On the third-order bounds of the effective shear modulus of two-phase composites

Formulation of variational bounds for properties of inhomogeneous media constitutes one of the most fundamental parts of mechanics. The earliest work on multiphase media is the so-called Voigt’s upper bound and Reuss’ lower bound, corresponding to the simple rule of mixture or first-order bounds. The second-order bounds were formulated by Hashin and Shtrikman for macroscopically isotropic random composites. The third-order bounds of the bulk modulus were derived by Beran, which contain a pair of third-order bulk parameters. The third-order bounds of the shear modulus first derived by McCoy were improved by Milton and Phan-Thien, which further involve a pair of third-order shear parameters. In this study, by applying the stochastic variational principle of Xu (2009) the third-order bounds of the shear modulus are derived in an analytically most extensive trial function space. By further modifying Milton’s definition of shear parameters, the third-order bounds are finalized into a symmetric form, exactly like the Beran’s bounds of the bulk modulus. Since the bounds of the shear modulus play an essential role in plasticity theory of composites, the finalization of the third-order bounds also paves the way for further formulation of variational principles and bounds of nonlinear composites.     

Expression of third-order effective nonlinear susceptibility for third-harmonic generation in crystals

Third-harmonic-generation processes in crystals are governed by the fourth-rank tensor ((3))(Xijkl), which reflects the crystal symmetry. In this case, the third-order nonlinear susceptibility tensor can be contracted to the compact matrix form ((3))(Xim). The matrices ((3))(Xim) for isotropic media and all 32 crystallographic point groups are presented. With these matrices, the analytic expressions of third-order effective nonlinear susceptibility can be easily derived.     

Homogenized free surface flow in porous media for wet‐out processing

This paper presents a novel porous media model for homogenized free surface flow, representing wet‐out composites processing. The model is derived from concepts of homogenization applied to a compressible two‐phase flow, accounting for capillary effects and the concept of relative permeability. Based on mass balance considerations, we obtain a nonlinear set of equations of convection‐diffusion type involving the mixture (fluid) pressure and the degree of saturation as primary fields. A staggered Galerkin finite element approach is employed to decouple the solution. Moreover, the streamline upwind/Petrov‐Galerkin technique is applied to attenuate the oscillations in the saturation solutions. The model accuracy and convergence of the finite element solutions are demonstrated through 1‐dimensional and 2‐dimensional examples, representing resin transfer molding flow processes.     

Bounds on effective magnetic permeability of three-phase composites with coated spherical inclusions

An effective model is developed to bound the effective magnetic permeability of three-phase composites with coated spherical inclusions. In the present model, the trial magnetic potential for the upper bound and the trial magnetic induction field for the lower bound are constructed to satisfy continuity interface conditions. According to the variational principle, the upper and lower bounds on the effective magnetic permeability of three-phase composites with coated spherical inclusions are derived. In this paper, trial magnetic potentials with different function forms are taken and the optimal upper bound is obtained for the trail magnetic potential corresponding to the third-order function. When the three-phase model degenerates into the composite spheres assemblage model [1], it is interesting that the optimal upper and lower bounds are the same. The effects of the volume fraction of coated spherical inclusions and the thickness and magnetic permeability of coated layers between the matrix and spherical inclusions on the effective magnetic permeability of composites are analyzed. The upper and lower bounds are finite non-zero values when the magnetic permeability of spherical inclusions tends to ∞$\infty$ and 0, respectively.     

Organic Materials for Third-Order Nonlinear Optics

The current status of organic low-molecular weight and polymeric materials for third-order nonlinear optics is reviewed. The importance of organic materials lies in their promise of large nonlinear optical figure of merit, high optical damage thresholds, ultrafast optical responses, architectural flexibility, and ease of fabrication. Organic materials exhibiting interesting third-order nonlinear optical properties are discussed to illustrate the importance of structure–property correlations. Results on emerging organic materials that include liquids, dyes, fullerenes, charge-transfer complexes, π-conjugated polymers, dye-grafted polymers, organometallic compounds, composites, and liquid crystals are presented. Organic nonlinear optical materials seem promising for a wide range of applications and their potential for integrated optics should be further explored.     

Micromechanics models for the effective nonlinear electro-mechanical responses of piezoelectric composites

The nonlinear behavior of piezoelectric composites becomes prominent when the composites are subjected to high electric fields, which is often the case in actuator applications. Understanding the nonlinear behavior of piezoelectric composites is crucial in designing structures comprising of these materials. This study presents micromechanics models for predicting nonlinear electro-mechanical responses of polarized piezoelectric composites, comprising of a linear non-piezoelectric homogeneous medium (matrix) reinforced by either nonlinear piezoelectric fibers or particles, subjected to high electric fields. The maximum electric field applied is within the coercive electric field limit. The constitutive relations for the polarized piezoelectric inclusions consist of the third- and fourth-order electro-mechanical coupling tensors and the second- and third-order electric permeability tensors. The Mori–Tanaka micromechanics and simplified unit-cell micromechanics models are formulated to predict the effective nonlinear electro-mechanical responses of piezoelectric fiber reinforced and particle reinforced composites, respectively. Linearized micromechanical relations are first used to provide trial solutions followed by iterative schemes in order to correct errors from linearizing the nonlinear responses. Numerical results are presented to illustrate the performance of each micromechanics model.     

Existence and uniqueness results for a nonlinear differential equation arising in stagnation point flow in a porous medium

We establish existence and uniqueness results over the semi-infinite interval [0, ∞) for a class of nonlinear third-order ordinary differential equations arising in the stagnation point flow in a porous medium.     

Higher-order Photon Counting Statistics of Gaussian-lorentzian Light

A method of arriving at the closed-form expression for an Nth-order generating function for gaussian-lorentzian light is described. Various aspects of the third-order photon counting statistics are derived explicitly for counting times of arbitrary duration.     

Optical properties of binary composite materials with two nonlinear components

Abstract

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.     

有机/无机复合相变储能材料的长期化学稳定性研究

根据傅立叶变换红外光谱图,对用真空浸渗法制备的多孔介质基复合相变储能材料的相变循环稳定性进行了研究。对其中的有机相变材料月桂酸的结构进行了分析,并与未经相变循环的纯月桂酸及月桂酸盐的红外谱图进行了对比。结果表明,月桂酸相变材料的各个基团和化学键的振动吸收峰仍然存在,与基体材料的化学相容性非常好。     

Ellipsoidal bounds and percolation thresholds of transport properties of composites

Classical Hashin-Shtrikman bounds physically correspond to spherical inclusions distributed in a self-similar pattern. For general ellipsoidal inclusions, new bounds named as ellipsoidal bounds are theoretically derived in this study. As there remains a major theoretical question on rigorous determination of percolation thresholds, this study also fills this major gap between lack of theoretical prediction and the gigantic amount of experimental results produced each year on percolation of composites. Formulae of percolation thresholds are for the first time universally presented for electrical, thermal, magnetic, and hydraulic properties of a variety of composites. New bounds of transport properties and percolation thresholds estimated enable the geometry of fillers or cavities, the most direct and obvious microstructure information, to be explicitly taken into account for both engineering composites and natural media (rocks, soils, and sands) containing spheroidal particles/voids, fibers, cracks, nanotubes, etc.     

A consistent u‐p formulation for porous media with hysteresis

This paper presents a continuum formulation based on the theory of porous media for the mechanics of liquid unsaturated porous media. The hysteresis of the liquid retention model is carefully modelled, including the derivation of the corresponding consistent tangent moduli. The quadratic convergence of Newton's method for solving the highly nonlinear system with an implicit finite element code is demonstrated. A u‐p formulation is proposed where the time discretisation is carried out prior to the space discretisation. In this way, the derivation of all consistent moduli is fairly straightforward. Time integration is approximated with the Theta and Newmark's methods, and hence the fully coupled nonlinear dynamics of porous media is considered. It is shown that the liquid retention model requires also the consistent second‐order derivative for quadratic convergence. Some predictive simulations are presented illustrating the capabilities of the formulation, in particular to the modelling of complex porous media behaviour. Copyright © 2014 John Wiley & Sons, Ltd.     

Flow modeling of linear and nonlinear fluids in two scale fibrous fabrics

The fundamental macroscopic material property needed to quantify the flow in a fibrous medium viewed as a porous medium is the permeability. Composite processing models require the permeability as input data to predict flow patterns and pressure fields. As permeability reflects both the magnitude and anisotropy of the fluid/fiber resistance, efficient numerical techniques are needed to solve linear and nonlinear homogenization problems online during the flow simulation. In a previous work the expressions of macroscopic permeability were derived in a double-scale porosity medium for both Newtonian and rheo-thinning resins. In the linear case only a microscopic calculation on a representative volume is required, implying as many microscopic calculations as representative microscopic volumes exist in the whole fibrous structure. In the non-linear case, and even when the porous microstructure can be described by a unique representative volume, microscopic calculation must be carried out many times because the microscale resin viscosity depends on the macroscopic velocity, which in turn depends on the permeability that results from a microscopic calculation. Thus, a nonlinear multi-scale problem results. In this paper an original and efficient offline-online procedure is proposed for the efficient solution of nonlinear flow problems in porous media.     

State space solution of non-axisymmetric Biot consolidation problem for multilayered porous media

In this paper, the non-axisymmetric Biot consolidation problem for multilayered porous media is studied. Taking stresses, pore pressure and displacements at layer interfaces as basic unknown functions, two sets of partial differential equations, which are independent each other, are formulated. Using Fourier expansion, Laplace transforms and Hankel transforms with respect to the circumferential, time and radial coordinates, respectively, the partial differential equations presented are reduced to the ordinary differential equations. Transfer matrices describing the transfer relation between the state vectors for a finite layer are derived explicitly in the transform space. Using the transfer matrices presented, three cases are studied for the lower surface: (1) permeable rough rigid base, (2) impermeable rough rigid base, and (3) poroelastic half space. The explicit solution in the transform space is presented. Considering the continuity condition at layer interfaces, the solutions of the non-axisymmetric Biot consolidation problems for multilayered semi-infinite porous media are presented in the integral form. The time histories of displacements, stresses and pore pressure are obtained by solving a linear equation system for discrete values of Laplace-Hankel transform inversions.     

Third-order optical nonlinearity of cadmium sulfide nanoparticles loaded in mesostructured silica materials

Cadmium sulfide nanoparticles have been successfully incorporated in three forms of CTAB-templated mesoporous silica materials: one is the mesoporous silica spheres suspended in ethanol solution, the other is the mesoporous silica spheres spin-coated on glass slide, and the third is the dip-coated mesoporous silica thin film. The mesostructures were characterized by XRD and TEM, respectively. Linear optical properties were investigated using UV-visible spectra, and the diameter of the incorporated CdS nanoparticles was measured to be around 3.1 nm. Z-scan technique manifested that these three composites exhibited distinct third-order optical nonlinearities due to the different preparation techniques. Reverse saturation absorption could be detected in the CdS-loaded mesoporous silica spheres suspended in solution, while those dispersed on glass slide presented saturation absorption. The difference in nonlinear absorption of the two mesoporous silica sphere samples could be attributed to defect-related transitions. On the contrary, the CdS-loaded mesoporous silica thin film showed self-defocusing behavior with no nonlinear absorption signals. Compared to that of the CdS nanoparticles with larger size previously reported, the intrinsic microscopic third-order nonlinear optical susceptibility of those incorporated in CTAB-templated mesoporous thin film was increased as predicted by the quantum theory, and the third-order optical nonlinearity was further determined to arise from intraband transitions induced by quantum confinement.     

多孔介质的一种流-固耦合动态边界理论

基于Biot理论,推导了考虑渗流作用的可变形多孔介质流-固耦合问题的基本方程,建立了本问题的渗流模型,给出了所考虑问题的流体动力弥散分布概率及系数表达式,进一步建立了多孔介质中微压液体位移场模型,讨论了流体动力弥散因素对多孔介质边界的影响,建立了描述多孔介质的动态边界方程,并分别对所建立的四种边界模拟了动态结果及算例。     

基于ANSYS软件对SiCp/A1(ZL102)复合材料铸造渗流过程多孔介质结构的有限元处理

采用统计平均的方法描述多孔介质微观结构的影响,根据多孔介质的连续介质模型及有限元方法对多孔介质内的铝液的渗流行为进行数值模拟,给出了可视化的瞬态温度场分布,并且初步预测了不同时刻的渗流有效高度.     

Physical Mechanisms of Nonlinear Optical Activity in Crystals

Possible mechanisms of nonlinear optical activity (NLOA) in crystals connected with electronic nonlinearities are considered in this work. Relations are obtained that describe nonlinear changes in the structure of light polarization in a medium having weak, arbitrary nonlinearity. It is shown that in media with third-order nonlinearity, NLOA may be linked either with the distribution of nonlinearity in the medium or with the anisotropy of nonlinear absorption-so-called absorptive and dispersive NLOA.     

Numerical analysis of strongly nonlinear extensional vibrations in elastic rods

In the framework of transduction, nondestructive testing, and nonlinear acoustic characterization, this article presents the analysis of strongly nonlinear vibrations by means of an original numerical algorithm. In acoustic and transducer applications in extreme working conditions, such as the ones induced by the generation of high-power ultrasound, the analysis of nonlinear ultrasonic vibrations is fundamental. Also, the excitation and analysis of nonlinear vibrations is an emergent technique in nonlinear characterization for damage detection. A third-order evolution equation is derived and numerically solved for extensional waves in isotropic dissipative media. A nine-constant theory of elasticity for isotropic solids is constructed, and the nonlinearity parameters corresponding to extensional waves are proposed. The nonlinear differential equation is solved by using a new numerical algorithm working in the time domain. The finite-difference numerical method proposed is implicit and only requires the solution of a linear set of equations at each time step. The model allows the analysis of strongly nonlinear, one-dimensional vibrations and can be used for prediction as well as characterization. Vibration waveforms are calculated at different points, and results are compared for different excitation levels and boundary conditions. Amplitude distributions along the rod axis for every harmonic component also are evaluated. Special attention is given to the study of high-amplitude damping of vibrations by means of several simulations. Simulations are performed for amplitudes ranging from linear to nonlinear and weak shock.     

Transient evolution of weakly nonlinear sloshing waves: an analytical and numerical comparison

The problem of water waves generated in a horizontally oscillating basin is considered, with specific emphasis on the transient evolution of the wave amplitude. A third-order amplitude evolution equation is solved analytically in terms of Jacobian elliptic functions. The solution explicitly determines the maximum amplitude and nonlinear beating period of the resonated wave. An observed bifurcation in the amplitude response is shown to correspond to the elliptic modulus approaching unity and the beating period of the interaction approaching infinity. The theoretical predictions compare favorably to fully nonlinear simulations of the sloshing process. Due to the omission of damping, the consideration of only a single mode, and the weakly nonlinear framework, the analytical solution applies only to finite-depth, non-breaking waves. The inviscid numerical simulations are similarly limited to finite depth.