Spherical Inhomogeneity Research Articles

Electromagnetic Scattering by Radially Inhomogeneous Dielectric Spheres

A method to determine the electric field scattered by a dielectric sphere with a permittivity and conductivity varying only in the radial direction is presented. In order to take advantage of the spherically symmetrical geometry, the interior electric field is expanded in terms of vector spherical harmonics which are orthogonal over a spherical surface. Using the orthogonality of those functions with the vector wave functions in the expanded form of the free space dyadic Green's function, one can reduce the three-dimensional (3-D) electric field integral equation into a system of one-dimensional (1-D) integral equations. The scalar coefficients of the series expansion for the interior electric field are then obtained by solving this system of equations. For the medium outside the sphere, the scattered field is evaluated through the same procedure. Comparisons with alternative methods for specific cases demonstrate that the method is reliable in determining the interior field and the scattered field, for spherical objects with layered or continuously varying radial profile.

Simulations of two sedimenting-interacting spheres with different sizes and initial configurations using immersed boundary method

Numerical investigations are carried out for the drafting, kissing and tumbling (DKT) phenomenon of two freely falling spheres within a long container by using an immersed-boundary method. The method is first validated with flows induced by a sphere settling under gravity in a small container for which experimental data are available. The hydrodynamic interactions of two spheres are then studied with different sizes and initial configurations. When a regular sphere is placed below the larger one, the duration of kissing decreases in pace with the increase in diameter ratio. On the other hand, the time duration of the kissing stage increases in tandem with the increase in diameter ratio as the large sphere is placed below the regular one, and there is no DKT interactions beyond threshold diameter ratio. Also, the gap between homogeneous spheres remains constant at the terminal velocity, whereas the gaps between the inhomogeneous spheres increase due to the differential terminal velocity.

A statistical descriptor based volume-integral micromechanics model of heterogeneous material with arbitrary inclusion shape

A continuing challenge in computational materials design is developing a model to link the microstructure of a material to its material properties in both an accurate and computationally efficient manner. In this paper, such a model is developed which uses image-based data from characterization studies combined with a newly developed self-consistent volume-integral micromechanics model (SVIM) for linear elastic material. It is observed that SVIM is able to capture the effective stress/strain distribution inside the inclusion, as well as effects of volume fraction and nearest inclusion distance on the effective properties of heterogeneous material. More importantly, SVIM can be applied to inclusions with arbitrary shape through discretizing the inclusion domain. For both 2-dimensional and 3-dimensional problems with circular and spherical inhomogeneities, SVIM's capability of predicting effective elastic properties is validated against experiments and direct numerical simulations using the finite element method. Finally, the effect of inclusion shape is predicted by SVIM.

New developments on the geometric nonholonomic integrator

In this paper, we will discuss new developments regarding the geometric nonholonomic integrator (GNI) (Ferraro et al 2008 Nonlinearity 21 1911–28; Ferraro et al 2009 Discrete Contin. Dyn. Syst. (Suppl.) 220–9). GNI is a discretization scheme adapted to nonholonomic mechanical systems through a discrete geometric approach. This method was designed to account for some of the special geometric structures associated to a nonholonomic motion, like preservation of energy, preservation of constraints or the nonholonomic momentum equation. First, we study the GNI versions of the symplectic-Euler methods, paying special attention to their convergence behaviour. Then, we construct an extension of the GNI in the case of affine constraints. Finally, we generalize the proposed method to nonholonomic reduced systems, an important subclass of examples in nonholonomic dynamics. We illustrate the behaviour of the proposed method with the example of the inhomogeneous sphere rolling without slipping on a table.

Third-order effect in magnetic small-angle neutron scattering by a spatially inhomogeneous medium

Magnetic small-angle neutron scattering (SANS) is a powerful tool for investigating nonuniform magnetization structures inside magnetic materials. Here, we consider a ferromagnetic medium with weakly inhomogeneous uniaxial magnetic anisotropy, saturation magnetization, and exchange stiffness, and derive, to second order in the amplitudes of the inhomogeneities, the micromagnetic solutions for the equilibrium magnetization textures. Further, we compute the corresponding magnetic SANS cross section up to the third order. For the special case of scattering geometry where the incident neutron beam is perpendicular to the applied magnetic field, twice the cross section along the direction orthogonal to both the field and the neutron beam cancels the cross section along the field direction in the second order. This cancellation does not depend on the defect shape and amplitudes of the exchange inhomogeneities. Hence, such a cross-section difference has only a third-order contribution in the amplitudes of the inhomogeneities. It provides a separate gateway for a deeper analysis of the sample's magnetic structure. We derive and analyze analytical expressions for the dependence of this difference on the scattering-vector magnitude for the case of spherical Gaussian inhomogeneities.

Energy-equivalent inhomogeneity approach to analysis of effective properties of nanomaterials with stochastic structure

A mathematical model based on the method of conditional moments combined with a new notion of the energy-equivalent inhomogeneity is presented and applied in the investigation of the effective properties of a material with randomly distributed nanoparticles. The surface effect is introduced via Gurtin–Murdoch equations describing the properties of the matrix/nanoparticle interface. The real system, consisting of the inhomogeneities and their surfaces possessing different properties and, possibly, residual stresses, is replaced by energy-equivalent inhomogeneities with modified bulk properties which incorporate the surface effects. The effective stiffness tensor of the material with so defined equivalent inhomogeneities is determined by the method of conditional moments. Closed-form expressions for the effective moduli of a composite consisting of a matrix and randomly distributed spherical inhomogeneities are derived for both the bulk and the shear moduli. Dependence of those moduli on the radius of nanoparticles is included in these expressions exhibiting analytically the nature of the size-dependence in nanomaterials. As numerical examples, nanoporous aluminum and nanoporous gold are investigated. The dependence of the normalized bulk and shear moduli of nanoporous aluminum (for two sets of surface properties) on the pore volume fraction (for different radii of nanopores) and on the radius of nanopores (for fixed volume fraction of nanopores) are compared to and discussed in the context of other theoretical predictions. Further, the normalized effective Young’s modulus of nanoporous gold as a function of void volume fraction for various ligament radii is analyzed.

Influence of Inhomogeneity on the Stress State of the Hemisphere Under the Locally Distributed Vertical Load

There is a solution of the problem of stress-strain state determining in the radially inhomogeneous hollow concrete sphere. The inhomogeneity in this axisymmetric problem is induced by the temperature field. The problem reduces to a differential equation system with variable coefficients. The allowance for the variable Young's modulus lets to arrive to a more accurate solution.

Effective bulk moduli of materials containing stochastically distributed nano‐inhomogeneities with surface stresses

AbstractIn the present contribution, a mathematical model for the investigation of the effective properties of a material with randomly distributed nano‐particles is proposed. The surface effect is introduced via Gurtin‐Murdoch equations describing properties of the matrix/nano‐particle interface. They are added to the system of stochastic differential equations formulated within the framework of linear elasticity. The homogenization problem is reduced to finding a statistically averaged solution of the system of stochastic differential equations. These equations are based on the fundamental equations of linear elasticity, which are coupled with surface/interface elasticity accounting for the presence of surface tension. Using Green's function this system is transformed to a system of statistically non‐linear integral equations. It is solved by the method of conditional moments. Closed‐form expressions are derived for the effective moduli of a composite consisting of a matrix with randomly distributed spherical inhomogeneities. The radius of the nano‐particles is included in the expression for the bulk moduli. As numerical examples, nano‐porous aluminum and nano‐porous gold are investigated assuming that only the influence of the interface effects on the effective bulk modulus is of interest. The dependence of the normalized bulk moduli of nano‐porous aluminum on the pore volume fraction (for certain radii of nano‐pores) are compared to and discussed in the context of other theoretical predictions. The effective Young's modulus of nano‐porous gold as a function of pore radius (for fixed void volume fraction) and the normalized Young's modulus vs. the pore volume fraction for different pore radii are analyzed. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Approach to approximating the pair distribution function of inhomogeneous hard-sphere fluids.

We introduce an approximation for the pair distribution function of the inhomogeneous hard sphere fluid. Our approximation makes use of our recently published averaged pair distribution function at contact, which has been shown to accurately reproduce the averaged pair distribution function at contact for inhomogeneous density distributions. This approach achieves greater computational efficiency than previous approaches by enabling the use of exclusively fixed-kernel convolutions and thus allowing an implementation using fast Fourier transforms. We compare results for our pair distribution approximation with two previously published works and Monte Carlo simulation, showing favorable results.

Heat conduction in a semi-infinite medium with time-periodic boundary temperature and a circular inhomogeneity

We solve the problem of heat conduction in a 2D homogeneous medium (of diffusivity α) below a boundary subjected to time-periodic temperature (of frequency ω), in the presence of a circular inhomogeneity (of radius R), whose center is at distance d > R (depth) from the boundary. This study is a continuation of a previous one which considers a 3D medium with a spherical inhomogeneity. The general solution depends on four dimensionless parameters: d/R, the heat conductivity ratio κ, the heat capacity ratio C and the displacement thickness δ/R=2α/(ωR2). An analytical solution is derived as an infinite series of eigenfunctions pertaining to the 2D Helmholtz equation. The solution converges quickly and is shown to be in agreement with a finite element numerical solution. The results are illustrated and analyzed for a given accuracy and for a few values of the governing parameters. A comparison is held with the previous 3D solution pointing out the differences between the two. To widen the range of possible applications, an extension of the solution to a domain of finite depth is also presented. The general solution can be simplified considerably for asymptotic values of the parameters. A first approximation, obtained for R/d≪1, pertains to an unbounded domain. A further approximate solution, for R/δ≪1, while κ and C are fixed, can be regarded as pertaining to a quasi-steady regime. However, its accuracy deteriorates for κ≪1, and a solution, coined as the insulated circle approximation, is derived for this case. Comparison with the exact solution shows that these approximations are accurate for a wide range of parameter values.

Propagation of Spherical Magnetogasdynamic Shock in a Radiative Gas

In this paper the propagation of explosion waves, in a conducting gas, produced on account of a point explosion into inhomogeneous self-gravitating gas sphere is considered. Radiation effects have been taken into account and the density of the gas is assumed to vary as r/sup -alpha/, r being the distance from the point of explosion. In order that the mass and pressure be positive in the equilibrium state, the choice of alpha is restricted between 1 and 3. The variation of Mach number of the shock and energy of the wave with time have been taken into consideration.

Transmission line approach to quantifying the resonance and transparency properties of electrically small layered plasmonic nanoparticles

The scattering and transparency properties of layered plasmonic nanoparticles are studied from the perspective of spherical transmission line theory. The advantage of this approach is that the interaction of the nanoparticle with its surroundings can be very conveniently represented as a combination of admittances. In this framework we reformulate the total impedance expression of a spherical nanoparticle, and from this we derive through a compact and intuitive methodology the two conditions that govern the resonance and transparency states of a nanoparticle. These conditions are satisfied when the particle’s input admittance becomes inductive and capacitive, respectively. The recursive relations that determine the TM admittance of an electrically small, radially inhomogeneous dielectric sphere are analyzed, and it is demonstrated that any degree of layering can be homogenized by a decomposition into successive binary mixtures. The appropriate material choice results in mixtures that exhibit multiple Lorentzian resonances that are directly mapped to the particle’s admittance, due to its small electrical size. Consequently, the particle’s admittance exhibits multiple inductive-to-capacitive switchings, and thus the resonance and transparency conditions are satisfied at multiple frequencies. This explains the well-known feature of multiple resonance–transparency pairs observed in the scattering signature of layered plasmonic nanoparticles.

AFM-based indentation stiffness tomography—An asymptotic model

The so-called indentation stiffness tomography technique for detecting the interior mechanical properties of an elastic sample with an inhomogeneity is analyzed in the framework of the asymptotic modeling approach under the assumption of small size of the inhomogeneity. In particular, it is assumed that the inhomogeneity size and the size of contact area under the indenter are small compared with the distance between them. By the method of matched asymptotic expansions, the first-order asymptotic solution to the corresponding frictionless unilateral contact problem is obtained. The case of an elastic half-space containing a small spherical inhomogeneity has been studied in detail. Based on the grid indentation technique, a procedure for solving the inverse problem of extracting the inhomogeneity parameters is proposed.

The structure and properties of a simple model mixture of amphiphilic molecules and ions at a solid surface.

We investigate microscopic structure, adsorption, and electric properties of a mixture that consists of amphiphilic molecules and charged hard spheres in contact with uncharged or charged solid surfaces. The amphiphilic molecules are modeled as spheres composed of attractive and repulsive parts. The electrolyte component of the mixture is considered in the framework of the restricted primitive model (RPM). The system is studied using a density functional theory that combines fundamental measure theory for hard sphere mixtures, weighted density approach for inhomogeneous charged hard spheres, and a mean-field approximation to describe anisotropic interactions. Our principal focus is in exploring the effects brought by the presence of ions on the distribution of amphiphilic particles at the wall, as well as the effects of amphiphilic molecules on the electric double layer formed at solid surface. In particular, we have found that under certain thermodynamic conditions a long-range translational and orientational order can develop. The presence of amphiphiles produces changes of the shape of the differential capacitance from symmetric or non-symmetric bell-like to camel-like. Moreover, for some systems the value of the potential of the zero charge is non-zero, in contrast to the RPM at a charged surface.

Nonaxisymmetric electroelastic vibrations of a hollow sphere made of functionally gradient piezoelectric material

The nonaxisymmetric problem of natural vibrations of a hollow sphere made of functionally gradient piezoelectric material is solved based on 3D electroelasticity. The properties of the material change continuously along a radial coordinate according to an exponential law. The external surface of the sphere is free of tractions and either insulated or short-circuited by electrodes. After separation of variables and representation of the components of the displacements and of the stress tensor in terms of spherical functions, the initially three-dimensional problem is reduced to a boundary-value problem for the eigenvalues expressed by ordinary differential equations. This problem is solved by a stable discrete-orthogonalization technique in combination with a step-by-step search method with respect to the radial coordinate. Moreover, a numerical investigation is performed based on the algorithm used for solving the problem. In particular, we investigate the influence of the geometric and electric parameters on the frequency spectrum at the nonaxisymmetry of natural vibrations of an inhomogeneous piezoceramic thick-walled sphere.

A new homogenization method based on a simplified strain gradient elasticity theory

Homogenization methods utilizing classical elasticity-based Eshelby tensors cannot capture the particle size effect experimentally observed in particle–matrix composites at the micron and nanometer scales. In this paper, a new homogenization method for predicting effective elastic properties of multiphase composites is developed using Eshelby tensors based on a simplified strain gradient elasticity theory (SSGET), which contains a material length scale parameter and can account for the size effect. Based on the strain energy equivalence, a homogeneous comparison material obeying the SSGET is constructed, and two sets of equations for determining an effective elastic stiffness tensor and an effective material length scale parameter for the composite are derived. By using Eshelby’s eigenstrain method and the Mori–Tanaka averaging scheme, the effective stiffness tensor based on the SSGET is analytically obtained, which depends not only on the volume fractions and shapes of the inhomogeneities (i.e., phases other than the matrix) but also on the inhomogeneity sizes, unlike what is predicted by the existing homogenization methods based on classical elasticity. To illustrate the newly developed homogenization method, sample cases are quantitatively studied for a two-phase composite filled with spherical, cylindrical, or ellipsoidal inhomogeneities (particles) using the averaged Eshelby tensors based on the SSGET that were derived earlier by the authors. Numerical results reveal that the particle size has a large influence on the effective Young’s moduli when the particles are sufficiently small. In addition, the results show that the composite becomes stiffer when the particles get smaller, thereby capturing the particle size effect.

Stress State of a Radial Inhomogeneous Semi Sphere under the Vertical Uniform Load

There is a solution of the problem of stress-strain state determining in the radially inhomogeneous concrete semi sphere. The inhomogeneity is induced by the temperature field. The problem reduces to a differential equation system with variable coefficients. The allowance for the variable Young's module lets to arrive to a more accurate solution.

Radar albedos and circular-polarization ratios for realistic inhomogeneous media using the discrete-dipole approximation

Electromagnetic scattering is simulated using the discrete-dipole approximation for inhomogeneous spheres and closely-packed aggregates of spherical particles with an emphasis on backscattering and planetary radar applications. The simulations are utilized to study how different parameters, including the composition and distribution of a homogeneous medium, affect the circular-polarization ratio and the radar albedo. Two different pairs of refractive indices are chosen, using values relevant for asteroids and meteorites of silicaceous and basaltic materials at microwave wavelengths. The results show substantial variance due to the geometry at backscattering, which is why analytic formulae cannot be deduced. However, some tendencies are visible: for example, the circular-polarization ratio increases when the structure becomes less isotropic as well as when the effective refractive index increases due to an increase in the phase shift. As for the radar albedo, the values computed for the homogeneous spheres with an equal effective refractive index correlate to some extent, but also differ from the radar albedos of the inhomogeneous spheres systematically. For the aggregates of spheres, the effect of the general shape and differences between the algorithms that dominate the effects caused by the inhomogenization. We find that the general scattering features remain regardless of the inhomogeneous internal structure.

Influence of interfaces on effective properties of nanomaterials with stochastically distributed spherical inclusions

The method of conditional moments is generalized to include evaluation of the effective elastic properties of particulate nanomaterials and to investigate the size effect in those materials. Determining the effective constants necessitates finding a stochastically averaged solution to the fundamental equations of linear elasticity coupled with surface/interface conditions (Gurtin–Murdoch model). To obtain such a solution the system of governing stochastic differential equations is first transformed to an equivalent system of stochastic integral equations. Using statistical averaging, the boundary-value problem is then converted to an infinite system of linear algebraic equations. A two-point approximation is considered and the stress fluctuations within the inclusions are neglected in order to obtain a finite system of algebraic equations in terms of component-average strains. Closed-form expressions are derived for the effective moduli of a composite consisting of a matrix and randomly distributed spherical inhomogeneities. As a numerical example a nanoporous material is investigated assuming a model in which the interface effects influence only the bulk modulus of the material. In that model the resulting shear modulus is the same as for the material without surface effects. Dependence of the bulk moduli on the radius of nanopores and on the pore volume fraction is analyzed. The results are compared to, and discussed in the context of other theoretical predictions.

Testing the Copernican principle by constraining spatial homogeneity

Abstract We present a new programme for placing constraints on radial inhomogeneity in a dark-energy-dominated universe. We introduce a new measure to quantify violations of the Copernican principle. Any violation of this principle would interfere with our interpretation of any dark-energy evolution. In particular, we find that current observations place reasonably tight constraints on possible late-time violations of the Copernican principle: the allowed area in the parameter space of amplitude and scale of a spherical inhomogeneity around the observer has to be reduced by a factor of 3 so as to confirm the Copernican principle. Then, by marginalizing over possible radial inhomogeneity we provide the first constraints on the cosmological constant which are free of the homogeneity prior prevalent in cosmology.