An analysis of cavity nucleation in superplasticity

Superplastic alloys possess either a quasi-single phase or a microduplex microstructure: in quasi-single phase alloys, cavities are observed to nucleate predominantly at coarse grain boundary particles whereas in microduplex alloys, cavities tend to form at interphase boundaries and at triple point junctions. A general analysis is presented for cavity nucleation, in both microstructures, under the stress concentrations caused by bursts of grain boundary sliding during superplastic deformation. In quasi-single phase alloys, calculations indicate the cavities nucleate at coarse particles located at grain boundaries because local interphase diffusion creep cannot accommodate the stress concentrations sufficiently rapidly. The analysis demonstrates that it is possible for cavities to nucleate at grain boundary ledges under some limited experimental conditions. It is demonstrated also that the present analysis is in agreement with the available experimental data on a quasi-single phase Cu-based superplastic alloy and a microduplex superplastic Zn-22% Al eutectoid alloy. Calculations show that small pre-existing cavities, if present, are likely to be sintered rapidly prior to superplastic déformation at elevated temperatures.     

The role of ledges in stress tensor-mediated surface processes for Si and GaAs

Calculation results of terrace and ledge reconstruction-induced surface stress tensors, using semiempirical potential energy functions for Si, GaAs, GaAs/Al, and GaAs/Si, are reviewed. Long-range (～50 to 100 Å) interaction effects between ledges and between interfaces are pre sented. Two-dimensional homogeneous and heterogeneous nucleation effects are also presented, showing that Actatom cluster populations are strongly influenced by these reconstruction events. The consequences of these reconstruction-induced surface fields on γ plots, ledge motion kinetic laws, surface Actatom transport, and solute partitioning at ledges are discussed. The numerical results were generated largelyvia Monte Carlo calculations and thus neglect entropy effects.     

A statistical analysis of cavity nucleation at particles in grain boundaries

Cavity nucleation on grain boundary particles during creep has been examined using a classical thermodynamic method. The particle sizes and spacings are assumed to obey a log-normal distribution. It is found that a threshold shear stress is needed for cavity nucleation to occur. When the resolved shear stress on a grain boundary segment reaches the threshold stress, a critical normal stress for cavity nucleation is produced on the panicle-matrix interface by grain boundary sliding and cavity nucleation occurs. The threshold stress is determined mainly by the concentration and distribution of grain boundary particles and falls in the stress range of engineering applications. A program has been developed to calculate the fraction of particles which can serve as nucleation sites. The model is used to predict the onset of cavitation in the oxide dispersion strengthened alloy Inconel MA 754. Implications for avoiding the nucleation of cavities in engineering alloys are discussed.     

The elastic strain energy of growth ledges on coherent and partially coherent precipitates

The formation rate of growth ledges on a faceted precipitate strongly affects the growth kinetics and the shape of the precipitate. An Eshelby-type model is used to compare the strain energy associated with the nucleation of a ledge on different facet planes of a body-centered cubic (bcc) precipitate in face-centered cubic (fcc) matrix. Ledge nucleation is only likely at facet areas where the interaction energy between the ledge and the precipitate is negative. The strain energy for ledge formation is not symmetric on any of the facet planes, but it is symmetric about the center of the precipitate. For coherent precipitates comparable to those observed in the Ni-Cr system, ledges form with the lowest strain energy on the broad facet of the precipitate implying that precipitate thickening should occur faster than lengthening and widening. A procedure for modifying the Eshelby model is suggested in order to allow strain-energy calculations of partially coherent precipitates. The strain energy for ledge formation on at least one type of partially coherent lath is lowest for a ledge located on the facet perpendicular to the crystallographic invariant line (IL). This situation favors precipitate lengthening in the invariant line direction.     

Grain boundary sliding and stress concentration during creep

Importance of grain boundary sliding to creep intergranular fracture is focussed. Previous metallographic and fractographic studies of creep intergranular fracture on metal bicrystals and polycrystals are briefly reviewed in order to show the close relationship between grain boundary sliding and fracture. Deformation ledge and migration irregularity are shown to be potential sites of stress concentration and crack nucleation on sliding grain boundaries without particles. The effect of grain boundary structure on creep intergranular fracture is discussed on the basis of the effect of grain boundary structure on sliding, the contribution of sliding to the overall creep deformation, and a sliding-fracture diagram. Recent observations of the effect of grain boundary structure on creep intergranular fracture on alpha iron-tin alloy polycrystals are shown. This paper is based on a presentation made at the symposium “The Role of Trace Elements and Interfaces in Creep Failure” held at the annual meeting of The Metallurgical Society of AIME, Dallas, Texas, February 14-18, 1982, under the sponsorship of The Mechanical Metallurgy Committee of TMS-AIME.     

Cavity nucleation at high temperatures involving pile-ups of grain boundary dislocations

The current concept of sliding-induced cavitation is first reviewed, with due consideration to the respective role of the remote and internal stresses. The results show that the transient stresses produced by sliding play only a secondary role in cavity nucleation and that at high temperatures, the effect of sliding is eliminated in less than 1 millisecond. It is thus concluded that sliding cannot be the cause for cavity nucleation. Next, a model involving pile-ups of grain boundary dislocations (GBDs) is proposed. Unlike the sliding mechanism, the high strain energy ahead of the pile-up is a steady state phenomenon during secondary creep. It helps to compensate the large capillarity forces in the formation of sub-micron sized cavities, thereby rendering cavity nucleation barrierless. However, a threshold stress exists below which the cavities cannot grow to effect fracture. The present model suggests that cavity nucleation is feasible in single phase metals and alloys at the intersections of cell and grain boundaries. Predominant cavity formation after the onset of steady state creep and an intermediate temperature ductility trough during hot tensile tests are also features of the model. Good agreements are found between the model's predictions and the experiments.     

Cavity nucleation at particles on sliding grain boundaries. A shear crack model for grain boundary sliding in creeping polycrystals

The paper summarizes the results of classical nucleation theory applied to the nucleation of creep cavities at hard second-phase particles. Stress concentrations at particles in sliding grain boundaries are analysed for the cases that the particles can be circumvented by diffusion and power-law creep and for purely elastic material response. The shear stress which is effectively transmitted through a sliding grain boundary in a polycrystal is calculated using a shear crack model. Comparison of the results with finite element calculations for a planar, hexagonal array of grains shows that the accuracy of the model is satisfactory. The duration of elastic transients is estimated and it is found that, in the numerical examples considered, stress relaxation times are shorter than the incubation time for cavity nucleation. From the calculated stress concentration factors it is concluded that cavities cannot be nucleated by the considered mechanism of vacancy condensation unless the nuclei have a narrow, rather than spherical, shape.     

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.     

Nucleation of grain boundary cavities under the combined influence of helium and applied stress

Modelling cavity nucleation at grain boundaries in structural alloys under the combined influence of helium and stress is the primary objective of this paper. The role of stress in cavity nucleation is analyzed using an extension of classical theory by taking into account grain boundary sliding to describe stress concentration buildup and relaxation at particles and triple-point junctions. Helium clustering in the matrix is modeled using rate theory. The helium flux to grain boundaries is determined by the application of sink strength theory which takes into account the various competing clustering mechanisms in the matrix. Helium clustering on grain boundaries is also theoretically investigated using rate theory. The work agrees with experimental observations showing that irradiation results in grain boundary bubble densities which are orders of magnitude larger than cavity populations observed in conventional creep experiments. It is shown that even if the total injected helium is as little as one part per million, it can result in grain boundary bubble densities on the order of 10¹³ m⁻². Such cavity population exceeds typical grain boundary cavity densities associated with creep experiments. Grain boundary bubble densities are shown to reach steady state for injected helium amounts on the order of 10 parts per million.     

Computer simulation of morphological changes of grain boundary precipitates growing by the ledge mechanism

A previously developed computer model was modified to simulate the growth of grain boundary precipitates which grow by the ledge mechanism. The ledges were assumed to be nucleated in the grain boundary region at constant, parabolically decreasing, and random rates and to grow under the control of volume diffusion of solute to or from the riser of ledges. At lower under coolings at which the motion of individual ledges is slow, late-nucleated ledges soon catch up with first-nucleated ones, and precipitates tend to extend along the grain boundary: the overall precipitate shape is essentially that of a grain boundary allotriomorph. At larger undercoolings, first-nucleated ledges move fast to form a protuberance similar to Widmanstätten sideplates, while late-nucleated ones stay near the grain boundary region. The transition of precipitate shape from one to the other occurs in a very narrow range of supersaturation. The results are compared with various characteristics of the growth of proeutectoid ferrite allotriomorphs and sideplates in Fe-C alloys documented in the literature.     

On the role of elastic interactions in the nucleation and growth of ledges

Elastic interactions among ledges on transformation interfaces have noticeable consequences when chemical and interfacial tension (capillary) forces are small, namely, near equilibrium. This occurs just at nucleation (unstable equilibrium) or during the slow coarsening regime. When the interface lies perpendicular to the misfit strain (as do the large faces of misfits in Al-Cu alloys), ledges of like sign repel one another, and nucleation of new ledges occurs as far as possible from existing ones. However, when the interface lies parallel to the misfit strain, ledges of like sign attract one another. We then expect the formation of superledges. Essentially, such an interface with ledges is elastically unstable. Expressions are derived for the kinetics of ledge amaleamation. This paper is based on a presentation made in the symposium “The Role of Ledges in Phase Transformations” presented as part of the 1989 Fall Meeting of TMS-MSD, October 1–5, 1989, in Indianapolis, IN, under the auspices of the Phase Transformation Committee of the Materials Science Division, ASM INTERNATIONAL.     

Stochastic aspects of creep cavitation in ceramics

Creep fracture of ceramic materials frequently occurs by the nucleation, growth, and coalescence of grain boundary cavities. Recent experimental studies of cavitation kinetics in compression crept ceramics, supported by micromechanical modeling, have identified a number of stochastic aspects of cavitation. The stochastic nature of cavitation arises primarily due to the dependence of both cavity nucleation and cavity growth on grain boundary sliding. A degree of randomness is also imposed by the nonuniform distribution of potential nucleation sites. Pertinent experimental results and micromechanical models are briefly presented and used to support the important role of grain boundary sliding. A stochastic model of grain boundary sliding is then proposed by considering the sliding events to occur as an inhomogeneous Poisson process. Implications of the stochastic nature of cavitation are then discussed in terms of the cavity nucleation, growth, and coalescence processes. This paper is based on a presentation made at the symposium “Stochastic Aspects of Fracture” held at the 1986 annual AIME meeting in New Orleans, LA, on March 2-6, 1986, under the auspices of the ASM/MSD Flow and Fracture Committee.     

Crack and cavity nucleation at interfaces during creep

Athermal nucleation of microcracks and thermal nucleation of cavities during creep deformation are reviewed with an emphasis on effects of solute segregation to grain boundaries and cavity surfaces. The magnitude and the duration of stress concentration at a triple grain junction or at a grain boundary inclusion are estimated for transient Coble creep and steady state power-law creep conditions. Stable configurations of wedge-type microcracks are predicted by a Griffith-like crack model. The rate for thermal nucleation of cavities is obtained by the Fokker-Planck equation for vacancy clusters. Cracks and cavities are interdependent, and cavity nucleation occurs continuously throughout the three creep stages. The local stress concentration enhances microcrack and cavity nucleation. The cavity nucleation rate is generally increased as a result of solute segregation to the surfaces and interfaces and/or gas precipitation into cavity volume. This enhanced nucleation is more effective in a system with mobile solutes than with immobile solutes. Immobile solute or trace elements may affect the nucleation rate also by changing the grain boundary diffusivity. Experimental techniques for quantitative analyses of cavity nucleation processes are discussed. This paper is based on a presentation made at the symposium “The Role of Trace Elements and Interfaces in Creep Failure” held at the annual meeting of The Metallurgical Society of AIME, Dallas, Texas, February 14-18, 1982, under the sponsorship of The Mechanical Metallurgy Committee of TMS-AIME.     

Cavitation at grain and phase boundaries during superplastic flow of an aluminum bronze

Cavities have been observed to form at grain and phase boundaries under certain strain rate conditions during superplastic tensile deformation of a Cu-9.5 pct Al-4 pct Fe aluminum-bronze. The cavities form preferentially at α-β interfaces or triple junctions involving both phases. The process of cavitation is associated with grain boundary sliding and cavity nucleation probably occurs at points of stress concentration in the sliding interfaces. The ductility is not markedly impaired by the cavities because the high strain-rate sensitivity of the material inhibits the interlinkage of cavities at high strains. A range of strains and strain rates for superplastic forming processes has been determined at which the volume fraction of cavities present was tolerable.     

Development of final creep failure in polycrystalline aggregates

The final stages of creep rupture in a polycrystalline metal, by the linking-up of grain boundary microcracks to form a macroscopic crack, is studied by the numerical analysis of plane strain unit cells containing many hexagonal grains. Power law creep and elasticity are accounted for inside the grains, while intergranular failure occurs by cavity nucleation and growth to coalescence or by grain boundary sliding. The pattern of damage development initiated by an initial microcrack in the centre of the unit cell is studied for different stress states and different amounts of grain boundary viscosity. Furthermore, the model analyses are used to estimate the fraction of the total life time spent in the final link-up process. The life times are compared with estimates based on simple models that cannot describe microcrack linking-up.     

The role of ledges in the proeutectoid ferrite and proeutectoid cementite reactions in steel

The influence of interphase boundary ledges on the growth and morphology of proeutectoid ferrite and proeutectoid cementite precipitates in steel is examined. After reviewing current theoretical treatments of growth by the ledge mechanism, investigations that clearly document the presence and motion of ledges with thermionic emission electron microscopy (THEEM) and transmission electron microscopy (TEM) are reviewed. A fundamental distinction is made between two types of ledges: (1) mobile growth ledges whose lateral migration displaces the inter-phase boundary and (2) misfit-compensating structural ledges. Both types of ledges strongly affect the apparent habit plane and aspect ratio of precipitate plates. Agreement between measured growth rates of proeutectoid ferrite and cementite (plates and allotriomorphs) and predicted growth kinetics assuming volume diffusion-controlled migration of ledge-free disordered boundaries is shown to be consistently poor. Physically realistic growth models should incorporate the ledge mechanism. More accurate comparisons of the growth models with experimental data will need to account for observed ledge heights, interledge spacings, and ledge velocities. In this vein, the sluggish growth kinetics of cementite allotriomorphs observed in an Fe-C alloy are shown to be quantitatively consistent with a strong increase in interledge spacing with time. This paper is based on a presentation made in the symposium “The Role of Ledges in Phase Transformations” presented as part of the 1989 Fall Meeting of TMS-MSD, October 1–5, 1989, in Indianapolis, IN, under the auspices of the Phase Transformations Committee of the Materials Science Division, ASM INTERNATIONAL.     

The effect of temperature and carbon content on the cavitation behavior of a 1.5 pct Cr-0.5 pct V steel

Constant strain rate tests at 10^-6 s^-1 have been carried out at 823 K and 923 K on a vacuum melted 1 1/2 pct Cr 1/2 pct V ferritic steel containing 3 different carbon contents. After straining to various elongation values specimens were unloaded, cooled and fractured at 77 K. This gave fracture surfaces consisting almost entirely of intergranular facets, enabling a quantitative study to be made of the different stages of cavity nucleation and growth. It was found that cavity growth rates were independent of carbon content but were higher at 923 K than at 823 K. Subsequent grain boundary sliding measurements, using a surface offset technique showed that sliding increased with increasing carbon content and that cavity nucleation occurred selectively at large grain boundary carbides. Formerly of the Department of Metallurgy, University of Manchester.     

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.     

Superledge Model for Interphase Precipitation During Austenite-to-Ferrite Transformation

A model for interphase precipitation has been developed based on the ledge mechanism of austenite-to-ferrite transformation. Carbide precipitation is considered on the migrating ferrite/austenite interface as an interaction of transformation and precipitation kinetics. The derived equations describe sheet spacing and particle spacing of interphase-precipitated carbides as well as the overall interface velocity which are related to the nucleation rates of carbides and ferrite ledges, respectively. The microstructure characteristics of interphase precipitation are predicted as a function of transformation temperature and steel composition and replicate trends observed experimentally.     

Thickening of grain-boundary α allotriomorphs in a Ti-Cr alloy by multiple sets of ledges

A computer model is developed to simulate the growth of grain-boundary allotriomorphs having more than one set of growth ledges at their interfaces. The growth is controlled by the volume diffusion of solute to or from the riser of a ledge. The time dependence of the growth rate of two orthogonal sets of ledges is found to be somewhat different from that of a single set of ledges. However, the operation of multiple sets of ledges is unlikely to alter significantly the growth kinetics of grain-boundary allotriomorphs from those predicted from the disordered growth theory, except at small ledge spacings or at short reaction times. Faster growth kinetics of proeutectoid α allotriomorphs than those of either planar or ellipsoidal disordered boundaries which have been reported in a Ti-6.6 at. pet Cr alloy are not likely to be accounted for with the heights and spacings of double sets of ledges actually observed on the interfaces of allotriomorphs. Hence, the grain-and interphase-boundary diffusion-assisted growth of precipitates, (rejector plate mechanism, RPM) appears to be operative during the growth of a allotriomorphs, as previously proposed on the basis of growth-rate measurements. This paper is based on a presentation made in the symposium “The Role of Ledges in Phase Transformations” presented as part of the 1989 Fall Meeting of TMS-MSD, October 1–5, 1989, in Indianapolis, IN, under the auspices of the Phase Transformations Committee of the Materials Science Division, ASM INTERNATIONAL.