JOHN P.HIRTH

By 1960 many of our present alloy systems had been developed as a result of the research effort during World War II. Thus, many alloy and stainless steels, nickel-base superalloys, brasses and bronzes, precipitation-hardened aluminum alloys, and titanium alloys were then available. The optical characterization of microstructures and of the phase transformations leading to them was extensive. Moreover, with the advent of transmission electron microscopy in 1956,^{1} more detailed microstructural characterization at the atomic level was under way.

Fundamental advances in the relationship between mechanical properties and microstructural defects had been made, and the properties of straight, single dislocations were developed.^{2}^{,}^{3} Irwin^{4} showed that the strain-energy release rate in crack propagation could be related to local crack tip stress-intensity measures, extending the treatment of a brittle crack.^{5} Moreover, Eshelby’s energy momentum tensor, the basis for the modern theory for crack extension, was available.^{6} Work had begun on more complex dislocation arrays, such as pileups,^{7} and simple models for work hardening had been proposed.^{8} Lastly, Orowan^{9} had already suggested the key equation relating flow stress to the inverse spacing of dislocation obstacles such as second-phase particles. Yet, there was little translation of these ideas into concepts that could provide guidelines for alloy design.

This chapter presents a discussion of advances and remaining problems in four areas pertinent to property-microstructure interrelationships, including dislocation theory, fracture theory, properties of complex alloys, and environmental effects. To illustrate the advances and the interplay between fundamental work and macroscopic properties, the discussion is restricted to low-temperature flow and fracture under monotonic tensile loading. Parallel

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Microstructure and Mechanical Properties of Metals
JOHN P.HIRTH
By 1960 many of our present alloy systems had been developed as a result of the research effort during World War II. Thus, many alloy and stainless steels, nickel-base superalloys, brasses and bronzes, precipitation-hardened aluminum alloys, and titanium alloys were then available. The optical characterization of microstructures and of the phase transformations leading to them was extensive. Moreover, with the advent of transmission electron microscopy in 1956,1 more detailed microstructural characterization at the atomic level was under way.
Fundamental advances in the relationship between mechanical properties and microstructural defects had been made, and the properties of straight, single dislocations were developed.2,3 Irwin4 showed that the strain-energy release rate in crack propagation could be related to local crack tip stress-intensity measures, extending the treatment of a brittle crack.5 Moreover, Eshelby’s energy momentum tensor, the basis for the modern theory for crack extension, was available.6 Work had begun on more complex dislocation arrays, such as pileups,7 and simple models for work hardening had been proposed.8 Lastly, Orowan9 had already suggested the key equation relating flow stress to the inverse spacing of dislocation obstacles such as second-phase particles. Yet, there was little translation of these ideas into concepts that could provide guidelines for alloy design.
This chapter presents a discussion of advances and remaining problems in four areas pertinent to property-microstructure interrelationships, including dislocation theory, fracture theory, properties of complex alloys, and environmental effects. To illustrate the advances and the interplay between fundamental work and macroscopic properties, the discussion is restricted to low-temperature flow and fracture under monotonic tensile loading. Parallel

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advances have been made in the areas of fatigue, creep, friction, and wear, but this broader range of topics is outside the scope of this discussion.
DISLOCATIONS
Elastic Theory
In 1939 Burgers10 developed a vector field theory for dislocations, including an expression for the displacement field of a dislocation loop in terms of line integrals over its length and an area integral over its enclosed area. In addition, Peach and Koehler11 presented an expression for the virtual thermodynamic force on a dislocation segment. Yet, in 1960 these concepts had been developed extensively only for straight dislocations. The theory of curved dislocations appeared during the past 25 years.
Mura12 transformed Burgers’s equation into a line integral for displacement gradients in terms of an integrand containing the elastic Green function tensor. This led to the Brown formula13–15 for the stress field at an arbitrary point, produced by a line segment of dislocation, in terms of elastic energy coefficients for an infinite straight dislocation pair passing through the point in question and the ends of the segment. The result applies for isotropic or anisotropic elasticity. Blin16 also presented an expression, based on Burgers’s equation, for the interaction energy of two dislocation loops. Manipulations of this expression and the Peach-Koehler result led to isotropic elastic expressions for the self- and interaction energies of dislocation segments,17,18 the interaction force between segments,19 and the displacement field of a segment.20 By approximating an arbitrary curved dislocation as a connected set of straight segments, one can use this set of relations to determine the elastic fields of complex dislocation arrays. Examples include stacking-fault tetrahedra, loops, double kinks, and double jogs.21
The elastic interaction of an arbitrarily inclined dislocation and a free surface was expressed by Lothe22 as an image interaction analogous to that in electrostatics. This interaction was also extended to the case of dislocation segments, with numerous subsequent applications.23,24
As an alternative to the eigenvalue sextic solution of Eshelby and co-workers,25 Stroh developed an explicit solution for the anisotropic elastic field of a straight dislocation.26 Another alternative formulation using Fourier analysis was presented by Willis.27 The Stroh theory was elaborated as an integral theory, facilitating numerical calculations, by Barnett and Lothe.28 Advanced anisotropic elastic calculations for complex dislocation configurations have now been initiated.29
Lattice Theory
With the advent of computers, atomic calculations have been used to estimate the Peierls stress and energy, the variation of energy with position

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FIGURE 1 View of a screw dislocation along a <110> direction in a bcc crystal showing nascent dissociation into fractional dislocations on {110} and {112} planes.
of a dislocation caused by the periodicity of the lattice. A review of such work is given by Puls.30 An important finding was that a dislocation is a center of dilatation producing an increase of about one atomic volume per plane cut by the dislocation, with about 40 percent residing in the highly nonlinear core region.31 The other major success of such calculations was in explaining the large Peierls stress of body-centered cubic (bcc) metals on the basis of atomic-scale dissociations of screw dislocations into fractional dislocations32,33 (see Figure 1). The threefold dissociated structure must be constricted, requiring stress-assisted thermal activation, for dislocation motion to occur. The dissociation also breaks the crystal symmetry, giving rise to new local defects called flips on the dislocation line where the rotation

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shown in Figure 1 changes sense. Lattice calculations of the energies of kinks, jogs, and flips have provided information on typical concentrations of these defects.34
Of interest in a broader context, the theory developed for dislocation motion by double-kink nucleation and growth35 corresponds exactly to the theory for creation and motion of a soliton pair;36 indeed, kinks can be regarded as solitons.
Grain Boundaries, Interfaces, and Partial Dislocations
The concept of grain boundary dislocations (GBD), with Burgers vector lengths unequal to those of lattice dislocations, and their geometrical description were presented by Bollmann37 and represent a topic of great current interest.38 Similar concepts have been applied to interphase interfaces.39 Observations of these defects38,40 suggest that they have important roles in phase transformations, creep, recrystallization, work hardening, and recovery, through their interaction with lattice dislocations and vacancies. In particular, nonuniform spacings of GBDs imply some degree of interface control in vacancy diffusion processes. Figure 2 shows an example of a GBD array.
Much of our knowledge of dislocation interactions has been gained from electron microscopy. In recent years, the weak-beam technique41 and direct lattice resolution42 have provided information at the near-atomic size scale. For face-centered cubic (fcc) metals, Figure 3 shows seven extended dislocation arrays, many of which provide important barriers to dislocation motion under ambient conditions. In alumina, for example, dislocations dissociated by climb into multiple partial dislocations illustrate a strong impediment to glide in ceramic crystals.43
Problem Areas
Computer simulations are most efficiently used when pair potentials are used to describe atom interactions, whereas fundamental interactions are not of a pair nature, other than in the sense of an empirical fit. An improved first-principles method is needed to treat atom interactions more accurately near highly strained dislocation cores. Cluster calculations can be used for this purpose, but at present are limited to a few tens of atoms, insufficient to model bulk dislocations. Thus, an accurate representation of core structure, crucial to many mechanical and physical properties influenced by dislocations, remains to be done.
Exact elastic field calculations for large numbers of dislocations have been performed only for special arrays such as pileups or grain boundaries. Multiple-dislocation calculations can only be approximated by smearing the dislocations into a continuous array of infinitesimal dislocations. Although

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FIGURE 2 Dark-field micrograph (g=[111]1) of dislocations in a ∑=31 (17.900º/ [111]1/2) related grain boundary in type 304 stainless steel, b=9/31[9,8,14]1:9/31 [8,9,14]2, and the scale marker is 2,500 nm. From W.A.T.Clark.

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FIGURE 3 Dislocations extended on {111} planes into partial dislocations bounding stacking faults in fcc crystals, (a) single dislocation, (b) bend, (c) Lomer-Cottrell barrier, (d) extended node, (e) Z-dipole, (f) Frank loop, and (g) stacking-fault tetrahedron.
many properties, such as lattice curvature, are provided by such a model, as well as a connection with continuum plasticity theory, many details are lost in averaging. Perhaps the greatest need in extending dislocation theory to describe macroscopic flow behavior is that of developing a statistical procedure to describe the interaction among many dislocation segments and thereby to predict constitutive behavior.
Nonlinear elastic theory has been applied to dislocations using only perturbation theory and small departures from linearity. Further work is needed to extend the theory to the nonlinear core region.
Dislocations, and jogs, kinks, and flips that are present on them, interact

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with both ionic and electronic defects in ionic, covalent, insulating, and semiconducting crystals. The role of dislocations is becoming better understood, but many problems remain, for example, in associating band-gap levels with dislocation defects.
CRACKS
J Integral
Independent of Eshelby,6 Rice44 derived a path-independent integral J. For a perfectly brittle crack, J exactly equals the strain-energy release rate derived by Irwin.4 Moreover, for small-scale yielding or deformation at the crack, the integral still represents the energy released by the external loading device and the system per unit of crack advance and, hence, represents the resistance of a material to crack propagation. The parameter is now measured extensively, and for ductile materials the critical value Jc replaces the critical stress intensity Kc for brittle crack initiation. For plastic (as opposed to nonlinear elastic) materials, J is significant only as a measure of the strength of the crack tip singularity in the continuum field before crack growth.
Attention in recent years has focused on crack configurations and crack-defect interactions at the atomic scale. A lattice-trapping barrier analogous to the Peierls barrier for dislocations exists for perfectly brittle cleavage cracking,45 and cracks can propagate by stress-assisted thermal activation of double kinks (solitons).46 The competition between brittle cleavage and crack blunting can be understood in terms of the probability of emission of a dislocation from a sharp crack, a problem solved in a circular dislocation loop approximation by Rice and Thomson.47
Screening
The screening of a crack tip by surrounding dislocations—that is, the cancellation of part of the externally applied field at the crack tip—has been extensively treated in two-dimensional calculations.48 Screened configurations of pileups emitted from cracks have been observed.49 Crack-defect interaction fields are available in both the isotropic47,50 and anisotropic elastic51,52 cases. Motion of other defects near the crack tip can also contribute to the energy release rate and hence to the apparent macroscopic value of J.53 Screening can also be provided by other types of defects, such as those that occur in the transformation toughening of ceramics by dispersed zirconia particles. The particles undergo a phase transformation under the influence of the crack-tip strain field, and the strain fields of the transformed particles in the wake of the crack screen the crack tip.54,55

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Problem Areas
The J formalism works well for any single mode among the opening mode I, the in-plane shear mode II, and the antiplane shear mode III, but even then, not when extensive crack growth precedes instability. However, except for the perfectly brittle crack, there is no detailed theoretical model for mixed-mode cracking, which is an important element in failure by shear instability. Continuum mechanical, elastic-plastic solutions are difficult to obtain for moving cracks, but such asymptotic and numerical results as are now available56 provide a promising framework for certain aspects of crack growth. Analogous results have been obtained for viscoplastic (for example, high-temperature creep) constitutive models.
Nearly all crack defect calculations are performed in two-dimensional approximations. Three-dimensional theoretical calculations are needed for cracks with curved fronts, crack interactions with curved dislocations, and crack interactions with compact second-phase particles of various shapes. Meandering and branched cracks, known to enhance toughness, represent another challenging three-dimensional problem, and work has begun in this area.57
Despite progress in treating the kinematics of rapid crack propagation,58 there remain ambiguities in understanding the problems of acceleration and inertia.
PROPERTIES OF COMPLEX ALLOYS
Flow
Two key relations are successful in qualitatively describing the relation of properties to microstructure on the basis of dislocation concepts. The first of these is the Hall-Petch relation between strength and grain size:
σ=σ0+Kd–1/2, (1)
where σ is the flow stress, d is the grain diameter, and σ0 and K are material constants. The expression dates from the 1950s, but extensive work has been performed recently to verify it.59,60 The expression follows directly from dislocation pileup theory.21 The flow stress in lamellar two-phase structures such as pearlite in steel also follows Equation (1), in which case d then represents the thickness of the metal lamellae.
The second relation is the Orowan-Friedel expression for breakaway of a dislocation from pinning particles (see Figure 4).
(2)

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FIGURE 4 Glide plane and bow-out of dislocation from (a), (b) pinning points or (c) around particles. Area swept, A, critical angles ϕ and β, bow-out segment lengths λ, particle diameter D, and standoff distance x are depicted.
where E is the energy per unit length of dislocation, b is the length of the Burgers vector, λ is the obstacle spacing, and cos ϕ/2 is a factor representing the obstacle strength; the dislocation breaks away from the obstacle at a critical angle ϕ.
The factor E has been calculated in a number of approximations, including the dislocation-segment, anisotropic elastic case.61 Statistical averaging effects have been sampled by computer simulations of dislocation motion through point obstacles of varying pinning strengths randomly distributed in the glide plane.62 The results agree with the form of Equation (2). Many experimental results for coherent or semicoherent precipitates or dispersed, equiaxed particles also agree with Equation (2).62 Solutes that interact strongly with dislocations, such as interstitial carbon in bcc metals, also follow Equation (2), with λ now the solute-solute spacing proportional to c1/2, where c is the solute concentration. Concentrated solid solutions interact primarily in the core region and give a strengthening increment linear in c. In other cases the situation is complicated in that simultaneous breakaway from multiple pinning points can occur. One theoretical estimate predicts a strengthening proportional to c2/3 for this case.63 In many solute cases the well-defined solution hardening is independent of temperature (except for the weak temperature dependence of the elastic constants) and gives rise to the so-called plateau stress over a limited temperature range.63

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For strong dispersoids ϕ approaches zero and dislocations can completely bow out and bypass particles, leaving behind loops of dislocation that encircle the particles. The loops can assume various configurations, readily understandable on the basis of Ashby’s concept of geometrically necessary dislocations.64 These dislocations give rise to rapid work hardening in dispersion-strengthened alloys, of great benefit in inhibiting plastic instability and in giving rise to long-range back stresses.65
Ductile Fracture
In recent years ductile tensile fracture has been classified in three types.66 The first is necking to a point or chisel point, as might occur for a pure fcc metal. The second is deformation or necking terminated by a shear instability leading to a mixed-mode crack following the shear trace, as might occur for a nominally pure bcc metal. The third involves necking or deformation leading to void formation at inclusions or second-phase particles and crack propagation by void linking through either local necking of ligaments or shear localization. The latter process is most pertinent to complex engineering alloys. Figure 5 shows the crack propagation process. Particles crack or decohere under the influence of the crack strain field and thereby nucleate a void. The void grows and limits the plastic flow to a region whose extent is of the order of the void spacing.67 Thus, the smaller the void spacing, the less the plastic flow, the lower the energy release rate, the lower Jc, and the less the toughness. Smaller void spacings are associated with weak interfacial cohesion, brittle particles, large particles, and small spacing of particles. The particle size enters because the nucleation of a crack or a decohesion becomes less probable as the particle size decreases. A rough estimate for spherical particles indicates that the critical local stress for decohesion is proportional to the inverse square root of the particle size and that decohesion should not occur below a critical size of about 20 nm. For very fine particles, of approximately 1 nm, the particles become ineffective as obstacles. These numbers would change somewhat for other particle shapes, particularly those with sharp salient features. Hence, a “window” of sizes exists for optimum dispersion strengthening and toughening.
Theoretical calculations, with some experimental support, also indicate that voids, once formed, increase the susceptibility of a material to failure by macroscopic shear instability.68 The susceptibility to shear instability is much greater under plane-strain conditions and when work hardening is low.69,70 Surface instability in the form of surface rumpling is also a precursor to bulk shear instability and is amenable to experimental study.70
The presence of a metastable phase that transforms in the presence of a local stress or strain concentration provides another means of improving toughness. In transformation-induced plasticity, when a material necks or

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FIGURE 5 A crack tip and the intensely deformed plastic region of a size approximating the spacing L of holes formed at second-phase particles.
undergoes an incipient shear instability, the larger local strain rate and stress can induce martensite transformation, selectively hardening the region and inhibiting continued strain localization.71 If the degree of metastability is properly adjusted so that these effects take place in the intensified stress region ahead of a crack, the toughness of the material can be improved.72
Contained ductile rupture can occur at low toughness values, as indicated by Kc or Jc.73 A thin continuous ductile layer between two harder regions can fail by a ductile mechanism with very little volume of plastic flow. This phenomenon occurs in precipitate-free zones near grain boundaries in improperly heat-treated, precipitation-hardened alloys.
Brittle Fracture
Brittle crack propagation occurs with little energy release below a ductile-to-brittle transition temperature (DBTT). Yet, experimental estimates of J

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give values for metals74 and ceramics,75 respectively, of at least 10 and 3 times the surface energy; thus, a dissipation process other than the creation of surface is still active. Progress has been made in understanding the DBTT for the critical mode I stress intensity KIc(MPa m1/2) that represents the resistance of a material to crack propagation. The intensified strain field of a crack causes a crack nucleation event that could be related to grain size at a critical distance ahead of the crack.74 The event could correspond to the cracking or decohering of a second-phase particle such that the crack then propagates in an unstable manner into the matrix. The problem then becomes statistical and involves the spacing, position, and crack nucleation probability of particles and the position-dependent stress field of the crack;75 some fractographic support exists for such a model.76 Near the DBTT, particle cracking may not be the critical event. Instead the crack may propagate across a single grain and then be arrested. Nonpropagating microcracks spanning grains have been observed in heat-treated steel, with a maximum density at the DBTT.77 The critical event would then be statistical as before but now involves unstable propagation of a favorably situated and oriented microcrack.
Dislocation pileups at the crystal plasticity level of description of plastic deformation, or shear instabilities at a more macroscopic level, could be important in providing stress concentrations to enhance crack nucleation. Even when a crack tip is stressed at the Griffith level, theoretical calculations show that there is usually enough of a stress concentration at the tip to move several dislocations, a tendency that is more likely the lower the strength of the material.
In ceramic materials where the matrix is normally completely resistant to dislocation motion, toughening can still be achieved by screening of the crack tip in a manner analogous to dislocation screening. In this case, the screening is provided by metastable dispersed particles (e.g., ZrO2 in Al2O3) that undergo a phase transformation in the stress field of the crack tip and produce transformation stresses that screen the tip in the sense of decreasing the local stress that would tend to extend the crack.78–80 Analogous toughening effects can be achieved if nonpropagating microcracks form in the region of the crack tip.80,81
Alloy Design
The concepts of flow and fracture in alloys are sufficiently sound and tested to provide at least qualitative guidelines for alloy design, a situation that did not exist in 1960. An ideal alloy should be strengthened primarily by hard dispersed second-phase particles, because these give large work hardening, resistance to failure by plastic instability, and thus some damage tolerance. The particles should be as fine as possible, but greater than the bypass size of about 1 nm, to minimize crack nucleation or decohesion. The

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volume fraction should be as large as possible (to minimize λ) without exceeding the percolation limit (approximately 18 volume percent) at which particle-particle contact becomes highly probable. Precipitation hardening, solid solution hardening, grain size minimization, and deformation each may promote secondary hardening. The cohesive strength of the particle interface should be high. Impurities and large inclusions should be avoided.
Many of these guidelines have been demonstrated in rapid solidification technology alloys.82 In such alloys, segregation is minimized and overaging of particles formed during solidification and cooling is suppressed. Thus, the fine dispersoids desirable for strength can be formed.
Problem Areas
The constant σ0 in Equation (1) is not well understood theoretically.
Pileup-pileup interactions can modify Equation (1).82 Therefore, three-dimensional calculations are needed.
Refinements needed in the interpretation of Equation (2) include consideration of screening effects caused by different elastic properties of obstacles; statistical averaging or computer simulation for finite-size obstacles and those with long-range strain fields; and further work on the weak, dilute solute case.
Experimentally derived values of cohesive energies of particles, which would be valuable in providing a guideline for alloy design, are sparse.
A detailed estimate of the decohesion probability for a dispersoid is required.
Three-dimensional models of dislocations would be beneficial in understanding particle decohesion and crack-void interactions.
More experiments are needed to test the statistical models for the DBTT.
IMPURITY AND ENVIRONMENTAL EFFECTS
Impurities
With the complications of multicomponent systems, adsorption, absorption, and diffusion, the roles of solute and environment cannot be discussed completely. However, several advances in the past 25 years are relevant to this discussion. Although there had been suggestions of such effects earlier, the role of impurities in enhancing embrittlement, specifically intergranular fracture, is now well established.83 The key was the development of Auger electron microscopy. Such microscopy showed, for example, that elements in Groups V and VI of the periodic table adsorb to prior austenite grain boundaries to cause temper embrittlement of nickel-chromium steels. Coad-

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sorption effects, with solute-solute interactions, are important in establishing the degree of embrittlement.
Hydrogen Embrittlement
Resolving an issue that had led to years of controversy, recent work has shown that hydrogen at moderate fugacity softens pure iron at room temperature and at low plastic strains.84,85 The effects of flow and internal friction indicate that the mechanism is one of enhanced double-kink nucleation on screw dislocations.86 Hydrogen also enhances the planarity of slip, on {110} planes, in iron.84
In ductile fracture, hydrogen degrades steels by enhancing shear instability under plane-strain conditions, lowering the critical strain by about a factor of 2.87 In uniaxial, smooth-bar, or notched-bar tension tests, hydrogen enhances void nucleation88 and growth.89 Hydrogen also enhances brittle fracture,90 lowering KIc by a factor of 2 to 3, for example. The embrittlement tendency is greater with greater hydrogen fugacity or strength of the material. Hydrogen also promotes intergranular fracture, alone or in combination with Group V and VI elements. Work on Fe-3%Si single crystals91 shows a monotonic decrease of crack opening angle with increased chemical potential of hydrogen, suggesting that hydrogen enhances “bond breaking” at the crack tip. However, the observation of enhanced dislocation motion by hydrogen85 has led to the alternative postulation that hydrogen enhances flow near the crack tip92 or near grain boundaries,85 weakening them for a mixed-mode fracture akin to that mentioned in the foregoing discussion of ductile fracture.
Another category of hydrogen embrittlement involves preferential hydride formation in the highly stressed crack-tip region, cracking of the hydride, blunting of the crack when it reaches the matrix, and repetition. This category, verified by transmission electron microscopy, occurs for many transition metals.93
Stress Corrosion Cracking
Most investigators agree that a hydrogen embrittlement mechanism causes the stress corrosion cracking of ferritic steels and ferritic stainless steel in acids.94 Hydrogen released within the crack in a local cathodic reaction diffuses to the crack tip and produces embrittlement analogous to that discussed above. Austenitic stainless steels crack in neutral and weak acids according to the film-rupture model:95 slip breaks the passivating film and, with an intermediate repolarization rate, local attack occurs before a new film forms and the process repeats itself. Some investigators suggest that the latter steels also fail by hydrogen embrittlement.96 Such embrittlement is

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possible because in the limited volume of the crack, depletion of hydroxyl ions and consequent acidification can occur. Similar controversy holds for other alloy systems, including aluminum and titanium alloys.97
Under stress corrosion conditions, the influence on toughness parallels the hydrogen case, with a lowering of Kc and a tendency for intergranular embrittlement. Moreover, the intergranular crack surfaces exhibit features resembling those for hydrogen embrittlement, so the same issue of the possibility of locally enhanced plastic flow is present.92 Indeed, a similar situation prevails in the case of liquid-metal embrittlement.
Problem Areas
Further research at the atomic scale is required to resolve the issue of a decohesion model versus a plastic-flow-weakening model for brittle intergranular cracking in hydrogen embrittlement, stress corrosion cracking, and liquid-metal embrittlement.
Finite-element/finite-difference solutions are needed to predict local chemistries, pH values, and electric potentials within a crack in the presence of an electrolyte.
All the problems that have been discussed for cracks in pure crystals, in particular those associated with interatomic potentials, apply for the present case as well and are exacerbated.
FUTURE PROSPECTS
Dislocations and Cracks
With an interdisciplinary approach involving crystal plasticity and continuum mechanics theory, observations at the atomic level, and improved abilities to make numerical calculations, the difficult problem of many-body dislocations should be solvable. Together with current results for simpler dislocation arrays, this would provide constitutive relations useful in both alloy design and structural design for metals and for ceramic materials at elevated temperatures.
Advances in the physical description of atomic interactions will improve our knowledge of the configuration and properties of dislocation cores and sharp crack tips. Together with three-dimensional elastic solutions for cracks, this would make possible a more detailed analysis of complex cracking problems, including many-body interactions. Such an advance would also be useful in analyzing fatigue crack propagation and eventually should be applicable in the more complex case of environmental interactions.

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Structural Materials
The application of fundamental principles in alloy design has begun and is still an area of great potential. A crucial problem that should be resolvable is the determination of the particle-size “window” for optimum dispersion hardening and toughening. Another factor that should be incorporated in design concepts is that of controlled transformation toughening. Many of the concepts developed for dispersed phases are also applicable to composite fibers, specifically the concept of an optimum size. Some new concepts related to toughening—for example, a slipping but very viscous interface— are specific to fibers.98 The critical problem of fiber-matrix interface properties can be solved to a large extent using current research techniques.
The discussion of dislocations and cracks is also applicable to crystalline polymers. For glassy polymers or amorphous metals, which may now become available in bulk form,99 disclinations, dispirations, and other defects are also important. Although these defects have been studied extensively, they are less well characterized than dislocations. Further study should provide improvements for the amorphous structures analogous to those for metals and ceramics.
New Materials
Advances in electronics and the concomitant miniaturization of electronic devices have led to the development of new materials and new materials problems. Elimination of dislocations is critical to the operation of many semiconductor devices. Thus an understanding of the properties of defects will also have impact on solid-state electronics. The role of defects may be critical in determining whether strained superlattices (alternating thin layers, 10 nm or less, of different semiconductors or compound semiconductors) will have the stability to be useful in device applications.
The structural understanding of liquid crystals in terms of dislocations and disclinations should prove important. For the new quasicrystals100 with fivefold symmetry (see Cahn and Gratias, in this volume), new types of defects may be discovered and may be necessary to describe mechanical and physical properties.
Finally, there are opportunities in what could be termed “micromaterials.” In very-large-scale integrated circuitry, for example, there are problems regarding mechanical properties and microstructural control at the micrometer-size scale (along with special problems such as electrotransport). Macroscopic concepts are often inapplicable at this scale and new phenomenology will be developed. A further example is the extraordinary modulus enhancement for fine metallic-layer structures at thicknesses of approximately 2 nm, an effect that is yet to be either explained or exploited.101

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SUMMARY
Many advances have been made during the period 1960–1985 in our understanding of the relations between microstructure and properties of materials. Theories of simple defect configurations have been developed, and new tools have emerged to view defects at the near-atomic scale. Qualitative guidelines for designing alloys apply to some of the new alloys produced by rapid solidification. Yet, our understanding is incomplete. Elastic fields, structures, and mechanisms in dislocation cores or crack tips represent unsolved problems. Statistical averaging methods are needed to make the connection between crystal plasticity and continuum plasticity.
ACKNOWLEDGMENTS
The support of this work by the National Science Foundation under grant DMR 8311620 is gratefully acknowledged. The author is pleased to acknowledge the helpful comments and discussions of A.S.Argon, M.Cohen, J.W.Hutchinson, W.D.Nix, and J.R.Rice.
NOTES
1.
P.B.Hirsch, R.W.Home, and M.J.Whelan, Philos. Mag. 1, 677 (1956); W.Bollmann, Phys. Rev. 103, 1588 (1956).
2.
C.S.Smith, editor, The Sorby Centennial Symposium on the History of Metallurgy (Gordon and Breach, New York, 1965).
3.
W.F.Flanagan, H.Margolin, and A.W.Thompson, editors, Symposium on 50th Anniversary of the Introduction of Dislocations, Metall. Trans. A 16A, 2085–2231 (1985).
4.
G.R.Irwin, J. Appl. Mech. 24, 361 (1957).
5.
A.A.Griffith, Philos. Trans. R. Soc. London, Ser. A 221, 163 (1920).
6.
J.D.Eshelby, Solid State Phys. 3, 79 (1956).
7.
J.D.Eshelby, F.C.Frank, and F.R.N.Nabarro, Philos. Mag. 42, 351 (1951).
8.
Reviewed in J.P.Hirth and J.Weertman, editors, Work Hardening (Gordon and Breach, New York, 1968).
9.
E.Orowan, in Symposium on Internal Stresses (Institute of Metals, London, 1947), p. 451.
10.
J.M.Burgers, Proc. Kon. Ned. Akad. Wetenschjap. 42, 293, 378 (1939).
11.
M.O.Peach and J.S.Koehler, Phys. Rev. 80, 436 (1950).
12.
T.Mura, Philos. Mag. 8, 843 (1963).
13.
J.Lothe, Philos. Mag. 15, 353 (1967).
14.
L.M.Brown, Philos. Mag. 15, 363 (1967).
15.
V.L.Indenbom and S.S.Orlov, Sov. Phys. Crystallogr. 12, 849 (1968).
16.
J.Blin, Acta Metall. 3, 199 (1955).
17.
T.Jossang, J.Lothe, and K.Skylstad, Acta Metall. 13, 271 (1965).
18.
J.D.Eshelby and T.Laub, Can. J. Phys. 45, 887 (1967).
19.
J.P.Hirth and J.Lothe, in Physics of Strength and Plasticity, edited by A.S.Argon (MIT Press, Cambridge, Mass., 1969), p. 39.
20.
J.P.Hirth and J.Lothe, Theory of Dislocations, 2nd ed. (Wiley, New York, 1982), p. 146.

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