Scattering of elastic waves from symmetric inhomogeneities at low frequencies

The problem of the scattering of elastic waves is treated in the low-frequency regime by a systematic expansion in powers of the frequency in the spatial domain of the scatterer. The zeroth and first degree scattering amplitudes vanish if the scatterer is localized in all directions. The second degree scattering amplitude corresponds to the so called Rayleigh regime in which the quasi-static result of Gubernatis et al. is valid. In general, the third and higher degree scattering amplitudes are nonvanishing. However, in the case where the scatterer has inversion symmetry about the origin, it is shown that the third degree scattering amplitude vanishes identically for all incident and scattered directions and for all polarizations. This result implies that the frequency derivative of the phase shift approaches zero at least quadratically as the frequency goes to zero. In other words, at sufficiently low frequencies the effective scattering center of a scatterer with inversion symmetry is its geometrical center. The use of this result in the processing of experimental scattering data is discussed.     

A melt rheometer with integrated small angle light scattering

A stress-controlled commercial rheometer has been equipped with small-angle light scattering (SALS) instrument working in a range of wave-vectors between 0.5 and 4.2 m^–1 By a specially designed optical system the scattered light is focused directly on the square chip of a CCD-detector with high dynamical range allowing quantitative two-dimensional (2 D) intensity measurements of polarized and depolarized scattering. The rheo SALS system is suited for simultaneous fast measurements of 2D SALS and rheological material functions of polymer melts and liquid crystals. As a first application results on the influence of shear rate on the late stage of spinodal decomposition of a polybutadiene/polyisoprene blend are reported. A second application concerns a creep experiment in the liquid crystalline phase of a lyotropic non-ionic surfactant with simultaneous measurement of the time-dependent compliance and associated changes of the 2D intensity patterns of the polarized and depolarized scattering.     

Asymptotic inverse scattering

Several inverse techniques are developed for determining the shape of an unknown scattering surface by analyzing backscattered acoustic or electromagnetic waves. These techniques are based on asymptotic high frequency representations of the fields and may be divided into three categories. The first one is the geometrical imaging method where the surface is reconstructed by means of a travel-time analysis which is here specified to the far field by utilizing Minkowski's support function. Furthermore, a geometrical method is developed for localizing edges from mid field data measured along a curve. The second category is called quasigeometrical imaging and uses geometric optics or higher order amplitudes for the reconstruction. It is shown that cross-polarized electromagnetic far field amplitude measurements permit one to deduce the complete quadratic approximation of the surface at the specular points from which the surface can be reconstructed pointwise. The third category may be subsumed under ‘asymptotic inverse scattering identities’. Here, asymptotic relations between scattered fields and distributions associated with the geometry of the scatterer are established. It is shown that the physical optics far field inverse scattering identity is only a leading order asymptotic relation but as such is also valid for non-convex scatterers. Furthermore, asymptotic inverse scattering identities are deduced which relate the singular function of a closed surface to the backscattered field data measured on a sphere enclosing the scatterer. This generalizes far field results of Cohen and Bleistein (Wave Motion 1 (1979), p. 153) to the mid field.     

A variational principle for waves in discrete random media

A new variational principle is derived for the Green's function of the linear harmonic response of a scalar wave field in a discretely heterogeneous medium. The variational principle is derived directly from exact multiple scattering integral equations and is stated in terms of a certain functional of trial configuration dependent fields. The functional is found to be stationary with respect to small variations in the fields when those fields have their correct configuration dependence. A certain trial dependence in the fields is shown to generate the Lax quasicrystalline multiple scattering equations. It is furthermore clear that these equations are optimal in the sense that any more realistic form for the trial fields unavoidably generates approximate equations entailing the generally unknown high order correlation functions. At its stationary point the proposed functional takes on a physically significant value. It becomes the change in the medium admittance due to the introduction of the scatterers. In a nonlossy medium this is related to the spectral density of modes. As errors in the trial fields cancel to first order, the final evaluation of the functional at its stationary point is especially accurate. Other related functionals are proposed and discussed as well.     

Second-order homogenisation of functionally graded materials

The homogenisation theory for periodic composites is generalised to the case of quasi-periodic composites. In quasi-periodic composites, the unit cell does not repeat throughout the medium but gradually changes along one or more directions of periodicity (grading directions). Quasi-periodic composites are thus to functionally graded materials (FGMs) what periodic composites are to statistically uniform composite materials. Contrarily to most of the homogenisation methods applied to FGMs, the proposed second-order homogenisation theory takes explicitly into account the grading at the micro-level. The derived equivalent material happens to be a particular second gradient material in which few components of the strain gradient (second gradient of the displacement) should be taken into account in addition to the classical strains (first gradient of displacement). The second gradient theory therefore appears as the natural framework to appropriately handle functionally graded materials at the macro-level. It is worth mentioning that the presented second-order homogenisation procedure is somehow analogous to the one developed for periodic composite materials submitted to rapidly varying macroscopic strain fields as in regions of high gradients. In fact, both are a generalisation of the first-order homogenisation theory for periodic media and lead to a second gradient equivalent material. However, besides their different domains of application, they exhibit further substantial differences, which are highlighted in the paper.     

Stabilization of a laminar boundary layer by an active surface-localized action

A method for rapidly damping instability waves is proposed as a means of actively controlling a perturbed gas boundary layer flow. The method is based on the use of an active body surface segment which reacts to an instantaneous local pressure variation by producing a proportional local wall displacement normal to the surface with a constant time lag calculated to result in the optimal suppression of unstable disturbances. It is shown that in the one-frequency case the wave number spectrum of the optimal control law contains multiple eigenvalues. The effectiveness of the method is demonstrated over a wide range of variation of the instability wave frequencies and directions. The propagation of an instability wave over an active segment of finite length is calculated using an integral-equation method based on solving the problem of boundary layer flow receptivity to surface vibration. Explicit formulas describing the process of scattering of the instability wave into stable modes at the junction point of the rigid and active surfaces are obtained using the Fourier method and the integral Cauchy theorem.     

HEAT TRANSFER CHARACTERISTICS OF CONDUCTIVE MATERIAL UNDER INNER NON-UNIFORM ELECTROMAGNETIC FIELDS

Electronic transport properties can be influenced by the applied electromagnetic fields in conductive materials. The change of the electron distribution function evoked by outfields obeys the Boltzmann equation. In this paper, a general law of heat conduction considering the non-uniform electromagnetic effect is developed from the Boltzmann equation. An analysis of the equation leads to the result that the electric field gradient and the magnetic gradient in the conductive material are responsible for the influences of electromagnetic fields on the heat conduction process. A physical model is established and finite element numerical simulation reveals that heat conduction can be increased or delayed by the different directions of the electric field gradient, and the existence of the magnetic gradient always hinders heat conduction.     

The Effects of Combined Horizontal and Vertical Heterogeneity on the Onset of Convection in a Porous Medium with Vertical Throughflow

The effects of hydrodynamic and thermal heterogeneity, for the case of variation in both the horizontal and vertical directions, on the onset of convection in a horizontal layer of a saturated porous medium uniformly heated from below, with weak vertical throughflow, are studied analytically for the case of weak heterogeneity. It is found that when the boundary conditions at the upper and lower boundaries are symmetric, the throughflow magnitude and the permeability and conductivity gradients enter the expression for the critical Rayleigh number at second order. The throughflow on its own is stabilizing but the combination of throughflow and heterogeneity may be either stabilizing or destabilizing.     

SH波绕界面孔的散射

用波函数展开方法研究了SH波绕界面孔的散射问题。由入射、反射和透射波组成的自由波场与孔的散射场叠加成总波场。按照一定方式将两个半平面散射波场延拓于全平面，通过Hankel-Fourier展开方法求得了任意形状孔散射场的级数解。以椭圆形孔为例计算了孔边缘的动应力集中系数。     

Propagation of Transient Acoustic Waves in Layered Porous Media: Fractional Equations for the Scattering Operators

Acoustic waves scattering from a rigid air-saturated porous medium is studied in the time domain. The medium is one dimensional and its physical parameters are depth dependent, i.e., the medium is layered. The loss and dispersion properties of the medium are due to the fluid-structure interaction induced by wave propagation. They are modeled by generalized susceptibility functions which express the memory effects in the propagation process. The wave equation is then a fractional telegraphist’s equation. The two relevant quantities are the scattering operators—transmission and reflection operators—which give the scattered fields from the incident wave. They are obtained from Volterra equations which are fractional equations for the scattering operators.     

The onset of convection in a shallow box occupied by a heterogeneous porous medium with constant flux boundaries

The effects of hydrodynamic and thermal heterogeneity, for the case of variation in both the horizontal and vertical directions, on the onset of convection in a horizontal layer of a saturated porous medium uniformly heated from below, are studied analytically using linear stability theory for the case of weak heterogeneity. Attention is focused on the case of constant flux upper and lower boundaries, a case for which the critical horizontal wavenumber is zero, and attention is also concentrated on the case of a shallow layer. It is found that the effect of such heterogeneity on the critical value of the Rayleigh number Ra based on mean properties is of second order if the properties vary in a piecewise constant or linear fashion. The effects of horizontal heterogeneity and vertical heterogeneity are then comparable once the aspect ratio is taken into account, and to a first approximation are independent. The combination of permeability heterogeneity and conductivity heterogeneity can be either stabilizing or destabilizing for the present case.     

The scattering of P-waves by a piezoelectric particle with FGPM interfacial layers in a polymer matrix

Propagation of P-wave in an unbounded elastic polymer medium which contains a set of nested concentric spherical piezoelectric inhomogeneities is formulated. The polymer matrix is made of Epoxy and is isotropic; each phase of the inhomogeneity is made of a different piezoelectric material and is radially polarized and has spherical isotropy. Note that the individual phases are homogeneous, and all interfaces are perfectly bonded. The scattered displacement and electric potentials in the matrix are expressed in terms of spherical wave vector functions and Legendre functions, respectively. The transmitted displacement and electric potentials within each phase of the piezoelectric particle are expressed in terms of Legendre functions. The equations of motion and electrostatics in each phase of the piezoelectric inhomogeneity lead to a system of coupled second order differential equations, which is solved using the generalized Frobenius series. The present theory is extended to the case where the core of the inhomogeneity is made of PZT-4 and its coating is made of functionally graded piezoelectric material (FGPM) whose microstructural composition varies smoothly from PZT-4 at the core–coating interface to Epoxy at the coating–matrix interface. The effects of different types of variation in the electro-mechanical properties of FGPM on scattering cross-section and other electro-mechanical fields are addressed. The present theory is valid for arbitrary coating thickness, and arbitrary frequencies.     

ON THE SECOND ORDER WAVE DIFFRACTION IN TWO LAYER FLUIDS

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Homotopy analysis solution for micropolar fluid flow through porous channel with expanding or contracting walls of different permeabilities

The flow of a micropolar fluid through a porous channel with expanding or contracting walls of different permeabilities is investigated. Two cases are considered, in which opposing walls undergo either uniform or non-uniform motion. In the first case,the homotopy analysis method (HAM) is used. to obtain the expressions for the velocity and micro-rotation fields. Graphs are sketched for some parameters. The results show that the expansion ratio and the different permeabilities have important effects on the dynamic characteristics of the fluid. Following Xu's model, in the second case which is more general, the wall expansion ratio varies with time. Under this assumption, the governing equations are transformed into nonlinear partial differential equations that can also be solved analytically by the HAM. In the process, both algebraic and exponential models are considered to describe the evolution of α(t) from the initial state α0 to the final state α1. As a result, the time-dependent solutions are found to approach the steady state very rapidly. The results show that the time-dependent variation of the wall expansion ratio can be ignored because of its limited effects.     

Equivalent multipolar point-source modeling of small spheres for fast and accurate electromagnetic wave scattering computations

In this paper, we develop reduced models to approximate the solution of the electromagnetic scattering problem in an unbounded domain which contains a small perfectly conducting sphere. Our approach is based on the method of matched asymptotic expansions. This method consists in defining an approximate solution using multi-scale expansions over outer and inner fields related in a matching area. We make explicit the asymptotics up to the second order of approximation for the inner expansion and up to the fifth order for the outer expansion. We validate the results with numerical experiments which illustrate theoretical orders of convergence for the asymptotic models requiring negligible computational cost.     

Elastic-plastic fields near crack tip growing step-by-step in a perfectly plastic solid

In this paper, based on the three-dimensional flow theory of plasticity, the fundametal equations for plane strain problem of elastic-perfectly plastic solids are presented. By using these equations the elastic-plastic fields near the crack tip growing step-by-step in an elastic incompressible-perfectly plastic solid are analysed.The first order asymptotic solutions for the stress field and velocity fields near the crack tip are obtained. The solutions show the evolution process of elastic unloading domain and the development process of central fan domain and reveal the possibility of the presence of the secondary plastic domain. The second order asymptotic solution for stress field is also presented.     

Off-axial acoustic scattering of a high-order Bessel vortex beam by a rigid sphere

In this paper, the off-axial acoustic scattering of a high-order Bessel vortex beam by a rigid immovable (fixed) sphere is investigated. It is shown here that shifting the sphere off the axis of wave propagation induces a dependence of the scattering on the azimuthal angle. Theoretical expressions for the incident and scattered field from a rigid immovable sphere are derived. The near- and far-field acoustic scattering fields are expressed using partial wave series involving the spherical harmonics, the scattering coefficients of the sphere, the half-conical angle of the wave number components of the beam, its order and the beam-shape coefficients. The scattering coefficients of the sphere and the 3D scattering directivity plots in the near- and far-field regions are evaluated using a numerical integration procedure. The calculations indicate that the scattering directivity patterns near the sphere and in the far-field are strongly dependent upon the position of the sphere facing the incident high-order Bessel vortex beam.     

网格与高精度差分计算问题

研究NS方程差分求解时来流雷诺数、计算格式精度和计算网格之间的关系．给出了判定空间三个方向上的粘性贡献在给定雷诺数、格式精度和网格下是否能够正确计入的估计方法．指出在NS方程的二阶差分方法的数值模拟中,由于物面法向采用了压缩网格技术,物面附近的网格间距很小,该方向上的粘性贡献可被计入．但是如果流向和周向的网格较粗,相应的差分方程中的粘性贡献可能落入截断误差相同的量级,因此在精度上等于仍是求解略去流向和周向粘性项的薄层近似方程．指出,高阶精度的差分计算格式,可以避免对网格要求苛刻的困难．并进一步讨论了建立高阶精度格式的问题,提出了建立高阶精度格式应该满足的原则：耗散控制原则以及色散控制原则．为了避免激波附近可能出现的微小非物理振荡,建议发展混合高阶精度格式,即在激波区,采用网格自适应的NND格式,在激波以外的区域,采用按上述原则发展的高阶格式．     

Compressional waves in fluid-saturated porous solid containing a penny-shaped crack

Diffraction of normal compression waves by a penny-shaped crack in a fluid-saturated porous medium is investigated. Two wave types are considered, namely, compressional wave of the first kind, and the second kind. The former, also known as fast wave, propagates primarily through the solid, whereas the latter or slow wave, propagates mainly in the fluid. Each wave propagates in the medium along with induced wave of the same type in the companion constituent of the material. Application of Biot’s theory in conjunction with integral transform technique reduces the problem to a mixed boundary-value problem whose solution is in turn governed by a Fredholm integral equation of the second kind. Near-field and far-field solutions are obtained in terms of the dynamic stress-intensity factor and the scattering cross section, respectively. They are of particular importance to the linear elastic fracture mechanics (LEFM) and in the scattering theory of elastic waves. The mode I stress-intensity factors are computed numerically for a set of selected material property values, and shown graphically for various mass density and viscosity-to-permeability ratios. The obtained results reveal significant impact of the presence of pore fluid upon the stress-intensity factors, both magnitudes and frequencies at their peak values. The influence of the fluid is also observed from the calculated scattering cross sections of the scattered far-field. Accuracy of the present solution procedure is verified by comparing the numerical results with existing results in the limiting case of dry elastic materials.