含椭圆夹杂二维各向异性电磁弹性介质的耦合问题

研究了无限大二维各向异性电磁弹性固体的椭圆夹杂问题.基于广义Stroh方法、保角变换和扰动概念,得到了基体和夹杂内电场、磁场和弹性场的封闭解.本文结果可用于研究电磁弹性复合材料的有效性能.     

半平面压电体的Green函数及其应用

本文研究半平面压电体在线力、电荷和位错作用下的弹性场和电场，即Green函数．基于各向异性弹性力学中的Stroh方法和解析延拓理论，推导了Green函数的封闭形式的解．作为解的应用，分析了含半无限裂纹的无限大压电介质的机电耦合场，给出了应力和电位移强度因子的解析表达式．     

Surface waves in electrostrictive materials under biasing fields

The theory of small incremental fields superposed on finite biasing fields in an electroelastic body is summarized first in this paper. The Stroh formalism for two-dimensional anisotropic elasticity and piezoelectricity is generalized to two-dimensional problems of an electroelastic body under biasing mechanical and electric fields. Application of this generalized Stroh formalism to study surface waves propagating in an electroelastic half-space under uniform initial fields indicates that there exist at most two surface waves even in the presence of biasing fields and these two surface waves correspond to the well-known Rayleigh and Bleustein waves in the mechanical nature. A simple procedure is established to distinguish these two types of waves. Application of the results to a model electrostrictive material is included for illustration. It is found that initial biasing fields can greatly influence propagation feature of surface waves.     

含界面电极压电介质的Green函数

研究了由两个不同压电材料和一半无限长电极组成的复合材料系统的广义二维问题·　基于Stroh公式,提供了当一个线力、线电荷和一个线电偶极子施加在电极端附近时,精确的Green函数解·　进一步地,获得了相应的场强度系数·　这些结果可作为边界元的基本解,以分析更加复杂的压电复合材料断裂问题·     

Fundamental elastic field in an infinite plane of two-dimensional piezoelectric quasicrystal subjected to multi-physics loads

By utilizing the extended Stroh formalism, the Green's function of infinite plane is obtained for the problem of two-dimensional decagonal quasicrystals with the piezoelectric effect subjected to multi-physics loads. By numerical computations, the piezoelectric effect of the two-dimensional decagonal quasicrystals is revealed; the changes of the stress and displacement fields with multi-physics loads are discussed. The variation laws of material constants in stress and displacement fields are investigated. The results show that the effect of the phason field on the generalized displacement is larger than that on the generalized stress; and the effects of material parameters are different in diverse field.     

Singular stress analysis near sharp corners in anisotropic notched plates subjected to bending loads

This paper presents an analytical approach to determine the singular stress fields near sharp corners in anisotropic notched plate subjected to bending. The first-order shear deformation plate theory (FSDPT) is used and the material anisotropy is modeled by the Stroh formalism. Based on the Mellin transform method, the stress singularity orders as well as the stress intensities are determined analytically. The results presented in this paper give us the singular behaviors for various plate angle α, material orientation angles ϕ and the anisotropic parameters ε. These new findings are helpful in reducing or even eliminating the singularities near the sharp corner.     

Averaging the resolvent with a space-correlated random potential

The problem of the quasi-particle spectrum in a binary disordered alloy with a space-correlated random potential is considered. The extended space formalism is used to represent the average resolvent. To calculate the mass operator, some self-consistent approximation procedures are suggested that coincide with the well-known self-consistent approximations for α=0 (where α is the short-range order parameter). The elaborated theory ensures the correct passage to the Green's function of a perfect crystal in the limits α→1 and α→−1 for any concentration and 50% concentration, respectively. The approximations possess the correct analytic properties for all values of the parameter α. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 114, No. 2, pp. 296–313, February, 1998.     

Green's functions for elliptic and parabolic equations with random coefficients II

This paper is concerned with linear parabolic partial differential equations in divergence form and their discrete analogues. It is assumed that the coefficients of the equation are stationary random variables, random in both space and time. The Green's functions for the equations are then random variables. Regularity properties for expectation values of Green's functions are obtained. In particular, it is shown that the expectation value is a continuously differentiable function in the space variable whose derivatives are bounded by the corresponding derivatives of the Green's function for the heat equation. Similar results are obtained for the related finite difference equations. This paper generalises results of a previous paper which considered the case when the coefficients are constant in time but random in space.

     

Green's function in plane anisotropic bimaterials with imperfect interface

^** Email: lsudak{at}ucalgary.ca The fundamental solutions or Green's functions for 2D or 3Danisotropic media with imperfect interface remain a challengingproblem. In this paper, a general method is presented for therigorous solution for the 2D Green's function in an anisotropicelastic bimaterial subject to a line force or a line dislocation.Most significant is the fact that the bonding along the bimaterialinterface is considered to be homogeneous imperfect. Specifically,the tractions are continuous but the displacements are discontinuousand proportional, in terms of interface stiffness parameters,to their respective traction components. Using complex variabletechniques, the basic boundary-value problem for two analyticvector functions is reduced to a coupled linear first-orderdifferential equation for a single analytic vector functiondefined in the lower half space. The coupled linear differentialequation for the single analytic vector function can be subsequentlydecoupled into three independent linear first-order differentialequations for three newly defined analytic functions. Closed-formsolutions for the 2D Green's function are derived in terms ofthe exponential integral. Unlike previous works which involvesome sort of inverse transform method to obtain the physicalquantities from the transform domain, the key feature of thepresent method is that the physical quantities can be readilycalculated without the need to perform any inverse transformoperations.     

压电压磁复合材料中二维散射问题的解析研究

用极化方法分析了含一二维夹杂的无限压电压磁基体中的波动散射问题．以此为目的,首先构建了二维压电压磁“相对体”的极化方法．当一般性波动退减为简谐振动时,极化方法的核心函数退减为二维谐波Green函数．利用氡变换的解析方法,首次求得了二维谐波Green函数的积分表达式,该表达式在低频初始波与小尺度椭圆柱夹杂物的假设下可得到进一步的简化,并最终求得解析解．推导针对同时具有压电以及压磁效应的一般性各向异性材料进行,然后将所得的结果简化到仅针对压电复合材料的情况．以此简化解析解为基础,提供了两个算例,讨论了影响含一二维椭圆柱夹杂的PZT-5H压电陶瓷复合材料的散射截面的各种不同因素（包括夹杂的尺寸、形状效应,材料常数的影响,以及压电效应等）．     

On interface waves in misoriented pre-stressed incompressible elastic solids

Some relationships, fundamental to the resolution of interfacewave problems, are presented. These equations allow for thederivation of explicit secular equations for problems involvingwaves localized near the plane boundary of anisotropic elastichalf-spaces, such as Rayleigh, Scholte, or Stoneley waves. Theyare obtained rapidly, without recourse to the Stroh formalism.As an application, the problems of Stoneley wave propagationand of interface stability for misaligned predeformed incompressiblehalfspaces are treated. The upper and lower half-spaces aremade of the same material, subject to the same prestress, andare rigidly bonded along a common principal plane. The principalaxes in this plane do not, however, coincide, and the wave propagationis studied in the direction of the bisectrix of the angle betweena principal axis of the upper half-space and a principal axisof the lower half-space.     

半空间中一类调和函数的例外集

利用Whitney方体的相关性质, 给出了一类调和函数在半空间中无穷远点处的增长估计, 且刻画了其 例外集的几何性质. 本文推广了张艳慧和邓冠铁在半空间中的相关结果.     

Green's function of the Brinkman equation in a 2D anisotropic case

The purpose of this paper is to obtain the Green's functionof the Brinkman equation in a 2D case of hydrodynamic anisotropywith respect to the permeability. The anisotropic nature ofthe permeability is assumed to be not space or time dependent.We use the method of Fourier transform which reduces the computationof the Green's function to the computation of the fundamentalsolution of a fourth-order partial differential equation. Thisresearch work has several applications in engineering and medicineto the motion of bodies in anisotropic porous media.     

压电介质内裂纹问题的精确解

在压电介质断裂力学分析中,人们常假定裂纹面上的电位移法向分量为零,可是实验表明,这一假设将导致错误的结果。本文基于精确的电边界条件,并应用Stroh公式的方法,导出了含裂纹压电介质在无限远处均匀外载作用下二维问题的精确解。结果表明:(ⅰ)应力强度因子与各向同性材料相同,而电位移强度因子取决于材料常数和机械载荷,但与电载荷无关;(ⅱ)能量释放率大于纯弹性各向异性材料内的值,即总是正的,且与电载荷无关;(ⅲ)裂纹内所含空气的介电常数对介质内的场强无影响。     

On an approach to the construction of a solution of the generalized coupling problem

We propose a method of solving generalized coupling problems by reducing them to an inhomogeneous differential equation in the space of generalized functions that can be solved using a normalized fundamental function or a Green's function.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 198–203.     

Molecular hydrodynamics of degenerate weakly nonideal bose gas. I. Generalized equations of two-fluid hydrodynamics

A degenerate weakly nonideal Bose gas is investigated at temperatures near zero. Relations connecting the irreducible Green's functions are used to obtain exact expressions for the two-time temperature-dependent Green's functions. In the case of weak nonideality an expression possessing interpolation properties with respect to the frequency and momentum is obtained for the density—density Green's function. At low frequencies, the results of two-fluid hydrodynamics are reproduced. At wavelengths less than the mean free path the energy of the elementary excitations and the damping obtained in the paper agree with the results of perturbation theory with respect to the coupling constant. An expression for the operator of the superfluid velocity is obtained.In memory of Nikolai Nikolaevich BogolyubovV. A. Steklov Mathematics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 93, No. 3, pp. 412–465, December, 1992.     

SH波对界面含有半圆形脱胶的圆柱形弹性夹杂的散射的近似分析

采用Green函数法、复变函数法研究了SH波对界面附近含有半圆形脱胶的圆柱形弹性夹杂的散射,并给出了动应力集中系数的数值结果．首先,界面将整个空间分成上下两部分．在下半空间,给出在含有半圆形凸起的圆柱形弹性夹杂的弹性半空间中,水平表面上任意一点承受时间谐和的出平面线源荷载作用时的位移函数．其次,取该位移函数作为Green函数．上下空间连接时在界面处满足连续性条件,构造出半圆形脱胶裂纹,进而求出应力和位移的表达式．最后作为算例,给出了动应力集中系数的数值结果,分析了介质参数和入射波参数对动应力集中的影响情况．     

Boundary value problems for higher order linear impulsive differential equations

In this paper higher order linear impulsive differential equations with fixed moments of impulses subject to linear boundary conditions are studied. Green's formula is defined for piecewise differentiable functions. Properties of Green's functions for higher order impulsive boundary value problems are introduced. An appropriate example of the Green's function for a boundary value problem is provided. Furthermore, eigenvalue problems and basic properties of eigensolutions are considered.     

Function theory for Laplace and Dirac-Hodge Operators in hyperbolic space

We develop basic properties of solutions to the Dirac-Hodge and Laplace equations in upper half space endowed with the hyperbolic metric. Solutions to the Dirac-Hodge equation are called hypermonogenic functions, while solutions to this version of Laplace's equation are called hyperbolic harmonic functions. We introduce a Borel-Pompeiu formula forC ¹ functions and a Green's formula for hyperbolic harmonic functions. Using a Cauchy integral formula, we introduce Hardy spaces of solutions to the Dirac-Hodge equation. We also provide new arguments describing the conformal covariance of hypermonogenic functions and invariance of hyperbolic harmonic functions and introduce intertwining operators for the Dirac-Hodge operator and hyperbolic Laplacian. Research supported by the National Science Foundation of China (Mathematics Tianyuan Foundation, No A324610) and Hebei Province (105129) Research supported by Academy of Finland     

Evaluation of singular integrals for anisotropic elastic boundary element analysis

To improve the numerical evaluation of weakly singular integrals appearing in the boundary element method, a logarithmic Gaussian quadrature formula is usually suggested in the literature. In this formula the singular function is expressed in terms of the distance between source point and field point, which is a real variable. When an anisotropic elastic solid is considered, most of the existing fundamental solutions are written in terms of complex variables. When the problems with holes, cracks, inclusions, or interfaces are considered, to suit for the shape of the boundaries usually a mapping function is introduced and then the solutions are expressed in terms of mapped complex variables. To deal with the trouble induced by the complex variables, in this study through proper change of variables we develop a simple way to improve the evaluation of weakly singular integrals, especially for the problems of anisotropic elastic solids containing holes, cracks, inclusions, or interfaces. By simple matrix expansion, the proposed method is extended to the problems with piezoelectric or magneto-electro-elastic solids. By using the dual reciprocity method, the proposed method employed for the elastostatic fundamental solution can also be applied to the elastodynamic analysis.