Chapter 6: Strengthening Mechanisms PDF

# Chapter 6: Strengthening Mechanisms ## 6-1 Introduction Chapter 4 considered the plastic deformation of single crystals in terms of movement of dislocations and the basic deformation mechanisms of slip and twinning. Single-crystal specimens represent the most ideal condition for study. The simpl...

# Chapter 6: Strengthening Mechanisms ## 6-1 Introduction Chapter 4 considered the plastic deformation of single crystals in terms of movement of dislocations and the basic deformation mechanisms of slip and twinning. Single-crystal specimens represent the most ideal condition for study. The simplifications which result from the single-crystal condition materially assist in describing the deformation behavior in terms of crystallography and defect structure. However, with the exception of solid-state electronic devices, single crystals are rarely used for engineering applications because of limitations involving their strength, size, and production. Commercial metal products invariably are made up of a tremendous number of individual crystals or grains. The individual grains of the polycrystalline aggregate do not deform in accordance with the relatively simple laws which describe plastic deformation in single crystals because of the restraining effect of the surrounding grains. Chapter 5 was concerned with the basic relationships that govern dislocation behavior. It should be clear that strength is inversely related to dislocation mobility and that high-purity single crystals have several factors that can affect their strength and mechanical behavior. The crystal structure determines the number and type of slip systems, fixes the Burgers vector, and determines the lattice friction stress (Peierls stress) which sets the base strength level and temperature dependence of strength. In close-packed structures, the stacking-fault energy determines the extent of dislocation dissociation, which influences the ease of cross slip and the subsequent strain-hardening rate. The purity and method of preparation determine the initial dislocation density and substructure. These limited variables introduce complexity so that mechanical behavior is not easily predicted. However, materials of highest strength and usefulness are produced by fine grain size, additions of solute atoms, fine particles, and phase transformations which increase strength. These various strengthening mechanisms and aspects of the deformation of polycrystalline materials are discussed in this chapter. ## 6-2 Grain Boundaries and Deformation The boundaries between grains in a polycrystalline aggregate are a region of disturbed lattice only a few atomic diameters wide. The ordinary high-angle grain boundary represents a region of random misfit between the adjoining crystal lattices. As the difference in orientation between grains on each side of the boundary decreases, the state of order in the boundary increases. For the limiting case of a low-angle boundary, the orientation difference across the boundary may be less than 1º, the boundary is composed of a regular array of dislocations. Figure 6-1a schematically illustrates the structure at a high-angle grain boundary. Note the unorganized structure, with a few atoms belonging to both grains, while most belong to neither. Those atoms that belong to both grains are called coincident sites. This grain boundary structure contains grain-boundary dislocations. These are not mobile dislocations producing extensive slip; rather, they group together within the boundary to form a step or grain-boundary ledge. High-angle grain boundaries are boundaries of rather high surface energy. For example, a grain boundary in copper has an interfacial surface energy of about 600 mJ m-2, while the energy of a twin boundary is only about 25 mJ m-2. Because of their high energy, grain boundaries serve as preferential sites for solid-state reactions such as diffusion, phase transformations, and precipitation reactions. The high energy of a grain boundary usually results in a higher concentration of solute atoms at the boundary than in the interior of the grains. This makes it difficult to separate the pure mechanical effect of grain boundaries on properties from an effect due to impurity segregation. When a single crystal is deformed in tension, it is usually free to deform on a single slip system. The lattice can rotate as extension takes place. However, individual grains in a polycrystalline specimen are not subjected to a single uniaxial stress system when the specimen is deformed in tension. In a polycrystal, continuity must be maintained, so that the boundaries between the deforming crystals remain intact. Although each grain tries to deform homogeneously in conformity with the deformation of the specimen as a whole, the constraints imposed by continuity cause considerable differences in deformation between neighboring grains and within each grain. Studies of coarse-grain aluminum showed that the strain in the vicinity of a grain boundary usually differs markedly from the strain at the center of the grain. Although the strain is continuous across boundaries, a steep strain gradient can exist in this region. As the grain size decreases and strain increases, deformation becomes more homogeneous. Because of the constraints imposed by the grain boundaries, several slip systems can operate even at low strains. This causes slip to occur on non-close-packed planes in regions near the grain boundary. Slip in polycrystalline aluminum has been observed on the {100}, {110}, and {113} planes. Different slip systems can operate in adjacent regions of the same grain, which result in complex lattice rotations forming deformation bands. Since more slip systems are usually operative near the grain boundary, the hardness usually will be higher near the boundary than in the center of a grain. As the grain diameter is reduced, more of the effects of grain boundaries will be felt at the grain center. Thus, the strain hardening of a fine grain size metal will be greater than in a coarse-grain polycrystalline aggregate. ## 6-3 Strengthening from Grain Boundaries Direct evidence for the mechanical strengthening of grain boundaries is provided by experiments on bicrystals in which the orientation difference between a longitudinal grain boundary was varied in a systematic manner. The yield stress of the bicrystals increased linearly with increasing misorientation across the grain boundary, and extrapolation to zero misorientation angle gave a value close to that of the yield stress of a single crystal. These results imply that a simple grain boundary has little inherent strength and that the strengthening due to grain boundaries results from mutual interference to slip within the grains. Several attempts have been made to calculate the stress-strain curve for a polycrystal from stress-strain curves for single crystals. The resolved shear stress in a single crystal was given by $τ = σ sin x cos x = σ/M$ (6-1) Where M is an orientation factor (the reciprocal of the Schmid factor). For a polycrystal the orientation factor M varies from grain to grain and it is necessary to determine some average orientation factor M. The best estimate for an fcc lattice is M = 3.1, obtained by G. I. Taylor based on the use of the von Mises compatibility condition and assuming that all grains undergo the same deformation as the overall deformation. The energy expended in deforming a polycrystal must be equal to the sum of the increments of work performed on each of the n slip systems. $σδε = Στ_i αγ_i$ (6-2) If we assume that the critical shear stress is the same on each slip system, then $σ = M τ/de$ (6-3) The value of M was determined by finding the combination of slip systems that minimized the value of |Edy1| but still satisfied the required continuity at the grain boundaries. To get reasonable agreement, single-crystal curves must be used which involve the same slip mechanisms as the polycrystalline specimen. Since polycrystals involve multiple slip, the single-crystal curves for fcc metals should have the (111) or (100) orientation for which easy glide is a minimum. The polycrystalline curve is calculated from the relations $σ = Mr$ and $ε = σ_y/M$. Quite reasonable agreement has been found. By combining these equations, the strain-hardening rate for an fcc polycrystal should be about 9.5 times that for a single crystal. $do/dT = M^2 * dε/dy$ (6-5) ## 6-4 Low-Angle Grain Boundaries A definite substructure can exist within the grains surrounded by high-energy grain boundaries. The subgrains are low-angle boundaries in which the difference in orientation across the boundary may be only a few minutes of arc or, at most, a few degrees. Because of this small orientation difference, special x-ray techniques are required to detect the existence of a substructure network. Subgrain boundaries are lower-energy boundaries than grain boundaries, and therefore they etch less readily than grain boundaries. However, in many metals they can be detected in the microstructure by metallographic procedures. A low-angle boundary contains a relatively simple arrangement of dislocations. The simplest situation is the case of a tilt boundary. Figure 6-4a illustrates two cubic crystals with a common [001] axis. The slight difference in orientation between the grains is indicated by the angle 8. In Fig. 6-4b the two crystals have been joined to form a bicrystal containing a low-angle boundary. Along the boundary the atoms adjust their position by localized deformation to produce a smooth transition from one grain to the other. However, elastic deformation cannot accommodate all the misfit, so that some of the atom planes must end on the grain boundary. Where the atom planes end, there is an edge dislocation. Therefore, low-angle tilt boundaries can be considered to be an array of edge dislocations. From the geometry of Fig. 6-4b the relationship between 𝜃 and the spacing between dislocations is given by $𝜃 = 2 tan-1 * (b/2D)$ (6-17) Where b is the magnitude of the Burgers vector of the lattice. The validity of the dislocation model of the low-angle boundary is found in the fact it is possible to calculate the grain-boundary energy as a function of the difference in orientation between the two grains. So long as the angle does not become greater than about 20°, good agreement is obtained between the measured values of grain-boundary energy and the values calculated on the basis of the dislocation model. Other evidence for the dislocation nature of low-angle boundaries comes from metallographic observations. If the angle is low, so that the spacing between dislocations is large, it is often possible to observe that the boundary is composed of a row of etch pits, with each pit corresponding to the site of an edge dislocation. Subboundaries or low-angle boundaries can be produced in a number of ways. They may be produced during crystal growth, during high-temperature creep deformation, or as the result of a phase transformation. The veining in ferrite grains is a well-known example of a substructure resulting from the internal stresses accompanying a phase transformation. Perhaps the most general method of producing a substructure network is by introducing a small amount of deformation (from about 1 to 10 percent prestrain) and following this with an annealing treatment to rearrange the dislocations into subgrain boundaries. The amount of deformation and temperature must be low enough to prevent the formation of new grains by recrystallization. This process has been called recrystallization in situ, or polygonization. ## 6-5 Yield-Point Phenomenon Many metals, particularly low-carbon steel, show a localized, heterogeneous type of transition from elastic to plastic deformation which produces a yield point in the stress-strain curve. Rather than having a flow curve with a gradual transition from elastic to plastic behavior, such as was shown in Fig. 3-1, metals with a yield point have a flow curve or, what is equivalent, a load-elongation diagram similar to Fig. 6-8. The load increases steadily with elastic strain, drops suddenly, fluctuates about some approximately constant value of load, and then rises with further strain. The load at which the sudden drop occurs is called the upper yield point. The constant load is called the lower yield point, and the elongation which occurs at constant load is called the yield-point elongation. The deformation occurring throughout the yield-point elongation is heterogeneous. At the upper yield point a discrete band of deformed metal, often readily visible with the eye, appears at a stress concentration such as a fillet, and coincident with the formation of the band the load drops to the lower yield point. The band then propagates along the length of the specimen, causing the yield-point elongation. In the usual case several bands will form at several points of stress concentration. These bands are generally at approximately 45° to the tensile axis. They are usually called Lüders bands, Hartmann lines, or stretcher strains, and this type of deformation is sometimes referred to as the Piobert effect. When several Lüders bands are formed, the flow curve during the yield-point elongation will be irregular, each jog corresponding to the formation of a new Lüders band. After the Lüders bands have propagated to cover the entire length of the specimen test section, the flow will increase with strain in the usual manner. This marks the end of the yield-point elongation. The yield-point phenomenon was found originally in low-carbon steel. A pronounced upper and lower yield point and a yield-point elongation of over 10 percent can be obtained with this material under proper conditions. More recently the yield point has come to be accepted as a general phenomenon, since it has been observed in a number of other metals and alloys. In addition to iron and steel, yield points have been observed in polycrystalline molybdenum, titanium, and aluminum alloys and in single crystals of iron, cadmium, zinc, alpha and beta brass, and aluminum. Usually the yield point can be associated with small amounts of interstitial or substitutional impurities. For example, it has been shown that almost complete removal of carbon and nitrogen from low-carbon steel by wet-hydrogen treatment will remove the yield point. However, only about 0.001 percent of either of these elements is required for a reappearance of the yield point. A number of experimental factors affect the attainment of a sharp upper yield point. A sharp upper yield point is promoted by the use of an elastically rigid (hard) testing machine, very careful axial alignment of the specimen, the use of specimens free from stress concentrations, high rate of loading, and, frequently, testing at subambient temperatures. If, through careful avoidance of stress concentrations, the first Lüders band can be made to form at the middle of the test specimen, the upper yield point can be roughly twice the lower yield point. However, it is more usual to obtain an upper yield point 10 to 20 percent greater than the lower yield point. The onset of general yielding occurs at a stress where the average dislocation sources can create slip bands through a good volume of the material. Thus, the general yield stress can be expressed as $σ_o = σ + σ_i$ (6-18) Where σ is the stress to operate the dislocation sources and σ_i is the friction stress representing the combining effect of all the obstacles to the motion of dislocations arising from the sources. If the stress to operate the sources is high, then the initial yield stress is high. The explanation of the yield-point phenomenon in terms of dislocation behavior arose originally from the idea that the dislocation sources were locked or pinned by solute atom interactions. The explanation of this behavior was one of the early triumphs of dislocation theory. Carbon or nitrogen atoms in iron readily diffuse to the position of minimum energy just below the extra plane of atoms in a positive edge dislocation. The elastic interaction is so strong that the impurity atmosphere becomes completely saturated and condenses into a row of atoms along the core of the dislocation. The breakaway stress required to pull a dislocation line away from a line of solute atoms is $τ_b ≈ A/r_o$ (6-19) Where A is given by Eq. (5-35) and r_o ≈ 0.2 nm is the distance from the dislocation core to the line of solute atoms. When the dislocation line is pulled free from the influence of the solute atoms, slip can occur at a lower stress. Alternatively, where dislocations are strongly pinned, such as by carbon and nitrogen in iron, new dislocations must be generated to allow the flow stress to drop. This explains the origin of the upper yield stress (the drop in load after yielding has begun). The dislocations released into the slip plane pile-up at grain boundaries. As discussed in Sec. 6-3, the pile-up produces a stress concentration at its tip which combines with the applied stress in the next grain to unlock sources (or create new dislocations) and in this way a Lüders band propagates over the specimen. The magnitude of the yield-point effect will depend on the interaction energy, Eq. (5-35), and the concentration of solute atoms at the dislocations, Eq. (5-36). ## 6-6 Strain Aging Strain aging is a type of behavior, usually associated with the yield-point phenomenon, in which the strength of a metal is increased and the ductility is decreased on heating at a relatively low temperature after cold-working. This behavior can best be illustrated by considering Fig. 6-9, which schematically describes the effect of strain aging on the flow curve of a low-carbon steel. Region A of Fig. 6-9 shows the stress-strain curve for a low-carbon steel strained plastically through the yield-point elongation to a strain corresponding to point X. The specimen is then unloaded and retested without appreciable delay or any heat treatment (region B). Note that on reloading the yield point does not occur, since the dislocations have been torn away from the atmosphere of carbon and nitrogen atoms. Consider now that the specimen is strained to point Y and unloaded. If it is reloaded after aging for several days at room temperature or several hours at an aging temperature like 400 K, the yield point will reappear. Moreover, the yield point will be increased by the aging treatment from Y to Z. The reappearance of the yield point is due to the diffusion of carbon and nitrogen atoms to the dislocations during the aging period to form new atmospheres of interstitials anchoring the dislocations. Support for this mechanism is found in the fact that the activation energy for the return of the yield point on aging is in good agreement with the activation energy for the diffusion of carbon in alpha iron. Nitrogen plays a more important role in the strain aging of iron than carbon because it has a higher solubility and diffusion coefficient and produces less complete precipitation during slow cooling. From a practical standpoint it is important to eliminate strain aging in deep-drawing steel because the reappearance of the yield point can lead to difficulties with surface markings or "stretcher strains" due to the localized heterogeneous deformation. To control strain aging, it is usually desirable to lower the amount of carbon and nitrogen in solution by adding elements which will take part of the interstitials out of solution by forming stable carbides or nitrides. Aluminum, vanadium, titaniun, col-umbium, and boron have been added for this purpose. While a certain amount of control over strain aging can be achieved, there is no commercial low-carbon steel which is completely non-strain aging. The usual industrial solution to this problem is to deform the metal to point X by roller leveling or a skin-pass rolling operation and use it immediately before it can age. The local plastic deformation by rolling produces sufficient fresh dislocations so that subsequent plastic flow can occur without a yield point. The occurrence of strain aging is a fairly general phenomenon in metals. In addition to the return of the yield point and an increase in the yield stress after aging, strain aging also produces a decrease in ductility and a low value of strain-rate sensitivity. Strain aging also is associated with the occurrence of serrations in the stress-strain curve (discontinuous or repeated yielding). This dynamic strain-aging behavior (Fig. 6-10) is called the Portevin-LeChatelier effect. The solute atoms are able to diffuse in the specimen at a rate faster than the speed of the dislocations so as to catch and lock them. Therefore, the load must increase and when the dislocations are torn away from the solute atoms there is a load drop. This process occurs many times, causing the serrations in the stress-strain curve. Dynamic strain aging is not the only phenomena that can cause a serrated stress-strain curve. Mechanical twinning occurring during deformation or stress-assisted martensitic transformations will produce the same effect. For plain carbon steel, discontinuous yielding occurs in the temperature region of 500 to 650 K. This temperature region is known as the blue brittle region because steel heated in this temperature region shows a decreased tensile ductility and decreased notched-impact resistance. This temperature range is also the region in which steels show a minimum in strain-rate sensitivity and a maximum in the rate of strain aging. These facts point to the realization that blue brittleness is not a separate phenomenon but is just an accelerated strain aging. ## 6-7 Solid-Solution Strengthening The introduction of solute atoms into solid solution in the solvent-atom lattice invariably produces an alloy which is stronger than the pure metal. There are two types of solid solutions. If the solute and solvent atoms are roughly similar in size, the solute atoms will occupy lattice points in the crystal lattice of the solvent atoms. This is called substitutional solid solution. If the solute atoms are much smaller than the solvent atoms, they occupy interstitial positions in the solvent lattice. Carbon, nitrogen, oxygen, hydrogen, and boron are the elements which commonly form interstitial solid solutions. The factors which control the tendency for the formation of substitutional solid solutions have been uncovered chiefly through the work of Hume-Rothery. If the sizes of the two atoms, as approximately indicated by the lattice parameter, differ by less than 15 percent, the size factor is favorable for solid-solution formation. When the size factor is greater than 15 percent, the extent of solid solubility is usually restricted to less than 1 percent. Metals which do not have a strong chemical affinity for each other tend to form solid solutions, while metals which are far apart on the electromotive series tend to form intermetallic compounds. The relative valence of the solute and solvent also is important. The solubility of a metal with higher valence in a solvent of lower valence is more extensive than for the reverse situation. For example, zinc is much more soluble in copper than is copper in zinc. This relative-valence effect can be rationalized to a certain extent in terms of the electron-atom ratio. For certain solvent metals, the limit of solubility occurs at approximately the same value of electron-atom ratio for solute atoms of different valence. Finally, for complete solid solubility over the entire range of composition the solute and solvent atoms must have the same crystal structure. The acquisition of fundamental information about the causes of solid-solution hardening has been a slow process. Early studies of the increase in hardness resulting from solid-solution additions showed that the hardness increase varies directly with the difference in the size of the solute and solvent atoms, or with the change in lattice parameter resulting from the solute addition. However, it is apparent that size factor alone cannot explain solid-solution hardening. An improvement in correlation of data results when the relative valence of the solute and solvent are considered in addition to the lattice-parameter distortion. The importance of valence is shown in Fig. 6-11, where the yield stress of copper alloys of constant lattice parameter is plotted against the electron-atom ratio. Further results show that alloys with equal grain size, lattice parameter, and electron-atom ratio have the same initial yield stress, but the flow curves differ at larger strains. The usual result of solute additions is to raise the yield stress and the level of the stress-strain curve as a whole. As was seen in Sec. 6-5, solute atoms also frequently produce a yield-point effect. Since solid-solution alloy additions also frequently produce a yield-point effect. ## 6-8 Strengthening from Fine Particles Small second-phase particles distributed in a ductile matrix are a common source of alloy strengthening. In dispersion hardening³ *the hard particles are mixed with matrix powder and consolidated and processed by powder metallurgy techniques*. However, very many alloy systems can be strengthened by precipitation reactions in the solid state. Precipitation hardening, or age hardening, is produced by solution treating and quenching an alloy in which a second phase is in solid solution at the elevated temperature but precipitates upon quenching and aging at a lower temperature. The age-hardening aluminum alloys and copper-beryllium alloys are common examples. For precipitation hardening to occur, the second phase must be soluble at an elevated temperature but must exhibit decreasing solubility with decreasing temperature. By contrast, the second phase in dispersion-hardening systems has very little solubility in the matrix, even at elevated temperatures. Usually there is atomic matching, or coherency, between the lattices of the precipitate and the matrix, while in dispersion-hardened systems there generally is no coherency between the second-phase particles and the matrix. The requirement of a decreasing solubility with temperature places a limitation on the number of useful precipitation-hardening alloy systems. On the other hand, it is at least theoretically possible to produce an almost infinite number of dispersion-hardened systems by mixing finely divided metallic powders and second-phase particles (oxides, carbides, nitrides, borides, etc.) and consolidating them by powder metallurgy techniques. Advantage has been taken of this method to produce dispersion-hardened systems which are thermally stable at very high temperatures. Because of the finely dispersed second-phase particles, these alloys are much more resistant to recrystallization and grain growth than single-phase alloys. ## 6-9 Strengthening from Fine Particles Small second-phase particles distributed in a ductile matrix are a common source of alloy strengthening. In dispersion hardening, hard particles are mixed with matrix powder and consolidated and processed by powder metallurgy techniques. *However, very many alloy systems can be strengthened by precipitation reactions in the solid state*. Precipitation hardening, or age hardening, is *produced by solution treating and quenching an alloy in which a second phase is in solid solution at the elevated temperature but precipitates upon quenching and aging at a lower temperature*. The age-hardening aluminum alloys and copper-beryllium alloys are common examples. For precipitation hardening to occur, the second phase must be soluble at an elevated temperature but must exhibit decreasing solubility with decreasing temperature. By contrast, the second phase in dispersion-hardening systems has very little solubility in the matrix, even at elevated temperatures. Usually, there is atomic matching, or coherency, between the lattices of the precipitate and the matrix, while in dispersion-hardened systems, there generally is no coherency between the second-phase particles and the matrix. The requirement of a decreasing solubility with temperature places a limitation on the number of useful precipitation-hardening alloy systems. On the other hand, it is at least theoretically possible to produce an almost infinite number of dispersion-hardened systems by mixing finely divided metallic powders and second-phase particles (oxides, carbides, nitrides, borides, etc.) and consolidating them by powder metallurgy techniques. Advantage has been taken of this method to produce dispersion-hardened systems which are thermally stable at very high temperatures. Because of the finely dispersed second-phase particles, these alloys are much more resistant to recrystallization and grain growth than single-phase alloys. ## 6-10 Fiber Strengthening Materials of high strength, and especially high strength-to-weight ratio, can be produced by incorporating fine fibers in a ductile matrix. The fibers must have high strength and high elastic modulus while the matrix must be ductile and nonreactive with the fibers. Because of their very high strength, whiskers of materials such as Al2O3 have been used with good results, but most fiber-strengthened materials use fibers of boron or graphite or metal wires such as molybdenum. The fiber-strengthened materials are usually used for high-temperature applications; however, several have found uses at lower temperatures (e.g., in composites for aircraft structures). Generally, the strengthening from fibers is due to the load transfer from the matrix to the fiber. The fiber is the strengthening element. As the load is applied, the matrix will yield before the fiber. The fiber is pulled until the stress concentration at the end or along the fiber is high enough to initiate slip in the matrix. ## 6-11 Strain Hardening Strain hardening or cold-working is an important industrial process that is used to harden metals or alloys that do not respond to heat treatment. The rate of strain hardening can be gauged from the slope of the flow curve. Generally, the rate of strain hardening is lower for hcp metals than for cubic metals. Increasing temperature also lowers the rate of strain hardening. For alloys strengthened by solid-solution additions the rate of strain hardening may be either increased or decreased compared with the behavior for the pure metal. However, the final strength of a cold-worked solid-solution alloy is almost always greater than that of the pure metal cold-worked to the same extent. Figure 6-29 shows the typical variation of strength and ductility parameters with increasing amount of cold-work. Since in most cold-working processes one or two dimensions of the metal are reduced at the expense of an increase in the other dimensions, cold-work produces elongation of the grains in the principal direction of working. Severe deformation produces a reorientation of the grains into a preferred orientation. ## 6-12 Annealing of Cold-Worked Metal The cold-worked state is a condition of higher internal energy than the undeformed metal. Although the cold-worked dislocation cell structure is mechanically stable, it is not thermodynamically stable. With increasing temperature the cold-worked state becomes more and more unstable. Eventually the metal softens and reverts to a strain-free condition. The overall process by which this occurs is known as annealing. Annealing is very important commercially because it restores the ductility to a metal that has been severely strain-hardened. Therefore, by interposing annealing operations after severe deformation it is possible to deform most metals to a very great extent. The process of annealing can be divided into three fairly distinct processes: recovery, recrystallization, and grain growth. Figure 6-30 will help to distinguish between these processes. Recovery is usually defined as the restoration of the physical properties of the cold-worked metal without any observable change in microstructure. Electrical conductivity increases rapidly toward the annealed value during recovery, and lattice strain, as measured with x-rays, is appreciably reduced. The properties that are most affected by recovery are those which are sensitive to point defects. The strength properties, which are controlled by dislocations, are not affected at recovery temperatures. An exception to this is single crystals of hcp metals which have deformed on only one set of planes (easy glide). For this situation it is possible to recover completely the yield stress of a strain-hardened crystal without producing recrystallization. Recrystallization is the replacement of the cold-worked structure by a new set of strain-free grains. Recrystallization is readily detected by metallographic methods and is evidenced by a decrease in hardness or strength and an increase in ductility. The density of dislocations decreases considerably on recrystallization, and all effects of strain hardening are eliminated. The stored energy of cold-work is the driving force for both recovery and recrystallization. If the new strain-free grains are heated at a temperature greater than that required to cause recrystallization, there will be a progressive increase in grain size. The driving force for grain growth is the decrease in free energy resulting from a decreased grain-boundary area due to an increase in grain size. Figure 6-31 shows the progression from a cold-worked microstructure to a fine recrystallized grain structure, and finally to a larger grain size by grain growth. The recrystallization process consists of the nucleation of a strain-free region whose boundary can transform the strained matrix into strain-free material as it moves. In the growth of the boundary out from the nucleus the dislocations are annihilated in the region swept through. This requires that the moving boundary be a high-angle boundary so that there is a high degree of "misfit" to accommodate the dislocations. At least two distinct nucleation mechanisms have been identified for recrystallization. The first is called strain-induced boundary migration, where a strain-free nucleus is formed when one of the existing grain boundaries moves into its neighbor, leaving a strain-free recrystallized region behind. The boundary moves into the grain which contains the higher dislocation density in the local region. In the second nucleation mechanism new grain boundaries are formed in regions of sharp lattice curvature through subgrain growth. ## 6-13 Cold-Worked Structure In Chap. 4 strain hardening was attributed to the interaction of dislocations with other dislocations and with other barriers to their motion through the lattice. So long as slip takes place on only a single set of parallel planes, as with single crystals of hcp metals, only a small amount of strain hardening occurs. However, even with single crystals extensive easy glide is not a general phenomenon, and with polycrystalline specimens it is not observed. Because of the mutual interference of adjacent grains in a polycrystalline specimen multiple slip occurs readily, and there is appreciable strain hardening. Plastic deformation which is carried out in a temperature region and over a time interval such that the strain hardening is not relieved is called cold-work. Plastic deformation produces an increase in the number of dislocations, which by virtue of their interaction results in a higher state of internal stress. An annealed metal contains about 104 to 106 dislocations per mm², while a severely plastically deformed metal contains about 1010 mm 2. Strain hardening or cold-work can be readily detected by x-ray diffraction, although detailed analysis of the x-ray patterns in terms of the structure of the cold-worked state is not usually possible. In Laue patterns cold-work produces a blurring, or asterism, of the spots. For Debye-Scherrer patterns the lines are broadened by cold-work. X-ray line broadening can be due to both a decrease in size of the diffraction unit, as would occur if the grains were fragmented by cold-work, and an increase in lattice strain due to dislocation interaction. Techniques for analyzing the entire peak profile of x-ray lines and separating out the contribution due to lattice strain and particle size have been developed. Considerable detailed knowledge on the structure of the cold-worked state has been obtained from thin-film electron microscopy. In the early stages of plastic deformation slip is essentially on primary glide planes and the dislocations form coplanar arrays. As deformation proceeds, cross slip takes place and multiplication processes operate. The cold-worked structure forms high-dislocation-density regions or tangles, which soon develop into tangled networks. Thus, the characteristic structure of the cold-worked state is a cellular substructure in which high-density-dislocation tangles form the cell walls. The cell structure is usually well developed at strains of around 10 percent. The cell size decreases with strain at low deformation but soon reaches a fixed size, indicating that as strain proceeds the dislocations sweep across the cells and join the tangle in the cell walls. The exact nature of the cold-worked structure will depend on the material, the strain, the strain rate, and the temperature of deformation. The development of a cell structure is less pronounced for low temperature and high strain-rate deformation and in materials with low stacking-fault energies (so that cross slip is difficult). Most of the energy expended in deforming a metal by cold-working is converted into heat. However, roughly about 10 percent of the expended energy is stored in the lattice as an increase in internal energy. Reported values of stored energy range from about 0.04 to 4 kJ kg ¹ of metal. The magnitude of stored energy increases with the melting point of the metal and with solute additions. For a given metal the amount of stored energy depends on the type of deformation process, e.g., wire drawing vs. tension. The stored energy increases with strain up to a limiting value corresponding to saturation. It increases with decreasing temperature of deformation. Very careful calorimeter measurements are required to measure the small amounts of energy stored by cold-working. The major part of the stored energy is due to the generation and interaction of dislocations during cold-working. Vacancies account for part of the stored energy for metals deformed at very low temperature. Stacking faults and twin faults are probably responsible for a small fraction of the stored energy. A reduction in short-range order during the deformation of solid solutions may also contribute to stored energy. Elastic strain energy accounts for only a minor part of the measured stored energy. ## 6-14 Strain Hardening Strain hardening or cold-working is an important industrial process that is used to harden metals or alloys that do not respond to heat treatment. The rate of strain hardening can be gaged from the slope of the flow curve. Generally, the rate of strain hardening is lower for hcp metals than for cubic metals. Increasing temperature also lowers the rate of strain hardening. For alloys strengthened by solid-solution additions the rate of strain hardening may be either increased or decreased compared with the behavior for the pure metal. However, the final strength of a cold-worked solid-solution alloy is almost always greater than that of the pure metal cold-worked to the same extent. Figure 6-29 shows the typical variation of strength and ductility parameters with increasing amount of cold-work. Since in most cold-working processes one or two dimensions of the metal are reduced at the expense of an increase in the other dimensions, cold-work produces elongation of the grains in the principal direction of working. Severe deformation produces a reorientation of the grains into a preferred orientation. ## 6-15 Annealing of Cold-Worked Metal The cold-worked state is a condition of higher internal energy than the undeformed metal. Although the cold-worked dislocation cell structure is mechanically stable, it is not thermodynamically stable. With increasing temperature, the cold-worked state becomes more and more unstable. Eventually, the metal softens and reverts to a strain-free condition. The overall process by which this occurs is known as annealing. Annealing is important commercially because it restores ductility to a metal that has been severely strain-hardened. Therefore, by interposing annealing operations after severe deformation, it is possible to deform most metals to a very great extent. The process of annealing can be divided into three fairly distinct processes: recovery, recrystallization, and grain growth. Figure 6-30 will help to distinguish between these processes. Recovery is usually defined as the restoration of the physical properties of the cold-worked metal without any observable change in microstructure. Electrical conductivity increases rapidly toward the annealed value during recovery, and lattice strain, as measured with x-rays, is appreciably reduced. The properties that are most affected by recovery are those which are sensitive to point defects. The strength properties, which are controlled by dislocations, are not affected at recovery temperatures. An exception to this is single crystals of hcp metals which have deformed on only one set of planes (easy glide). For this situation it is possible to recover completely the yield stress of a strain-hardened crystal without producing recrystallization. Recrystallization is the replacement of the cold-worked structure by a new set of strain-free grains. Recrystallization is readily detected by metallographic methods and is evidenced by a decrease in hardness or strength and an increase in ductility. The density of dislocations decreases considerably on recrystallization, and all effects of strain hardening are eliminated. The stored energy of cold-work is the driving force for both recovery and recrystallization. If the new strain-free grains are heated at a temperature greater than that required to cause recrystallization, there will be a progressive increase in grain size. The