Nonlinear Composites Research Articles

Effective AC Response of Nonlinear Spherical Coated Composites**The project supported by National 863 Project of China under Grant No. 2002AA639270, National Natural Science Foundation of China under Grant No. A10374026, and the 100-Talent Project of Chinese Academy of Sciences

Under alternating current electric field, effective response of granular nonlinear composites with spherical coated inclusions is investigated in the dilute limit by using the perturbation approach. For an external sinusoidal applied field with finite frequency ω, the local fields and potentials of composites in general consist of components at all harmonics for cubic nonlinear constitutive relationships. We derive the local potentials of spherical coated composites at harmonics. Moreover, we give the formulae of the nonlinear effective AC susceptibility at the third harmonic frequency.

Selective Oxidation of Organic Sulfides to Sulfoxides by Hydrogen Peroxide with Trirutile-Type Solid Oxide Catalyst HNbMoO6

Selective oxidation of organic sulfides to sulfoxides has been the subject of extensive research because of their importance of sulfoxides as intermediates in various organic reactions. One of the simplest methods for oxidation of sulfides to sulfoxides is the use of hydrogen peroxide as an oxidant in the presence of a catalyst. Though a number of catalysts including soluble metal complexes have been used for the reaction, there are a few reports employing a heterogeneous catalyst despite the advantage in separation and reuse of the catalyst. In this paper, we introduce a new heterogeneous catalyst, HNbMoO6, which shows an excellent catalytic activity in the oxidation of sulfides. Trirutile-type solid oxide, AMM’O6, was first synthesized in 1970. Since then, many applications of the solid oxide in various fields such as non-linear optics and composites have been reported. However, the catalytic activity of the oxide in organic reactions has not been well studied in spite of the potentials expected from the two different transition metals contained in the solid oxide. Recently, we reported a selective sulfoxidation of allylic sulfides catalyzed by the trirutile-type solid oxide, LiNbMoO6 and successfully applied the oxidation method to the synthesis of Lansoprazole. Because of the good catalytic activity shown by the LiNbMoO6 in the sulfoxidation of sulfides, we decided to test other trirutile-type solid oxides, AMM’O6 (A=H, Li, M=Nb, Ta, M’=Mo, W), as a catalyst in the hydrogen peroxide oxidation of sulfides to sulfoxides (Eq. 1).

A theory of nonlinear AC response in coated composites

A method for determining effective dielectric responses of Kerr-like coated nonlinear composites under the alternating current (AC) electric field is proposed by using perturbation approach. As an example, we have investigated the composite with coated cylindrical inclusions randomly embedded in a host under an external sinusoidal field with finite frequency omega. The local field and potential of composites in general consists of components with all harmonic frequencies. The effective nonlinear AC responses at all harmonics are induced by the coated nonlinear composites because of the nonlinear constitutive relation. Moreover, we have derived the formulae of effective nonlinear AC responses at the fundamental frequency and the third harmonic in the dilute limit.

Effective dielectric response of nonlinear composites with external ac and dc electric field

The perturbation method is developed to investigate the effective nonlinear dielectric response of Kerr composites when the external ac and dc electric field is applied. Under the external ac and dc electric field Eapp=Ea(1+sin ωt), the effective coupling nonlinear response can be induced by the cubic nonlinearity of Kerr nonlinear materials at the zero frequency, the finite basic frequency ω, the second and the third harmonics, 2ω and 3ω, and so on. As an example, we have investigated the cylindrical inclusions randomly embedded in a host and derived the formulas of the effective nonlinear dielectric response at harmonics in dilute limit. For a higher concentration of inclusions, we have proposed a nonlinear effective-medium approximation by introducing the general effective nonlinear response. With the relationships between the effective nonlinear response at harmonics and the general effective nonlinear response, we have derived a set of formulas of the effective nonlinear dielectric responses at harmonics for a larger volume fraction.

Researching Language Learning Processes in Open CALL Settings for Advanced Learners

This article reports on a project (electronic role-play) run at Nottingham Trent University. It investigates how knowledge can be constructed collaboratively in an open setting based on computer-mediated communication (CMC) and the internet as primary source for material. The analysis will concentrate on evidence of (1) focus on language form and (2) acquisition of content. Furthermore, it reports about observations made regarding the way in which students utilize the electronic environment during the learning process, namely (3) the flow of information within an open CALL framework and (4) the non-linear composition of text.

Choreographic Cognition

The process of inception, development and refinement during the creation of a new dance work is described and explored. The account is based on annotated video of the professional choreographer and dancers as they create and sequence new movement material, as well as weekly journal entries made by one of the dancers. A 24-week chronology is reported. We analyse the choreographic process using the Geneplore model of creative cognition as an organising framework and identify generative and exploratory processes including problem finding and problem solving, metaphorical thinking, non-linear composition, and multi-modal imagery. An analytical tool adapted from the discipline of music analysis is used to explore relationships between recurring themes and visual, visceral, spatial and tactile images. Ideas for experimental work relating to choreographic cognition are discussed.

Effective medium approximation for optical bistability in nonlinear metal-dielectric composites

A general theory based on the spectral representation method and effective medium approximation is adopted to investigate the optical bistable behavior in a nonlinear two-phase composite with symmetrical microstructure, in which the metal particles of the volume fraction p and the dielectric particles of the volume fraction 1− p are randomly dispersed but oriented with respect to one another. The relation between the spatial average of local field squared and the external applied field is established through the spectral density function m( x), obtained from the modified Bruggeman effective medium approximation. We find that the optical bistability (OB) is dependent on the depolarization factor L of the components and the volume fraction p. For a given p, we predict that OB can be observed only when L is larger than the critical value L c, and bistable behavior is more pronounced at large L. Moreover, numerical results show that both the upper threshold field and the width of OB region increase monotonically as L increases. The field-dependent reflectance at normal incidence R in random composites is also investigated.

Effective ac response in weakly nonlinear composites

The perturbation method is developed to deal with the problem of determining the effective nonlinear conductivity of Kerr-like nonlinear media under an external ac electric field. As an example, we have considered the cylindrical inclusion embedded in a host under the sinusoidal external field E1 sin (ωt) + E3 sin (3ωt) with frequencies ω and 3ω. The potentials of composites at higher harmonics are derived in both local inclusion particle and host regions. The effective responses of bulk nonlinear composites at basic frequency and harmonics are given for cylindrical composites in the dilute limit. Moreover, the relationships between the nonlinear effective responses at the basic frequency and the third harmonics are derived.

Optical bistability and tristability in nonlinear metal/dielectric composite media of nonspherical particles.

Based on a spectral representation method and a self-consistent mean field theory, we present a general framework to investigate the optical bistability in a nonlinear two-phase composite, where spheroidal metallic inclusions are randomly oriented and embedded in a dielectric host. The relation between the spatial average of the local field squared <[absolute value of E](2)>(i) (i=1,2) and the external field squared E20 is obtainable through the spectral density function which is predicted from our recently derived Maxwell-Garnett approximation. In addition to single optical bistability (OB), the appearance of double OB and optical tristability (OT) is reported, and the corresponding phase diagram is given. We find that the regions of the single OB, the double OB, and the OT are dependent on the shape and volume fraction of the metallic particles. Our method allows us to take one step forward to study some field-dependent effective optical properties, such as the refractive index, extinction coefficient as well as reflectance. The general framework is also applied to investigate exactly the solvable composites consisting of nonlinear spheroidal inclusions and linear dielectric host in the dilute limit. To this end, the present method is shown to be in excellent agreement with the exact solution. In addition, the present method predicts a larger threshold intensity than the variational approach.

Giant enhancement of optical nonlinearity in mixtures of graded particles with dielectric anisotropy

To investigate the effective linear dielectric constante and third-order nonlinear susceptibil- ity χe of composite media, in which graded inclusions with radial dielectric anisotropy are randomly embedded in a linear isotropic matrix, we develop an nonlinear anisotropic differential effective dipole approximation (NADEDA). Alternatively, based on a first-principles approach, the exact expressions for � e and χe are also derived for the linear dielectric profiles with small slopes. Then, excellent agreement between the two methods is numerically demonstrated. As an application, we further apply the NADEDA to a nonlinear metal-dielectric composite, in which the metal particles possess spatially varying radial dielectric anisotropy, in an attempt to study the nonlinearity enhancement and the figure of merit of the composite. To this end, it is shown that the presence of gradation in the radial dielectric constant plays a crucial role in enhancing the optical nonlinearity as well as the figure of merit.

On the lower bounds for critical loads under large deformations in non-linear hyperelastic composites with imperfect interlaminar adhesion

The present paper investigates a mechanism of compressive fracture for heterogeneous incompressible non-linear materials with special kinds of defects of interfacial adhesion under large deformations. The analysis finds the lower bounds for the critical load. In order to calculate the bounds, the problem of the internal instability is considered within the scope of the exact statement based on the application of the model of a piecewise-homogeneous medium and the equations of the 3-D stability theory. The solution of the 3-D problem is found for the most general case accounting for large deformations and the biaxiality of compressive loads. The characteristic determinants are derived for the first four modes, which are more commonly observed. Special attention is given to the calculation of critical loads for hyperelastic layers described by a simplified version of Mooney's potential, namely the neo-Hookean potential.

Effects of fiber distribution on elastic–viscoplastic behavior of long fiber-reinforced laminates

In this paper, effects of transverse fiber distribution on the elastic–viscoplastic behavior of long fiber-reinforced laminates subjected to in-plane tensile loading are studied using a homogenization theory. To this end, a unit cell with random fiber distribution is arranged point-symmetrically with respect to cell facet centers and also Y-periodically. By discussing these two fiber arrangements in terms of standard deviations and a radial distribution function, it is demonstrated that the point-symmetric cell arrangement can enhance the randomness of fiber distribution in comparison with the Y-periodic arrangement. Then, on the assumption that the random fiber distribution generated by the point-symmetric cell arrangement prevails on the transverse section in each lamina, the in-plane elastic–viscoplastic deformation of carbon fiber/epoxy laminates is analyzed using a homogenization theory of nonlinear time-dependent composites. The analysis based on the perfectly periodic hexagonal fiber distribution in laminae is additionally performed for comparison. It is thus shown that the transverse randomness of fiber distribution in laminae has negligible influence on the macroscopic elastic–viscoplastic behavior of laminates, though it markedly affects the microscopic distribution of stress and strain. It is also shown that the analysis predicts very well the macroscopic behavior observed in the corresponding experiments.

Optical bistability in composite media with nonlinear coated inclusions

We study the intrinsic optical bistability in nonlinear composites of coated spherical particles with a metallic core and a nonlinear dielectric shell. By means of self-consistent mean field approximation, the relation between the local field in the core of inclusions and the external applied field is established. We demonstrate numerically that the optical bistability exists only when the interfacial parameter λ is less than a critical one λ c . By the suitable adjustment of λ in the range (0, λ c ), the threshold fields can be reduced substantially, and thus improve the potential applicability of this kind of material. The dielectric loss is found to reduce the optical bistability region due to the increase in the lower threshold field. Furthermore, the reflectance at normal incidence R is calculated, and the hysteresis loops are found also.

Intraphase strain heterogeneity in nonlinear composites: a computational approach

Recent progress in the analysis of nonlinear composites or polycrystals suggests that higher-order moments of the local fields in individual phases could play a significant role on the effective properties of such composites. The present study is devoted to a detailed computational investigation of the first and second moment of the strain fields. It shows that the predictive schemes belonging to the class of secant methods underestimate significantly the second moments of the strain field. As a consequence, their predictions are too stiff in comparison with numerical simulations.

Field fluctuations and macroscopic properties for nonlinear composites

A recently introduced nonlinear homogenization method [J. Mech. Phys. Solids 50 ( 2002) 737–757] is used to estimate the effective behavior and the associated strain and stress fluctuations in two-phase, power-law composites with aligned-fiber microstructures, subjected to anti-plane strain, or in-plane strain loading. Using the Hashin–Shtrikman estimates for the relevant “linear comparison composite,” results are generated for two-phase systems, including fiber-reinforced and fiber-weakened composites. These results, which are known to be exact to second-order in the heterogeneity contrast, are found to satisfy all known bounds. Explicit analytical expressions are obtained for the special case of rigid-ideally plastic composites, including results for arbitrary contrast and fiber concentration. The effective properties, as well as the phase averages and fluctuations predicted for these strongly nonlinear composites appear to be consistent with deformation mechanisms involving shear bands. More specifically, for the case where the fibers are stronger than the matrix, the predictions appear to be consistent with the shear bands tending to avoid the fibers, while the opposite would be true for the case where the fibers are weaker.

Optical properties of binary composite materials with two nonlinear components

A new formulation is presented for the calculation of effective dielectric magnitudes of two-component composites in which both components (the host matrix particles and the embedded particles) exhibit nonlinear behaviour of the Kerr type. It is predicted that, under certain conditions, two nonlinear component composites can exhibit optical bistable behaviour as a function of the shape and concentration of the embedded particles, the dielectric contants of the components, the intensity of the external electric field (power density) and the intrinsic third-order nonlinear optical susceptibilities χ(3) p and χ(3) m of the nonlinear components. It is also deduced that, as the power density increases, the effective third-order nonlinear optical susceptibility χ(3) of the composite exhibits a clear transition from values close to χ(3) p (low power density) to χ(3) m (high power density). Therefore, it is shown that the optical response of binary composites dramatically changes at moderate and high power densities. A comparison is performed between the optical response of a real two nonlinear component composite and that of a composite with a single nonlinear component.

Effective medium approximation for two-component nonlinear composites withshape distribution

The effective medium approximation (EMA) is derived to investigate theeffective linear and nonlinear responses of two-component composites in which onecomponent is nonspherical and distributed in shape. Both components with the volumefractions pand qare assumed to obey a current–field J–Erelation of the form J = σiE + χi|E|2E, where σi and χi are the linearconductivity and nonlinear response of the componenti(i = 1, 2) respectively. As thepercolation threshold pc(or qc)is approached from above (or below), the effective linear conductivity σe and effective nonlinearresponse χebehave as σe ∼ [p − pc(Δ)]t and χe ∼ [p − pc(Δ)]t2in the conductor/insulator (C/I) limit, and σe ∼ [qc(Δ) − q]−s and χe ∼ [qc(Δ) − q]−s2in the superconductor/conductor (S/C) limit, where the exponents are found to bet = s = 1 andt2 = s2 = 2,independent of the shape variance parameter Δ, and pc(Δ) (orqc(Δ))is a monotonically decreasing (or increasing) function with Δ.For a finite-conductivity ratio h = σ1/σ2, numerical resultsshow that σemay be increased or decreased with increasing Δ, dependent on whether the firstcomponent is a good or a poor conductor, while χe can exhibita monotonic increase, monotonic decrease and nonmonotonic behaviour. Therefore, χe canbe greatly enhanced by the adjustment of the shape variance parameter, and therebyprovides an alternative way to achieve large enhancement of effective nonlinearresponse. The results of EMA with shape distribution are also compared withexact solutions in the dilute limit and reasonable agreement is found.

Maxwell-Garnett type approximation for nonlinear composites with shape distribution

Maxwell-Garnett type approximation is derived to investigate the effective linear and nonlinear responses of nonlinear composites in which randomly-oriented nonspherical granular inclusions with shape variance distribution are embedded in the host medium. The granular inclusions with the volume fraction f and the host medium with the volume fraction 1− f are assumed to obey a current–field J – E relation of the form J =σ i E +χ i| E | 2 E , where σ i and χ i are the linear conductivity and nonlinear response of component i ( i=1,2). Within the mean-field approximation, we numerically calculate the effects of the shape variance parameter Δ on the effective linear conductivity σ e and effective nonlinear response χ e . We find that σ e is a monotonically increasing or decreasing function with Δ, dependent on whether the granular inclusions are a good or poor conductor. For the granular inclusions being nonlinear, χ e exhibits monotonic increase; but for the host medium being nonlinear, χ e can take on monotonic increase, monotonic decrease and even nonmonotonic behavior. In both nonlinear cases, large enhancement of the effective nonlinear response may be observed when the nonlinear component possesses a poor conductivity. In the dilute limit, an exact formula for the effective nonlinear response is derived analytically. Exact results are in comparison with those from Maxwell-Garnett type approximation with shape variance distribution, reasonable agreement is found.

Optical bistability in nonlinear composites with coated ellipsoidal nanoparticles

Nonlinear composite structures show great promise for use in optical switching, signal processing, etc. We derive an effective nonlinear dielectric permittivity of composite structures where coated ellipsoidal nonlinear particles are imbedded in a linear host medium. The derived expression for the effective dielectric permittivity tensor follows the Clasius–Mossotti approximation. We observe conditions for the existence of the optical bistability effect in a coated ellipsoidal particle with a nonlinear core and a metallic shell. Our numerical results show stronger bistability effects in more dense suspensions of nonlinear heterogeneous ellipsoids.

Geometrically derived anisotropy in cubically nonlinear dielectric composites

We consider an anisotropic homogenized composite medium (HCM) arising from isotropic particulate component phases based on ellipsoidal geometries. For cubically nonlinear component phases, the corresponding zeroth-order strong-permittivity-fluctuation theory (SPFT) (which is equivalent to the Bruggeman homogenization formalism) and second-order SPFT are established and used to estimate the constitutive properties of the HCM. The relationship between the component phase particulate geometry and the HCM constitutive properties is explored. Significant differences are highlighted between the estimates of the Bruggeman homogenization formalism and the second-order SPFT estimates. The prospects for nonlinearity enhancement are investigated.