212 Modern Physical Metallurgy and Materials Engineering distinguish those parts of the specimen that have interstitial sites below the half-plane. Thus, where both yielded, A, from those which have not, B. Arrival at dislocations and solute atoms are present in the lattice the upper yield point is indicated by the formation interactions of the stress field can occur, resulting in passes through the stage of the yield point elongation provides a driving force tending to attract solute atoms these bands spread along the specimen and coalesce dislocations and if the necessary time for diffusion is until the entire gauge length has been covered. At this allowed, a solute atom'atmosphere'will form around stage the whole of the material within the gauge length each dislocation has been overstrained, and the yield point elongation When a stress is applied to a specimen in which the is complete. The growth of a Luders band is shown dislocations are locked by carbon atoms the dislo a diagrammatically in Figure 7. 24b. It should be noted tions are not able to move at the stress level at which that the band is a macroscopic band crossing all free dislocations are normally mobile. With increas the grains in the cross-section of a polycrystallin ing stress yielding occurs when dislocations suddenly specimen, and thus the edges of the band are not become mobile either by breaking away from the car necessarily the traces of individual slip planes. A bon atmosphere or by ucleating fresh dislocations second point to observe is that the rate of plastic flow at stress concentrations. At this high stress level the in the edges of a band can be very high even in pparently slow test; this is because the zones, marked mobile dislocation dens C in Figure 7. 24b, are very narrow compared with the yield stress is then the stress at which free dislocations gauge length continue to move and produce plastic flow. The over- These Luders bands frequently occur in drawing strained condition corresponds to the situation where the mobile dislocations, brought to rest by unloading and stamping operations when the surface markings the specimen, are set in motion again by reloading in relief are called stretcher strains. These markings are unsightly in appearance and have to be avoided before the carbon atmospheres have time to develop on many finished products. The remedy consists in by diffusion. If, however, time is allowed for diffu- overstraining the sheet prior to pressing operations, by sion to take place, new atmospheres can re-form and immobilize the dislocations again. This is the strain hat the yield phenomenon is eliminated. It is essential. aged condition when the original yield characteristics once this operation has been performed, to carry out reappear pressing before the sheet has time to strain-age: the The upper yield point in conventional experiments use of a non-ageing steel is an alternative remedy on polycrystalline materials is the stress at which These yielding effects are infuenced by the pres- initially yielded zones trigger yield in adjacent grains ence of small amounts of carbon or nitrogen atoms As more and more grains are triggered the yield zones interacting with dislocations. The yield point can spread across the specimen and form a Liders band be removed by annealing at 700C in wet-hydrogen The propagation of yield is thought to occur when a atmosphere, and cannot subsequently be restored by dislocation source operates and releases an avalanche ny strain-ageing treatment. Conversely, exposing the of dislocations into its slip plane which eventually pile decarburized specimen to an atmosphere of dry hydro- up at a grain boundary or other obstacle. The stress gen containing a trace of hydrocarbon at 700C for as concentration at the head of the pile-up acts with the tle as one minute restores the yield on applied stress on the dislocations of the next grain and nitrogen atoms can also be removed from solt and operates the nearest source, so that the proce such elements as molybdenum, manganese, chromium, a, at which yielding propagates is g/reB. ear stress vanadium. niobium or titanium which have a strong affinity for forming carbides or nitrides in steels. For σ=01+(0r12)d-l/2 (78) his reason, these elements are particularly effective in removing the yield point and producing a non-strain where r is the distance from the pile-I source, 2d is the grain diameter and o is the stress The carbon/nitrogen atoms are important in yielding required to operate a source which involves unpinning process because they interact with the dislocations and dislocation te at that temperature. Equation(7. 8 immobilize them. This locking of the dislocations is reduces to the Hall-Petch equation a, =0+kd-1/2 brought about because the strain energy due to the where o; is the'friction'stress term and ky the grain distortion of a solute atom can be relieved if it fits into a structural region where the local lattice parameter size dependence parameter(=m-ter/)discussed in approximates to that of the natural lattice parameter of Section 7.4. II the solute. Such a condition will be brought about by the segregation of solute atoms to the dislocations, with 7.4.5 Yield points and crystal structure large substitutional atoms taking up lattice positions in the expanded region, and small ones in the compressed The characteristic feature of discontinuous yielding region; small interstitial atoms will tend to segregate to is that at the yield point the specimen goes from
212 Modern Physical Metallurgy and Materials Engineering distinguish those parts of the specimen that have yielded, A, from those which have not, B. Arrival at the upper yield point is indicated by the formation of one or more of these bands and as the specimen passes through the stage of the yield point elongation these bands spread along the specimen and coalesce until the entire gauge length has been covered. At this stage the whole of the material within the gauge length has been overstrained, and the yield point elongation is complete. The growth of a Ltiders band is shown diagrammatically in Figure 7.24b. It should be noted that the band is a macroscopic band crossing all the grains in the cross-section of a polycrystalline specimen, and thus the edges of the band are not necessarily the traces of individual slip planes. A second point to observe is that the rate of plastic flow in the edges of a band can be very high even in an apparently slow test; this is because the zones, marked C in Figure 7.24b, are very narrow compared with the gauge length. These Liiders bands frequently occur in drawing and stamping operations when the surface markings in relief are called stretcher strains. These markings are unsightly in appearance and have to be avoided on many finished products. The remedy consists in overstraining the sheet prior to pressing operations, by means of a temper roll, or roller levelling, pass so that the yield phenomenon is eliminated. It is essential, once this operation has been performed, to carry out pressing before the sheet has time to strain-age; the use of a 'non-ageing' steel is an alternative remedy. These yielding effects are influenced by the presence of small amounts of carbon or nitrogen atoms interacting with dislocations. The yield point can be removed by annealing at 700~ in wet-hydrogen atmosphere, and cannot subsequently be restored by any strain-ageing treatment. Conversely, exposing the decarburized specimen to an atmosphere of dry hydrogen containing a trace of hydrocarbon at 700~ for as little as one minute restores the yield point. The carbon and nitrogen atoms can also be removed from solution in other ways: for example, by adding to the iron such elements as molybdenum, manganese, chromium, vanadium, niobium or titanium which have a strong affinity for forming carbides or nitrides in steels. For this reason, these elements are particularly effective in removing the yield point and producing a non-strain ageing steel. The carbon/nitrogen atoms are important in yielding process because they interact with the dislocations and immobilize them. This locking of the dislocations is brought about because the strain energy due to the distortion of a solute atom can be relieved if it fits into a structural region where the local lattice parameter approximates to that of the natural lattice parameter of the solute. Such a condition will be brought about by the segregation of solute atoms to the dislocations, with large substitutional atoms taking up lattice positions in the expanded region, and small ones in the compressed region; small interstitial atoms will tend to segregate to interstitial sites below the half-plane. Thus, where both dislocations and solute atoms are present in the lattice, interactions of the stress field can occur, resulting in a lowering of the strain energy of the system. This provides a driving force tending to attract solute atoms to dislocations and if the necessary time for diffusion is allowed, a solute atom 'atmosphere' will form around each dislocation. When a stress is applied to a specimen in which the dislocations are locked by carbon atoms the dislocations are not able to move at the stress level at which free dislocations are normally mobile. With increasing stress yielding occurs when dislocations suddenly become mobile either by breaking away from the carbon atmosphere or by nucleating fresh dislocations at stress concentrations. At this high stress level the mobile dislocation density increases rapidly. The lower yield stress is then the stress at which free dislocations continue to move and produce plastic flow. The overstrained condition corresponds to the situation where the mobile dislocations, brought to rest by unloading the specimen, are set in motion again by reloading before the carbon atmospheres have time to develop by diffusion. If, however, time is allowed for diffusion to take place, new atmospheres can re-form and immobilize the dislocations again. This is the strainaged condition when the original yield characteristics reappear. The upper yield point in conventional experiments on polycrystalline materials is the stress at which initially yielded zones trigger yield in adjacent grains. As more and more grains are triggered the yield zones spread across the specimen and form a Liiders band. The propagation of yield is thought to occur when a dislocation source operates and releases an avalanche of dislocations into its slip plane which eventually pile up at a grain boundary or other obstacle. The stress concentration at the head of the pile-up acts with the applied stress on the dislocations of the next grain and operates the nearest source, so that the process is repeated in the next grain. The applied shear stress ty,. at which yielding propagates is given by ~v -- cri -k (~crl/2)d -1/2 (7.8) where r is the distance from the pile-up to the nearest source, 2d is the grain diameter and o% is the stress required to operate a source which involves unpinning a dislocation rc at that temperature. Equation (7.8) reduces to the Hall-Petch equation o% = oi + kyd -I/2, where O" i is the 'friction' stress term and k,, the grain size dependence parameter (-m2rcr Z/2) discussed in Section 7.4.11. 7.4.5 Yield points and crystal structure The characteristic feature of discontinuous yielding is that at the yield point the specimen goes from
Mechanical behaviour of materials 213 a condition where the availability of mobile dislo- cations is limited to one where they are in abun- dance, the increase in mobile density largely aris- ocations ing from dislocation multiplication at the high stress ated 0.6 level. a further feature is that not all the dislo. cations have to be immobilized to observe a yield drop. Indeed, this is not usually possible because pecimen handling, non-axial loading, scratches, etc give rise to stress concentrations that provide a small g local density of mobile dislocations (i.e. pre-yiel For materials with a high Peierls-Nabarro(P-n yield drops 02 common example is that observed in silicon; this is an extremely pure material with no impurities to locl lislocations, but usually the dislo modest(10 m/m)and possesses a high P-N stress When these materials are pulled in a tensile test the overall strain rate y imposed on the specime Percentage elongation by the machine has to be matched by the motion of lislocations according to the relation y= pbv. How- Figure 7. 25 yield point in a copper whisker ever, because p is small the individual dislocations are forced to move at a high speed v, which is only attained at a high stress level (the upper yield stress because of the large P-N stress. As the dislocations lide at these high speeds, rapid multiplication occurs and the mobile dislocation density increases rapid Because of the increased value of the term pv, a lower average velocity of dislocations is then required to 02 maintain a constant strain rate, which means a lower glide stress. The stress that can be supported by the tmen thus dr int. an tion-dislocation interactions caused by the p produce a significant work-hardening In the fcc metals, the P-n stress is quite small and the stress to move a dislocation is almost indepen- dent of velocity up to high speeds. If such metals are to show a yield point, the density of mobile disloc- Figure 7.26 Calculated stress-strain curves showin achieved as shown in Figure 7.25 by the tensile test- inn foren 35 with 100 me"z ( o cm =2.op ing of whisker crystals which are very perfect. Yielding (ii)105 cm-2 and (iv)107 cm-2(after Hahn,1962; begins at the stress required to create dislocations in the courtesy of Pergamon Press) perfect lattice, and the upper yield stress approaches the theoretical yield strength, Following multiplica- tion, the stress for glide of these dislocations is several It is evident that discontinuous yielding can be orders of magnitude lower produced in all the common metal structures provided Bcc transition metals such as iron are intermediate the appropriate solute elements are present and cor- in their plastic behaviour between the fcc metals and rect testing procedure adopted. The effect is particu- diamond cubic Si and Ge. Because of the significant larly strong in the bcc metals and has been observed P-N stress these bcc metals are capable of exhibit- in a-iron, molybdenum, niobium, vanadium and B- dislocation density is not zero, as shown by the cal- solute element. The hexagonal metals(e. g.cadmium culated curves of Figure 7. 26. However, in practice, and zinc) can also show the phenomenon provided the dislocation density of well-annealed pure metals interstitial nitrogen atoms are added. The copper is about 100 m/mand too high for any significant and aluminium-based fcc alloys also exhibit yielding yield drop without an element of dislocation locking behaviour but often to a lesser degree. In this case it is substitutional atoms(e.g. zinc in a-brass and copper in
Mechanical behaviour of materials 213 a condition where the availability of mobile dislocations is limited to one where they are in abundance, the increase in mobile density largely arising from dislocation multiplication at the high stress level. A further feature is that not all the dislocations have to be immobilized to observe a yield drop. Indeed, this is not usually possible because specimen handling, non-axial loading, scratches, etc. give rise to stress concentrations that provide a small local density of mobile dislocations (i.e. pre-yield microtrain). For materials with a high Peierls-Nabarro (P-N) stress, yield drops may be observed even when they possess a significant mobile dislocation density. A common example is that observed in silicon; this is an extremely pure material with no impurities to lock dislocations, but usually the dislocation density is quite modest (107 m/m 3) and possesses a high P-N stress. When these materials are pulled in a tensile test the overall strain rate ~, imposed on the specimen by the machine has to be matched by the motion of dislocations according to the relation ~, = pbv. However, because p is small the individual dislocations are forced to move at a high speed u, which is only attained at a high stress level (the upper yield stress) because of the large P-N stress. As the dislocations glide at these high speeds, rapid multiplication occurs and the mobile dislocation density increases rapidly. Because of the increased value of the term pv, a lower average velocity of dislocations is then required to maintain a constant strain rate, which means a lower glide stress. The stress that can be supported by the specimen thus drops during initial yielding to the lower yield point, and does not rise again until the dislocation-dislocation interactions caused by the increased p produce a significant work-hardening. In the fcc metals, the P-N stress is quite small and the stress to move a dislocation is almost independent of velocity up to high speeds. If such metals are to show a yield point, the density of mobile dislocations must be reduced virtually to zero. This can be achieved as shown in Figure 7.25 by the tensile testing of whisker crystals which are very perfect. Yielding begins at the stress required to create dislocations in the perfect lattice, and the upper yield stress approaches the theoretical yield strength. Following multiplication, the stress for glide of these dislocations is several orders of magnitude lower. Bcc transition metals such as iron are intermediate in their plastic behaviour between the fcc metals and diamond cubic Si and Ge. Because of the significant P-N stress these bcc metals are capable of exhibiting a sharp yield point even when the initial mobile dislocation density is not zero, as shown by the calculated curves of Figure 7.26. However, in practice, the dislocation density of well-annealed pure metals is about 10 l~ m/m 3 and too high for any significant yield drop without an element of dislocation locking by carbon atoms. 70[ i l I ~ [ / ~ ~" Distocations / 601 I created ]06 o.4 % f30 I = '~ t, 1 ,, I 1 1 / 0 2 4 6 8 + -10 Percent age etongation Figure 7.25 Yield point in a copper whisker. 30 25 "E ~2o I/I 10 (i) Oi) ..... Ptasttc strain i (,i+) (iv) 0 05*/. 0.2 0.1 l N z t.9 Figure 7.26 Calculated stress-strain curves showing influence of initial dislocation density on the yield drop in iron for n -- 35 with (i) 101 cm -2, (ii) lO s cm -2, (iii) 10 5 cm -2 and (iv) 10 7 cm -2 (after Hahn, 1962; courtesy of Pergamon Press). It is evident that discontinuous yielding can be produced in all the common metal structures provided the appropriate solute elements are present, and correct testing procedure adopted. The effect is particularly strong in the bcc metals and has been observed in c~-iron, molybdenum, niobium, vanadium and /~- brass each containing a strongly interacting interstitial solute element. The hexagonal metals (e.g. cadmium and zinc) can also show the phenomenon provided interstitial nitrogen atoms are added. The copperand aluminium-based fcc alloys also exhibit yielding behaviour but often to a lesser degree. In this case it is substitutional atoms (e.g. zinc in c~-brass and copper in
214 Modern Physical Metallurgy and Materials Engineering aluminium alloys) which are responsible for the phe- APB-locking me nomenon(see Section 7.4.7) ng because the m eld by the lead dislo 7. 4.6 Discontinuous yielding in ordered alloys released by the trailing dislocation malf banthat Discontinuous yield points have been observed in a the APB-model and weak-beam electron mici wide variety of A3 B-type alloys. Figure 7. 27 shows (see Figure 7.28)shows that the superdislocatie the development of the yield point in Ni] Fe on ageing. aration for a shear APB corresponds to an energy of The addition of Al speeds up the kinetics of ordering 48+5 mJ/m, whereas a larger dislocation separation and therefore the onset of the yield point. Ordered corresponding to an APB energy of 25+ 3 mJ/m'was materials deform by superdislocation motion and the observed for a strained and aged cu3Al link between yield points and superdislocations is con firmed by the observation that in Cu3 Au, for example 7 4.7 Solute-dislocation interaction a transition from groups of single dislocations to more Iron containing carbon or nitrogen shows very marked randomly arranged superdislocation pairs takes at s=0.7(see Chapter 4)and this coincides yield point effects and there is a strong elastic interac- the onset of a large yield drop and rapid rise in work tion between these solute atoms and the dislocations The solute atoms occupy interstitial sites in the lat tice and produce large tetragonal distortions as well Sharp yielding may be explained by at less oslo- interact with both shear and hydrostatic stresses and mechanisms, namely(1)cross-slip of the super effectively pinning it and (2)dislocation locking by yielding behaviour is also expected in other bcc met- rearrangement of the APB on ageing. The shear APB als, provided they contain interstitial solute elements between a pair of superdislocations is likely to be On the other hand, in the case of fcc metals the energetically unstable since there are many like bonds arrangement of lattice positions around either intersti across the interface and thermal activation will mod- tial or substitutional sites is too symmetrical to allow a ify this sharp interface by atomic rearrangement, This solute atom to produce an asymmetrical distortion, and 100 2 2 +5%A gure 7. 27 Development of a yield point with ageing at 490C for the times indicated. (a)Ni Fe,(b) Ni3 Fe +5%Al: the
214 Modern Physical Metallurgy and Materials Engineering aluminium alloys) which are responsible for the phenomenon (see Section 7.4.7). 7.4.6 Discontinuous yielding in ordered alloys Discontinuous yield points have been observed in a wide variety of A3B-type alloys. Figure 7.27 shows the development of the yield point in Ni3Fe on ageing. The addition of AI speeds up the kinetics of ordering and therefore the onset of the yield point. Ordered materials deform by superdislocation motion and the link between yield points and superdislocations is confirmed by the observation that in Cu3Au, for example, a transition from groups of single dislocations to more randomly arranged superdislocation pairs takes place at ~S = 0.7 (see Chapter 4) and this coincides with the onset of a large yield drop and rapid rise in work hardening. Sharp yielding may be explained by at least two mechanisms, namely (1)cross-slip of the superdislocation onto the cube plane to lower the APB energy effectively pinning it and (2)dislocation locking by rearrangement of the APB on ageing. The shear APB between a pair of superdislocations is likely to be energetically unstable since there are many like bonds across the interface and thermal activation will modify this sharp interface by atomic rearrangement. This APB-locking model will give rise to sharp yielding because the energy required by the lead dislocation in creating sharp APB is greater than that released by the trailing dislocation initially moving across diffuse APB. Experimental evidence favours the APB-model and weak-beam electron microscopy (see Figure 7.28) shows that the superdislocation separation for a shear APB corresponds to an energy of 48 4- 5 mJ/m 2, whereas a larger dislocation separation corresponding to an APB energy of 25 4- 3 mJ/m 2 was observed for a strained and aged Cu3Au. 7.4.7 Solute-dislocation interaction Iron containing carbon or nitrogen shows very marked yield point effects and there is a strong elastic interaction between these solute atoms and the dislocations. The solute atoms occupy interstitial sites in the lattice and produce large tetragonal distortions as well as large-volume expansions. Consequently, they can interact with both shear and hydrostatic stresses and can lock screw as well as edge dislocations. Strong yielding behaviour is also expected in other bcc metals, provided they contain interstitial solute elements. On the other hand, in the case of fcc metals the arrangement of lattice positions around either interstitial or substitutional sites is too symmetrical to allow a solute atom to produce an asymmetrical distortion, and A O 200 150 1 O0 50 -,.,1% ./' as quenched t ii | ! , J / /V 2 4 / r j/ r 2-7 2-5 2-4 i 2-3 .,, i 1"- Ni3Fe Ni3Fe + 5% AI Figure 7.27 Development of a yield point with ageing at 490~ for the times indicated. (a) Ni3 Fe, (b) Ni3 Fe + 5 % Al: the tests are at room temperature
Mechanical behaviour of materials 215 Figure 7. 28 Weak-beam micrographs showing separation of superdislocation partials in Cu3 Au (a)As deformed, (b) after ageing at 225C (after Morris and Smallman, 1975 the atmosphere locking of screw dislocations, which v), and substituting K= 2u(1 +D)/3(1-2v, where requires a shear stress interaction, would appear to be u is the shear modulus and v Poisson's ratio, we get possible. Then by this argument, since the screw the expressi ield point should not be observed. Nevertheless, yield V(R8=b(l v)uAv sin e/3TR(I-v) points are observed in fcc materials and one reason =A sin e/R for this is that unit dislocations in fcc metals dissoci. This is the ate into pairs of partial dislocations which are elasti energy at a point whose polar cally coupled by a stacking fault. Moreover, since their ct to the centre of the disloc areR andθ Burgers vectors intersect at 120 there is no orienta- side (0<8 that V is positive on the upper tion of the line of the pair for which both can be pure the dislocation for a large atom screws. At least one of them must have a substantial (Av>0), and negative on the lower side, which ure of a large atom bein by hydrostatic interactions should cause a locking of repelled from the compressed region and attracted into the pair although it will undoubtedly be weaker In its quantitative form the theory of solute ato It is expected that the site for the strongest bind locking has been applied to the formation of an ing energy Vmax will be at a point 0= 3T/2,R atmosphere around an edge dislocation due to hydro- ro -b; and using known values of A, v and Av in static interaction. Since hydrostatic stresses are scalar equation(7.10)we obtain A =3 x 10-29N m2 and the orientation of the dislocation with respect to the value is almost certainly too high because of the lima quantities, no knowledge is required in this case of max =1 ev for carbon or nitrogen in a iron. Th lating shear stresses interactions. Cottrell and Bilby have shown that if the introduction of a solute atom realistic value obtained from experiment(e. g. internal causes a volume change Av at some point in the lattice friction experiments)is Vmax i to ev. For a sub- where the hydrostatic pressure of the stress field is p, stitutional solute atom such as zinc in copper Av is the interaction energy is ot only smaller but also easier to calculate from lat- ce parameter measurements. Thus, if r and r(1 +e) (7.9) are the atomic radii of the solvent and solute, respec- where K is the bulk modulus and e is the local dilata. tively, where e is the misfit value, the volume change tion strain. The dilatation strain at a point(R, 0) from a Av is 4rr'E and equation(7.10)becomes positive edge dislocation is b(l- 2v)x sin 8/2TR( V=4(1+v)ubEr'sin 0/3(1-v)R ITo a first approximation a solute atom does not interact Asin 0/R he screw: a second-orde there is no dilatation around Taking the known values u=40 GN/m, v=0.36, tation exists however, which b=2.55x 10-10 m, ro and 6=0.06, we find gives rise to a non-zero interaction falling off with distance 5 x 10-30 N m2, which gives a much lower binding om the dislocation according to I/r2. In real crystals isotropic elasticity will lead to first-order size effects even energy, Vr ith screw dislocations and hence a substantial interacti The yield phenomenon is particularly strong in iron because an additional effect is important; this
Mechanical behaviour of materials 215 Figure 7.28 Weak-beam micrographs showing separation of superdislocation partials in Cu3Au. (a) As deformed, (b) after ageing at 225~ (after Morris and Smallman, 1975). the atmosphere locking of screw dislocations, which requires a shear stress interaction, would appear to be impossible. Then by this argument, since the screw dislocations are not locked, a drop in stress at the yield point should not be observed. Nevertheless, yield points are observed in fcc materials and one reason for this is that unit dislocations in fcc metals dissociate into pairs of partial dislocations which are elastically coupled by a stacking fault. Moreover, since their Burgers vectors intersect at 120 ~ there is no orientation of the line of the pair for which both can be pure screws. At least one of them must have a substantial edge component, and a locking of this edge component by hydrostatic interactions should cause a locking of the pair although it will undoubtedly be weaker. In its quantitative form the theory of solute atom locking has been applied to the formation of an atmosphere around an edge dislocation due to hydrostatic interaction. Since hydrostatic stresses are scalar quantities, no knowledge is required in this case of the orientation of the dislocation with respect to the interacting solute atom, but it is necessary in calculating shear stresses interactions. ~ Cottrell and Bilby have shown that if the introduction of a solute atom causes a volume change A v at some point in the lattice where the hydrostatic pressure of the stress field is p, the interaction energy is V =pAv = K| (7.9) where K is the bulk modulus and tO is the local dilatation strain. The dilatation strain at a point (R, 0) from a positive edge dislocation is b(l - 2v) x sin O/2rrR(1 - 1To a first approximation a solute atom does not interact with a screw dislocation since there is no dilatation around the screw; a second-order dilatation exists however, which gives rise to a non-zero interaction falling off with distance from the dislocation according to l/r 2. In real crystals, anisotropic elasticity will lead to first-order size effects even with screw dislocations and hence a substantial interaction is to be expected. v), and substituting K = 2/z(1 + v)/3(1 -2v), where /z is the shear modulus and v Poisson's ratio, we get the expression V(R.0) = b(1 + v)/zAv sinO/3~rR(1 - v) = AsinO/R (7.10) This is the interaction energy at a point whose polar coordinates with respect to the centre of the dislocation are R and 0. We note that V is positive on the upper side (0 < 0 < rr) of the dislocation for a large atom (Av > 0), and negative on the lower side, which agrees with the qualitative picture of a large atom being repelled from the compressed region and attracted into the expanded one. It is expected that the site for the strongest binding energy Vm~ will be at a point 0 = 3zr/2, R = r0--~ b; and using known values of/z, v and Av in equation (7.10) we obtain A ~ 3 • 10 -29 N m 2 and Vm,x ~- 1 eV for carbon or nitrogen in s-iron. This value is almost certainly too high because of the limitations of the interaction energy equation in describing conditions near the centre of a dislocation, and a more realistic value obtained from experiment (e.g. internal friction experiments) is Vma x ~ ..~ ~ 3 to eV. For a substitutional solute atom such as zinc in copper A v is not only smaller but also easier to calculate from lattice parameter measurements. Thus, if r and r(1 + e) are the atomic radii of the solvent and solute, respectively, where e is the misfit value, the volume change A v is 4rrr3e and equation (7.10) becomes V = 4(1 + v)#ber 3 sin 0/3(1 - v)R = A sin O/R (7.11) Taking the known values /z = 40 GN/m 2, v = 0.36, b = 2.55 x 10 -l~ m, r0 and e = 0.06, we find A _~ 5 • l0 -30 N m 2, which gives a much lower binding l energy, Vmax = ~ eV. The yield phenomenon is particularly strong in iron because an additional effect is important; this concerns
216 Moderm Physical Metallurgy and Materials Engineering the type of atmosphere a dislocation gathers round and from equation(7. 13 )it can be shown that a 0.1 at. itself which can be either condensed or dilute. Dur- alloy has a condensation temperature T=250 K ing the strain-ageing process migration of the solute Copper-based alloys, on the other hand, usually form atoms to the dislocation occurs and two important extensive solid solutions, and, consequently, concen- cases arise. First, if all the sites at the centre of the trated alloys may exhibit strong yielding phenomena dislocation become occupied the atmosphere is then The best-known example is a-brass and, because said to be condensed; each atom plane threaded by the Vmax i ev, a dilute alloy containing I at. zinc dislocation contains one solute atom at the position has a condensation temperature Te 300 K. At low of maximum binding together with a diffuse cloud of zinc concentrations(1-10%)the yield point in brass other solute atoms further out, If, on the other hand probably solely due to the segregation of zinc atoms quilibrium is established before all the sites at the to dislocations. At higher concentrations,however,it centre are saturated, a steady state must be reached may also be due to short-range order in which the probability of solute atoms leaving the The steady-state distribution of solute atoms around 7.4. 8 Dislocation locking and temperature the dislocations is then given by the relation The binding of a solute atom to a dislocation is short range in nature, and is effective only over an atomic distance or so( Figure 7. 29). Moreover, the dislocation where co is the concentration far from a dislocation, k line is flexible and this enables yielding to begin by and c the local impurity concentration at a point near only a few atomic spacings long, beyond the position is known as the dilute or Maxwellian atmosphere. of the dislocation line from its anchopdrates the res the dislocation where the binding energy is V. This marked x2. The applied stress then sep by pu Clearly, the form of an atmosphere will be governed the sides of this loop outward along the dislocation by the concentration of atoms at the sites of line, i.e. by double kink movement. Such a breakaway maximum binding energy, V max and for a given alle process would lead to a yield stress which depends (i.e. Co and Vmax fixed) this concentration will be sensitively on temperature, as shown in Figure 7.30a Cymat= Co exp(vmax/kT) (7.12) dence parameter in the Hall-Petch equation, in most as long as cym is less than unity. The annealed bcc metals is almost independent of tem- depends only on the temperature, and as the temper- perature down to the range(<100 K)where twinning ature is lowered cvma will eventually rise to unity. occurs, and that practically all the large temperat By definition the atmosphere will then have passed dependence is due to o(see Figure 7.30b).To explain this observation it is argued that when locked disloc hich this occurs is known as the condensation tem- tions exist initially in the material, yielding starts by erature Tc, and can be obtained by substituting the unpinning them if they are weakly locked(this corre- alue Cvma l in equation(7. 12) sponds to the condition envisaged by Cottrell-Bilby ) Tc=Vmax/kIn(1/co) Substituting the value of Vmax for iron, i.e.i eV in thi eL (7. 13) but if they are strongly locked it starts instead by uation we find that only a very small concentration of carbon or nitrogen is necessary to give a condensed atmosphere at room temperature, and with the usual concentration strong yielding behaviour is expected up to temperatures of about 400C. In the fcc structure although the locking between a olute atom and a dislocation is likely to be weaker, condensed atmospheres are still possible if this weak ness can be compensated for by sufficiently increasing Displacement the concentration of the solution. This may be why Dislocation line xamples of yielding in fcc materials have been mainly obtained from alloys, Solid solution alloys of alt minium usually contain less than 0 I at. %o of solute element, and these show yielding in single crystals Unstable position ture,-196.C)whereas supersaturated alloys show evi- dence of strong yielding even in polycrystals at room F 7.29 Stress-displacem temperature; copper dissolved in aluminium has a mis- of a dislocation from its atmosphere(after Cottrell,1957. fit value e 20.12 which corresponds to Vmax=4 ev, courtesy of the Institution of Mechanical Engineers)
216 Modem Physical Metallurgy and Materials Engineering the type of atmosphere a dislocation gathers round itself which can be either condensed or dilute. During the strain-ageing process migration of the solute atoms to the dislocation occurs and two important cases arise. First, if all the sites at the centre of the dislocation become occupied the atmosphere is then said to be condensed; each atom plane threaded by the dislocation contains one solute atom at the position of maximum binding together with a diffuse cloud of other solute atoms further out. If, on the other hand, equilibrium is established before all the sites at the centre are saturated, a steady state must be reached in which the probability of solute atoms leaving the centre can equal the probability of their entering it. The steady-state distribution of solute atoms around the dislocations is then given by the relation C(R,o) = Co exp [V(R.o)/k T] where co is the concentration far from a dislocation, k is Boltzmann's constant, T is the absolute temperature and c the local impurity concentration at a point near the dislocation where the binding energy is V. This is known as the dilute or Maxwellian atmosphere. Clearly, the form of an atmosphere will be governed by the concentration of solute atoms at the sites of maximum binding energy, Vmax and for a given alloy (i.e. co and Vmax fixed) this concentration will be CVmax = CO exp (Vmax/k T) (7.12) as long as CVmax is less than unity. The value of CVmax depends only on the temperature, and as the temperature is lowered CVmax will eventually rise to unity. By definition the atmosphere will then have passed from a dilute to a condensed state. The temperature at which this occurs is known as the condensation temperature Tc, and can be obtained by substituting the value CVmax = 1 in equation (7.12) when Tc = Vmax/kln(1/co) (7.13) l Substituting the value of Vmax for iron, i.e. ~ eV in this equation we find that only a very small concentration of carbon or nitrogen is necessary to give a condensed atmosphere at room temperature, and with the usual concentration strong yielding behaviour is expected up to temperatures of about 400~ In the fcc structure although the locking between a solute atom and a dislocation is likely to be weaker, condensed atmospheres are still possible if this weakness can be compensated for by sufficiently increasing the concentration of the solution. This may be why examples of yielding in fcc materials have been mainly obtained from alloys. Solid solution alloys of aluminium usually contain less than 0.1 at. % of solute element, and these show yielding in single crystals only at low temperature (e.g. liquid nitrogen temperature, -196~ whereas supersaturated alloys show evidence of strong yielding even in polycrystals at room temperature; copper dissolved in aluminium has a mis- 1 fit value e "~ 0.12 which corresponds to Vmax -- ~ eV, and from equation (7.13) it can be shown that a 0.1 at. % alloy has a condensation temperature Tc = 250 K. Copper-based alloys, on the other hand, usually form extensive solid solutions, and, consequently, concentrated alloys may exhibit strong yielding phenomena. The best-known example is c~-brass and, because ,-,., l Vmax ~ eV, a dilute alloy containing 1 at. % zinc has a condensation temperature Tc--~ 300 K. At low zinc concentrations (1-10%) the yield point in brass is probably solely due to the segregation of zinc atoms to dislocations. At higher concentrations, however, it may also be due to short-range order. 7.4.8 Dislocation locking and temperature The binding of a solute atom to a dislocation is short range in nature, and is effective only over an atomic distance or so (Figure 7.29). Moreover, the dislocation line is flexible and this enables yielding to begin by throwing forward a small length of dislocation line, only a few atomic spacings long, beyond the position marked X 2. The applied stress then separates the rest of the dislocation line from its anchorage by pulling the sides of this loop outward along the dislocation line, i.e. by double kink movement. Such a breakaway process would lead to a yield stress which depends sensitively on temperature, as shown in Figure 7.30a. It is observed, however, that ky, the grain-size dependence parameter in the Hall-Petch equation, in most annealed bcc metals is almost independent of temperature down to the range (< 100 K) where twinning occurs, and that practically all the large temperaturedependence is due to cri (see Figure 7.30b). To explain this observation it is argued that when locked dislocations exist initially in the material, yielding starts by unpinning them if they are weakly locked (this corresponds to the condition envisaged by Cottrell-Bilby), but if they are strongly locked it starts instead by O'rnax t~ ffl / Displacement ~ / . . . I / """%/~DIslocahon hne / /~--" Unstable position O Xl X2 Figure 7.29 Stress-displacement cuta, e for the breakaway of a dislocation from its atmosphere (after Cottrell, 1957; courtesy of the h~stitution of Mechanical Engineers)