Microstructure evolution in three-dimensional inhomogeneous elastic media

On the unified approach to anisotropic and isotropic elasticity for singularity, interface and crack in dissimilar media

The effect on dislocation density of elastic anisotropy (relation of growth direction to elastic stiffness tensor of cubic symmetry) and of plastic anisotropy (relation of growth direction to crystallographic slip systems of {111} ≪110≫ type) is investigated for the growth from the melt of shaped, single-crystal semiconductors of III–V compounds (typically GaAs, InP) by the Czochralski method. The thermal stresses are determined numerically by using three-dimensional finite element techniques for growth along the ≪100≫, ≪111≫ or ≪110≫ directions. The equivalent shear stress is calculated and it is assumed that the dislocation density is proportional to the sum over all slip systems of the excess of the resolved thermal shear stress over the crystal's yield stress in shear. It is shown that proper account of both elastic and plastic anisotropy is necessary in order to correlate numerical estimates of dislocation density to experimentally determined patterns. In particular, it is shown that the effect of elastic anisotropy is to produce significantly higher dislocation densities, especially for the case of ≪111≫ and ≪110≫ growth. For moderate levels of elastic anisotropy, the dislocation density levels during ≪111≫ and ≪110≫ growth become significantly larger than density, during ≪100≫ growth, at specific locations within the crystal. The correlation between dislocation density and equivalent shear stress is presented, and the effect of plastic anisotropy is thus discussed. It is shown that the assumption of elastic isotropy leads to dislocation density levels that considerably underestimate the levels but, for a given growth direction, are qualitatively similar to the dislocation density patterns corresponding to the three-dimensional numerical solutions in which elastic anisotropy is accounted in full.

Nanoporous superalloy membranes are a functional extension of the use of nickel-based alloys. The material, which is usually used for high-temperature applications, consists mainly of the two phases γ and γ′. Through coarsening of the precipitates and thus forming of a bicontinuous γ/γ′ network, membranes can be produced by removing either of these phases. From the single-crystalline alloy CMSX-4, the bicontinuous network can be formed either thermo-mechanically by directional coarsening of coherent precipitates or by load-free coalescence of incoherent precipitates. Recent investigations have shown that membranes also can be produced from polycrystalline starting material in both ways. In this article, the process route for membranes by load-free coarsening of incoherent γ′ precipitates from a carbon-free version of the polycrystalline alloy Nimonic 115 is presented. This manufacturing method has the advantage of its simplicity and in comparison to single-crystalline membranes it can be realized in larger scales. We discuss the microstructure and show the mechanical properties by means of tensile tests. Despite the grain boundaries as a mechanical weak link, polycrystalline membranes show promising mechanical properties. Their strength even exceeds that of the single-crystalline membranes despite the significantly higher pore volume content.

Development and performance analysis of novel in situ Cu–Ni/Al2O3 nanocomposites

The elastic interaction energies between a calcium-cation vacancy pair and both a screw and an edge dislocation in a sodium chloride lattice are calculated using isotropic, semi-anisotropic, and anisotropic elasticity theory. The defect pairs considered have their axes along the 〈100〉 directions, the cation vacancy occupying a next-nearest neighbor position to the calcium impurity. They produce a tetragonal strain field in the lattice which interacts elastically with the stress field of both kinds of dislocations. The semi-anisotropic method introduces the single-crystal elastic constants by considering the defect pair as a force dipole. The interaction of the stress arising from this force dipole with the strain field of a dislocation as determined by isotropic elasticity theory is then calculated. For the other two methods of calculation the stress field of a dislocation in an isotropic and in an anisotropic medium are used. Dislocations in sodium chloride lie along certain high-symmetry directions thus enabling closed-form solutions for their stress fields in anisotropic materials to be obtained. Results of the calculations indicate good agreement between the isotropic and anisotropic cases and poor agreement between these two cases and the semi-anisotropic case. A comparison is made between the results of these calculations and the analogous situation of the interaction of dislocations with carbon interstitials in iron for which calculations have been made previously using isotropic and semi-anisotropic elasticity theories.

Atomistic insights into continuous and discontinuous precipitation in high-entropy alloys: Role of interfacial and elastic anisotropy

Severe plastic deformation of commercially pure titanium, leading to ultrafine-grained microstructure, can be used as an alternative to alloying to obtain better mechanical properties without the reduction of biocompatibility and corrosion resistance. However, the strong texture, induced by plastic deformation, results in anisotropy of mechanical properties, understanding of which is crucial to design and model perspective structural parts and devices. In this work, ultrafine-grained grade 2 Ti, obtained by equal channel angular pressing, is investigated. A detailed study of the texture and the elastic anisotropy of the material, heat-treated at several annealing temperatures, is conducted, complemented by ex-situ and indirect in-situ thermal experiments, which shed light on the stability and evolution of microstructure and material properties. The results show that the elastic properties only slightly evolve during annealing in contrast to other properties, such as thermal expansion, internal friction or hardness. The elastic anisotropy is directly linked to the (0001) texture, thus outlining the importance and possibilities of texture-based design and tailoring of mechanical properties.

The effect of dispersed-phase elasticity on droplet behavior and morphology of immiscible blends in simple shearing flow is investigated for blends of “Boger” fluids and high-molecular-weight polymer melt blends. For blends of “Boger” fluids consisting of polybutadiene (PBd) “Boger” fluids as droplet phase and poly(dimethyl siloxane) (PDMS) as matrix phase, the effect of dispersed-phase elasticity on steady state deformation and breakup of isolated droplets in simple shearing flow is investigated systematically for values of the dispersed-phase Weissenberg number (Wid) ranging up to around 3, where the Weissenberg number is defined as the ratio of the first normal stress difference to twice the shear stress at the imposed shear rate. The dependence on droplet elasticity of steady-state morphology for 10% or 20%- dispersed phase blends is also studied. The polybutadiene droplet phase is an elastic “Boger” fluid prepared by dissolving a high-molecular-weight polybutadiene into low-molecular-weight Newtonian polybutadiene. To isolate the contribution of droplet elasticity, all experiments were carried out on a fixed viscosity ratio of around unity, achieved by adjusting the temperature appropriately for each blend. When the droplet elasticity increases, the steady-state deformation of isolated droplets decreases for a given capillary number. The critical capillary number for breakup (Cacrit) increases linearly with the Weissenberg number of the droplet phase (Wid) up to a value of Wid of around unity. When Wid is greater than unity, Cacrit seems to approach an asymptotic value of 0.95 for high values of Wid. For 10% or 20%-dispersed phase blends, the steady-state capillary number (Cass) calculated from a volume-averaged droplet diameter is less than the Cacrit for isolated droplets for the same blend. Cass increases monotonically with the first normal stress difference of the droplet phase (Nid). For high-molecular-weight polymer melt blends consisting of polystyrene (PS) as a dispersed phases, and high density polyethylene (HDPE) as a matrix phase under the condition of both fluids have the same viscosity, the isolated viscoelastic droplets initially deform in the flow direction after startup of steady shear, but then begin reverting to a spherical shape, and, for the more elastic blend, eventually deform in the vorticity direction. With increasing capillary number, the droplet deforms increasingly along the vorticity direction, and above a critical capillary number Cacrit, breakup occurs when two ends of a drop situated on widely separated streamlines with significantly different velocities are displaced from each other under flow. The transition from alignment in the flow direction for Newtonian or slightly elastic droplets to alignment in the vorticity direction for highly elastic droplets can lead to large increases of the critical capillary number for droplet breakup, up to a factor of thirty greater than for Newtonian liquids. For concentrated high-molecular-weight polymer melt blends containing 20%-dispersed phase, the influence of elasticity of the blend constituent components on the steady-state size and size distribution of dispersed-phase droplets is investigated. The role played by the ratio of drop to matrix elasticity at a fixed viscosity ratio was investigated at which the viscosity ratios are roughly equal to three different values: 0.5, 1, and 2. The correlation between Cass and the elasticity contrast, defined as the ratio, Nlr = Nld/Nlm, of the first normal stress difference of dispersed (Nld) to that of matrix (Nlm) phase, is proposed. For the blend systems with viscosity ratio 0.5, 1 and 2, the values of Cass were found to monotonically increase with Nir and followed a power law with scaling exponents varying between 1.7 and 1.9.

An explicit expression of the elastic potential energy, W p, stored in a thin foil by a dissociated dislocation running parallel to the free surfaces is obtained in isotropic elasticity. It is used to discuss the metastable elastic equilibrium of a 60° dislocation in an ultrathin silicon foil when the fault plane is slightly inclined with respect to the free surfaces. The energy W p depends in particular on the positions of the two partials in the foil and on the thickness h, a situation not fully considered previously. For such ultrathin foils, propitious to the observation of partial dislocations cores at near atomic resolution, the theory predicts rapid changes with h of the separation distance S. This result is in accordance with previous experimental observations of S realized with the so-called “forbidden-reflection lattice imaging” technique.

Firstly, the elastic displacement and stress fields of an edge dislocation located at the corner of a dihedral interface separating two media + and ‐ are discussed using isotropic elasticity. A fully analytical solution is derived and particular examples are calculated for the shear stress field when the ratio of both shear moduli μ+/μ− is 2 and 1/2, and the dihedral angle is 60°. Secondly, an example which illustrates the shear stress field of a dihedral edge dislocation at the corner of two {112}facets of a twin boundary in gold is presented using anisotropic elasticity.

Oriented cracks and/or oriented joints in rocks produce anisotropic elasticity. When the rock has anisotropic elasticity, one must know more than two elastic constants to describe fully the elastic behavior of rocks under stress. In an anisotropic body, there are twenty-one elastic constants with respect to an arbitrary coordinate system. When a material has some sort of symmetry, a number of non-zero elements, with respect to the symmetry axes, can be reduced. In general, the full information of elastic symmetry has to be known, because the elastic constants of rock-forming minerals and rocks are determined by measuring elastic properties with respect to the elastic symmetry. In this study, it was tried to determine twenty-one elastic constants of anisotropic rocks, and to find symmetry axes of the rock, by using elastic wave velocities propagating in various directions. The calculated elastic constants and the axes of symmetry of Oshima granite agreed well with the published results which were determined in the assumption of orthorhombic elasticity with respect to the planes of oriented cracks.

This study presents an integrated phase field-crystal plasticity modeling framework for predicting the microstructure evolution during aging heat treatment and the subsequent mechanical properties for single crystal Ni-based superalloys. The phase field model, based on the Kim-Kim-Suzuki model, accounts for the anisotropic growth and coarsening of precipitates during aging. Specifically, the interfacial anisotropy in the phase field model is used to simulate the morphological evolution of precipitates from an initial spherical/circular shape to a cuboidal/square shape, which is a characteristic feature of the γ ′ precipitate growth in superalloys. Application of the model has been demonstrated in the form of three different case studies. The first case study simulates the two stage aging of a commercial single crystal superalloy, CMSX-4, followed by the prediction of the temperature-dependent tensioncompression asymmetry of the aged microstructures. These results highlight the capability of the model to predict processing-microstructure-mechanical property correlations. The second case study simulates the microstructure evolution during long-term aging of CMSX-4 for relatively larger microstructures, highlighting the capability of the model to predict precipitate coarsening. Further, the effect of varying the cooling rate on the microstructure evolution is also demonstrated. The third case study simulates the microstructure evolution during aging at relatively lower temperatures for Ni-Al alloys with different compositions. These results highlight the capability of the model to predict the change in precipitate morphology from circular to square shape during the initial stages of aging. Model predictions of the precipitate size, volume fraction and mechanical properties have been compared with relevant literature data. Overall, this integrated phase field-crystal plasticity framework offers a powerful tool for the computational design of single crystal superalloys.

Finite element analysis of void growth behavior in nickel-based single crystal superalloys

An anisotropic elastoplastic Cosserat continuum model for shear failure in stratified geomaterials

Microstructural Evolution in Inhomogeneous Elastic Media