’s coeffi- (2) Measure the height ‘H2’ of the deformed ring. cient of friction, l, for upset ring test with outer diameter, inner diameter (3) Measure the final inner diameter ‘ID2’ of the deformed and thickness in a ratio of 6:3:2 (Male and Cockcroft 1964–1965) ring, or, measure the final outer diameter ‘OD2’, from (5) Calculate the percentage change in the inside diameter which calculate the final inner diameter ‘ID2’ of the of the deformed ring, i.e. compressed ring, using the principle of constancy in  volume during plastic deformation. ID1 À ID2 Â 100 ¼ %DID ðsayÞ: (4) Calculate the percentage reduction in height of the ID1 deformed ring, i.e. For lower values of ‘µ’ or ‘m’, the ‘ID’ of the ring  increases after deformation, i.e. ID2 [ ID1, and so % H1 À H2 Â 100 ¼ %DH ðsayÞ: ΔID will give a negative value, while for higher values H1 of ‘µ’ or ‘m’, the ‘ID’ of the ring decreases after deformation, i.e. ID2 \\ ID1, and so a positive value of % ΔID will be shown. (6) Determine the friction factor ‘µ’ or ‘m’, from the cal- culated values of % ΔID and % ΔH, with the help of the

434 10 Fundamentals of Mechanical Working Decrease in internal diameter of ring, % 70 0.5 The disadvantages of the ring test are as follows: 1.0 Interface • If machining or processing required to make ring speci- 60 friction 0.3 men introduces anisotropy in it, then the shape of ring after compression can become oval, in which case an factor, m average diameter has to be used. 50 • The test cannot be applied to forming operations that 40 involve considerable relative motion between the work-piece and the tools under high contact pressures, 30 such as extrusion operation. 20 10.3.4 Adverse Effects of Friction 0.2 The greater the friction at the job–tool interface, the more is 10 0.15 the work required to overcome it. Consequently, lubricants 0 are used in working processes to minimize the friction at the 0.1 interface. Influence of friction, which is not desirable in forming operations, is mainly as follows. -10 1. The greater the friction force due to increase in the -20 coefficient of friction l, or the interface friction factor m, the higher is the load required to produce a given -30 deformation. For an average 40% reduction in area, 0.05 friction causes to increase the deformation load by 10% if the coefficient of friction l % 0:05 and 20% if l % 0 10 20 30 40 50 60 70 0:1: It is known that the purpose of lubricants employed Reduction in heoght, % in the working processes is to reduce friction forces, which in turn result in some other beneficial effects. Fig. 10.20 Friction calibration curve in terms of friction factor, m, for upset ring test with outer diameter, inner diameter and thickness in a 2. Frictional stress influences the flow of material and ratio of 6:3:2 (Lee and Altan 1972) produces inhomogeneity of deformation, as well as sur- face cracks and other defects. The frictional drag appre- calibration curves for the upset ring test. Note that for a ciably retards the flow of the surface layers of the given reduction, the higher the algebraic value of % work-piece even in the presence of a lubricant, while the ΔID, the higher is the value of ‘µ’ or ‘m’. flow of the interior remains unhindered. So the fast moving interior imposes a secondary tensile stress on the To measure ‘µ’ or ‘m’ at the contact surfaces between the surface that sometimes leads to formation of cracks on work-piece and the tools, the material and surface roughness the surface. Changes in friction may completely alter the of the ring and the compression plates in ring-compression general pattern of deformation. For example, unlubri- test must be, respectively, similar to those of the work-piece cated extrusion having high container-wall friction often and the tools in the working condition. The temperature and progresses with the formation of a dead zone of stagnant strain rate used in this test must also correspond to those in material. Extrusion with a dead zone usually leaves a the actual working operation. poor surface on the product. Good lubrication producing low friction can entirely eliminate the formation of dead The advantages of the ring test are as follows: zone. This has been discussed in Chap. 13. High friction at the interface of the dies and the job in open-die forging • To evaluate coefficient of friction ‘µ’, or friction factor may produce barreling in the forgings. This barreling can ‘m’, it is not necessary to measure the deformation load introduce secondary circumferential tensile stresses, or to know the flow stress of the ring specimen or the which can ultimately restrict the allowable deformation work-piece at temperature and strain rate of the test or the or develop internal cracks during the upsetting of a working condition. cylindrical or a round work-piece. • If deformation pressure is measured during compression of the ring, it is possible to calculate the flow stress (Male et al. 1973; Douglas and Altan 1973) apart from deter- mining ‘µ’ or ‘m’. • The test is simulative and easy to perform and can be applied to a variety of temperatures and strain rates.

10.3 Friction 435 3. Rough tool surfaces, inadequate lubrication or break- stage, which reduces the yield. The lower the friction at the down of lubrication causing material-to-material contact face of the pressure pad, the easier is the inward flow of the between the tools and the work-piece results in high material and the sooner the extrusion defect begins to form. interfacial friction. Under such circumstances, ‘material This is particularly noticed where a plain graphite dummy pickup on tools’ occurs, which must be avoided in block, having low friction, is used as the pressure pad to working processes. This has been discussed subsequently achieve a complete extrusion without a butt (discard) left in in Sect. 10.4.1. the chamber. This may produce considerably long rear-end pipe. So to avoid the formation of the rear-end pipe, the 4. An increase in the coefficient of friction (COF) increases friction is intentionally increased by placing asbestos pads the tool wear, although there is no direct relation between between the surfaces of the billet and dummy block. the COF and the amount of wear. Wear can be defined as 5. In deep drawing of a cup, there must be high friction on the unintentional gradual removal of solid material from the punch and low friction against the die so that the rubbing surfaces. onset of necking in the walls of the cup is prevented. 6. In push-bench manufacture of tube on a mandrel, the dies 5. Tool life can be prolonged both by reduction of interfa- are lubricated with a graphite material and the mandrel is cial friction and by prevention of material-to-material kept unlubricated so that a reasonably high friction is contact between the tools and the work-piece. maintained on the mandrel. The higher friction on the mandrel is advantageous, because part of the drawing 10.3.5 Beneficial Effects of Friction force is carried by the mandrel. In this way, the tensile stress on the leading end of the finished tube is reduced Although high friction is undesirable in mechanical working and it becomes possible to achieve greater reduction in processes, in some cases, friction between the work-piece area without fracture. and the deforming tools is deliberately increased to have the 7. In open-die forging, interfacial friction must be sufficient beneficial effects of friction, which are summarized below. to prevent the outflow of the work-piece material from between the tools. 1. Increasing coefficient of friction, l, at the job–die inter- 8. In closed-die forging, friction is one of the reasons for face increases maximum draft, Dhmax, and maximum creation of high back pressure in the flash to ensure reduction in area, rmax, per pass in rolling according to filling of all recesses of the die cavity. the following respective relations given by (12.17b) and 9. It is desired to have high friction in many varieties of tensile grips used in working processes. (12.18) in Chap. 12: The above instances show that the reduction of friction Dhmax ¼ 4R sin2 f and rmax ¼ 4R sin2 f ; coefficient to a minimum is not always desirable, but care 2 h1 2 must be taken to avoid ‘material pickup on tools’ in cases where high friction is desirable. where R ¼ the roll radius, f ¼ the angle of friction ¼ tanÀ 1 l, and h1 ¼ ingoing stock thickness. 10.4 Lubrication 2. Since ingoing stock thickness h1 ¼  sin2ðf à The major function of a lubricant used in working processes 4R =2Þ rmax ; is to prevent or minimize material transference from the work-piece, then secondly to reduce tool wear and thirdly to therefore, for the same roll radius R; and for a given minimize friction. Let us first have a brief introduction on material transfer. maximum reduction rmax; a higher coefficient of friction, l; will permit a thicker work-piece to enter the roll throat. 10.4.1 Material Transfer Hence, the angle of bite in rolling increases with The transfer of work-piece material to the tools is called ‘material pickup on tools’ or simply ‘pickup’, which can increasing the coefficient of friction, l: limit the range of possible deformation in a pass. The ‘pickup’ occurs due to inadequate lubrication. Transfer of 3. In rolling, if the neutral point does not lie within the arc material may occur in two main ways: of contact of the rolls, the rolls skid instead of deforming the work-piece, which may occur if the friction is too low. 4. During the final stage in extrusion, when the deformation pressure becomes too high, the back end of the work-piece flows over the face of the pressure pad. The rearward sur- face of the work-piece thus becomes entrained along the axis of an extruded product, and even a hollow pipe may form at the rear end of the product, which is an extrusion defect. This requires cutting off the defective material with saw or termination of the extrusion operation at an early

436 10 Fundamentals of Mechanical Working • One is primarily associated with rough tool surfaces. If 3. To prevent or minimize material pickup on tools. the lubricant film is depleted at the interface, work-piece 4. To reduce tool wear and prolong tool life, caused by material is forced into crevices in the tool. Subsequent tangential motion along the tool faces shears off the reduction in friction. projecting soft material and results in tool pickup. This 5. To obtain desirable surface finish of the final product. usually results in a poor surface finish of the product, but 6. To act as thermal insulator for the work-piece and the is not disastrous. tools. • The other type of transfer is adhesive and is much more 7. To cool the work-piece and/or the tools by dissipating the serious. If lubrication breaks down under high pressure or if small-scale or rust particle scours away the protec- heat generated in the working process. tive surface film on the work-piece, leaving bare material, 8. To restrict rise in temperature by reducing heat generated the tool and work-piece cold weld together at their bare contact under high pressure. Subsequent shearing results due to friction. in a fragment of material being torn away from the work-piece and left firmly adhering to the tool. The act of The characteristics of a good metalworking lubricant are tearing exposes fresh clean surfaces, which is even more such that it must susceptible to cold welding. Thus, when pickup begins, it often becomes progressively worse and the tools have to • Have capability to function over a wide range of tem- be reconditioned after terminating the working operation. perature, pressure and speed of deformation. The adhesion depends on the nature of material used to make the tool or die; e.g., steels are much less liable to • Be sufficiently viscous so that it is not squeezed out from pick up on tungsten carbide than on tool steel surface. the gap between the tools and the work-piece. The interposition of non-metallic films, such as phos- phate coating on work-piece, also helps to eliminate or • Possess good spreading and wetting properties so that it reduce pickup in the event of lubrication breakdown. If is able to adhere strongly to the surface of the any adhesion occurs, it may seriously affect the life of the work-piece. This characteristic may be enhanced by tools as well as the quality of the product. surfactants (wetting agents that reduce surface tension) such as fatty acids (Likhtman et al. 1962), capable of 10.4.2 Functions and Characteristics adsorbing rapidly and strongly to the surface of the of a Lubricant work-piece. • Possess good thermal stability. • Be non-corrosive to tools and products. • Have capability to completely burn and produce harmless residue that causes no staining on subsequent welding or heat treatment, and which can easily be removed. • Not produce harmful fumes. • Be non-toxic, free of fire hazard and inexpensive. Whenever possible, a sufficiently thick film of lubricant that 10.4.3 Lubrication Mechanism causes a complete separation between the surfaces of the work-piece and the tools must be maintained. In the presence Lubrication is effected by several mechanisms; though the of very thick film of lubricant, the individual crystals of the distinctions among them are not clear, it is usual to consider material deform freely, in accordance with their crystallo- lubrication under separate headings as follows: graphic orientation, and produce surface with a dull or matte appearance. Although metallurgical or mechanical properties 10.4.3.1 Hydrodynamic or Full-Fluid of the end product are not affected by this surface appear- or Thick-Film Lubrication ance, visual inspection of flaws becomes difficult which is undesirable. On the other hand, at the risk of thinning the In this case, the sliding surfaces are completely separated by lubricant film close to the breakdown level, surface with a a thick layer of liquid lubricant, so that there is no direct highly burnished or shiny appearance is produced in cold contact between the surfaces of the work-piece and tools. working where the surface asperities are flattened by a The finished product will have a dull or matte surface in the smooth-polished tool. However, the functions of an ideal presence of a very thick film of lubricant, as mentioned lubricant in metalworking process are as follows: earlier. In ideal case, the liquid film is continuous and the true area of contact is zero. An ideal thick fluid film is about 1. To reduce deformation load and consumption of power, ten times greater the surface roughness. For this condition, that is caused by reduction in friction. the coefficient of friction (COF) becomes very low, ranges from 0.001 to 0.01 and is determined by the viscosity of the 2. To increase the deformation limit prior to fracture; this is also due to reduction in friction.