A dislocation model for the plastic relaxation of the transformation strain energy of a misfitting spherical particle

A dislocation approach is developed to explain the dependency of the plastic zone size on precipitate size and misfit for a spherical misfitting particle. The dislocation model is predicated upon the punching and mutual interaction of dislocation prismatic loops and is used to explain the presence of both prismatic loops and dense tangles of dislocations surrounding the precipitate. The plastic zone is modeled as a series of concentric dislocation loop shells which can climb and glide in such a way as to minimize the energy of the system; the extent of climb and glide being limited by the critical resolved shear stress of the matrix phase. It is found that plastic zone size is a strong function of particle size. As particle size increases, the ratio of plastic zone radius to particle radius increases. For large particles, plastic zone radii approach the predictions of earlier work in which the three dimensional elasto-plastic problem for a misfitting sphere was solved exactly using the Prandtl-Reuss equations and von Mises yield criterion.     

Equilibrium position of misfit dislocations at planar interfaces

The equilibrium position of misfit dislocations at interfaces is analysed within linear elasticity theory. Calculations are performed for an isolated dislocation as well as for an infinite array of dislocations in an infinite bicrystal. Analytic expressions are obtained for the image forces on the dislocation array due to the elasticity discontinuity. The position of the dislocations is determined by the balance between image forces and coherency forces on the dislocations. An equilibrium position is obtained in crystal I with smaller shear modulus G₁. The equilibrium stand-off distance is proportional to the inverse of the lattice misfit and increases with the ratio of the shear moduli, G₂/G₁, and of the Poisson's ratios, ν₂/ν₁, respectively. The calculated stand-off distance of misfit dislocations in Nb adjacent to a NbAl₂O₃ interface is smaller by a factor of 2 than the experimentally observed distance. This discrepancy can be explained qualitatively by the higher core energy of a dislocation in the vicinity to the elastically rigid Al₂O₃ compared to the core energy of a dislocation in bulk Nb.     

Internal stresses due to dislocation walls around second phase particles

Nickel base superalloys with large γ′ volume fractions exhibit a high threshold stress during a tensile test, and a large internal stress when tested in high temperature creep. Arrays of regularly spaced edge dislocations which develop during the first stages of creep have been observed. They lie in the γ-γ′ interfaces and form dipolar arrangements on opposite sides of the γ′ cuboids. The various components of the stress tensor are calculated for such a dipolar wall configuration and mapped by drawing equistress lines. The stress field resulting from such dislocation configurations has a very strong component along the tensile axis and opposes the applied stress. The shear component in the glide plane of similar dislocations is also very large between the precipitates and would tend to repel any new dislocation, in the absence of an external stress. The structural instability of these alloys under creep strain is also interpreted by this model.     

A study on coherency strain and precipitate morphologyvia a discrete atom method

Morphological evolution of coherent precipitates is studied by means of a discrete atom method under a plane strain condition with a purely dilatational misfit. The method is predicated upon Hookean atomic interactions and Monte Carlo diffusion and makes no assumption of a specific precipitate shape. Precipitates having elastic constants different from those of the matrix phase are treated in both isotropic and anisotropic elastic systems. Shape evolution is examined under the condition of a constant precipitate size and an isotropic interfacial energy. The results show that in general, an elastically soft precipitate tends to have an equilibrium morphology of low symmetry such as a plate, whereas a hard particle tends to take up a shape of high symmetry such as a circle. Morphological evolution proceeds through dynamic activities of coherency-induced interfacial waves whose wavelength depends upon the difference in elastic constants, precipitate geometry, anisotropy, and diffusion temperature. Coherency-induced interfacial waves seem to be responsible for the protrusions often observed along elastically hard directions in γ′ particles of Ni-base superalloys and also to be a source for fresh ledges for growthvia the ledge mechanism. For a highly nonequilibrium precipitate, first splitting followed by coalescence appears to be a common feature in achieving its equilibrium morphology. This article is based on a presentation made during TMS/ASM Materials Week in the symposium entitled “Atomistic Mechanisms of Nucleation and Growth in Solids,” organized in honor of H.I. Aaronson’s 70th Anniversary and given October 3–5, 1994, in Rosemont, Illinois.     

An analysis of equilibrium dislocation distributions

Equilibrium distributions of collections of discrete dislocations are analyzed, with the dislocations modelled as line defects in a linear elastic medium. The dislocated equilibrium configuration is determined by finding a minimum potential energy configuration, with respect to variations in the dislocation positions, for a fixed number and type of dislocations. Numerical results are presented for finite and infinite bodies with distributions of edge dislocations under plane strain conditions. Calculations involving doubly periodic arrays of cells, within which there is a single set of parallel slip planes, show a strong tendency for sharp dislocation walls to form. Perturbations of the wall structure due to the presence of pinned dislocations, vacant slip planes and free surfaces are illustrated. The stress fields due to the dislocation walls are calculated and large shear stress values are found away from any dislocation core. Pileups involving dislocations on two sets of intersecting slip planes are found to give rise to equilibrium configurations involving dislocation free regions. The response of dislocation patterns in an infinite medium to an imposed shear stress is also analyzed.     

Heterogeneous nucleation of σ′ on dislocations in a dilute aluminum-lithium alloy

The nucleation of S on dislocations with small undercooling in binary aluminum-lithium alloys has been examined. The study of related microstructures was performed using transmission electron microscopy (TEM), which demonstrates that σ′ preferentially nucleates on dislocations with a strong edge character and locates at the side where the stress field is compressive without destroying the dislocation core structure. This qualitatively justifies the theoretical prediction by Larché on coherent heterogeneous nucleation on edge dislocations. Following the evaluation of the volume free energy change for the binary system by the ideal solution model and the mean-field model by Khachaturyan, the nucleation barrier and the nucleation rate were calculated and compared with experimentally determined data based on Larché's model. Specifically, the back-calculated interfacial energies from the experimentally determined nucleation rate data are in good agreement with the interfacial energy temperature dependence predicted by the related interfacial energy model. The effects of misfit strain, volume diffusion, interfacial energy, and nucleation sites are discussed. Formerly Graduate Student This article is based on a presentation made during TMS/ASM Materials Week in the symposium entitled “Atomistic Mechanisms of Nucleation and Growth in Solids,” organized in honor of H.I. Aaronson’s 70th Anniversary and given October 3–5. 1994 in Rosemont, Illinois.     

Diffusionally modified dislocation-particle elastic interactions

The effects of diffusion on the elastic interactions between dislocations and incoherent second phase particles is examined. In the absence of diffusion, the particle-matrix interface is stressed in the presence of a dislocation. These stresses are related, in part, to the elastic requirements that both the tractions and displacements are continuous across the interface. Diffusion in the interface over length scales comparable to the width of the interface leads to a viscous-like relaxation of the shear tractions resulting in a sliding interface. When interfacial diffusion occurs over distances of order the particle radius, normal stress gradients along the particle-matrix interface may also be relaxed. A solution of the elastic problem of an edge dislocation interacting with a cylindrical particle is obtained in the limit that both of these diffusional relaxation processes have gone to completion. As a result of the diffusional relaxation, a dislocation on any glide plane that intersects the particle is always attracted toward the particle. These results differ from the diffusionless interaction, where attraction occurs only when the shear modulus of the matrix exceeds that of the particle.     

Interfacial ledge structures in Ni3 Al-Ni3Cb eutectic composites

The interfacial structure of Ni₃Al-Ni₃Cb directionally solidified eutectic composites has been investigated by transmission electron microscopy. These interfaces contain at least three distinguishable arrays of features. Two of the arrays, misfit dislocations, have been discussed previously by Nakagawa and Weatherly. The third set, ledges which can fulfill both structural and kinetic growth functions, may interact with the dislocation arrays through strain-energy mechanisms. The interaction is manifested both as a local alteration of the line vector of the dislocation in certain circumstances, and as a change in the response of the dislocation image to ±g electron-microscope image-contrast experiments. A simple model of the strain field of a ledge based on that of an edge dislocation is formulated to rationalize the behavior of a misfit dislocation lying in close proximity to a ledge. The interaction of ledges and dislocation segments is expected to have significance in physical processes of practical interest such as production of matrix slip dislocations, misfit dislocation rearrangement, boundary sliding, and coarsening, and these processes are discussed in some detail.     

Analysis of the interfacial stresses produced by a pile-up of discrete edge dislocations in two phase materials

The local stresses associated with a pile-up of discrete edge dislocations against a welded interface at a second phase have been calculated using the elasticity solution of Dundurs. The calculations permit a study of the effects of applied stress, pile-up length, and the elastic properties of the two phases on the distribution of dislocations in the pile-up and on the stresses at the interface. The equilibrium positions of the dislocations and their discrete contributions to the stresses at the interface have been investigated for cases in which the shear modulus of the second phase is greater than and less than (or equal to) that of the matrix. For the latter case, the leading dislocation is assumed to be locked at the interface. The local stress is found to increase with applied stress and pile-up length, as expected. For a given applied stress and pile-up length, the stresses at the interface increase with decreasing shear modulus of the second phase and also depend on the Poisson's ratios of the two phases. The analysis is used to study the effects of microstructure on the plastic deformation of two phase alloys.     

Growth of δ′ on dislocations in a dilute Al-Li alloy

The quantitative results of δ′ growth kinetics on dislocations in an Al-2.27 wt pct Li alloy demonstrate that at temperatures greater than 0.5 of the homologous melting temperature, T _m , volume diffusion is the dominant mechanism. However, a small contribution, approximately one atom in 300, is made by pipe diffusion through the dislocation. This can be established by careful examination of dislocation climb associated with precipitate growth. An analysis based on the δ′ growth kinetics and diffusion equation gives an activation energy of 0.56 eV for Li pipe diffusion. At temperatures <0.5 T _m , δ′ precipitate growth is faster when associated with dislocations, and here, pipe diffusion is necessary to account for the kinetics observed. This article is based on a presentation made in the symposium “Kinetically Determined Particle Shapes and the Dynamics of Solid:Solid Interfaces,” presented at the October 1996 Fall meeting of TMS/ASM in Cincinnati, Ohio, under the auspices of the ASM Phase Transformations Committee.     

A high resolution transmission electron microscope study of early stage precipitation on dislocation lines in Al-Zn-Mg

The weak beam transmission electron microscopy technique was employed to study the early stages of precipitation on dislocation lines in Al-3.87 wt pct Zn-1.79 wt pct Mg. The heterogeneous precipitation sequence was found to follow the homogeneous sequence in this alloy. The interaction between the initial coherent precipitate particles and the strain fields of the catalyzing dislocations produced “gaps” of background intensity at precipitate locations along the otherwise continuous weak beam images of the dislocation lines. A simple model was developed to relate a distribution of measured weak beam gap lengths to a particle size distribution at a given aging treatment. In this manner the growth kinetics of the initial precipitate phase was observed; it was found that the precipitation followed the Cottrell-Bilbyt ^2/3law, suggesting that matrix dislocations may assist the growth of heterogeneous precipitates in a manner analogous to grain boundary “collector plates.” Weak beam microscopy was found to be superior to standard bright field microscopy for the current study. Particles too small to be visible in bright field were revealed in weak beam. Weak beam observations also indicated that the coherent precipitate particles were positioned asymmetrically about the dislocation cores.     

Atomic mechanisms of diffusional nucleation and growth and comparisons with their counterparts in shear transformations

An integrated overview is presented of a viewpoint on the present understanding of nucleation and growth mechanisms in both diffusional and shear (martensitic) transformations. Special emphasis is placed on the roles played by the anisotropy of interphase boundary structure and energy and also upon elastic shear strain energy in both types of transformation. Even though diffusional nucleation is based on random statistical fluctuations, use of the time reversal principle shows that interfacial energy anisotropy leads to accurately reproducible orientation relationships and hence to partially or fully coherent boundaries, even when nucleation at a grain boundary requires an irrational orientation relationship to obtain. Since the fully coherent boundary areas separating most linear misfit compensating defects are wholly immobile during diffusional growth because of the improbability of moving substitutional atoms even temporarily into interstitial sites under conditions normally encountered, partially and fully coherent interphase boundaries should be immovable without the intervention of growth ledges. These ledges, however, must be heavily kinked and usually irregular in both spacing and path if they, too, are not to be similarly trapped. On the other hand, the large shear strain energy usually associated with martensite requires that its formation be initiated through a process which avoids the activation barrier associated with nucleation, perhaps by the Olson-Cohen matrix dislocation rearrangement mechanism. During growth, certain ledges on martensite plates serve as transformation dislocations and perform the crystal structure change (Bain strain). However, the terraces between these ledges in martensite (unlike those present during diffusional growth) are also mobile during non-fcc/hcp transformations; glissile dislocations on these terraces perform the lattice invariant deformation. Growth ledges operative during both diffusional and shear growth probably migrate by means of kink mechanisms. However, diffusional kinks appear to be nonconservative and sessile (and therefore resist immediate transmission of elastic shear strain energy), whereas those associated with martensitic growth must be conservative and glissile (and fully transmit such strain energy). The broad faces of both diffusionally and martensitically formed plates contain an invariant line, as emphasized by Dahmen and Weatherly. However, in the diffusional case, minimization of growth ledge formation kinetics seems to be the main role thereby played, whereas in martensitic growth, the main purpose of such an interface is to minimize elastic shear strain energy. The latter minimization requires that martensite forms as plates (or perhaps as laths) enclosed by a pair of invariant line-containing interfaces. During diffusional transformations, on the other hand, other interfaces at which growth ledge formation kinetics are not too much faster than those at the invariant line interface can also comprise a significant portion of the interfacial area, thereby leading to the formation of other, quite different morphologies, such as intragranular idiomorphs and grain boundary allotriomorphs. Critical problems remaining unsolved in diffusional transformations include calculation of critical nucleus shapes when the crystal structures of the two phases are significantly different, highly accurate calculation of the energies of the interphase boundaries thus formed, and direct observation of atomic scale kinks on the risers of growth ledges by means of a yet-to-be-invented three-dimensional (3-D) atomic-resolution form of transmission electron microscopy. Experimental identification and characterization of transformation dislocations and experimental testing of “nucleation” mechanisms are now of special importance in fundamental studies of martensitic transformations.     

Precipitation on low-angle grain boundaries in Al-Zn-Mg

Weak beam microscopy was used to examine the precipitation of the 17 phase (MgZn₂) on low angle grain boundaries which had formed in an Al-2.84 wt pct Zn-1.95 wt pct Mg alloy. The low angle boundaries observed in the partially recovered structure can be separated into two categories, planar and nonplanar. The planar boundaries form either lozenge or hexagonal configurations. The nonplanar boundary dislocation intersections form a “chair”-shaped figure. The geometry of the dislocation strain fields in the boundaries controls the precipitate growth and the size, shape, and position of the precipitates on each boundary type. The lozenge boundary can have both equiaxed and lath shaped precipitates growing upon it. The equiaxed precipitates form only at four dislocation junctions because at this position the strain field distribution is symmetric. The lath precipitates are restricted to grow along dislocations in the boundary whose line direction is coincident with the long axis of the precipitate since in these pinned dislocation segments there are high line tension forces present to resist dislocation bowing. The “chair” boundary contains only lath precipitates, which begin to grow from alternate three dislocation intersections and continue growth along the dislocation line. The contrast observed bordering the lath on each side is produced by lattice dislocations. A proposed explanation for the above observations involves assuming that the dislocations in the chair boundary dissociate into Lomer-Cottrell locks which have constricted-extended node pairs. A lath can begin growth from the constricted node forcing the stair rod dislocation to dissociate. The reaction products of the stair rod can combine with the partials in the Lomer-Cottrell lock to form the observed lattice dislocations.     

Dislocation interactions with internal coherency stresses in age-hardened Cu-Ni-Fe alloys

The effects of internal coherency stresses on dislocation configurations and dislocation radii of curvature were investigated for age-hardened Cu-Ni-Fe alloys. Biaxial compression and tensile stresses were calculated from tetragonal crystal structure data for the two coherent precipitate phases, which were assumed to be elastically strained cubic phases. Shear stresses on slip planes were resolved from the biaxial coherency stresses and they correctly predicted the dislocation configurations observed in a transmission electron micrograph. Dislocations oriented perpendicular to the platelets were wavy because the shear stress on the slip plane abruptly changed sign (direction) at the platelet boundaries. Dislocation radii of curvature in the wavy sections were both calculated and observed to be about 300å. The analysis indicated both dislocations with Burgers vectors parallel to the platelets and wavy dislocation sections would move freely under small applied loads, but dislocations trapped between platelets would not.     

Dislocation interactions with internal coherency stresses in age-hardened Cu-Ni-Fe alloys

The effects of internal coherency stresses on dislocation configurations and dislocation radii of curvature were investigated for age-hardened Cu-Ni-Fe alloys. Biaxial compression and tensile stresses were calculated from tetragonal crystal structure data for the two coherent precipitate phases, which were assumed to be elastically strained cubic phases. Shear stresses on slip planes were resolved from the biaxial coherency stresses and they correctly predicted the dislocation configurations observed in a transmission electron micrograph. Dislocations oriented perpendicular to the platelets were wavy because the shear stress on the slip plane abruptly changed sign (direction) at the platelet boundaries. Dislocation radii of curvature in the wavy sections were both calculated and observed to be about 300&#x00E5;. The analysis indicated both dislocations with Burgers vectors parallel to the platelets and wavy dislocation sections would move freely under small applied loads, but dislocations trapped between platelets would not.      

Microstructure and mechanisms of cyclic deformation of aluminum single crystals at 77 K

Aluminum single crystals were cyclically deformed in single slip at small strain amplitudes at 77 K to presaturation. The observed mechanical behavior is consistent with recent work by other investigators. The dislocation substructure can be described as consisting of dense bundles or veins of edge dislocation dipoles, of a single Burgers vector, separated by lower dislocation density regions or channels where substantial debris is evident. This debris was determined as almost exclusively relatively short edge-dipole segments. Screw dislocations with the same Burgers vector span the channels. In situ cyclic reverse (shear) deformation experiments in the high-voltage transmission electron microscope (HVEM) were successfully performed using the X-Y technique where thin foils are stressed in alternating perpendicular directions. Our experiments indicate that loops frequently expand from the dipole bundles into the channel and the edge component is absorbed by nearby bundles, leaving screw segments behind. The screw dislocations that span the channel move easily and reverse direction with shear reversal. Screws may move first with a strain reversal. A comparable fraction of the strain during each cycle appears to be provided by screw and edge dislocations. Dipole “flipping” was not observed. There is no obvious evidence for internal backstresses that assist plastic deformation on reversal of the applied shear.     

The effect of orientation and sense of applied uniaxial stress on the morphology of coherent gamma prime precipitates in stress annealed nickel-base superalloy crystals

The coarsening of coherent γ′[Ni₃(Al, Ti)] precipitates in single crystals of a representative nickel-base superalloy, Udimet-700, is shown to be affected by a uniaxial stress applied during annealing. Depending on the sense of the applied stress and its crystallographic orientation, stress annealing results in oriented cuboidal, plate, or parallelepiped shaped γ′ precipitates. A general thermodynamic analysis of the effect of stress annealing on precipitate morphology is presented that takes into account free energy changes due to changes in bulk precipitation strain, effective modulus, coherency strain energy, and the total interphase boundary area. The analysis correctly predicts the observed γ′ precipitate morphologies as a function of stress axis orientation, stress sense, the lattice misfit of the precipitate phase, and the elastic constants of the matrix and precipitate phases. The analysis also shows that stress induced morphological changes can be completely precluded, as may be desired to optimize mechanical behavior, only if the elastic constants of the matrix and precipitate phases are equal. Changes in morphology due to changes in bulk precipitation strain, which in the case of Udimet-700 is shown to be the dominant effect, can be eliminated by alloying for zero lattice misfit or, in single crystals, by stressing parallel to < 111> . Applications to long-term creep behavior and to the fabrication of composite structures are discussed. Formerly with Advanced Materials Research and Development Laboratory, Pratt & Whitney Aircraft     

Dislocation structure behind a shock front in fcc perfect crystals: Atomistic simulation results

Large-scale molecular dynamics simulations are used to investigate the dislocation structure behind a shock front in perfect fcc crystals. Shock compression in both the 〈100〉 and 〈111〉 directions induces dislocation loop formation via a sequential emission of partial dislocations, but in the 〈100〉 case, this process is arrested after the first partial, resulting in stacking-fault loops. The large mobility of the bounding partial dislocations results in a plastic wave that is always overdriven in the 〈100〉 direction; the leading edges of the partials are traveling with the plastic front, as in the models of Smith and Hornbogen. In contrast, both partials are emitted in 〈111〉 shock compression, resulting in perfect dislocation loops bounded only by thin stacking fault ribbons due to the split partial dislocations. These loops grow more slowly than the plastic shock velocity, so new loops are periodically nucleated at the plastic front, as suggested by Meyers. This article is based on a presentation given in the symposium “Dynamic Deformation: Constitutive Modeling, Grain Size, and Other Effects: In Honor of Prof. Ronald W. Armstrong,” March 2–6, 2003, at the 2003 TMS/ASM Annual Meeting, San Diego, California, under the auspices of the TMS/ASM Joint Mechanical Behavior of Materials Committee.     

Effect of uniaxial stress on coarsening of precipitate clusters

The influence of moderate applied uniaxial stresses (σ_app/C ₄₄ ≈ 10⁻³) on the coarsening behavior of misfitting coherent precipitates in binary alloys has been studied. Three-dimensional (3-D) computer simulations of the coarsening have been performed for elastically homogeneous systems with tetragonal misfit strain and elastically heterogeneous systems with dilatational misfit strain. Precipitate shapes are restricted to spheres. Results depend on the sign of the misfit strain, the sign of the applied field, and the character of the elastic heterogeneity: precipitates softer than the matrix phase with positive (negative) misfit strain align along the direction of the applied stress for compressive (tensile) fields and arrange in planes perpendicular to it for tensile (compressive) fields. Precipitates harder than the matrix behave in the opposite way. This article is based on representation made during TMS/ASM Materials Week in the symposium entitled “ldAtomistic Mechanisms of Nucleation and Growth in Solids,” organized in honor of H.I. Aaronson’s 70th Anniversary and given October 3–5, 1994, in Rosemont, Illinois.     

Atomic arrangement and the formation of partially coherent interfaces in the Ti-V-N system

The precipitation of (V,Ti) (bcc structure) in a (Ti,V)N (NaCl structure) matrix is considered in the current study. The lattice parameter ratio of this system, a _f /a _b =1.34, is quite different from most previous studies (a _f /a _b ∼ 1.26) and provides an opportunity to test recent models proposed for the formation of precipitate morphology and the interface structure. Like many other fcc:bcc precipitation systems, the Ti-V-N system involves an invariant line transformation strain. In this system, the invariant line is associated with a high-index orientation relationship (OR). The observed OR is in good agreement with a predicted relationship based upon a geometric matching criterion proposed by Ryan et al. The Burgers vectors for the interfacial defects were determined directly by making high-resolution transmission electron microscope (HRTEM) observations along three different directions. The observations confirm that the formation of the precipitate facets, the spacings of misfit dislocations, and the direction of interfacial defects all agree with atom-matching considerations. This article is based on a presentation made in the symposium “Kinetically Determined Particle Shapes and the Dynamics of Solid:Solid Interfaces,” presented at the October 1996 Fall meeting of TMS/ASM in Cincinnati, Ohio, under the auspices of the ASM Phase Transformations Committee.