Elastic Interaction Energy Research Articles

An Investigation of the Microstructural Evolution of a Service-Exposed Turbine Blade

A field-emission scanning electron microscope with an energy-dispersive X-ray spectroscopy system was used to study the microstructural evolution of nickel-based superalloy K465 during service, including γ′ degeneration, decomposition of MC carbides, and formation of creep cavities. Results show that the primary γ′ precipitates having regular and cubic morphologies are agglomerated together by elastic interaction energy and gradually evolve to more stable γ′ plates. The γ′ particles in the interdendritic region have higher coarsening rates than those in the dendrite core. The sizes of γ′ precipitates at various locations of the blade, characterized by quantitative metallography, are employed to estimate the temperature distribution of the airfoil. MC carbides react with the surrounding matrix, forming M6C carbide and γ′ film. The interfaces between the carbides, both at grain boundaries and in grains, and the γ′ films are preferential sites for the nucleation and fast propagation of creep cavities.

On variant distribution and coarsening behavior of the α phase in a metastable β titanium alloy

The stereology, variant distribution and coarsening behavior of semicoherent α(hcp) precipitates in a β(bcc) matrix of a Ti5553 alloy has been analyzed, and a dominant 3-variant cluster has been observed in which the variants are related to each other by an axis-angle pair <112¯ 0 >/ 60°. Shape and spatial distribution independent elastic self and interaction energies for all pairwise and triplet combinations of α have been calculated and it is found that the 3-cluster combination that is experimentally observed most frequently has the lowest energy for the semicoherent state. The coarsening behavior of the delta distribution follows LSW kinetics after an initial transient, and has been modeled by phase field methods.

Dissolution behaviour of the γ′ precipitates in two kinds of Ni-based superalloys

The effects of high heating temperature and short dwell time on the evolution of γ′ precipitates were investigated in two kinds of Ni-based superalloys. The microstructural evolutions of the samples were characterised by qualitative and quantitative metallography during various heating cycles, and the dissolution kinetics were then evaluated. The results illustrated that the heating temperature plays a more predominant role in the dissolution of γ′ precipitates than the dwell time. Meanwhile, the γ′ morphology in the dissolution process was strongly influenced by the elastic strain energy associated with the lattice mismatch between γ and γ′ phases. For IN 100 alloy, the γ′ evolution was dominated by the elastic interaction energy between adjacent precipitates, but for DS Rene 125 alloy, the γ′ precipitates kept a regular arrangement along the [001] softest directions in the dissolution process. On increasing the dwell time for the two kinds of superalloys, the dissolution rate of γ′ gradually decreased, and the dissolution activation energy gradually increased. However, the dissolution activation energy for IN 100 alloy was different from that of DS Rene 125 alloy under the same heating conditions.

Assessment of the dislocation bias in fcc metals and extrapolation to austenitic steels

A systematic study of dislocation bias has been performed using a method that combines atomistic and elastic dislocation-point defect interaction models with a numerical solution of the diffusion equation with a drift term. Copper, nickel and aluminium model lattices are used in this study, covering a wide range of shear moduli and stacking fault energies. It is found that the dominant parameter for the dislocation bias in fcc metals is the width of the stacking fault ribbon. The variation in elastic constants does not strongly impact the dislocation bias value. As a result of this analysis and its extrapolation, the dislocation bias of the widely applied austenitic stainless steels of 316 type is predicted to be about 0.1 at temperature close to the swelling peak (815 K) and typical dislocation density of 1014 m−2. This is in line with the bias calculated using the elastic interaction model, which implies that the prediction method can be used readily in other fcc systems even without EAM potentials. By comparing the bias values obtained using atomistic- and elastic interaction energies, about 20% discrepancy is found, therefore a more realistic bias value for the 316 type alloy is 0.08 in these conditions.

Effect of external stress on γ nucleation and evolution in TiAl alloys

We have shown in a previous paper that twin boundary area fraction in lamellar γ-TiAl based alloys depends on collective or correlated nucleation induced by long-range elastic interactions among nucleating precipitates or between pre-existing and nucleating precipitates. In this study we investigate the effects of applied stresses on γ nucleation and evolution by elastic interaction energy calculations and phase field simulations. It is found that the elastic interaction energy reaches minimum when two variants are twin related because of their self-accommodation of the coherency strain. When external stresses are present, such autocatalysis during nucleation could be either enhanced or suppressed. Compression perpendicular or tension parallel to the basal plane of the α phase is found to enhance the internal elastic interactions between nucleating precipitates, promote twin variant selection and reduce the overall nucleation rate, while isostatic pressing, compression parallel, tension perpendicular to the basal plane does the opposite. Shear along the interface always reduce the twin boundary area fraction, regardless of the shear direction. These findings could shed light on optimizing processing routes to increase twin boundary area fraction in lamellar microstructures of γ-TiAl based alloys.

Fabrication and microstucture of spray formed powder metallurgy superalloy FGH4095M

Spray forming is a kind of near-net-shaped rapid solidification process based on powder metallurgy gas atomization technology. In this work, the FGH4095M is fabricated by spray forming. The pre-alloy is prepared by vacuum induction melting and vacuum arc remelting techniques. Then the alloy is sprayed by SK2 facility with atomization gas nitrogen at University of Bremen in Germany. In this paper we study the density and microstructure of the spray-formed billet, especially the special morphology of γ’ phase. The results show that density is associated with different parts of the deposited billet. The relative density of the bottom part is higher (99.63%) than those in the other parts. The relative density of top part (98.91%) is lowest. After hot isostatic pressing, the relative density can be up to 100%. Uniform and fine equiaxed grains are the remarkable morphology of spray-formed alloy without prior particle boundary. The sizes of grains are in a range of about l0-40 μm and the grains at bottom part of billet are finest. The grain sizes of primary γ’ phase are in a range of about 0.6-0.8 μm, and the grain sizes of secondary γ’ phase in a range of about 0.1-0.5 μm as well as dispersed spherical tertiary γ’ particles with the sizes of 10-20 nm. The special morphology of secondary γ’ phase occurs with the splitting of γ’ particle, which is related to the low cooling rate of the depositing process. The splitting behavior reduces the total energy of γ’ particle. Total energy of γ’ particle includes elastic interaction energy, elastic strain energy and surface energy, among which the elastic strain energy is invariable. The surface energy increases with the splitting process and the elastic interaction energy reduces. The effect of elastic interaction energy on particles is the major reason why the total energy is reduced. The trend of splitting behavior is analyzed by calculating the equivalent diameter of splitting γ’ particle. It indicates that when the equivalent diameter is over 0.40 μm, there is the possibility to split. Subsequently, spray-formed FGH4095M billet is treated by hot isostatic pressing, isothermal forging and heat treatment process to obtain the FGH4095M alloy turbine disk. The research of tensile property of FGH4095M alloy turbine disk shows an excellent property either at room temperature or at high temperature for the optimized alloy. The relationship between special morphology of γ’ phase and excellent property needs further investigating.

Hillock formation by surface drift-diffusion driven by the gradient of elastic dipole interaction energy under compressive stresses in bi-crystalline thin films

We investigated surface drift diffusion induced grain boundary (GB) grooving and ridge (hillock) formation and growth, under the combined actions of the capillary forces and applied uniaxial compressive stresses, in bi-crystal thin films with dynamical computer simulations. In the present theory, the generalized driving force for the stress induced surface drift diffusion includes not only the usual gradient of the elastic strain energy density, but also the elastic dipole tensor interaction energy. During the morphological evolution of GB ridge formation and growth, triple junction (TJ) displacement and its velocity are continuously tracked down in order to resolve precisely the crossover time and depth at which velocity sign inversion takes place. An incubation time for the onset of the ridge growth stage coupled to the GB-TJ displacement velocity inversion is defined and its dependence on the stress is investigated. This analysis implies that the ridge growth stage is not controlled by Ziegler’s ‘maximum entropy production principle’ but rather Prigogine’s ‘minimum entropy production hypothesis’ for the stationary non-equilibrium states in complex systems, which are exposed to external applied body forces and surface tractions. Keywords: Surfaces and interfaces; Grain boundary grooving; Non-equilibrium thermodynamics; Surface/grain boundary diffusion; Compressive stresses; Thin films DOI: 10.17350/HJSE19030000010 Full Text:

084 マルテンサイト組織形成に及ぼす弾性相互作用エネルギー場の影響

084 マルテンサイト組織形成に及ぼす弾性相互作用エネルギー場の影響

Three-Dimensional Phase-Field Simulation and Experimental Validation of β-Mg17Al12 Phase Precipitation in Mg-Al-Based Alloys

A three-dimensional (3D) phase-field model has been developed to simulate the formation of lath-shaped β-Mg17Al12 phase during hcp→bcc transformation in Mg-Al-based alloys. The model considers the synergistic effects of the elastic strain energy associated with the lattice rearrangements that accompany the phase transformation, and the interface anisotropy (both in interfacial energy and interface mobility coefficient). By using the proposed model, the essential features of 3D morphology of the β phase precipitate have been successfully predicted and experimentally validated using high-resolution transmission electron microscopy and atomic force microscopy. Furthermore, the spatial distribution of anisotropic elastic interaction field around a pre-existing β precipitate has been quantitatively determined using 3D phase-field simulation, and the effects of the anisotropic elastic interaction energy on subsequent nucleation of β phase near a pre-existing precipitate have been revealed. The results suggest that the anisotropic elastic interaction energy can promote the formation of new nucleus near the lozenge ends of the pre-existing precipitate, as explicitly substantiated by the experimental observations. The influence of different combinations of interface anisotropy and elastic strain energy on the thickness of β phase precipitate has been elucidated. The correlation between microstructural design during precipitation and the alloy-strengthening mechanisms has also been discussed in terms of dislocation motion. Based on these results, possible strategies for strengthening Mg-Al-based alloys are proposed for magnesium alloy development and microstructural design.

Role of applied uniaxial stress during creep testing on precipitation in Mg–Nd alloys

The coupled effects of an applied stress and temperature, during creep testing at 90MPa/177°C, on intragranular second phase (orthorhombic β′ and fcc β1 phases) precipitation in a binary Mg–0.5at% Nd alloy were investigated by comparing and contrasting with second phase precipitation under isothermal (177°C) annealing conditions. In the early stages of creep deformation, after 5h, homogeneous precipitation of fine scale β′ was observed within the α-Mg matrix, while on the other hand, only small pockets of Nd enrichment were observed in the isothermally annealed sample after 5h, indicating that β′ precipitation is accelerated during creep testing. With respect to β1 precipitation, these precipitates appeared to decorate dislocation lines primarily, which is indicative of a dislocation stress-field assisted precipitation mechanism, in both creep tested and isothermally annealed samples. Interestingly, the β1 precipitates appear to grow and coarsen more rapidly in the case of the isothermally annealed sample than in the creep tested sample. The differences between the precipitation behavior of β′ and β1 phases under the influence of stress and temperature (creep testing) vs. only under isothermal annealing conditions have been rationalized based on the crystal structures of these phases and their elastic interaction energies with the stress-fields associated with dislocations as well as bulk applied stress.

Perovskite Ti3AlC Carbide Splitting in High Nb Containing TiAl Alloys

ABSTRACTTransmission electron microscopy has been used to investigate the morphological development of the perovskite (P-) Ti3AlC carbides in the γ matrix of a Ti-45Al-5Nb-0.75C alloy during annealing. P-Ti3AlC carbides in the γ matrix initially have a needle-like shape but during annealing at 800 °C they change to a plate-like shape. In the needle-like shape the carbides are orientated parallel to the [001] direction of the matrix. They extend along the [100]γ or [010]γ direction into plates later and subsequently split into sub particles after extended annealing. It is proposed that the elastic interaction energy between the split sub domains may be the reason that this decomposition into sub-particles is energetically favorable.

Locating the Focus of a Starting Earthquake

This article describes a method of locating the focus of a starting earthquake based on the use of the elastic interaction energy. The method allows determining the focus location and its energy class as well as evaluating the stresses caused by it and observing its evolution.

Elastic interaction of point defects in crystals with cubic symmetry

The energy of elastic mechanical interaction between point defects in cubic crystals is analyzed numerically. The finite-element complex ANSYS is used to investigate the character of interaction between point defects depending on their location along the crystallographic directions 〈100〉, 〈110〉, 〈111〉 and on the distance from the free boundary of the crystal. The numerical results are compared with the results of analytic computations of the energy of interaction between two point defects in an infinite anisotropic medium with cubic symmetry. The interaction between compressible and incompressible defects of general type is studied. Conditions for onset of elastic attraction between the defects, which leads to general relaxation of the crystal elastic energy, are obtained.

Line-integral representations for the elastic displacements, stresses and interaction energy of arbitrary dislocation loops in transversely isotropic bimaterials

The elastic displacements, stresses and interaction energy of arbitrarily shaped dislocation loops with general Burgers vectors in transversely isotropic bimaterials (i.e. joined half-spaces) are expressed in terms of simple line integrals for the first time. These expressions are very similar to their isotropic full-space counterparts in the literature and can be easily incorporated into three-dimensional (3D) dislocation dynamics (DD) simulations for hexagonal crystals with interfaces/surfaces. All possible degenerate cases, e.g. isotropic bimaterials and isotropic half-space, are considered in detail. The singularities intrinsic to the classical continuum theory of dislocations are removed by spreading the Burgers vector anisotropically around every point on the dislocation line according to three particular spreading functions. This non-singular treatment guarantees the equivalence among different versions of the energy formulae and their consistency with the stress formula presented in this paper. Several numerical examples are provided as verification of the derived dislocation solutions, which further show significant influence of material anisotropy and bimaterial interface on the elastic fields and interaction energy of dislocation loops.

Size-dependent reversal of the elastic interaction energy between misfit nanostructures

By exploiting a fully three-dimensional finite-element modeling of strain fields, we investigate the spatial dependence of the elastic interaction energy between misfitting nanostructures beyond the point-dipole approximation. When interacting islands are finite in size, the detailed shape of the elastic strain field around and under the islands may convert the repulsive interactions, usually experienced between equal-sized islands, into an attractive basin between a large island and a population of neighboring clusters smaller than a critical size. The results of the simulations applied to large Ge islands grown on a Si(111) substrate have significant implications for the understanding of the strain-mediated coarsening of quantum dots around the islands.

Influence of the low interfacial density energy on the coarsening resistivity of the nano-oxide particles in Ti-added ODS material

The ODS materials are composed of very fine (Y,Ti,O) nano-particles. During annealing at 1573K, the nano-clusters evolve into a fully cubical shape Y2Ti2O7-type particles. This evolution is caused by the elastic interaction energy at the nano-particle/matrix interface. Indeed, the cube-on-cube orientation relationship of the daughter Y2Ti2O7 phase is likely linked to be a coherent configuration of the parent nano-clusters. The elastic driven morphology allows for an estimate of the interfacial energy of these nano-clusters of σ=290mJm−2. This low interfacial energy value could be part of the explanation of the very high resistivity to coarsening of the (Y,Ti,O) coherent nano-cluster system, which is confirmed by observations of particle coarsening when the nano-size particles lose their coherency after recrystallisation or grain growth.

Reasons for self ordering in multilayer quantum dots Part II: Interaction energy

Two ways to calculate the elastic interaction between quantum dots (QDs) in the framework of linear elasticity are introduced and shown to vary in a similar way as the hydrostatic pressure. It is shown that the hydrostatic stress is the potential for the elastic interaction energy. The approach was used to estimate quantitatively the interaction energy between QDs in material systems that may form vertical order, anticorrelated order and FCC superlattice. The vertical interaction energy is very small compared with the thermal energy, nevertheless it is just enough to induce vertical ordering of QDs between layers. The attractive lateral interaction is an order of magnitude smaller than the vertical interaction, therefore ordering of multilatered QDs is made possible only by the vertical interactions. Several experimental observations are explained based on this understanding. The lateral interactions are small due to the small stresses set in the substrate after relief of the misfit strains in the free space. The vertical interactions are larger due to the large tensile stresses set up in the thin layer of matrix that separates an embedded QD from the free surface.

Formation of nanometer coherent structures during spinodal decomposition and ordering coexistence phase transformation in Fe-24Al alloys

The nanometer coherent structure evolution of spinodal decomposition and ordering coexistence phase transformation in Fe-24Al alloys is investigated by the microscopic phase field kinetic model. The results show that the concentration and long-range order parameters all continuously change towards to their equilibrium values during phase transformation. With the increase of elastic interaction energy, the anisotropy along [01] or [10] elastic soft direction is more obvious and the time reaching equilibrium state is also shortened. According to the results, the formation of nanometer coherent structures during phase transformation is composed of the initial decreasing stage of order degree stage, the incubation stage, the continuous increasing stage of concentration order parameter and long-range order parameter, and the later stable stage. The spinodal decomposition and ordering is interaction; the initial ordering stage is a necessary condition of the coexistence phase transformation. The nanometer coherent structures are not found to grow during the whole phase transformation. The simulation results are in accordance with the results in experiment obtained by the aging treatment in Fe-24Al alloys.

Guided aggregation of three-dimensional nanostructures in stressed thin films

Stress fields induced by external loads can alter the energy landscape in alloy systems and direct precipitation to form organized nanostructures. The aggregation of periodically patterned nanostructures via surface indentation on thin films is investigated using a phase-field model, which includes chemical, interfacial and elastic energies coupled with externally imposed stress fields. Both cylindrical and spherical indenters are considered, which lead to the formation of nanorods and nanodots, respectively, in the film, and the effects of loading geometry and material properties are systematically studied through 3D simulations. Nanostructures can be formed with varying precipitate morphologies. The results are shown to be consistent with estimates of elastic interaction energies associated with transformation strain, contrast in elastic properties and external loading.

P-phase precipitation and its effect on martensitic transformation in (Ni,Pt)Ti shape memory alloys

A new precipitate phase named P-phase has recently been identified in (Ni,Pt)Ti high temperature shape memory alloys. In order to understand the roles played by the fine coherent P-phase precipitates in determining the martensitic transformation temperature (Ms), strength of the B2 matrix phase, dimensional stability and shape memory effect of the alloys, a phase field model of P-phase precipitation is developed. Model inputs, including lattice parameters, precipitate–matrix orientation relationship, elastic constants and free energy data, are obtained from experimental characterization, ab initio calculations and thermodynamic databases. Through computer simulations, the shape and spatial distribution of the P-phase precipitates, as well as the compositional and stress fields around them, are quantitatively determined. On this basis, the elastic interaction energy between the P-phase precipitates and a martenstic nucleus is calculated. It is found that both the chemical non-uniformity and stress field associated with the P-phase precipitates are in favor of the martensitic transformation. Their relative contributions to the increase in Ms temperature are quantified as a function of aging time and the result seems to agree with the experimental measurements. The shape and spatial distribution of the P-phase precipitates predicted by the simulations also agree well with experimental observations.