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Mechanical behaviour of materials 231 Grain boundaries affect work-hardening by acting as barriers to slip from one grain to the next. In addition, the continuity criterion of polycrystals enforces com- plex slip in the neighbourhood of the boundaries which spreads across the grains with increasing deformation. This introduces a dependence of work-hardening rate on grain size which extends to several per cent elonga- tion. After this stage, however, the work-hardening rate is independent of grain size and for fcc polycrystals is about /40, which, allowing for the orientation factors, is roughly comparable with that found in single crys- tals deforming in multiple slip. Thus from the relations D m and ε D /m the average resolved shear stress on a slip plane is rather less than half the applied tensile stress, and the average shear strain parallel to the slip plane is rather more than twice the tensile elongation. The polycrystal work-hardening rate is thus related to the single-crystal work-hardening rate by the relation d/dε D m 2 d/d (7.28) For bcc metals with the multiplicity of slip systems and the ease of cross-slip m is more nearly 2, so that the work-hardening rate is low. In polycrystalline cph metals the deformation is complicated by twinning, but in the absence of twinning m ³ 6.5, and hence the work-hardening rate is expected to be more than an order of magnitude greater than for single crystals, and also higher than the rate observed in fcc polycrystals for which m ³ 3. 7.6.2.4 Dispersion-hardened alloys On deforming an alloy containing incoherent, non- deformable particles the rate of work-hardening is much greater than that shown by the matrix alone (see Figure 8.10). The dislocation density increases very rapidly with strain because the particles produce a turbulent and complex deformation pattern around them. The dislocations gliding in the matrix leave loops around particles either by bowing between the particles or by cross-slipping around them; both these mechanisms are discussed in Chapter 8. The stresses in and around particles may also be relieved by activating secondary slip systems in the matrix. All these dislo- cations spread out from the particle as strain proceeds and, by intersecting the primary glide plane, hinder primary dislocation motion and lead to intense work- hardening. A dense tangle of dislocations is built up at the particle and a cell structure is formed with the particles predominantly in the cell walls. At small strains 1% work-hardening probably arises from the back-stress exerted by the few Orowan loops around the particles, as described by Fisher, Hart and Pry. The stress–strain curve is reasonably linear with strain ε according to D i C ˛f 3/2 ε with the work-hardening depending only on f, the vol- ume fraction of particles. At larger strains the ‘geomet- rically necessary’ dislocations stored to accommodate the strain gradient which arises because one component deforms plastically more than the other, determine the work-hardening. A determination of the average den- sity of dislocations around the particles with which the primary dislocations interact allows an estimate of the work-hardening rate, as initially considered by Ashby. Thus, for a given strain ε and particle diameter d the number of loops per particle is n ¾ εd/b and the number of particles per unit volume N v D 3f/4r 2 , or 6f/d 3 The total number of loops per unit volume is nN v and hence the dislocation density D nN v d D 6fε/db. The stress–strain relationship from equation (7.27) is then D i C ˛fb/d 1/2 ε 1/2 (7.29) and the work-hardening rate d/dε D ˛ 0 f/d 1/2 b/ε 1/2 (7.30) Alternative models taking account of the detailed struc- ture of the dislocation arrays (e.g. Orowan, prismatic and secondary loops) have been produced to explain some of the finer details of dispersion-hardened mate- rials. However, this simple approach provides a useful working basis for real materials. Some additional fea- tures of dispersion-strengthened alloys are discussed in Chapter 8. 7.6.2.5 Work-hardening in ordered alloys A characteristic feature of alloys with long-range order is that they work-harden more rapidly than in the disordered state.  11 for Fe-Al with a B2 ordered structure is ³/50 at room temperature, several times greater than a typical fcc or bcc metal. However, the density of secondary dislocations in Stage II is relatively low and only about 1/100 of that of the primary dislocations. One mechanism for the increase in work-hardening rate is thought to arise from the generation of antiphase domain boundary (apb) tubes. A possible geometry is shown in Figure 7.43a; the superdislocation partials shown each contain a jog produced, for example, by intersection with a forest dislocation, which are nonaligned along the direction of the Burgers vector. When the dislocation glides and the jogs move nonconservatively a tube of apbs is generated. Direct evidence for the existence of tubes from weak-beam electron microscope studies was first reported for Fe-30 at.% Al. The micrographs show faint lines along h111i, the Burgers vector direction, and are about 3 nm in width. The images are expected to be weak, since the contrast arises from two closely spaced overlapping faults, the second effectively cancelling the displacement caused by the first, and are visible only when superlattice reflections 232 Modern Physical Metallurgy and Materials Engineering Figure 7.43 Schematic diagram of superdislocation (a) with non-aligned jogs, which, after glide, produce an apb-tube and (b) cross-slipped onto the cube plane to form a Kear–Wilsdorf (K–W) lock. are excited. APB tubes have since been observed in other compounds. Theory suggests that jogs in superdislocations in screw orientations provide a potent hardening mecha- nism, estimated to be about eight times as strong as that resulting from pulling out of apb tubes on non-aligned jogs on edge dislocations. The major contributions to the stress to move a dislocation are (1) s , the stress to generate point defects or tubes, and (2) the interaction stress i with dislocations on neighbouring slip planes, and s C i D 3 4 ˛ s f / p ε. Thus, with ˛ s D 1.3and provided f / p is constant and small, linear hardening with the observed rate is obtained. In crystals with A 3 B order only one rapid stage of hardening is observed compared with the normal three-stage hardening of fcc metals. Moreover, the temperature-dependence of  11 / increases with tem- perature and peaks at ¾0.4T m . It has been argued that the apb tube model is unable to explain why anomalously high work-hardening rates are observed for those single crystal orientations favourable for sin- gle slip on f111g planes alone. An alternative model to apb tubes has been proposed based on cross-slip of the leading unit dislocation of the superdislocation. If the second unit dislocation cannot follow exactly in the wake of the first, both will be pinned. For alloys with L1 2 structure the cross-slip of a screw superpartial with b D 1 2 [10 1] from the primary 111 plane to the 010 plane was first proposed by Kear and Wilsdorf. The two 1 2 [10 1] superpartials, one on the 111 plane and the other on the 010 plane, are of course dissociated into h112i-type partials and the whole configuration is sessile. This dislocation arrangement is known as a Kear–Wilsdorf (K–W) lock and is shown in Figure 7.43b. Since cross-slip is thermally activated, the number of locks and there- fore the resistance to 111 glide will increase with increasing temperature. This could account for the increase in yield stress with temperature, while the onset of cube slip at elevated temperatures could account for the peak in the flow stress. Cube cross-slip and cube slip has now been observed in a number of L1 2 compounds by TEM. There is some TEM evidence that the apb energy on the cube plane is lower than that on the 111 plane (see Chapter 9) to favour cross-slip which would be aided by the torque, arising from elastic anisotropy, exerted between the components of the screw dislocation pair. 7.6.3 Development of preferred orientation 7.6.3.1 Crystallographic aspects When a polycrystalline metal is plastically deformed the individual grains tend to rotate into a common orientation. This preferred orientation is developed gradually with increasing deformation, and although it becomes extensive above about 90% reduction in area, it is still inferior to that of a good single crystal. The degree of texture produced by a given deformation is readily shown on a monochromatic X-ray transmis- sion photograph, since the grains no longer reflect uni- formly into the diffraction rings but only into certain segments of them. The results are usually described in terms of an ideal orientation, such as [u, v, w]forthe fibre texture developed by drawing or swaging, and fhklghu vwi for a rolling texture for which a plane of the form (hkl) lies parallel to the rolling plane and a direction of the type hu vwi is parallel to the rolling direction. However, the scatter about the ideal orienta- tion can only be represented by means of a pole-figure which describes the spread of orientation about the ideal orientation for a particular set of (hkl) poles (see Figure 7.44). Mechanical behaviour of materials 233 Figure 7.44 (111)polefigures from (a) copper, and (b) ˛-brass after 95% deformation (intensities in arbitrary units). In tension, the grains rotate in such a way that the movement of the applied stress axis is towards the operative slip direction as discussed in Section 7.3.5 and for compression the applied stress moves towards the slip plane normal. By considering the deformation process in terms of the particular stresses operating and applying the appropriate grain rotations it is possible to predict the stable end-grain orientation and hence the texture developed by extensive deformation. Table 7.3 shows the predominant textures found in different metal structures for both wires and sheet. For fcc metals a marked transition in deformation texture can be effected either by lowering the defor- mation temperature or by adding solid solution alloy- ing elements which lower the stacking fault energy. The transition relates to the effect on deformation modes of reducing stacking fault energy or thermal energy, deformation banding and twinning becoming more prevalent and cross-slip less important at lower temperatures and stacking fault energies. This texture transition can be achieved in most fcc metals by alloy- ing additions and by altering the rolling temperature. Al, however, has a high fault energy and because of the limited solid solubility it is difficult to lower by alloying. The extreme types of rolling texture, shown by copper and 70/30 brass, are given in Figures 7.44a and 7.44b. In bcc metals there are no striking examples of solid solution alloying effects on deformation texture, the preferred orientation developed being remarkably insensitive to material variables. However, material variables can affect cph textures markedly. Variations in c/a ratio alone cause alterations in the orientation developed, as may be appreciated by consideration of the twinning modes, and it is also possible that solid solution elements alter the relative values of critical resolved shear stress for different deformation modes. Processing variables are also capable of giving a degree of control in hexagonal metals. No texture, stable to further deformation, is found in hexagonal metals and the angle of inclination of the basal planes to the sheet plane varies continuously with deforma- tion. In general, the basal plane lies at a small angle <45 ° to the rolling plane, tilted either towards the rolling direction (Zn, Mg) or towards the transverse direction (Ti, Zr, Be, Hf). The deformation texture cannot, in general, be elim- inated by an annealing operation even when such a treatment causes recrystallization. Instead, the forma- tion of a new annealing texture usually results, which is related to the deformation texture by standard lattice rotations. 7.6.3.2 Texture-hardening The flow stress in single crystals varies with orienta- tion according to Schmid’s law and hence materials with a preferred orientation will also show similar plastic anisotropy, depending on the perfection of the texture. The significance of this relationship is well illustrated by a crystal of beryllium which is cph and capable of slip only on the basal plane, a compres- sive stress approaching ³2000 MN/m 2 applied normal to the basal plane produces negligible plastic defor- mation. Polycrystalline beryllium sheet, with a texture such that the basal planes lie in the plane of the sheet, Table 7.3 Deformation textures in metals with common crystal structures Structure Wire (fibre texture) Sheet (rolling texture) bcc [110] f112gh1 10i to f100gh011i fcc [1 1 1], [1 0 0] double fibre f110gh112i to f3 51gh112i cph [210] f0001gh1000i 234 Modern Physical Metallurgy and Materials Engineering shows a correspondingly high strength in biaxial ten- sion. When stretched uniaxially the flow stress is also quite high, when additional (prismatic) slip planes are forced into action even though the shear stress for their operation is five times greater than for basal slip. Dur- ing deformation there is little thinning of the sheet, because the h11 20i directions are aligned in the plane of the sheet. Other hexagonal metals, such as tita- nium and zirconium, show less marked strengthening in uniaxial tension because prismatic slip occurs more readily, but resistance to biaxial tension can still be achieved. Applications of texture-hardening lie in the use of suitably textured sheet for high biaxial strength, e.g. pressure vessels, dent resistance, etc. Because of the multiplicity of slip systems, cubic metals offer much less scope for texture-hardening. Again, a con- sideration of single crystal deformation gives the clue; for whereas in a hexagonal crystal m can vary from 2 (basal planes at 45 ° to the stress axis) to infinity (when the basal planes are normal), in an fcc crystal m can vary only by a factor of 2 with orientation, and in bcc crystals the variation is rather less. In extending this approach to polycrystalline material certain assump- tions have to be made about the mutual constraints between grains. One approach gives m D 3.1fora random aggregate of fcc crystals and the calculated orientation dependence of / for fibre texture shows that a rod with h111i or h110i texture / D 3.664 is 20% stronger than a random structure; the cube tex- ture / D 2.449 is 20% weaker. If conventional mechanical properties were the sole criterion for texture-hardened materials, then it seems unlikely that they would challenge strong precipitation- hardened alloys. However, texture-hardening has more subtle benefits in sheet metal forming in optimizing fabrication performance. The variation of strength in the plane of the sheet is readily assessed by tensile tests carried out in various directions relative to the rolling direction. In many sheet applications, however, the requirement is for through-thickness strength (e.g. to resist thinning during pressing operations). This is more difficult to measure and is often assessed from uniaxial tensile tests by measuring the ratio of the strain in the width direction to that in the thickness direction of a test piece. The strain ratio R is given by R D ε w /ε t D lnw 0 /w/ lnt 0 /t D lnw 0 /w/ lnwL/w 0 L 7.31 where w 0 , L 0 , t 0 are the original dimensions of width, length and thickness and w, L and t are the corre- sponding dimensions after straining, which is derived assuming no change in volume occurs. The average strain ratio R, for tests at various angles in the plane of the sheet, is a measure of the normal anisotropy, i.e. the difference between the average properties in the plane of the sheet and that property in the direction normal to the sheet surface. A large value of R means that there is a lack of deformation modes oriented to provide Figure 7.45 Schematic diagram of the deep-drawing operations indicating the stress systems operating in the flange and the cup wall. Limiting drawing ratio is defined as the ratio of the diameter of the largest blank which can satisfactorily complete the draw D max to the punch diameter (d) (after Dillamore, Smallman and Wilson, 1969; courtesy of the Canadian Institute of Mining and Metallurgy). strain in the through-thickness direction, indicating a high through-thickness strength. In deep-drawing, schematically illustrated in Figure 7.45, the dominant stress system is radial tension combined with circumferential compression in the drawing zone, while that in the base and lower cup wall (i.e. central stretch-forming zone) is biaxial tension. The latter stress is equivalent to a through- thickness compression, plus a hydrostatic tension which does not affect the state of yielding. Drawing failure occurs when the central stretch-forming zone is insufficiently strong to support the load needed to draw the outer part of the blank through the die. Clearly differential strength levels in these two regions, leading to greater ease of deformation in the drawing zone compared with the stretching zone, would enable deeper draws to be made: this is the effect of increasing the R value, i.e. high through-thickness strength relative to strength in the plane of the sheet will favour drawability. This is confirmed in Figure 7.46, where deep drawability as determined by limiting drawing ratio (i.e. ratio of maximum drawable blank diameter to final cup diameter) is remarkably insensitive to ductility and, by inference from the wide range of materials represented in the figure, to absolute strength level. Here it is noted that for hexagonal metals slip occurs readily along h11 20i thus contributing no strain in the c-direction, and twinning only occurs on the f10 12g when the applied stress nearly parallel to the c-axis is compressive for c/a > p 3 and tensile for c/a < p 3. Thus titanium, c/a < p 3, has a high strength in through-thickness compression, whereas Mechanical behaviour of materials 235 Figure 7.46 Limiting draw ratios (LDR) as a function of average values of R and of elongation to fracture measured in tensile tests at 0 ° ,45 ° and 90 ° to the rolling direction (after Wilson, 1966; courtesy of the Institute of Metals). Zn with c/a < p 3 has low through-thickness strength when the basal plane is oriented parallel to the plane of the sheet. In contrast, hexagonal metals with c/a > p 3 would have a high R for f10 10g parallel to the plane of the sheet. Texture-hardening is much less in the cubic met- als, but fcc materials with f111gh110i slip system and bcc with f110gh111i are expected to increase R when the texture has component with f111g and f110g parallel to the plane of the sheet. The range of values of R encountered in cubic metals is much less. Face-centred cubic metals have R ranging from about 0.3 for cube-texture, f100gh001i, to a maximum, in textures so far attained, of just over 1.0. Higher values are sometimes obtained in body-centred cubic metals. Values of R in the range 1.4 ¾ 1.8 obtained in aluminium-killed low-carbon steel are associated with significant improvements in deep-drawing per- formance compared with rimming steel, which has R-values between 1.0 and 1.4. The highest values of R in steels are associated with texture components with f111g parallel to the surface, while crystals with f100g parallel to the surface have a strongly depress- ing effect on R. In most cases it is found that the R values vary with testing direction and this has relevance in rela- tion to the strain distribution in sheet metal forming. In particular, ear formation on pressings generally devel- ops under a predominant uniaxial compressive stress at the edge of the pressing. The ear is a direct con- sequence of the variation in strain ratio for different directions of uniaxial stressing, and a large variation in R value, where R D R max R min generally cor- relates with a tendency to form pronounced ears. On this basis we could write a simple recipe for good deep-drawing properties in terms of strain ratio mea- surements made in a uniaxial tensile test as high R and low R. Much research is aimed at improving forming properties through texture control. 7.7 Macroscopic plasticity 7.7.1 Tresca and von Mises criteria In dislocation theory it is usual to consider the flow stress or yield stress of ductile metals under simple conditions of stressing. In practice, the engineer deals with metals under more complex conditions of stress- ing (e.g. during forming operations) and hence needs to correlate yielding under combined stresses with that in uniaxial testing. To achieve such a yield stress criterion it is usually assumed that the metal is mechanically isotropic and deforms plastically at constant volume, i.e. a hydrostatic state of stress does not affect yield- ing. In assuming plastic isotropy, macroscopic shear is allowed to take place along lines of maximum shear stress and crystallographic slip is ignored, and the yield stress in tension is equal to that in compression, i.e. there is no Bauschinger effect. A given applied stress state in terms of the princi- pal stresses 1 , 2 , 3 which act along three principal axes, X 1 , X 2 and X 3 , may be separated into the hydro- static part (which produces changes in volume) and the deviatoric components (which produce changes in shape). It is assumed that the hydrostatic component has no effect on yielding and hence the more the stress state deviates from pure hydrostatic, the greater the tendency to produce yield. The stresses may be rep- resented on a stress–space plot (see Figure 7.47a), in which a line equidistant from the three stress axes rep- resents a pure hydrostatic stress state. Deviation from this line will cause yielding if the deviation is suf- ficiently large, and define a yield surface which has sixfold symmetry about the hydrostatic line. This arises because the conditions of isotropy imply equal yield stresses along all three axes, and the absence of the Bauschinger effect implies equal yield stresses along 1 and 1 . Taking a section through stress space, perpendicular to the hydrostatic line gives the two simplest yield criteria satisfying the symmetry require- ments corresponding to a regular hexagon and a circle. The hexagonal form represents the Tresca criterion (see Figure 7.47c) which assumes that plastic shear takes place when the maximum shear stress attains a critical value k equal to shear yield stress in uniaxial tension. This is expressed by max D 1 3 2 D k (7.32) where the principal stresses 1 > 2 > 3 . This crite- rion is the isotropic equivalent of the law of resolved shear stress in single crystals. The tensile yield stress Y D 2k is obtained by putting 1 D Y, 2 D 3 D 0. 236 Modern Physical Metallurgy and Materials Engineering Figure 7.47 Schematic representation of the yield surface with (a) principal stresses 1 , 2 and 3 , (b) von Mises yield criterion and (c) Tresca yield criterion. The circular cylinder is described by the equation 1 2 2 C 2 3 2 C 3 1 2 D constant 7.33 and is the basis of the von Mises yield criterion (see Figure 7.47b). This criterion implies that yielding will occur when the shear energy per unit volume reaches a critical value given by the constant. This constant is equal to 6k 2 or 2Y 2 where k is the yield stress in simple shear, as shown by putting 2 D 0, 1 D 3 , and Y is the yield stress in uniaxial tension when 2 D 3 D 0. Clearly Y D 3k compared to Y D 2k for the Tresca criterion and, in general, this is found to agree somewhat closer with experiment. In many practical working processes (e.g. rolling), the deformation occurs under approximately plane strain conditions with displacements confined to the X 1 X 2 plane. It does not follow that the stress in this direction is zero, and, in fact, the deformation condi- tions are satisfied if 3 D 1 2 1 C 2 so that the ten- dency for one pair of principal stresses to extend the metal along the X 3 axis is balanced by that of the other pair to contract it along this axis. Eliminating 3 from the von Mises criterion, the yield criterion becomes 1 2 D 2k and the plane strain yield stress, i.e. when 2 D 0, given when 1 D 2k D 2Y/ p 3 D 1.15Y For plane strain conditions, the Tresca and von Mises criteria are equivalent and two-dimensional flow occurs when the shear stress reaches a critical value. The above condition is thus equally valid when written in terms of the deviatoric stresses 0 1 , 0 2 , 0 3 defined by equations of the type 0 1 D 1 1 3 1 C 2 C 3 . Under plane stress conditions, 3 D 0 and the yield surface becomes two-dimensional and the von Mises criterion becomes 2 1 C 1 2 C 2 2 D 3k 2 D Y 2 (7.34) which describes an ellipse in the stress plane. For the Tresca criterion the yield surface reduces to a hexagon Figure 7.48 The von Mises yield ellipse and Tresca yield hexagon. inscribed in the ellipse as shown in Figure 7.48. Thus, when 1 and 2 have opposite signs, the Tresca crite- rion becomes 1 2 D 2k Y and is represented by the edges of the hexagon CD and FA. When they have the same sign then 1 D 2k D Y or 2 D 2k D Y and defines the hexagon edges AB, BC, DE and EF. 7.7.2 Effective stress and strain For an isotropic material, a knowledge of the uniaxial tensile test behaviour together with the yield func- tion should enable the stress–strain behaviour to be predicted for any stress system. This is achieved by defining an effective stress–effective strain relation- ship such that if D Kε n is the uniaxial stress–strain relationship then we may write. D Kε n (7.35) for any state of stress. The stress–strain behaviour of a thin-walled tube with internal pressure is a typi- cal example, and it is observed that the flow curves obtained in uniaxial tension and in biaxial torsion coincide when the curves are plotted in terms of effec- tive stress and effective strain. These quantities are defined by: D p 2 2 [ 1 2 2 C 2 3 2 C 3 1 2 ] 1/2 7.36 Mechanical behaviour of materials 237 and ε D p 2 3 [ε 1 ε 2 2 C ε 2 ε 3 2 C ε 3 ε 1 2 ] 1/2 7.37 where ε 1 , ε 2 and ε 3 are the principal strains, both of which reduce to the axial normal components of stress and strain for a tensile test. It should be empha- sized, however, that this generalization holds only for isotropic media and for constant loading paths, i.e. 1 D ˛ 2 D ˇ 3 where ˛ and ˇ are constants inde- pendent of the value of 1 . 7.8 Annealing 7.8.1 General effects of annealing When a metal is cold-worked, by any of the many industrial shaping operations, changes occur in both its physical and mechanical properties. While the increased hardness and strength which result from the working treatment may be of importance in certain applications, it is frequently necessary to return the metal to its original condition to allow further forming operations (e.g. deep drawing) to be carried out of for applications where optimum physical properties, such as electrical conductivity, are essential. The treatment given to the metal to bring about a decrease of the hardness and an increase in the ductility is known as annealing. This usually means keeping the deformed metal for a certain time at a temperature higher than about one-third the absolute melting point. Cold working produces an increase in dislocation density; for most metals increases from the value of 10 10 –10 12 lines m 2 typical of the annealed state, to 10 12 –10 13 after a few per cent deformation, and up to 10 15 –10 16 lines m 2 in the heavily deformed state. Such an array of dislocations gives rise to a substantial strain energy stored in the lattice, so that the cold-worked condition is thermodynamically unstable relative to the undeformed one. Consequently, the deformed metal will try to return to a state of lower free energy, i.e. a more perfect state. In general, this return to a more equilibrium structure cannot occur spontaneously but only at elevated temperatures where thermally activated processes such as diffusion, cross- slip and climb take place. Like all non-equilibrium processes the rate of approach to equilibrium will be governed by an Arrhenius equation of the form Rate D A exp [Q/kT] where the activation energy Q depends on impurity content, strain, etc. The formation of atmospheres by strain-ageing is one method whereby the metal reduces its excess lattice energy but this process is unique in that it usually leads to a further increase in the structure- sensitive properties rather than a reduction to the value characteristic of the annealed condition. It is necessary, therefore, to increase the temperature of the deformed metal above the strain-ageing temperature before it recovers its original softness and other properties. The removal of the cold-worked condition occurs by a combination of three processes, namely: (1) recovery, (2) recrystallization and (3) grain growth. These stages have been successfully studied using light microscopy, transmission electron microscopy, or X-ray diffraction; mechanical property measurements (e.g. hardness); and physical property measurements (e.g. density, electrical resistivity and stored energy). Figure 7.49 shows the change in some of these prop- erties on annealing. During the recovery stage the decrease in stored energy and electrical resistivity is accompanied by only a slight lowering of hard- ness, and the greatest simultaneous change in proper- ties occurs during the primary recrystallization stage. However, while these measurements are no doubt striking and extremely useful, it is necessary to under- stand them to correlate such studies with the structural changes by which they are accompanied. 7.8.2 Recovery This process describes the changes in the distribution and density of defects with associated changes in phys- ical and mechanical properties which take place in worked crystals before recrystallization or alteration of orientation occurs. It will be remembered that the structure of a cold-worked metal consists of dense dis- location networks, formed by the glide and interaction of dislocations, and, consequently, the recovery stage of annealing is chiefly concerned with the rearrange- ment of these dislocations to reduce the lattice energy and does not involve the migration of large-angle boundaries. This rearrangement of the dislocations is Figure 7.49 (a) Rate of release of stored energy P, increment in electrical resistivity and hardness (VPN) for specimens of nickel deformed in torsion and heated at 6 k/min (Clareborough, Hargreaves and West, 1955). 238 Modern Physical Metallurgy and Materials Engineering assisted by thermal activation. Mutual annihilation of dislocations is one process. When the two dislocations are on the same slip plane, it is possible that as they run together and annihilate they will have to cut through intersecting dislocations on other planes, i.e. ‘forest’ dislocations. This recovery process will, therefore, be aided by ther- mal fluctuations since the activation energy for such a cutting process is small. When the two dislocations of opposite sign are not on the same slip plane, climb or cross-slip must first occur, and both processes require thermal activation. One of the most important recovery processes which leads to a resultant lowering of the lattice strain energy is rearrangement of the dislocations into cell walls. This process in its simplest form was originally termed polygonization and is illustrated schematically in Figure 7.50, whereby dislocations all of one sign align themselves into walls to form small-angle or sub- grain boundaries. During deformation a region of the lattice is curved, as shown in Figure 7.50a, and the observed curvature can be attributed to the formation of excess edge dislocations parallel to the axis of bend- ing. On heating, the dislocations form a sub-boundary by a process of annihilation and rearrangement. This is shown in Figure 7.50b, from which it can be seen that it is the excess dislocations of one sign which remain after the annihilation process that align themselves into walls. Polygonization is a simple form of sub-boundary formation and the basic movement is climb whereby the edge dislocations change their arrangement from a horizontal to a vertical grouping. This process involves the migration of vacancies to or from the edge of the half-planes of the dislocations (see Section 4.3.4). The removal of vacancies from the lattice, together with the reduced strain energy of dislocations which results, can account for the large change in both electrical resis- tivity and stored energy observed during this stage, while the change in hardness can be attributed to the rearrangement of dislocations and to the reduction in the density of dislocations. The process of polygonization can be demonstrated using the Laue method of X-ray diffraction. Diffrac- tion from a bent single crystal of zinc takes the form of continuous radial streaks. On annealing, these aster- isms (see Figure 5.10) break up into spots as shown in Figure 7.50c, where each diffraction spot originates from a perfect polygonized sub-grain, and the distance between the spots represents the angular misorienta- tion across the sub-grain boundary. Direct evidence for this process is observed in the electron microscope, where, in heavily deformed polycrystalline aggregates at least, recovery is associated with the formation of sub-grains out of complex dislocation networks by a process of dislocation annihilation and rearrange- ment. In some deformed metals and alloys the disloca- tions are already partially arranged in sub-boundaries forming diffuse cell structures by dynamical recovery (see Figure 7.41). The conventional recovery process is then one in which these cells sharpen and grow. In other metals, dislocations are more uniformly dis- tributed after deformation, with hardly any cell struc- ture discernible, and the recovery process then involves formation, sharpening and growth of sub-boundaries. The sharpness of the cell structure formed by defor- mation depends on the stacking fault energy of the metal, the deformation temperature and the extent of deformation (see Figure 7.42). Figure 7.50 (a) Random arrangement of excess parallel edge dislocations and (b) alignment into dislocation walls; (c) Laue photograph of polygonized zinc (after Cahn, 1949). Mechanical behaviour of materials 239 7.8.3 Recrystallization The most significant changes in the structure-sensitive properties occur during the primary recrystallization stage. In this stage the deformed lattice is completely replaced by a new unstrained one by means of a nucle- ation and growth process, in which practically stress- free grains grow from nuclei formed in the deformed matrix. The orientation of the new grains differs con- siderably from that of the crystals they consume, so that the growth process must be regarded as incoherent, i.e. it takes place by the advance of large-angle bound- aries separating the new crystals from the strained matrix. During the growth of grains, atoms get transferred from one grain to another across the boundary. Such a process is thermally activated as shown in Figure 7.51, and by the usual reaction-rate theory the frequency of atomic transfer one way is exp F kT s 1 (7.38) and in the reverse direction exp F Ł C F kT s 1 (7.39) where F is the difference in free energy per atom between the two grains, i.e. supplying the driving force for migration, and F Ł is an activation energy. For each net transfer the boundary moves forward a distance b and the velocity is given by D MF (7.40) where M is the mobility of the boundary, i.e. the velocity for unit driving force, and is thus M D b kT exp S Ł k exp E Ł kT (7.41) Generally, the open structure of high-angle bound- aries should lead to a high mobility. However they are Figure 7.51 Variation in free energy during grain growth. susceptible to the segregation of impurities, low con- centrations of which can reduce the boundary mobility by orders of magnitude. In contrast, special bound- aries which are close to a CSL are much less affected by impurity segregation and hence can lead to higher relative mobility. It is well known that the rate of recrystallization depends on several important factors, namely: (1) the amount of prior deformation (the greater the degree of cold work, the lower the recrystallization tempera- ture and the smaller the grain size), (2) the tempera- ture of the anneal (as the temperature is lowered the time to attain a constant grain size increases exponen- tially 1 ) and (3) the purity of the sample (e.g. zone- refined aluminium recrystallizes below room tempera- ture, whereas aluminium of commercial purity must be heated several hundred degrees). The role these vari- ables play in recrystallization will be evident once the mechanism of recrystallization is known. This mecha- nism will now be outlined. Measurements, using the light microscope, of the increase in diameter of a new grain as a function of time at any given temperature can be expressed as shown in Figure 7.52. The diameter increases linearly with time until the growing grains begin to impinge on one another, after which the rate necessarily decreases. The classical interpretation of these observations is that nuclei form spontaneously in the matrix after a so-called nucleation time, t 0 , and these nuclei then proceed to grow steadily as shown by the linear rela- tionship. The driving force for the process is provided by the stored energy of cold work contained in the strained grain on one side of the boundary relative to that on the other side. Such an interpretation would suggest that the recrystallization process occurs in two distinct stages, i.e. first nucleation and then growth. During the linear growth period the radius of a nucleus is R D Gt t 0 ,whereG, the growth rate, is Figure 7.52 Variation of grain diameter with time at a constant temperature. 1 The velocity of linear growth of new crystals usually obeys an exponential relationship of the form D 0 exp [Q/RT]. 240 Modern Physical Metallurgy and Materials Engineering dR/dt and, assuming the nucleus is spherical, the vol- ume of the recrystallized nucleus is 4/3G 3 t t 0 3 . If the number of nuclei that form in a time increment dt is N dt per unit volume of unrecrystallized matrix, and if the nuclei do not impinge on one another, then for unit total volume f D 4 3 NG 3 t 0 t t 0 3 dt or f D 3 G 3 t 4 (7.42) This equation is valid in the initial stages when f − 1. When the nuclei impinge on one another the rate of recrystallization decreases and is related to the amount untransformed 1 f by f D 1 exp 3 NG 3 t 4 (7.43) where, for short times, equation (7.43) reduces to equation (7.42). This Johnson–Mehl equation is expected to apply to any phase transformation where there is random nucleation, constant N and G and small t 0 . In practice, nucleation is not random and the rate not constant so that equation (7.43) will not strictly apply. For the case where the nucleation rate decreases exponentially, Avrami developed the equation f D 1 expkt n (7.44) where k and n are constants, with n ³ 3 for a fast and n ³ 4 for a slow, decrease of nucleation rate. Provided there is no change in the nucleation mechanism, n is independent of temperature but k is very sensitive to temperature T; clearly from equation (7.43), k D NG 3 /3 and both N and G depend on T. An alternative interpretation is that the so-called incubation time t 0 represents a period during which small nuclei, of a size too small to be observed in the light microscope, are growing very slowly. This lat- ter interpretation follows from the recovery stage of annealing. Thus, the structure of a recovered metal consists of sub-grain regions of practically perfect crystal and, thus, one might expect the ‘active’ recrys- tallization nuclei to be formed by the growth of certain sub-grains at the expense of others. The process of recrystallization may be pictured as follows. After deformation, polygonization of the bent lattice regions on a fine scale occurs and this results in the formation of several regions in the lattice where the strain energy is lower than in the surrounding matrix; this is a necessary primary condition for nucleation. During this initial period when the angles between the sub-grains are small and less than one degree, the sub-grains form and grow quite rapidly. However, as the sub-grains grow to such a size that the angles between them become of the order of a few degrees, the growth of any given sub-grain at the expense of the others is very slow. Eventually one of the sub-grains will grow to such a size that the boundary mobility begins to increase with increasing angle. A large angle boundary,  ³ 30–40 ° , has a high mobility because of the large lattice irregularities or ‘gaps’ which exist in the boundary transition layer. The atoms on such a boundary can easily transfer their allegiance from one crystal to the other. This sub-grain is then able to grow at a much faster rate than the other sub- grains which surround it and so acts as the nucleus of a recrystallized grain. The further it grows, the greater will be the difference in orientation between the nucleus and the matrix it meets and consumes, until it finally becomes recognizable as a new strain-free crystal separated from its surroundings by a large-angle boundary. The recrystallization nucleus therefore has its origin as a sub-grain in the deformed microstructure. Whether it grows to become a strain-free grain depends on three factors: (1) the stored energy of cold work must be sufficiently high to provide the required driving force, (2) the potential nucleus should have a size advantage over its neighbours, and (3) it must be capable of con- tinued growth by existing in a region of high lattice curvature (e.g. transition band) so that the growing nucleus can quickly achieve a high-angle boundary. In situ experiments in the HVEM have confirmed these factors. Figure 7.53a shows the as-deformed substruc- ture in the transverse section of rolled copper, together with the orientations of some selected areas. The sub- grains are observed to vary in width from 50 to 500 nm, and exist between regions 1 and 8 as a transition band across which the orientation changes sharply. On heating to 200 ° C, the sub-grain region 2 grows into the transition region (Figure 7.53b) and the orientation of the new grain well developed at 300 ° C is identical to the original sub-grain (Figure 7.53c). With this knowledge of recrystallization the influ- ence of several variables known to affect the recrys- tallization behaviour of a metal can now be under- stood. Prior deformation, for example, will control the extent to which a region of the lattice is curved. The larger the deformation, the more severely will the lat- tice be curved and, consequently, the smaller will be the size of a growing sub-grain when it acquires a large-angle boundary. This must mean that a shorter time is necessary at any given temperature for the sub- grain to become an ‘active’ nucleus, or conversely, that the higher the annealing temperature, the quicker will this stage be reached. In some instances, heavily cold-worked metals recrystallize without any signif- icant recovery owing to the formation of strain-free cells during deformation. The importance of impurity content on recrystallization temperature is also evident from the effect impurities have on obstructing disloca- tion sub-boundary and grain boundary mobility. The intragranular nucleation of strain-free grains, as discussed above, is considered as abnormal sub- grain growth, in which it is necessary to specify that some sub-grains acquire a size advantage and [...]... metals and alloys is the presence of many straight-sided bands that run across grains These 244 Modern Physical Metallurgy and Materials Engineering Figure 7.58 Formation and growth of annealing twins (from Burke and Turnbull, 195 2; courtesy of Pergamon Press) Figure 7.57 Relation between grain size, deformation and temperature for aluminium (after Buergers, courtesy of Akademie-Verlags-Gesellschaft) bands... Argon, A ( 196 9) The Physics of Strength and Plasticity MIT Press, Cambridge, MA Cottrell, A H ( 196 4) Mechanical Properties of Matter John Wiley, Chichester Cottrell, A H ( 196 4) The Theory of Crystal Dislocations Blackie, Glasgow Dislocations and Properties of Real Metals ( 198 4) Conf Metals Society Evans, R W and Wilshire, B ( 199 3) Introduction to Creep Institute of Materials, London Freidel, J ( 196 4) Dislocations... the tensile axis, and these gradually grow and coalesce Second-phase particles play an important part in the nucleation of cracks and cavities by concentrating stress in sliding boundaries and at the intersection of slip bands with particles but these stress concentrations 2 49 are greatly reduced by plastic deformation by powerlaw creep and by diffusional processes Cavity formation and early growth... are weaker than those resisting further dislocation motion, and the pile-up stress causes it to slip back under a reduced load in the reverse direction The other important feature is Figure 7.71 Stress–strain curves for copper after increasing amounts of fatigue testing (after Broom and Ham, 195 9) 256 Modern Physical Metallurgy and Materials Engineering that the temperature-dependence of the hardening... rate-controlling mechanism, and diffusional creep with either grain-boundary 252 Modern Physical Metallurgy and Materials Engineering Figure 7.68 Deformation-mechanism maps for (a) nickel, (b) nickel-based superalloy (after M F Ashby) diffusion or lattice diffusion being important In a particular range of temperature, one of these mechanisms is dominant and it is therefore useful in engineering application... boundary, and are thus able to give rise to grain boundary migration, when sliding has temporarily ceased, which is proportional to the overall deformation A second creep process which also involves the grain boundaries is one in which the boundary acts 248 Modern Physical Metallurgy and Materials Engineering Figure 7.65 Grain boundary sliding on a bi-crystal tin (after Puttick and King, 195 2) Figure... concept 254 Modern Physical Metallurgy and Materials Engineering of cumulative damage, is illustrated in Figure 7.69b This hypothesis states that damage can be expressed in terms of the number of cycles applied divided by the number to produce failure at a given stress level Thus, if a maximum stress of value S1 is applied to a specimen for n1 cycles which is less than the fatigue life N1 , and then the... decreases 242 Modern Physical Metallurgy and Materials Engineering with increasing deformation The finer dispersions tend to homogenize the microstructure (i.e dislocation distribution) thereby minimizing local lattice curvature and reducing nucleation The formation of nuclei becomes very difficult when the spacing of second-phase particles is so small that each developing sub-grain interacts with a particle... on the particle size Small particles ³0.1 µm can have beneficial effects by homogenizing the slip pattern and delaying fatigue-crack nucleation Larger particles reduce the fatigue life by both facilitating crack nucleation by slip band/particle interaction and increasing crack growth rates by interface decohesion and voiding within the plastic zone at the crack tip The formation of voids at particles... represent the successive positions of the propagation front and are spaced further apart the higher the Figure 7.76 A schematic fatigue fracture 258 Modern Physical Metallurgy and Materials Engineering Figure 7.77 Schematic illustration of the formation of fatigue striations velocity of propagation They are rather uninfluenced by grain boundaries and in metals where cross-slip is easy (e.g mild steel or . 0. 236 Modern Physical Metallurgy and Materials Engineering Figure 7.47 Schematic representation of the yield surface with (a) principal stresses 1 , 2 and 3 , (b) von Mises yield criterion and. deformed in torsion and heated at 6 k/min (Clareborough, Hargreaves and West, 195 5). 238 Modern Physical Metallurgy and Materials Engineering assisted by thermal activation. Mutual annihilation of dislocations. polygonization by sub-boundary migration. Particle-stimulated nucleation occurs above a critical particle size which decreases 242 Modern Physical Metallurgy and Materials Engineering with increasing deformation.
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