The influence of particle shape on zener drag

Practically all papers concerned with grain refinement based on pinning by particle dispersions have considered spherical particles. In this paper we relax this restriction and Zener pinning equations are developed for ellipsoidal particles of any shape, orientation and degree of coherency. Theoretical predictions are verified using soap-bubble model experiments. It is found that within well defined conditions of particle orientation, disk or needle-shaped particles may offer greater pinning resistance, or Zener drag to migrating boundaries than spherical ones. It is shown, however, that the pinning resistance of non-spherical particles is very sensitive to particle orientation, and that their use in alloys is best utilised in systems based on mono-orientation dispersions.     

The crystallography of hydride formation in zirconium: I. The δ → γ transformation

The phenomenological theory of martensitic transformations has been applied to the transformation from δ (fcc) to γ (fct) zirconium hydride. The possibility of hydrogen diffusion prior to the transformation has been investigated by extrapolation of the published lattice parameters to lower hydrogen contents. The habit plane, orientation relationship, and lattice invariant shear were determined using transmission electron microscopy. Surface tilts were measured by interference microscopy and the direction and magnitude of the shape strain were determined from measurements of fiducial scratch displacements. The observed surface relief was indicative of an invariant plane strain. Good agreement between the predictions of the theory and the measured crystallographic features was obtained. The measured values of the direction of the shape strain and the habit plane were within 3.5 deg of the predicted values and the magnitude of the shape strain was in excellent agreement with the value predicted. The orientation relationship was near an identity, and the predicted and observed lattice invariant shear was twinning on (101)_γ. Formerly with the University of Illinois.     

Computer simulation of microstructure

The microstructure that results from a martensitic transformation is largely determined by the elastic strain that develops as martensite particles grow and interact. To study the development of microstructure, it is useful to have computer simulation models that mimic the process. One such model is a finite-element model in which the transforming body is divided into elementary cells that transform when it is energetically favorable to do so. Using the linear elastic theory, the elastic energy of an arbitrary distribution of transformed cells can be calculated, and the elastic strain field can be monitored as the transformation proceeds. In the present article, a model of this type is developed and evaluated by testing its ability to generate the preferred configurations of isolated martensite particles, which can be predicted analytically from the linear elastic theory. Both two- and three-dimensional versions of the model are used. The computer model is in good agreement with analytic theory when the latter predicts single-variant martensite particles. The three-dimensional model also generates twinned martensite in reason- able agreement with the analytic predictions when the fractions of the two variants in the particle are near 0.5. It is less successful in reproducing twinned martensites when one variant is dom- inant; however, in this case, it does produce unusual morphologies, such as “butterfly mar- tensite,” that have been observed experimentally. Neither the analytic theory nor the computer simulation predicts twinned martensites in the two-dimensional transformations considered here, revealing an inherent limitation of studies that are restricted to two dimensions.     

Influences of Fibers on Drying Shrinkage of Fiber-Reinforced Cementitious Composite

An analytical model has been formulated for the drying shrinkage performance of a fiber-reinforced cementitious composite. The model is based on the assumption that shear stress is produced between the fiber and surrounding matrix as the matrix shrinks. This shear stress in turn influences the matrix deformation behavior resulting in macroscopic composite shrinkage lower than that of a pure cement-based matrix. Through systematic derivation, a free shrinkage expression, which reflects the influences of the matrix and fiber properties as well as fiber orientation characteristics, is presented. A parametric study, including the influence of elastic moduli of the fiber and matrix, fiber dimension, and fiber content in the case of 3D fiber distribution, is carried out. The model results indicate that shrinkage of a fiber-reinforced cement-based composite is significantly influenced by elastic moduli of fiber and matrix as well as fiber length and thickness (i.e., diameter for fiber with a circular cross section). Model predictions based on independent parametric inputs compare favorably with experimental measurements of free shrinkage of fiber-reinforced mortar and concrete.     

Fracture in Equiaxed Two Phase Alloys: Part I. Fracture in Alloys with Isolated Elastic Particles

Fracture in equiaxed two phase alloys containing isolated elastic particles has been analyzed from the viewpoint of a recently proposed model for fracture initiation and propagation in such materials. This model predicts fracture toughness parameters in terms of the microstructural geometry, relative phase volume fractions, and tensile properties of the materials. Predictions of the model are tested experimentally for two phase Co-CoAl alloys over a wide range of compositions, and the results indicate good agreement between predicted and observed fracture toughnesses. M. A. PRZYSTUPA, formerly Graduate Research Assistant, Department of Metallurgical Engineering, Michigan Technological University.     

NiTi and NiTi-TiC composites: Part IV. Neutron diffraction study of twinning and shape-memory recovery

Neutron diffraction measurements of internal elastic strains and crystallographic orientation were performed during compressive deformation of martensitic NiTi containing 0 vol pct and 20 vol pct TiC particles. For bulk NiTi, some twinning takes place upon initial loading below the apparent yield stress, resulting in a low apparent Young's modulus; for reinforced NiTi, the elastic mismatch from the stiff particles enhances this effect. However, elastic load transfer between matrix and reinforcement takes place above and below the composite apparent yield stress, in good agreement with continuum mechanics predictions. Macroscopic plastic deformation occurs by matrix twinning, whereby (1 0 0) planes tend to align perpendicular to the stress axis. The elastic TiC particles do not alter the overall twinning behavior, indicating that the mismatch stresses associated with NiTi plastic deformation are fully relaxed by localized twinning at the interface between the matrix and the reinforcement. For both bulk and reinforced NiTi, partial reverse twinning takes place upon unloading, as indicated by a Bauschinger effect followed by rubberlike behavior, resulting in very low residual stresses in the unloaded condition. Shape-memory heat treatment leads to further recovery of the preferred orientation and very low residual stresses, as a result of self-accommodation during the phase transformations. It is concluded that, except for elastic load transfer, the thermal, transformation, and plastic mismatches resulting from the TiC particles are efficiently canceled by matrix twinning, in contrast to metal matrix composites deforming by slip.     

Modelling of the anisotropy of Young's modulus in polycrystals

The crystallographic orientation distribution of cold rolled and recrystallized low-carbon deep drawing steel is quantitatively investigated by means of the series expansion method and the technique of model component fit. The results of both methods are utilized to describe the elastic properties of the inspected sheets. The application of the model components allows a very good approximation of the course of Young's modulus and corresponds to experimental results as well as to the predictions stemming from the series expansion method. In addition to the influence of peak-type texture components also the impact of complete and incomplete fibre-type texture components on the course of Young's modulus is inspected.     

On composite-structure weaknesses: Part I. Simulation,properties, and numerical approach

Composite material samples were created by means of computer simulation to duplicate short-fiber-reinforced metal-matrix composites (MMCs). Each sample contains a fairly large number of Voronoi grains and ellipsoidal short fibers, which orient and distribute in a random manner, to mimic composite microstructures for investigating the coherent interconnections of composite-structure weaknesses (CSWs) with local microstructure. It is supposed that the samples are subjected to coupled boundary traction due to mechanical loading and thermal cycling. A Kr?ner-Kneer structure-based model and Waldvogel-Rodin algorithm were used for numerical computations of the mesoscopic stress distribution in constituent grains. The computations are based on the grain-volume average of local fields. Polycrystal elastic/thermal properties and effective elastic/thermal properties of simulated MMC samples were predicted, respectively, in terms of micromechanics models, in favor of incorporating the influences of macroscopic material properties on the formation of CSWs. An analytically-numerically-based approach is proposed for analyzing peak mesoscopic stress and strain distributions in short fibers. Three crucial aspects constitute a kernel of the approach, i.e., (1) segmentation of short fibers, (2) establishment of the geometric relations of a short fiber to the surrounding grains, and (3) the local nature of micromechanics. The analytically-numerically-based approach takes into account the grain orientation, fiber orientation, grain geometry, fiber geometry, and macroscopic properties of simulated MMC samples. The Numerical Assessment of Computer-Imitated Weaknesses-MMCs (NACIW-MMCs) software program has been developed for performing simulation of the microstructure of short-fiber-reinforced MMCs and executing all involved numerical computations.     

Characterization of anisotropie elastic constants of silicon-carbide participate reinforced aluminum metal matrix composites: Part I. Experiment

The anisotropic elastic properties of silicon-carbide particulate (SiC _p ) reinforced Al metal matrix composites were characterized using ultrasonic techniques and microstructural analysis. The composite materials, fabricated by a powder metallurgy extrusion process, included 2124, 6061, and 7091 Al alloys reinforced by 10 to 30 pct ofα-SiC _p by volume. Results were presented for the assumed orthotropic elastic constants obtained from ultrasonic velocities and for the microstructural data on particulate shape, aspect ratio, and orientation distribution. All of the composite samples exhibited a systematic anisotropy: the stiffness in the extrusion direction was the highest, and the stiffness in the out-of-plane direction was the lowest. Microstructural analysis suggested that the observed anisotropy could be attributed to the preferred orientation of SiC _p . The ultrasonic velocity was found to be sensitive to internal defects such as porosity and intermetallic compounds. It has been observed that ultrasonics may be a useful, nondestructive technique for detecting small directional differences in the overall elastic constants of the composites since a good correlation has been noted between the velocity and microstructure and the mechanical test. By incorporating the observed microstructural characteristics, a theoretical model for predicting the anisotropic stiffnesses of the composites has been developed and is presented in a companion article (Part II). Formerly with the Department of Aerospace Engineering and Engineering Mechanics, Iowa State University Formerly with Westinghouse Science & Technology Center     

The shape and orientation of the minimum strain energy of coherent ellipsoidal precipitate in an anisotropic cubic material

The elastic strain energy of perfectly coherent ellipsoid of revolution, which has the cube-cube orientation relationship with the matrix, has been calculated as a function of the orientation of the axis of revolution and of shape factor in anisotropic cubic crystalline materials. The minimum strain energy condition occurs at four different shapes and orientations, i.e. sphere, rod along 〈001〉 axis, disc on 001 plane and disc on 111 plane, depending on the two shear moduli of precipitate, i.e. μ*₁((C*₁₁—C*₁₂)/2) and μ* (C*₄₄). This is true regardless of the elastic property of the matrix phase when its anisotropy factor is larger than 1. The conditions of the occurrence of each shape and orientation are greatly affected by the difference in the misfit accommodation behavior depending on the shape of precipitate. A review of the experimental observations indicates the presence of all four different shapes and orientations in the case of GP zones in Al alloys. The conditions of their appearance are in good agreement with the prediction of the present calculation.     

A mathematical and physical representation of the raceway region in the iron blast furnace

Through the establishment of a macroscopic momentum balance a mathematical representation has been developed for predicting the size of the raceway region in blast furnaces. The formulation involved the establishment of a force balance where the radial momentum transfer from the incoming gas was balanced against the weight of the bed. The predictions based on the analysis were compared to experimental measurements on both two and three dimensional systems using solid particles of varying shape, size and density. Reasonable agreement was obtained between measurements and predictions and it was found that for small and light particles and for high blast rates the size of the raceway could be predicted from first principles without recourse to empirical correction factors. The predictions based on the analysis were also consistent with previous measurements of Elliott and Wagstaff on both laboratory scale models and operating blast furnaces. It was thus shown that the analysis developed in the paper may be used for estimating the size of the raceway in blast furnaces from first principles without recourse to empirical correlations. POVEROMO, formerly Graduate Student in the Center for Process Metallurgy, is now a Research Engineer, Bethlehem Steel Corp., Homer Research Laboratory, Bethlehem, Pa. 18016.     

Modeling the effects of stress state and crystal orientation on the stress-induced transformation of NiTi single crystals

A model that combines the phenomenological theory of martensite with a generalized Schmid’s law has been used to predict the principal stress combinations required to induce the martensitic transformation in unconstrained NiTi shape memory alloy (SMA) single crystals. The transformation surfaces prescribed by the model are anisotropic and asymmetric, reflecting the unidirectional character of shear on individual martensite habit planes. Model predictions of the transformation strain as a function of stress axis orientation for a uniaxial applied stress further demonstrate the anisotropy of the stress-induced transformation in NiTi single crystals. Model results for the uniaxial stress case compare favorably with previously published experimental observations for aged NiTi single crystals.     

Determination of single crystals' elastic constants from the measurement of ultrasonic velocity in the polycrystalline material

In order to determine the crystal elastic properties and to investigate the influence of external factors such as applied stresses, magnetic fields and temperature on the elastic properties, sufficiently large single crystals are required. However, for some materials, such large single crystals are difficult to grow. In view of this difficulty, an attempt has been made to determine the elastic constants of cubic crystals from ultrasonic velocities measured in polycrystalline textured materials. Based on the ultrasonic-velocity measurement and the orientation distribution function (ODF) of crystallines in the polycrystalline aggregate, the single crystals' elastic constants can be obtained.     

Densification of porous materials by power-law creep

The densification of a porous intermetallic alloy (Ti-14wt%Al-21wt%Nb) during the final stage of densification has been investigated under various states of stress and compared with the predictions of current models for densification by power-law creep. The experiments generally confirm the model predictions that the densification rate is a sensitive function of the stress state. Experimentally, unconstrained uniaxial compression resulted in the largest densification rates, while constrained uniaxial compression resulted in the lowest. Hydrostatic loading resulted in a densification rate similar (but slightly higher) than that of constrained compression. This ordering of the densification rates agreed well with the model predictions. However, the magnitudes of the measured densification rates are found not to be accurately predicted. A number of factors, including pore shape, pore spatial distribution and matrix microstructure have been observed to affect the densification rate, and the significance of each of these factors to predictive modelling of creep consolidation processes is assessed.     

Suitability of Micromechanical Model for Elastic Analysis of Masonry

A micromechanical model is proposed for determining the overall linear elastic mechanical properties of simple-texture brick masonry. The model, originally developed for long-fiber composites, relies on the exact solution due to Eshelby and describes brickwork as a mortar matrix with insertions of elliptical cylinder-shaped bricks. Macroscopic elastic constants are derived from the mechanical properties of the constituent materials and phase volume ratios. Conformity of the suggested model to real brickwork behavior has been verified by performing uniaxial compression tests on masonry panels composed of fired bricks and mud mortar. Composite masonry panels of varying phase percentages were then constructed and tested by replacing several of the fired bricks with mud bricks. Comparison of experimental results with theoretical predictions demonstrates that the model is suitable even in the presence of strongly differentiated phases, and is moreover able to predict different behavior as a function of phase concentration. The model fits experimental results more closely than the micromechanical models previously reported in the literature.     

Deformation and fracture of a particle-reinforced aluminum alloy composite: Part II. Modeling

In this article, the tensile and fracture properties of a discontinuously reinforced aluminum (DRA) alloy composite are modeled to determine the influence of constituent parameters on material behavior. Comparison of the elastic-modulus calculations to the experimental data suggest that the angular particles are more effective in load transfer than spherical particles, and that a unit cylinder geometry is a good representation of the particles under elastic conditions. This same geometry is used in the finite element-based elastic-plastic model of Bao et al., and reasonably good agreement is obtained between the experimental and predicted yield strengths. A fracture-mechanics model is proposed for predicting the elongation to failure. The model assumes the existence of particle cracks, and criticality is based on the strain required for matrix rupture between cracked particles. The damage criterion of Cockcroft and Latham is utilized, and model predictions are compared to data from different investigations. It is shown that the volume fraction of particles and the work-hardening coefficient of the matrix have a strong influence on the strain to failure. Fracture toughness modeling one again exposes the limitations of existing zero-degree crack-propagation models, such as that of Hahn and Rosenfield, which predict increased toughness with yield strength rather than a decrease, which is observed experimentally. A shear-failure model along a 45-deg direction is proposed for the higher-strength conditions, where concentrated slip bands were observed. The model exhibits the inverse toughness dependence on strength and better correlation to peak-aged (PA) data, but shows poorer agreement with underaged (UA) data. Thus, a transition from zero-degree propagation to 45-deg propagation with increasing strength is suggested. A simplified method for extracting particle stresses is illustrated and is used to estimate a Weibull modulus of 4.9 and a Weibull strength of 2450 MPa for the SiC particles of an average diameter of 10 μm. This article is based on a presentation made in the Symposium “Mechanisms and Mechanics of Composites Fracture” held October 11–15, 1998, at the TMS Fall Meeting in Rosemont, Illinois, under the auspices of the TMS-SMD/ASM-MSCTS Composite Materials Committee.     

Kinetics of strain-induced morphological transformation in cubic alloys with a miscibility gap

Morphological evolutions controlled by a transformation-induced elastic strain during a solid state precipitation are systematically investigated using a prototype binary alloy as a model system. A computer simulation technique based on a microscopic kinetic model including the elastic strain effect is developed. Without any a priori assumptions concerning shapes, concentration profiles and mutual positions of new phase particles, various types of coherent two-phase morphologies such as basket-weave structures, sandwich-like multi-domain structures, precipitate macrolattices and GP zones are predicted. A wide variety of interesting strain-induced kinetic phenomena are observed during development of the above microstructures, including selective and anisotropic growth, reverse coarsening, particle translational motion, particle shape transition and splitting. In spite of all simplifications of the model, most of the simulation results are confirmed by experimental observations in various alloy systems, indicating that this kinetic model can be efficiently used for understanding, interpreting and predicting structural evolutions in real alloys.     

Computational modeling of stationary gastungsten-arc weld pools and comparison to stainless steel 304 experimental results

A systematic study was carried out to verify the predictions of a transient multidimensional computational model by comparing the numerical results with the results of an experimental study. The welding parameters were chosen such that the predictions of the model could be correlated with the results of an earlier experimental investigation of the weld pool surface temperatures during spot gas-tungsten-arc (GTA) welding of Type 304 stainless steel (SS). This study represents the first time that such a comprehensive attempt has been made to experimentally verify the predictions of a numerical study of weld pool fluid flow and heat flow. The computational model considers buoyancy and electromagnetic and surface tension forces in the solution of convective heat transfer in the weld pool. In addition, the model treats the weld pool surface as a truly deformable surface. Theoretical predictions of the weld pool surface temperature distributions, the cross-sectional weld pool size and shape, and the weld pool surface topology were compared with corresponding experimental measurements. Comparison of the theoretically predicted and the experimentally obtained surface temperature profiles indicated agreement within ±8 pct for the best theoretical models. The predicted surface profiles were found to agree within ±20 pct on dome height and ±8 pct on weld pool diameter for the best theoretical models. The predicted weld cross-sectional profiles were overlaid on macrographs of the actual weld cross sections, and they were found to agree very well for the best theoretical models.     

The influence of elastic strain energy on the formation of coherent hexagonal phases

The elastic strain energy function,Y (n), of coherent hexagonal phases has been derived for arbitrary directions,n, in a parent phase with arbitrary crystal structure. These calculations indicate that in all casesY (n) exhibits transverse isotropy about thec axis. As a result,Y (n) has a pronounced effect on the morphology of the precipitating structures. In the case of hexagonal inclusions, three possible optimum shapes in reciprocal space are identified: 1) a rod parallel to thec axis, 2) a plate perpendicular to thec axis, and 3) a hollow conical shape with the axis of revolution parallel toc. The precise precipitate shape can be predicted by identifying the directionn _o which minimizes the strain energy function,Y (n). Evaluation ofY (n) for η and ή MgZn2 precipitates in the ternary Al-Mg-Zn system correctly predicts the orientation and morphology of the particles. This method has also been extended to explore the morphology of the microstructure of hexagonal spinodal alloys. It is shown that the gradient energy term is generally anisotropic, and that together with the strain energy function,Y(n), has a strong influence on composition fluctuations. It is predicted that a one-dimensional periodic compositional variation along the [001] direction should be observed whenY [001] is a global minimum. In all other cases, the microstructure is complex and lacks periodicity.     

Mathematical modeling of sulfide flash smelting process: Part I. Model development and verification with laboratory and pilot plant measurements for chalcopyrite concentrate smelting

A mathematical model has been developed to describe the various processes occurring in a flash furnace shaft. The model incorporates turbulent fluid dynamics, chemical reaction kinetics, and heat and mass transfer. The key features include the use of thek-ε turbulence model, incorporating the effect of particles on the turbulence, and the four-flux model for radiative heat transfer. The model predictions were compared with measurements obtained in a laboratory flash furnace and a pilot plant flash furnace. Good agreement was obtained between the predicted and measured data in terms of the SO₂ and O₂ concentrations, the amount of sulfur remaining in the particles, and the gas temperature. Model predictions show that the reactions of sulfide particles are mostly completed within about 1 m of the burner, and the double-entry burner system with radial feeding of the concentrate particles gives better performance than the singleentry burner system. The model thus verified was used to further predict various aspects of industrial flash furnace operation. The results indicate that from the viewpoint of sulfide oxidation, smelting rate can be substantially increased in most existing industrial flash furnaces. Formerly Graduate Student, Department of Metallurgical Engineering, University of Utah.