Magnetic field generation during shear motion of conducting layers

This paper considers the generation of a longitudinal magnetic field between rigid conducting layers of different electrical conductivity which are in shear motion at constant velocity across the flux lines of the initial field. The amplification limits for the generated field due to the finite thickness of the conducting layers are established. The evolution of the magnetic field under various conditions of generation was described using the models of two layers of finite thickness, a semiinfinite layer and a layer of finite thickness, and two semiinfinite layers.     

Bending analysis of five-layer curved functionally graded sandwich panel in magnetic field: closed-form solution

In this paper,an exact closed-form solution for a curved sandwich panel with two piezoelectric layers as actuator and sensor that are inserted in the top and bottom facings is presented.The core is made from functionally graded(FG)material that has heterogeneous power-law distribution through the radial coordinate.It is assumed that the core is subjected to a magnetic field whereas the core is covered by two insulated composite layers.To determine the exact solution,first characteristic equations are derived for different material types in a polar coordinate system,namely,magneto-elastic,elastic,and electro-elastic for the FG,orthotropic,and piezoelectric materials,respectively.The displacement-based method is used instead of the stress-based method to derive a set of closed-form real-valued solutions for both real and complex roots.Based on the elasticity theory,exact solutions for the governing equations are determined layer-by-layer that are considerably more accurate than typical simplified theories.The accuracy of the presented method is compared and validated with the available literature and the finite element simulation.The effects of geometrical and material parameters such as FG index,angular span along with external conditions such as magnetic field,mechanical pressure,and electrical difference are investigated in detail through numerical examples.     

Effect of slip and magnetic field on composite systems

The influence of slip velocity and transverse magnetic field on a horizontal composite (porous/ electrically conducting fluid) layer is investigated analytically and numerically. The analytical method is based on a perturbation technique while the numerical simulation is based on a finite difference scheme. Several important characteristics of the conducting flow as well as the concentration fields in the composite layer are reported. Also, the dependence of these characteristics on the dimensionless parameters of the problem are described graphically. The results of the present analysis are compared with similar results of viscous flow in the absence of a magnetic field and good agreement is found.     

Dynamic effective properties of matrix composite materials with high volume concentrations of particles

A new approach is proposed to investigate the propagation of a plane compressional wave in matrix composite materials with high volume concentrations of particles. The theory of quasicrystalline approximation and Waterman’s T matrix formalism are employed to treat the multiple scattering resulting from the particles in composites. The addition theorem for spherical Bessel functions is used to accomplish the translation between different coordinate systems. The Percus–Yevick correlation function widely applied in the molecular theory of liquids is employed to analyze the interaction of the densely distributed particles. The analytical expression for the Percus–Yevick correlation function is also given. The closed form solution for the effective propagation constant is obtained in the low frequency limit. Only numerical solutions are obtained at higher frequencies. Numerical examples show that the phase velocities in the composite materials with low volume concentration are in good agreement with those in previous literatures. The effects of the incident wave number, the volume fraction and the material properties of the particles and matrix on the phase velocity are also examined.     

Elastic field due to an edge dislocation in a multilayered composite

A numerical procedure is presented for the analysis of the elastic field due to an edge dislocation in a multilayered composite. The multilayered composite consists of n perfectly bonded layers having different material properties and thickness, and two half-planes adhere to the top and bottom layers. The stiffness matrices for each layer and the half-planes are first derived in the Fourier transform domain, then a set of global stiffness equations is assembled to solve for the transformed components of the elastic field. Since the singular part of the elastic field corresponding to the dislocation in the full-plane has been extracted from the transformed components, regular numerical integration is needed only to evaluate the inverse Fourier transform. Numerical results for the elastic field due to an edge dislocation in a bimaterial medium are shown in fairly good agreement with analytical solutions. The elastic field and the Peach–Kohler image force are also presented for an edge dislocation in a single layered half-plane, a two-layered half-plane and a multilayered composite made of alternating layers of two different materials.     

瞬态波在层合板中传播的粘弹比拟法

本文发展了粘弹比拟理论,并将之用于求解半无限空间三层复合材料在垂直层合方向传播的瞬态波问题。对于层合板中应力波的传播问题,寻找到了一等效粘弹体,并用一种较好的Laplace变换的数值反演法求得了等效松弛函数和其它一些必要的辅助函数。用特征线法求得了等效粘弹体的应力和速度,进而得到了三层复合材料中心的应力、速度,进一步就得到了层中任意点的应力和速度。对于一个可由精确理论(射线理论)给出计算结果的问题,将粘弹比拟理论的结果和射线理论的结果进行了比较,结果表明,粘弹比拟理论对三层复合材料的瞬态波传播问题是相当成功的。     

恒磁场作用下压磁/压电半导体复合圆柱壳的耦合响应分析

本文针对由压磁材料和压电半导体材料组成的无限长复合圆柱壳结构,理论研究了其在径向恒磁场作用下结构内的多场耦合力学响应问题。为解析求解方便,文中分别采用单向耦合法和线性全耦合法,导出复合圆柱壳内位移场、电场、载流子等物理量的解析表达式。利用导出的解析表达式,数值分析了磁场大小和半导体层厚度比对结构内电势、电场和载流子的影响。计算结果表明：磁场和厚度比均可用来有效调控半导体层内的电学量,复合圆柱壳结构的厚度比有一个最优区间。     

恒磁场作用下压磁/压电半导体复合圆柱壳的耦合响应分析

本文针对由压磁材料和压电半导体材料组成的无限长复合圆柱壳结构,理论研究了其在径向恒磁场作用下结构内的多场耦合力学响应问题。为解析求解方便,文中分别采用单向耦合法和线性全耦合法,导出复合圆柱壳内位移场、电场、载流子等物理量的解析表达式。利用导出的解析表达式,数值分析了磁场大小和半导体层厚度比对结构内电势、电场和载流子的影响。计算结果表明：磁场和厚度比均可用来有效调控半导体层内的电学量,复合圆柱壳结构的厚度比有一个最优区间。     

Magnetothermoelastic stress and perturbation of magnetic field vector in a functionally graded hollow sphere

In this article, a closed-form solution for one-dimensional magnetothermoelastic problem in a functionally graded material (FGM) hollow sphere placed in uniform magnetic and temperature fields subjected to an internal pressure is obtained using the infinitesimal theory of magnetothermoelasticity. Hyper-geometric functions are employed to solve the governing equation. The material properties through the graded direction are assumed to be nonlinear with an exponential distribution. The nonhomogeneity of the material in the radial direction is assumed to be exponential. The temperature, displacement and stress fields and the perturbation of magnetic field vector are determined and compared with those of the homogeneous case. Hence, the effect of inhomogeneity on the stresses and the perturbation of magnetic field vector distribution are demonstrated. The results of this study are applicable for designing optimum FGM hollow spheres.     

Three-dimensional analytical solution for a rotating disc of functionally graded materials with transverse isotropy

Based on the basic equations for axisymmetric problems of transversely isotropic elastic materials, the displacement components are expressed in terms of polynomials of the radial coordinate with the five involved coefficients, named as displacement functions in this paper, being undetermined functions of the axial (thickness) coordinate. Five equations governing the displacement functions are then derived. It is shown that the displacement functions can be found through progressive integration by incorporating the boundary conditions. Thus a three-dimensional analytical solution is obtained for a transversely isotropic functionally graded disc rotating at a constant angular velocity.The solution can be degenerated into that for an isotropic functionally graded rotating disc. A prominent feature of this solution is that the material properties can be arbitrary functions of the axial coordinate. Thus, the solution for a homogeneous transversely isotropic rotating disc is just a special case that can be easily derived. An example is finally considered for a special functionally graded material, and numerical results shows that the material inhomogeneity has a remarkable effect on the elastic field.     

Fluid capacity distributions of random porous media

As a quantitative measure of the microstructure in a statistically homogeneous porous material, we introduce the notion of thefluid capacity at a specified length scale . In two dimensions, fluid capacity is the void space per unit area for a square of side and in three dimensions it is the void space per unit volume for a cube of side . The most random distribution of fluid capacity, for a prescribed mean fluid capacity, corresponds to an exponential distribution. The distribution of fluid capacity is important during unstable fluid displacements in porous media where viscous fingering occurs. For a material with an exponential fluid capacity distribution, an unstable displacement process can be simulated by simple stochastic algorithms related to diffusion-limited aggregation. We measure the two-dimensional fluid capacity distributions of published cross-section photomicrographs of sandstone, salt, and packed beds of glass beads, for various length scales A. The form of the distribution depends upon the magnitude of the length scale . For the sandstone and salt packs, appropriate length scales are found on which the fluid capacity has, to a good approximation, an exponential distribution. An exponential distribution appears to be inappropriate for the packed bed of glass beads on all length scales.     

Spontaneous rotation in the exact solution of magnetohydrodynamic equations for flow between two stationary impermeable disks

Magnetohydrodynamic (MHD) flow of a viscous electrically conducting incompressible fluid between two stationary impermeable disks is considered. A homogeneous electric current density vector normal to the surface is specified on the upper disk, and the lower disk is nonconducting. The exact von Karman solution of the complete system of MHD equations is studied in which the axial velocity and the magnetic field depend only on the axial coordinate. The problem contains two dimensionless parameters: the electric current density on the upper plate Y and the Batchelor number (magnetic Prandtl number). It is assumed that there is no external source that produces an axial magnetic field. The problem is solved for a Batchelor number of 0–2. Fluid flow is caused by the electric current. It is shown that for small values of Y, the fluid velocity vector has only axial and radial components. The velocity of motion increases with increasing Y, and at a critical value of Y, there is a bifurcation of the new steady flow regime with fluid rotation, while the flow without rotation becomes unstable. A feature of the obtained new exact solution is the absence of an axial magnetic field necessary for the occurrence of an azimuthal component of the ponderomotive force, as is the case in the MHD dynamo. A new mechanism for the bifurcation of rotation in MHD flow is found.     

Plane acoustic wave propagation through a composite of elastic and Kelvin–Voigt viscoelastic material layers

The problem of plane wave propagation through a plane composite layer of thickness h is considered. The composite consists of periodically repeated elastic and Kelvin–Voigt viscoelastic material layers, and all layers are either parallel or perpendicular to the incident wave front. Moreover, it is assumed that the thickness of each separate layer of the composite is much less than the acoustic wave length and the thickness h of the entire composite. We study the problem by using a homogenized model of the composite, which allows us to find the reflection and transmission factors and the variation in the sound intensity level as it propagates though the composite layer of thickness h.     

Multiple scattering of elastic waves in metal-matrix composite materials with high volume concentration of particles

A new approach is proposed to investigate the propagation of compressional (P) and shear (SV) waves in metal-matrix composite materials with high volume concentration of particles. The theory of quasicrystalline approximation and Waterman's T matrix formalism are employed to treat the multiple scattering resulting from the particles in composites. The addition theorem for spherical Bessel functions is used to accomplish the translation between different coordinate systems. The analytical expression of the Percus–Yevick correlation function is also given. Closed form solutions for the effective propagation constants and the dynamic effective elastic modulus of materials are obtained in the low frequency limit. At higher frequencies, only numerical results of them are presented. Numerical examples show that the phase velocities of P and SV waves in the composite materials with low volume concentration in the low frequency are in good agreement with the results in previous literatures. The effects of the incident wave number, the volume fraction of particles and the material properties of the particles and matrix on the dynamic effective elastic modulus are also examined.     

Electrode layers in flows of an inviscid plasma with good conductivity

The structure of the electromagnetic electrode layers that are produced in flows across a magnetic field by a completely ionized and inviscid plasma with good conductivity and a high magnetic Reynolds number is examined in a linear approximation. Flow past a corrugated wall and flow in a plane channel of slowly varying cross section with segmented electrodes are taken as specific examples. The possibility is demonstrated of the formation of nondissipative electrode layers with thicknesses on the order of the Debye distance or electron Larmor radius and of dissipative layers with thicknesses on the order of the skin thickness, as calculated from the diffusion rate in a magnetic field [2].In plasma flow in a transverse magnetic field, near the walls, along with the gasdynamie boundary layers, which owe their formation to viscosity, thermal conductivity, etc. (because of the presence of electromagnetic fields, their structures may vary considerably from that of ordinary gasdynamic layers), proper electromagnetic boundary layers may also be produced. An example of such layers is the Debye layer in which the quasi-neutrality of the plasma is upset. No less important, in a number of cases, is the quasi-neutral electromagnetic boundary layer, in which there is an abrupt change in the frozen-in parameter k=B/p (B is the magnetic field and p is the density of the medium). This layer plays a special role when we must explicitly allow for the Hall effect and the related formation of a longitudinal electric field (in the direction of the veloeiryv of the medium). We will call this the magnetic layer. The magnetic boundary layer can be dissipative as well as noudissipative (see below). The dissipative magnetic layer has been examined in a number of papers: for an incompressible medium with a given motion law in [1], for a compressible medium with good conductivity in [2], and with poor conductivity in [3]. In the present paper, particular attention will be devoted to nondissipative magnetic boundary layers.     

Solution of the heat-transfer equation for equilibrium turbulent boundary layers when the temperature distribution of the streamlined surface is arbitrary

We give an approximate solution of the heat-transfer equation for equilibrium turbulent boundary layers for which the velocity distribution and the coefficient of turbulent viscosity can be described by functions of two parameters. In [1–4] equilibrium turbulent boundary layers characterized by a constant dimensionless pressure gradient were investigated. The $$\beta = \frac{{\delta ^{* \circ } }}{{\tau _w ^ \circ }}\left( {\frac{{dP}}{{dx^ \circ }}} \right)$$ profile of the velocity defect was calculated in [4] for such layers throughout the whole range ?0.5≤β≤∞, while a method was indicated in [5] for combining the defect velocity profiles with the universal profiles of the wall law, and a composite function defining the coefficient of turbulent viscosity was proposed. In this paper we construct the solution of the heat-transfer equation for equilibrium boundary layers under the assumption that the velocity distribution in the layer and the coefficient of turbulent viscosity are described by functions, obtained in [4, 5], of the dimensionless coordinateη=y/Δ, depending on two parametersβ and Re_*, while the turbulent Prandtl number Pr_t is either constant or is also a known function of η and the parametersβ and Re_*. The temperature of the surface T_w(x) is assumed to be an arbitrary function of the longitudinal coordinate and the solution is constructed in the form of series in the form parameters containing the derivatives of T_w(x). These form parameters are similar to those used in [6–9] to construct exact solutions of the equations of the laminar boundary layer.     

Diffusion of chemically reactive species of a non-Newtonian fluid immersed in a porous medium over a stretching sheet

The influence of reaction rate on the transfer of chemically reactive species in the laminar visco-elastic fluid flow immersed in a porous medium over a stretching sheet is considered. The flow is caused solely by the linearly stretching sheet and the reactive species is emitted from this sheet and undergoes an isothermal and homogeneous one-stage reaction as it diffuses into the surrounding fluid. A similarity transformation is introduced, which reduces the concentration conservation equation to an ordinary differential equation. An exact analytical solution due to Siddappa and Abel (Z. Angew. Math. Phys. 36 (1985) 890) is adopted for velocity, where as the concentration equation is obtained numerically for higher-order reactions. The numerical computations show that the effect of destructive chemical reaction is to reduce the thickness of concentration boundary layer and increase the mass transfer rate from the sheet to the surrounding fluid. This effect is more effective for zero- and first-order reaction than second- and third-order reactions.     

Unsteady hydromagnetic pipe flow at small Hartmann number

Summary In this paper we have studied the problem of the unsteady flow of an electrically conducting incompressible viscous fluid through a circular pipe under the influence of a uniform applied transverse magnetic field when the walls are non-conducting. It has been assumed that the velocity vanishes on the non-conducting walls and initially the fluid is at rest. The velocity field and the induced magnetic field are calculated by an iteration procedure and have been found up to second order terms inM (Hartmann number) which is taken to be small. We have also neglected the term involving (= 4/) the self inductance of the fluid, which is valid for small values ofM.Since the pressure gradient is not necessarily a constant in unsteady flow, we have assumed it to be an arbitrary function of time. A particular case when the pressure gradient is an exponential function of time has also been investigated in detail. For a constant pressure gradient the curves for the velocity field and the induced magnetic field have been drawn (at different values of Hartmann number and time). It has been found that for small values of time the fluid is accelerated near the centre in contrast to the case of a non-conducting fluid.     

Finite element analysis for steady-state hydride-induced fracture in metals by composite model

Delayed hydride cracking, which is observed in hydride-forming metals, due to the precipitation of hydrides near the crack tip, is investigated under conditions of constant temperature and crack velocity, plane strain and small-scale hydride-precipitation. The coupling of the operating physical processes of hydrogen-diffusion, hydride precipitation and material deformation is taken into account. The material is assumed to be an elastic composite made of hydrides and solid solution, with properties depending locally on the volume fraction of the hydrides. In the present analysis, the composite elastic properties have been derived by a generalized self consistent model for particulate composites. With respect to hydride-precipitation, two cases have been considered: (i) precipitation in a homogeneous medium with elastic properties, equal to the effective properties of the composite and (ii) precipitation in an inhomogeneous medium, where the expanding hydride has different elastic properties than those of the surrounding solid solution. The differences between the near-tip field distributions, produced by the two precipitation models, are relatively small. The effect of the hydrogen concentration far from the crack tip, on the near-tip field is also studied. It is shown that for small crack growth velocities, near the threshold stress intensity factor, the remote hydrogen concentration weakly affects the normalized stress distribution in the hydride-precipitation zone, which is controlled by the thermodynamically required hydrostatic stress, under hydrogen chemical equilibrium. However, for values of the applied stress intensity factor and the crack tip velocity, away from the threshold stress intensity factor and crack arrest, the effect of remote hydrogen concentration on the normalized near-tip stress field is strong. Reduction of the remote hydrogen concentration generally leads to reduction of the hydride-precipitation zone and increase of the near-tip stresses. Also reduction of the remote hydrogen concentration leads to distributions closer to those under hydrogen chemical equilibrium.