Statistics of inhomogeneous media formed by nucleation and growth

Important statistical properties of inhomogeneous microstructures formed by nucleation and growth are established using line transects. A fundamental time-dependent equation has been derived for the probability of sampling only matrix phase by a random line transect in a system containing growing Poisson-distributed spherical nuclei.It is established that the probability of sampling matrix phase only measures simultaneously the product of two fundamental characteristics of inhomogeneity: the volume fraction of the matrix phase and the probability of existence in the matrix of a free path of specified minimum length. Increasing the number density of the nuclei and decreasing the size of the mean projection of the nuclei on a plane perpendicular to the line transect by the same factor does not change the mean and the variance of the free paths.It is also demonstrated that the distribution of the intercepts from the weaker phase is an important indicator regarding the risk of poor properties. Accordingly, a new approach is suggested for setting MFFOP reliability requirements which minimise the risk from premature fracture.     

Chord intercepts in a two-dimensional cell growth model

Nuclei are Poisson-distributed within a plane. Out of the nuclei, grains begin to grow instantaneously, circularly and at a constant rate. The forming microstructure at each fraction transformed, consisting of grains and untransformed regions, is characterized in this work by use of a straight line (Rosiwal's line), which passes arbitrarily through the plane. Along this line we obtain chord intercepts of grains and of untransformed regions of different lengths, independent of the position of the straight line. The distributions of these lengths are essentially determined by the distribution of the nuclei and from the conditions of growth. From these assumptions we derive, by use of probability theory, the distributions of the chord intercepts. In Part 1, only the distribution density of the chord intercept of the untransformed regions is studied. From these results several other statistical quantities of the microstructure are also deduced.     

On the determination of grain-size distributions from intercept distributions

A matrix formulation for determining the spatial grain-size distribution of tetrakaidecahedral grains from linear-intercept data is developed. The truncation effect (which stems from the fact that the intersection of single-size grains by a test line gives rise to intercepts of different length) and the sampling effect (which results as a consequence of bigger grains being intersected more frequently than smaller ones) are separately taken into account. The derivation procedure of this formulation is applicable to any other convex shape, provided the linear intercept distribution for single-size grains of the corresponding shape is known. The percentage spatial grain-size distributions obtained by the formulation derived here are similar to those estimated by the Spektor's chord method for spherical grains.     

Comparison of pure and “Latinized” centroidal Voronoi tessellation against various other statistical sampling methods

A recently developed centroidal Voronoi tessellation (CVT) sampling method is investigated here to assess its suitability for use in statistical sampling applications. CVT efficiently generates a highly uniform distribution of sample points over arbitrarily shaped M-dimensional parameter spaces. On several 2-D test problems CVT has recently been found to provide exceedingly effective and efficient point distributions for response surface generation. Additionally, for statistical function integration and estimation of response statistics associated with uniformly distributed random-variable inputs (uncorrelated), CVT has been found in initial investigations to provide superior points sets when compared against latin-hypercube and simple-random Monte Carlo methods and Halton and Hammersley quasi-random sequence methods. In this paper, the performance of all these sampling methods and a new variant (“Latinized” CVT) are further compared for non-uniform input distributions. Specifically, given uncorrelated normal inputs in a 2-D test problem, statistical sampling efficiencies are compared for resolving various statistics of response: mean, variance, and exceedence probabilities.     

Energetic–statistical size effect simulated by SFEM with stratified sampling and crack band model

The paper presents a model that extends the stochastic finite element method to the modelling of transitional energetic–statistical size effect in unnotched quasibrittle structures of positive geometry (i.e. failing at the start of macro‐crack growth), and to the low probability tail of structural strength distribution, important for safe design. For small structures, the model captures the energetic (deterministic) part of size effect and, for large structures, it converges to Weibull statistical size effect required by the weakest‐link model of extreme value statistics. Prediction of the tail of extremely low probability such as one in a million, which needs to be known for safe design, is made feasible by the fact that the form of the cumulative distribution function (cdf) of a quasibrittle structure of any size has been established analytically in previous work. Thus, it is not necessary to turn to sophisticated methods such as importance sampling and it suffices to calibrate only the mean and variance of this cdf. Two kinds of stratified sampling of strength in a finite element code are studied. One is the Latin hypercube sampling of the strength of each element considered as an independent random variable, and the other is the Latin square design in which the strength of each element is sampled from one overall cdf of random material strength. The former is found to give a closer estimate of variance, while the latter gives a cdf with smaller scatter and a better mean for the same number of simulations. For large structures, the number of simulations required to obtain the mean size effect is greatly reduced by adopting the previously proposed method of random property blocks. Each block is assumed to have a homogeneous random material strength, the mean and variance of which are scaled down according to the block size using the weakest‐link model for a finite number of links. To check whether the theoretical cdf is followed at least up to tail beginning at the failure probability of about 0.01, a hybrid of stratified sampling and Monte Carlo simulations in the lowest probability stratum is used. With the present method, the probability distribution of strength of quasibrittle structures of positive geometry can be easily estimated for any structure size. Copyright © 2007 John Wiley & Sons, Ltd.     

Synthesis and microstructural characterization of Al–Mg alloy–SiC particle composite

Al–Mg-alloy was reinforced with 10 vol.% SiC particles size of 3 μm diameter by vortex technique. Stiffener distribution, particle interaction with metal-matrix and mechanical properties in as-cast condition was studied. The resulting as-cast composite structures were analyzed using X-ray diffraction (XRD) and scanning electron microscopy (SEM). The AlMg–SiCp composite microstructure showed excellent SiCp distribution into AlMg matrix. In addition, no evidence of secondary chemical reactions has been observed. Hence, mechanical properties are highly sensitive to the microstructure and these are indirectly related to solidification parameters and processing conditions. Al–Mg alloy possess lightweight and excellent properties as structural materials which can be optimized with SiCp addition and a good fabrication technique.     

Scale‐related topology optimization of cellular materials and structures

The integrated optimization of lightweight cellular materials and structures are discussed in this paper. By analysing the basic features of such a two‐scale problem, it is shown that the optimal solution strongly depends upon the scale effect modelling of the periodic microstructure of material unit cell (MUC), i.e. the so‐called representative volume element (RVE). However, with the asymptotic homogenization method used widely in actual topology optimization procedure, effective material properties predicted can give rise to limit values depending upon only volume fractions of solid phases, properties and spatial distribution of constituents in the microstructure regardless of scale effect. From this consideration, we propose the design element (DE) concept being able to deal with conventional designs of materials and structures in a unified way. By changing the scale and aspect ratio of the DE, scale‐related effects of materials and structures are well revealed and distinguished in the final results of optimal design patterns. To illustrate the proposed approach, numerical design problems of 2D layered structures with cellular core are investigated. Copyright © 2006 John Wiley & Sons, Ltd.     

Combustion Synthesis and Thermal Stress Analysis of TiC–Ni Functionally Graded Materials

Simultaneous combustion synthesis reaction and compaction of Ti, C, and Ni powders under a hydrostatic pressure was carried out to fabricate dense TiC–Ni functionally graded materials (FGMs) in a single processing operation. Scanning-electron microscope (SEM) and microprobe analysis (EPMA) was employed to investigate the microstructure and composition distribution. Experimental results demonstrate that Ni and Ti composition varies continuously and gradually along the thickness of the reacted sample, remarkably different from stepwise type prior to combustion synthesis. The constituents are continuous in microstructure everywhere and no distinct interaction occurs in TiC–Ni FGM. Moreover, the thermal physical and mechanical properties were measured as a function of composition. It was found that the properties of the FGMs were dependent on Ni content. The residual thermal stress of TiC–Ni FGM and dual-laminate non-FGM cooled to room temperature after combustion synthesis has been analyzed by finite element method. TiC–Ni FGM shows distortion and thermal stress relaxation, which is in striking contrast to the layered TiC–Ni non-FGM.     

Simulating crack growth and failure in a composite: A vector model for a layered composite material

Summary A model has been proposed for a layered composite having an inhomogeneous microstructure. It is represented as the union of vector spaces for the mechanical characteristics of the material and the volume contents of the layers. The elastic parameters for each layer are considered as random functions of an arbitrarily defined volume. For a microscopically inhomogeneous material, one can consider a base volume such that the elastic properties of the material are retained and for which one can apply effective elastic moduli.The state of stress and strain in such a material is described by the components of the macrostress and macrostrain tensors with respect to a certain (structural) volume, which exceeds the base volume. The justification for that approach is demonstrated for a microscopically inhomogeneous orthotropic composite.Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 26, No. 1, pp. 22–26, January–February, 1990.     

Quantitative evaluation of fiber distributions in a continuously reinforced aluminium alloy using automatic image analysis

An automatic image analysis system has been used to quantitatively evaluate fiber distributions in a cast aluminium alloy reinforced with 3-μm diameter aligned continuous fibers of Al₂O₃. Two techniques have been employed to evaluate fiber distributions, namely, an area fraction variance analysis and a matrix intercept length analysis. In addition, fiber distributions on a plane transverse to the axis of the fibers have been compared with a computer-generated random distribution of circular disks. The variance analysis has shown that, when compared with a random disk distribution of equal area (volume) fraction, the fiber distributions are clearly nonrandom. The deviation from randomness appears to indicate fiber clustering as opposed to ordering. Using the matrix intercept length analysis, it has also been found that when fiber and random distributions of equal area fraction have been compared, they are distinctly different. Once again there is evidence for fiber clustering in that the number of matrix intercept lengths in the 0–1 μm range is significantly larger for fibers.     

Microstructurally inhomogeneous composites: Is a homogeneous reinforcement distribution optimal?

Since the 1960s, it has been a common practice worldwide to pursue a homogeneous distribution of reinforcements within a matrix material, discontinuous metal matrix composites (DMMCs) in particular. Taking an overview of the worldwide activities in DMMC research, despite many favourable attributes such as improved specific strength, stiffness and superior wear resistance, DMMCs with a homogeneous microstructure tend to exhibit a very low room temperature damage tolerance even with a highly ductile matrix material such as aluminium. In this review, a range of uniquely multi-scale hierarchical structures have been successfully designed and fabricated by tailoring reinforcement distribution for DMMCs in order to obtain superior performance. A variety of specific microstructures that were developed in Al, Mg, Cu, Fe, Co and TiAl matrices indicate that there must be adequate plastic regions among the reinforcements to blunt or deflect cracks if one wants to toughen DMMCs. Following this path, aided by theoretical analyses, the most recent success is the design and fabrication of a network distribution of in situ reinforcing TiB whiskers (TiBw) in titanium matrix composites (TMCs), where a tailored three-dimensional (3D) quasi-continuous network microstructure displays significant improvements in mechanical properties. This resolves the brittleness surrounding TMCs fabricated by powder metallurgy. It is the large reinforcement-lean regions that remarkably improve the composite’s ductility by bearing strain, blunting the crack and decreasing the crack-propagation rate. The fracture, strengthening and toughening mechanisms are comprehensively elucidated in order to further understand the advantages of such an inhomogeneous microstructure, and to justify the development of novel techniques to produce such inhomogeneous microstructures. This approach opens up a new horizon of research and applications of DMMCs and can be easily extended to general multi-phase composites with enhanced physical and mechanical properties.     

Statistical dispersion of the field emission parameters of multipoint cathodes based on a polymer-carbon nanotube composite

Statistical properties of the characteristics of a field-electron emitter based on a polystyrene-carbon nanotube composite have been studied. Experimental data have been used to plot the slope versus intercept of I-U curve in the Fowler-Nordheim coordinates and a histogram of the effective heights of emission centers. Numerical estimations show that this statistics obeys the lognormal distribution law. A series of experiments have been performed for determining the influence of the level of emission current on the character of distribution, and a model describing this dependence is proposed.     

Microstructure variability and macroscopic composite properties of high performance fiber reinforced cementitious composites

High performance fiber reinforced cementitious composites have made major advances in recent years, to the point where they are being adopted in building and bridge constructions. The most significant advantage of HPFRCC over conventional concrete is their high tensile ductility. However, the tensile strain capacity has been observed to vary, most likely as a result of the variability of the microstructure derived from the processing of these materials.This paper describes the composite property variability, as well as the variability of the material microstructure. Scale linkage is discussed. In particular, the tensile stress–strain curves, and the crack pattern on uniaxially loaded specimens are presented. The treatment of random fibers in micromechanical models, and tailoring of matrix flaw size distribution for saturated multiple cracking are examined. It is suggested that robust composite properties can be achieved by deliberate control of microstructure variability. Some open issues concerning the randomness of microstructures and possibly related macroscopic behavior are also identified. Further gains in composite property control may be expected from improvements in characterization and modeling of the microstructure randomness.     

Numerical and experimental investigations in electromagnetic riveting with different rivet dies

In this work, the numerical simulations and electromagnetic riveting (EMR) experiments were conducted to investigate microstructure evolution and the forming mechanism of adiabatic shear bands (ASBs). And the effects of rivet dies on microstructure distributions in formed heads and mechanical properties of riveted structures were systematically explored. The impact velocity and deformation distribution results demonstrated that the proposed numerical method was accurate and reliable. The simulation results showed the slope angle of rivet dies notably affected the plastic flow of materials, and then determined the microstructure distribution in formed heads. The combined effects of inhomogeneous plastic flow and thermal softening were accounted for the forming of ASBs. The formed heads had two obvious ASBs (upper and lower ASB) for the 40° rivet die and flat rivet die. The formed heads only had the lower ASB and no clear upper for the 60° rivet die and 80° rivet die. The pull-out test results showed that the specific rivet die could improve the mechanical properties of the EMR joints, which contribute to the engineering applications of EMR riveted structures.     

Microstructure design of a two phase composite using two-point correlation functions

Two-point distribution functions are used here as to introduce “Microstructure Sensitive Design” in two-phase composites. Statistical distribution functions are commonly used for the representation of microstructures and also for homogenization of materials properties. The use of two-point statistics allows the composite designer to include the morphology and distribution in addition to the properties of the individual phases and components. Statistical continuum mechanics is used to make a direct link between the microstructure and properties (elastic and plastic) in terms of these two-point statistical functions. An empirical form of the two-point statistical function is used which allows the construction of a composite hull. Two different composites (isotropic and anisotropic) are considered and the effect of anisotropy for the prediction of the elastic properties is discussed     

Relationship between micro- and macrostresses in metals

The author describes mathematically the stress distribution in the microstructure of a polycrystal under any type of stress state in the macrovolume. Under the conditions of the triaxial macrostress state, the tensor of the fourth rank of moments of microstresses has an orthotropic symmetry and under a uniaxial stress state a hexagonal symmetry and is characterized correspondingly by 9 or 5 nonzero components. The determined analytical relationships between the statistics of the microstresses and macrostresses make it possible to obtain, for all solved problems of the classic theory of elasticity, additional information on the distribution of stresses in the microstructure (from the know values of the main macro-stresses and the elasticity properties of the poly-and single crystal).Translated from Problemy Prochnosti, No. 4, pp. 75–83, April, 1994.     

Observation of sub-kilohertz resonance in Rf-optical double resonance experiment in rare earth ions in solids

Abstract

We have observed kilohertz and sub-kilohertz resonance structures in RF-optical double resonance experiments of rare-earth-doped solids, when the frequency of the RF field is scanned across the hyperfine transitions while monitoring the resonant optical absorption of a CW laser. The effect is observed only when the laser spectral width is broad compared to the hyperfine structure. The observed line widths are apparently free of the inhomogeneous widths of hyperfine levels and the line shape has peculiar double peak structure. The effect is modelled with a resonance involving three atomic levels interacting with three electromagnetic fields, two optical and one RF, in a triangular or “delta’ configuration. While the ordinary optical-RF two-field resonance is limited by spin inhomogeneous width, the simultaneous excitation of three coupled transitions leads to narrow and highly nonlinear resonance structures that are not averaged by the inhomogeneous distribution of hyperfine transition.     

Anisotropic corrosion resistance of TiC reinforced Ni-based composites fabricated by selective laser melting

Electrochemical measurements on three planes of TiC/Inconel 718 composites fabricated by selective laser melting (SLM) were performed to study the corrosion property. The results showed that the YZ-plane with dense and fine columnar structures possessed high microhardness and superior corrosion resistance in 3.5 wt% NaCl solution. For the XZ-plane, a decreased anti-corrosion property was observed owing to its inhomogeneous microstructures. While the XY-plane with large irregular pores and clustered ring-like structures was more susceptible to corrosion compared with the other two planes. Comparative analysis suggested that the anisotropic corrosion behaviors were significantly dependent on the surface defects, microstructure characteristics and added reinforcements.     

The relationships between tensile properties and hole expansion property of C-Mn steels

In order to investigate the effects of the microstructure and chemical compositions on the hole expansion property of C-Mn steels, four C-Mn steels were used and heat treated into different structures. The influences of the tensile properties on the hole expansion property were also investigated. It has been found in this paper that C-Mn steels with a high ratio of yield strength to ultimate tensile strength usually have a good hole expansion property. A high silicon content in solid solution can improve the hole expansion property. Carbon has a significant detrimental effect on the hole expansion property.