Satish I. Rao

Discrete dislocation simulations of precipitation hardening in inverse superalloys

The low-temperature yield stress of a γ′ (Ni3Al) matrix–γ (Ni) precipitate ‘inverse’ superalloy, containing 40% Ni precipitates (γ), is calculated by discrete-dislocation simulations. Two different precipitate sizes and two anti-phase boundary energies are considered. The results of these simulations are compared with corresponding results from γ–γ′ superalloys (S. Rao, T.A. Parthasarathy, D. Dimiduk, et al., Phil. Mag. 84 3195 (2004)). In general, the results show that precipitation hardening in inverse superalloys is weaker than for regular superalloys. Similar to studies of superalloys, many of these results can be rationalized from the results of simulations on simple homogenized precipitate structures. The Hirsch, Kelly and Ardell precipitation-strengthening model (Metall. Trans. A 16 2131 (1985); Phil. Mag. 12 881 (1965); Trans. Jpn. Inst. Metals 9 1403 (1968).), developed for low-stacking-fault-energy spherical precipitates in a high-stacking-fault-energy matrix, adapted for inverse superalloys, shows qualitative agreement with the simulation results for spherical γ precipitates.

X-ray diffraction analysis of concentration and residual stress gradients in nitrogen-implanted niobium and molybdenum

Large biaxial residual strains are developed after a 5‐at.% implantation of N into Nb and Mo. The results indicate that the dominant source of internal strain arises from N located in interstitial sites. For Nb implanted at liquid‐nitrogen temperature, the N atoms are located in octahedral sites. However, the data allow for some clustering as di‐ or tri‐interstitials at the highest concentration (∼5 at. % N). Radiation damage is present as small vacancy and interstitial loops. Since vacancies and self‐interstitials are present in nearly equal concentrations, the overall bulk dilatation cancels. However, because of their small size, a lesser core expansion has been included as a correction to the overall residual strain. Although one can obtain an estimate of the N distribution from trim, a more accurate description must include the distribution of knock‐on energy. The latter has an important influence on the redistribution of N relative to that predicted by trim. Both host lattices (Nb and Mo) behave like...

Residual stress gradients along ion implanted zones — cubic crystals

Large internal strains and stresses can be produced by low temperature implantation over small distances from the free surface in a thick substrate. These are typically non-uniform and have large composition gradients. In dilute bcc solutions, containing unclustered interstitial implants, the residual macroscopic strains may be treated as isotropic. The calculation of residual strain (or stress) is based upon anisotropic elasticity theory and internal stress is given in terms of the dipole tensor for individual defects in single crystal films. In a completely elastic zone, forces act to maintain a rigid outside surface and cause the strain distribution to be zero along directions parallel to the free surface. This produces a strain magnification along the perpendicular direction from Poisson contractions. If the implanted zone is completely relaxed by plastic deformation, the strains are described by the free expansion strains due to both implants and lattice damage. There is no angular dependence of the free expansion strain in this extreme condition. One can determine whether a zone is completely elastic, completely relaxed by plastic deformation, or in some intermediate state from plots of strain against sin2χ, where χ is the angle of tilt relative to the surface normal. These results may be obtained from X-ray Bragg intensity data by measuring shifts and line broadening from (hkl) planes at different tilt angles. Theoretical results are given for both single crystal and polycrystalline materials in terms of residual strain and stress.

X-ray diffuse scattering from a nitrogen-implanted niobium film

A 2500‐A niobium single‐crystal film was deposited onto a sapphire substrate and subsequently implanted with nitrogen to an average concentration of 0.5 at. %. Synchrotron radiation was used to measure the difference between the implanted and an unimplanted film to isolate the diffuse scattering from the implanted film near two Bragg reflections. This diffuse intensity arises mainly from elastic displacement fields about radiation‐damage‐related loops located on (211) planes. A small contribution of the scattering is calculated from the displacements about single interstitial nitrogen in octahedral sites. The Burgers vector of the loops is along the [111] direction and makes an angle of 62° with the loop plane giving a dominant shear component. Vacancy loops have a radius ∼5 A while interstitials are somewhat larger ranging from 10 to 15 A. The number of vacancies and interstitials are nearly the same.

Derivation of many-body potentials for b.c.c. metals

Abstract A method of deriving the shape of pair potentials describing both the long- and short-wavelength behaviour of b.c.c. metals within the Finnis-Sinclair many-body-potential model is described. The method guarantees a fit to both the elastic constants and zone-boundary vibrational frequencies of b.c.c. metals using the five nearest-neighbour shells in the b.c.c. structure. The application of the method to niobium is briefly discussed and compared with previous work.

The measurement of elastic stresses and energy in cubic single‐crystal films by x‐ray diffraction

Anisotropic elasticity calculations have been made for use in conjunction with strain measurements by x‐ray diffraction for sputtered single‐crystal films. Only the cubic case has been treated. Data from InSb films with (100) and (111) orientations on similarly oriented GaAs substrates are given. It was found that nearly alike planar strains e yield lower planar stresses σ′1 and σ′2 and stored energy density U for the (100) orientation. The (100) films exhibit a relatively large strain perpendicular to the film e3.

The Interaction Between Dislocations and Lamellar Grain Boundaries in Pst γ Tiai

In lamellar TiAl the flat-plate geometry of the grains, the barriers to deformation across the grain boundaries and the coherency stresses all contribute to a marked anisotropy in the yield and fracture stresses of the material. Both yield and fracture occur at low stresses when the deformation is within the lamellae (soft mode) and they occur at high stresses when the deformation crosses the lamellae (hard mode). The anisotropy is enhanced by a new effect which can soften the soft mode and harden the hard mode: the geometry of the lamellar boundary produces degeneracies in the planar fault energies at the interfaces which enhance the mobilities of dislocations on these interfaces. These degeneracies modify the core structure of dislocations on or near the interfaces, consequently soft mode dislocations can dissociate widely and move more easily when their glide plane is contained in the interface. Hard mode dislocations can substantially reduce their core energies when intersecting a γ/γ interface, and subsequently become immobilized, by cross slipping on to the interface plane. This paper presents a discussion of the geometry and relative energies of the γ/γ interfaces using elements of Bollman O-lattice theory. In order to investigate the influence of the interfaces on dislocation core structure we have fit an empirical Embedded Atom Method (EAM) potential to the structural and elastic properties of bulk L10 TiAl. The mobility and core structure of the twinning dislocation at the 180° interface and the perfect, 1/2

Estimates of internal stresses about interstitials in a bcc lattice

The atomic volume of interstitial cubic and hexagonal interstitial compounds has been found to correlate with the size of the filled octahedron in corresponding dilute solutions. Data from binary systems containing N, C, and O in V, Nb, Ta, Cr, Mo, W, and Fe lattices were examined. Systematic correlations enable estimates to be made of the components of the dipole tensor for interstitials in octahedral sites within selected bcc lattices. The volume change in filling an interstitial site is used to determine the sum of the diagonal tensor components, while their ratio is obtained from the second-neighbour displacement of a lattice atom about a filled site. Estimates from crystallographic data are obtained using either a volume correlation or a common shape factor along with the second-neighbour displacement. A filled octahedron tends to give equal first and second-neighbour distance parameters making it nearly regular. Estimates are made from isotropic and anisotropic elasticity. Lattice Green function calculations support the use of the second neighbour as a core displacement parameter. As expected, core displacements obtained from lattice theory can differ greatly from those obtained by elastic calculations except for the second-neighbour displacement. The influence of crystal anisotropy on the long range elastic field is examined. In niobium the first neighbour is displaced along an elastically soft direction and one finds the largest displacements of any system examined.

On the Escaig obstacle hypothesis for cross-slip in face-centered-cubic materials

There is a significant body of literature wherein a linear approximation of Escaigs model is used to justify the large experimentally measured activation-volumes for cross-slip in face-centered-cubic copper. Here, by examining the error between the linear approximation and the original theory, we show that this explanation is not satisfactory. The calculated value for activation volume in copper, using Escaigs original equations, yields ∼60b 3 (b = Burgers vector) while the linear approximation yields 200b 3, the latter result fortuitously matching the experimental values.

A simplified procedure for obtaining relative x‐ray intensities when a texture and atomic displacements are present

The orientation problem in polycrystalline cubic materials has been simplified, using fundamental relationships, so that the determination of a quantitative interrelationship between the various Bragg peak intensities is no longer a formidable task. This is demonstrated with a cubic Cu‐Be‐Co alloy having a fiber texture, and a conventional focusing diffractometer. Because data are required which extend over a larger range in d spacings, extinction, thermal, and static atomic displacements must be included into the analysis of intensities. The displacement terms and the extinction parameters may be of primary interest or used as a correction. Seventeen diffraction peaks are used in the example. These must be internally consistent with a crystallite orientation function, the cubic symmetry of the sample, extinction effects influencing the two strongest peaks, and attenuation due to atomic displacements. Tabulated coefficients are presented which greatly reduce the task of calculating the orientation functio...