Volume Concentration Of Inclusions Research Articles

Development of a mathematical model of a chiral metamaterial based on a cylindrical helical elements accounting for the dispersion and concentration of elements

Background. Interest in the study of microwave metamaterials is associated with the possibility of using them to achieve the required frequency and polarization selective properties of interaction with electromagnetic radiation, which can’t be obtained for structures based on homogeneous media. Aim. The mathematical model of a chiral metamaterial based on a periodic matrix of arbitrarily oriented conducting thin-wire cylindrical helices located in a homogeneous isotropic container creation is considered. Unlike known models, it takes into account the explicit form of the dependence of the effective permittivity and the relative chirality parameter on the helices concentration. Methods. The heterogeneity of a chiral metamaterial based on the Maxwell Garnett formula, which makes it possible to determine the effective dielectric permittivity from the permeabilities of the container and the region occupied by conducting mirror asymmetric inclusions is taken into account when creating a mathematical model. The dispersion of permittivity using the quadratic Lorentz formula and the dispersion of the chirality parameter based on the Condon model are taken into account. Results. Analytical frequency-dependent expressions for the effective permittivity and the chirality parameter taking into account the concentration of helices and their geometric parameters, were obtained in the work. The expression for the relationship between the dimensionless volume concentration of inclusions and the distance between adjacent elements is obtained. The quasi-static approach is used to calculate the resonant frequency of conducting thin-wire cylindrical helices. Conclusion. The proposed method for constructing a mathematical model can be applied to chiral metamaterials based on periodic matrices of conductive elements of an arbitrary mirror asymmetric spatial configuration.

New insight into relationship between casting speed and inclusion removal in the tundish

The tundish is the last refractory metallurgical reactor during the continuous casting process, and the fluid flow in the tundish is one of the key factors to control inclusion collision-coalescence and its removal. Thus, numerical method is applied to obtain the flow field, the RTD curve and the inclusion field. Numerical results showed that the predicted RTD curve and the inclusion number density agree well with the experimental data. In the case of different fluid velocity at tundish inlet, the volume fraction of plug zone, dead zone and well-mixed zone keep unchanged, but the dimensionless tundish inclusion volume concentration, the dimensionless tundish inclusion radius and the inclusion removal ratio have the exponential relation to the fluid velocity at tundish inlet. Further, there is maximum inclusion removal rate for the specific tundish. Another interesting phenomenon is about the inclusion behavior and the characteristic time of the RTD curve. The inclusion removal ratio and the dimensionless tundish inclusion radius have the linear relation to the minimum residence time, the peak concentration time and the average residence time.

Collision-coalescence among inclusions with bubble attachment and transport in molten steel of RH

Inclusions can lead to various defects in the final steel products, and RH vacuum refining is an effective and vital technology to remove inclusions from molten steel. By the computational fluid dynamics (CFD) method, the inclusion mass/population conservation model is developed to have a deep insight into the inclusion transfer behavior in the argon molten steel flow of RH. Particularly, the argon slip velocity is introduced into the inclusion convection velocity in the mathematical model, and the inclusions attached by argon bubbles can return to the molten steel when bubbles escape from the free surface of the vacuum chamber. The numerical results of the fluid flow and the inclusion transfer behavior are validated by the measured data in the actual experiment. The argon bubbles significantly enhance the inclusion transport and collision-coalescence process. Besides, the variations of the inclusion volume concentration and the inclusion number density in the up snorkel are about 2 times those in the down snorkel. The number of inclusions with diameters less than 10 μm decreases with the vacuum treatment time, while the number of inclusions with diameters greater than 10 μm increases first and then decreases gradually. Near the free surface of the ladle, there are the greatest inclusion volume concentration, the greatest inclusion number density and the lowest inclusion characteristic radius.

Two-dimensional Stokes flows in porous medium composed of a large number of circular inclusions

A new approximate approach to solve the two-dimensional boundary-value problems for the Stokes flows in a porous medium composed of a set of circular inclusions is proposed. The idea of the approach, based on the boundary element method (BEM), lies in Fourier series representations of unknown functions on the surfaces of inclusions. It is shown that for small volume concentrations of inclusions, a good accuracy is achieved by taking into account only the first three terms in the Fourier decomposition. Simultaneously, the total number of unknowns, which is equal to the product of the number of inclusions and that of unknowns on every inclusion, is reduced dramatically in comparison with the traditional BEM. The approach is tested by investigating the flow around a circular cylinder in a rectangular periodic cell, the flow in a porous medium with a large number of uniformly or randomly located inclusions. To demonstrate the capacity of the approach, we have calculated the Stokes flows in the porous medium which is modelled by locating 5000 inclusions of two different radii. It is shown that the method can also be used to estimate the properties of the fluid flows in highly porous media formed by non-circular inclusions.

Numerical Simulation of Collision-Coalescence and Removal of Inclusions in a Tundish

Knowledge on the behavior of inclusions in molten steel is crucial to study the fluid flow in the tundish. In this study, an inclusion mass/population conservation model and fluid mass/momentum/energy conservation model were coupled to investigate the spatial distribution of the inclusion volume concentration and inclusion size in the tundish. In this model, collision-coalescence among inclusions occurs due to turbulent collisions and Stokes collisions. The numerical results predict an inclusion size distribution that agrees with experimental data. The inclusion volume concentration decreases while the inclusion characteristic radius increases along the flow path of the molten steel. Furthermore, it is easy for inclusions to collide and aggregate with each other at the guide hole of the baffle, because there is strong turbulent flow there. Also, the inclusion characteristic radius in the distributing chamber is greater than that in the receiving chamber, while the inclusion volume concentration in the distributing chamber is less than that in the receiving chamber.

Study of complex morphology of agglomerations formed by graphite nanoinclusions and its effect on the mechanical properties of the composite materials

The effective mechanical properties of composites are affected by the distribution and formation of agglomerations of graphite nanoparticles in a polyester resin matrix. The samples were made varying the volume concentration of inclusions and their maximum tensile strength and morphology of agglomerations were measured. The morphology of agglomerations was analyzed by measuring their average area, fractal dimension and lacunarity from the micrographs. A predictive model of the effective mechanical properties based on the rule-of-mixtures (ROM) was proposed, the parameters related to the complex microstructure were incorporated into the ROM model. This modified model proved to be a good tool for describing the effect of inclusions on effective mechanical properties by changing their volume concentration. Also, a statistical model based on multiple linear regression of observable variables was proposed, which is better for predicting the effective mechanical property.

Mathematical Model for Collision–Coalescence Among Inclusions in the Bloom Continuous Caster with M-EMS

Mathematical simulation is an effective tool to analyze the fluid flow and the inclusion behavior in the bloom continuous caster with mold electromagnetic stirring (M-EMS). The mathematical model is applied to the modeling of magnetic field, flow field, and inclusion field. Due to the introduction of Archimedes force, the collision mechanism and inclusion’s slipping velocity should be modified in the inclusion mass and population conservation model. Numerically predicted magnetic field, flow field, and the inclusion spatial distribution conform to the experimental results in the existing literature. Lorentz force plays an important role in the fluid flow, and Archimedes force plays an important role in the inclusion distribution in the continuous caster. Due to Brownian collision, Stokes collision, Archimedes collision, and turbulent collision, the coalescence among inclusions occurs in the bloom continuous caster with M-EMS. Among the four types of collisions, turbulent collision occurs most frequently, followed by Archimedes collision and Stokes collision. The frequency of Brownian collision is several orders of magnitudes smaller and is therefore negligible. The inclusion volume concentration, number density, and characteristic radius exhibit a U-shape in the continuous caster without M-EMS. However, with M-EMS, they exhibit an inverted U-shape.

Existence result of an effective stress for an isotropic visco-plastic composite

We define and present some theoretical characterizations of associated constitutive equations for a visco-plastic deformation of a composite. Introducing of an adequate topological framework for some visco-plastic state parameters, when considered a nonlinear stress–strain relations of Prandtl–Reuss type, we obtain the existence and uniqueness results of a steady state for a composite reinforced with other inclusions. The time-dependent plastic strains of these Al based composites reinforced with SiC inclusions are then calculated as a function of inclusion volume concentration, effective stress and effective visco-plastic strain, following an dynamical yield function of power type. Viewed as a Drucker–Prager material, a selection criterion for visco-plastic strain rate, permit us a formulation as an isotropic hardening problem and their reduction to a sweeping process problem for an elastic component of the strain. In this paper a family of models for the composite deformation, defined by Halphen and Nguyen Quoc Son, enlarges the classical class of generalized standard materials, ant they are characterized by monotone-gradient flow function. These models improve the tensile process of Al based composite, in order to allow new practical solutions in civil engineering.

Resonance dispersion of the longitudinal waves in disperse composites

Vibrations and longitudinal waves in a composite formed by a viscoelastic matrix and solid-state inclusions are considered. Vibration-wave processes in the long-wave approximation are described on the basis of the complex dynamic density taking into account inertial-elastic-viscous interaction of the matrix and inclusion-oscillators upon their forward vibrations. Resonance dependences for the dynamic density and translational viscosity of the composite at the moderate volume concentration of inclusions are presented. The formulas obtained are specified for composites with spherical and cylindrical inclusions and are compared with the known experimental results.

Acoustical and optical branches of wave propagation in random particulate composites

The work is dedicated to the analysis of acoustical and optical branches of longitudinal elastic wave propagation in the medium with a random set of spherical inclusions. The effective field method and quasicrystalline approximation are used for the construction of the dispersion equation for the wave number of the mean (coherent) wave field propagating in the composite. This dispersion equation serves for all frequencies of the incident field, properties and volume concentrations of inclusions. Different branches of the solutions of this equation are obtained and analyzed. Each of these branches may be interpreted as a specific mode of wave propagation, and its input in the mean (coherent) wave field is essential only in a certain frequency region. The predictions of the method are compared with some experimental data existing in the literature.

Propagation of longitudinal elastic waves in composites with a random set of spherical inclusions (effective field approach)

The work is devoted to the problem of plane monochromatic longitudinal wave propagation through a homogeneous elastic medium with a random set of spherical inclusions. The effective field method and quasicrystalline approximation are used for the calculation of the phase velocity and attenuation factor of the mean (coherent) wave field in the composite. The hypotheses of the method reduce the diffraction problem for many inclusions to a diffraction problem for one inclusion and, finally, allow for the derivation of the dispersion equation for the wave vector of the mean wave field in the composite. This dispersion equation serves for all frequencies of the incident field, properties and volume concentrations of inclusions. The long and short wave asymptotics of the solution of the dispersion equation are found in closed analytical forms. Numerical solutions of this equation are constructed in a wide region of frequencies of the incident field that covers long, middle, and short wave regions of propagating waves. The phase velocities and attenuation factors of the mean wave field are calculated for various elastic properties, density, and volume concentrations of the inclusions. Comparisons of the predictions of the method with some experimental data are presented; possible errors of the method are indicated and discussed.

Random polycrystals of grains containing cracks: Model of quasistatic elastic behavior for fractured systems

A model study on fractured systems was performed using a concept that treats isotropic cracked systems as ensembles of cracked grains by analogy to isotropic polycrystalline elastic media. The approach has two advantages: (a) Averaging performed is ensemble averaging, thus avoiding the criticism legitimately leveled at most effective medium theories of quasistatic elastic behavior for cracked media based on volume concentrations of inclusions. Since crack effects are largely independent of the volume they occupy in the composite, such a non-volume-based method offers an appealingly simple modeling alternative. (b) The second advantage is that both polycrystals and fractured media are stiffer than might otherwise be expected, due to natural bridging effects of the strong components. These same effects have also often been interpreted as crack-crack screening in high-crack-density fractured media, but there is no inherent conflict between these two interpretations of this phenomenon. Results of the study are somewhat mixed. The spread in elastic constants observed in a set of numerical experiments is found to be very comparable to the spread in values contained between the Reuss and Voigt bounds for the polycrystal model. Unfortunately, computed Hashin-Shtrikman bounds are much too tight to be in agreement with the numerical data, showing that polycrystals of cracked grains tend to violate some implicit assumptions of the Hashin-Shtrikman bounding approach. However, the self-consistent estimates obtained for the random polycrystal model are nevertheless very good estimators of the observed average behavior.

Seismic attenuation due to wave-induced fluid flow in a porous rock with spherical heterogeneities

SUMMARY Most natural porous rocks have heterogeneities at nearly all scales. Heterogeneities of mesoscopic scale—that is, much larger than the pore size but much smaller than wavelength—can cause significant attenuation and dispersion of elastic waves due to wave-induced flow between more compliant and less compliant areas. Analysis of this phenomenon for a saturated porous medium with a small volume concentration of randomly distributed spherical inclusions is performed using Waterman-Truell multiple scattering theorem, which relates attenuation and dispersion to the amplitude of the wavefield scattered by a single inclusion. This scattering amplitude is computed using recently published asymptotic analytical expressions and numerical results for elastic wave scattering by a single mesoscopic poroelastic sphere in a porous medium. This analysis reveals that attenuation and dispersion exhibit a typical relaxation-type behaviour with the maximum attenuation and dispersion corresponding to a frequency where fluid diffusion length (or Biot’s slow wavelength) is of the order of the inclusion diameter. In the limit of low volume concentration of inclusions the effective velocity is asymptotically consistent with the Gassmann theory in the low-frequency limit, and with the solution for an elastic medium with equivalent elastic inclusions (no-flow solution) in the low-frequency limit. Attenuation (expressed through inverse quality factor 1/Q) scales with frequency ω in the lowfrequency limit and with ω −1/2 in the high-frequency limit. These asymptotes are consistent with recent results on attenuation in a medium with a periodic distribution of poroelastic inclusions, and in continuous random porous media.

Role of interfacial charge in the piezoelectric properties of ferroelectric 0-3 composites

We investigated the effects of compensating charges (at the inclusion-matrix interface) on the piezoelectric properties of ferroelectric 0-3 composites. Our previously developed model [C. K. Wong, Y. M. Poon, and F. G. Shin, J. Appl. Phys. 90, 4690 (2001)] has been extended to include the additional contribution from the deformation of the inclusion particles due to the applied stress in the piezoelectric measurement. The relative significance of this contribution is mainly determined by the amount of compensating interfacial charge, which is significantly governed by the degrees of poling of the constituent materials in the composite sample. This model provides an explanation to an anomaly in the piezoelectric coefficients of 0-3 composite samples with the matrix and inclusion phases polarized in opposite directions. Explicit expressions in closed form have been derived for the effective d33, d31, and dh coefficients. After taking into consideration the degree of poling of the constituents and the effects of the compensating interfacial charges, theoretical predictions show good agreement with published experimental data. Goodness of fit is not limited to low volume concentration of inclusions.

The acoustic signature of fluid flow in complex porous media

Effective medium approximations for the frequency-dependent and complex-valued effective stiffness tensors of cracked/porous rocks with multiple solid constituents are developed on the basis of the T-matrix approach (based on integral equation methods for quasi-static composites), the elastic–viscoelastic correspondence principle, and a unified treatment of the local and global flow mechanisms, which is consistent with the principle of fluid mass conservation. The main advantage of using the T-matrix approach, rather than the first-order approach of Eshelby or the second-order approach of Hudson, is that it produces physically plausible results even when the volume concentrations of inclusions or cavities are no longer small. The new formulae, which operates with an arbitrary homogeneous (anisotropic) reference medium and contains terms of all order in the volume concentrations of solid particles and communicating cavities, take explicitly account of inclusion shape and spatial distribution independently. We show analytically that an expansion of the T-matrix formulae to first order in the volume concentration of cavities (in agreement with the dilute estimate of Eshelby) has the correct dependence on the properties of the saturating fluid, in the sense that it is consistent with the Brown–Korringa relation, when the frequency is sufficiently low. We present numerical results for the (anisotropic) effective viscoelastic properties of a cracked permeable medium with finite storage porosity, indicating that the complete T-matrix formulae (including the higher-order terms) are generally consistent with the Brown–Korringa relation, at least if we assume the spatial distribution of cavities to be the same for all cavity pairs. We have found an efficient way to treat statistical correlations in the shapes and orientations of the communicating cavities, and also obtained a reasonable match between theoretical predictions (based on a dual porosity model for quartz–clay mixtures, involving relatively flat clay-related pores and more rounded quartz-related pores) and laboratory results for the ultrasonic velocity and attenuation spectra of a suite of typical reservoir rocks.

Dielectric properties of matrix composite materials with high volume concentrations of inclusions (effective field approach)

The effective field method is applied to the calculation of the overall dielectric permittivities of composite materials consisting of a homogeneous matrix and a set of spherical inclusions (two-phase media). The main hypothesis of the classical version of the method is the so-called quasicrystalline approximation. This hypothesis is the assumption that the field acting on every inclusion in the composite is constant and the same for all the inclusions. The predictions of this version of the method are in agreement with experimental data and numerical solutions for various effective properties of composites if the volume concentration of inclusions does not exceed 0.3. For higher volume concentrations of the inclusions, the method needs correction. In this work the version of the effective field method is developed for the improvement of the predictions of the dielectric permittivities of the composites in the region of high volume concentrations of inclusions. This version was outlined in previous works of the author, and is based on the detailed description of pair interactions between inclusions in the composite. In order to describe such interactions the hypothesis of the quasicrystalline approximation has to be changed for a more complex one. It gives an integral equation for the field that acts on a pair of inclusions in the composite medium. The kernel of the integral operator in this equation depends on a specific three-point correlation function of the random field of inclusions. The influence of this function on the overall dielectric permittivities of the composites is analyzed in 3D and 2D-cases. The predictions of the method are compared with numerical calculations of the effective dielectric permittivities of the two-phase composites in 2D-case.

Flow patterns around rigid inclusions in a multiple inclusion system undergoing bulk simple shear deformation

During deformation of an inclusion-matrix system, the velocity fields around individual inclusions mutually interfere with one another. Such interacting inclusions rotate at slower rates than non-interacting, single inclusions. This paper presents a theoretical model that describes the flow pattern of matrix (viscous) material around interacting rigid inclusions of spherical shape in bulk simple shear deformation. Numerical simulations based on the velocity functions reveal that the volume concentration of inclusions is a crucial parameter controlling the flow pattern around rotating inclusions under interacting conditions. At low volume concentrations ( ρ v<0.01, where ρ v=( a/ b) 3, 2 a: inclusion diameter and 2 b: mean inter-inclusion distance) the flow is characterized by an eye-shaped separatrix, which, with increase in volume concentration ( ρ v>0.1), transforms into a pattern with a bow-tie shaped separatrix. At a large volume concentration ( ρ v=0.4) the separatrix assumes the geometry of a super-ellipse. We also present numerical models that illustrate the influence of volume concentration on the (1) nature of strain distribution, (2) distortion patterns of passive foliations ,and (3) mantle structures around inclusions in an interacting state. Based on this theory, it is shown that the rotational retardation of the inclusions slightly enhances the bulk viscosity of the inclusion-matrix system.

Self-consistent schemes in the problem of electromagnetic wave propagation through dielectric media with isolated inhomogeneities

Two main self-consistent schemes (the effective field and effective medium methods) are applied for the calculation of phase velocities and attenuation factors of plane monochromatic electromagnetic waves in dielectric media with random sets of spherical inclusions. The results of the calculations obtained in the frameworks of the methods are investigated and compared in a wide region of volume concentrations of inclusions and frequencies of exciting fields. This analysis together with experimental data given in literature allows to evaluate the region of application of the methods for the solution of the considered problem.

A homogenization theory for the overall creep of isotropic viscoplastic composites

Summary. A field-fluctuation method is introduced into the secant-viscosity framework to evaluate the homogenized effective stress of the heterogeneously deformed elastic-viscoptastic matrix in an isotropic composite. Two microgeometries are considered here: one is reinforced with spherical particles and the other with randomly oriented thin discs. The time-dependent creep strains of the elastic-viscoplastic composites are then calculated as a function of inclusion volume concentration. As these two microgeometries are known to provide the lower and upper bounds of the effective moduli in elasticity, the creep strains associated with these two inclusion shapes are believed to set the upper and lower ranges of the overall creep strain for all inclusion shapes. Detailed comparison with a previously developed direct work-rate method is also made. While the effective stress of the ductile matrix - and therefore the overall creep strain of the composite - are higher by the direct work-rate method, the difference between the two is found to be small. However, when the Laplace inversion of the effective secant viscosities of the viscoelastic comparison composite can be cast in an explicit form, the field-fluctuation method will provide a complete evaluation of the effective stress, and is also substantially simpler than the direct work-rate method.

Effective Field Method in the Problem of Electromagnetic Wave Propagation Through a Medium With Isolated Inclusions

Effective field method is applied for the calculation of the phase velocities and attenuation factors of plane monochromatic electromagnetic waves in a dielectric medium with an array of spherical inclusions. The results of calculation are investigated in a wide region of the volume concentrations of inclusions and frequencies of the exciting field. The influence of the space distribution of the inclusions on the effective velocities and attenuation factors is studied. The obtained results together with experimental data given in literature allow to evaluate the region of the application of the effective field method for the solution of the considered problem.