A rigid inclusion in an elastic space under the action of a uniform heat flow in the inclusion plane

A solution is presented for the three dimensional static thermoelastic problem of an absolutely rigid inclusion (anticrack) in the case when a uniform heat flow is directed along the inclusion plane. By using the potential method and the Fourier transform technique, the problem is reduced to a system of coupled two-dimensional singular integral equations for the shear stress jumps across the inclusion. As an illustration, a typical application to the circular anticrack is presented. Explicit expressions for the thermal stresses in the inclusion plane are obtained and discussed from the point of view of material failure.     

Elastodynamic wave scattering by an incompressible elastic inclusion

The problem of scattering of time-harmonic elastodynamic waves by an incompressible elastic inclusion is solved by means of the null field approach. The solution is obtained both directly and as a limit of the solution to the corresponding problem for a compressible inclusion. It is also demonstrated that the null field solution to the problem of scattering by a rigid movable scatterer can be obtained from a null field solution for the incompressible scatterer by taking the limit of infinite shear modulus. Some numerical results for spherical and spheroidal inclusions are given.     

On the investigation of elasticity equation for orthotropic materials and the solution of the associated scattering problem

The present work concerns the investigation of the two-dimensional direct scattering problem of time-harmonic elastic waves from bounded anisotropic components of isotropic media. We obtain a Fourier series expansion for the elastic field in the interior of the anisotropic inclusion based on a suitable diagonalization applied to the underlying differential system and a plane wave expansion of the sought field, provided that the inclusion exhibits orthotropic symmetry. This expansion is then exploited to acquire a semi-analytical solution to the associated elastic transmission scattering problem. Numerical results for several geometric configurations and varying degree of anisotropy are presented revealing the pronounced effect of the specific anisotropic character on the scattering mechanism.     

Scattering of plane monochromatic waves from a heterogeneous inclusion of arbitrary shape in a poroelastic medium: An efficient numerical solution

Scattering of plane longitudinal monochromatic waves from a heterogeneous inclusion of arbitrary shape in an infinite poroelastic medium is considered. Wave propagation in the medium is described by Biot’s equations of poroelasticity. The scattering problem is formulated in terms of the volume integral equations for displacements of the solid skeleton and fluid pressure in the pore space in the region occupied by the inclusion. An efficient numerical method is applied to solve these equations. In the method, Gaussian approximating functions are used for discretization of the problem. For regular node grids, the matrix of the discretized problem has Toeplitz’s properties, and the Fast Fourier Transform technique can be used for the calculation of matrix–vector products. The latter accelerates substantially the process of iterative solution of the discretized problem. For material parameters of typical sedimentary rocks, the system of differential equations of poroelasticity contains a differential operator with a small parameter. As the result, the wave field in the inclusion region is split up into a slowly changing part, and boundary layer functions concentrated near the inclusion interface. The method of matched asymptotic expansions is used for the numerical solution in this case. For a spherical inclusion, the results of the numerical and matched asymptotic expansion methods are compared with a semi-analytical series solution. For a non-spherical heterogeneous inclusion, an example of the numerical solution is presented.     

Scattering of rayleigh lamb waves from a 2d-cavity in an elastic plate

The T-matrix method is applied to the problem of scattering of Rayleigh-Lamb modes from a twodimensional cavity in an elastic plate. A formal solution is obtained which is valid also for non-planar surfaces. Explicit expressions and numerical results are given for a plate with plane surfaces.     

Scattering far field solution of SH-wave by movable rigid cylindrical interface inclusion

The Green's function is used to solve the scattering far field solution of SH-wave by a movable rigid cylindrical interface inclusion in a linear elastic body. First, a suitable Green's function is developed, which is the fundamental displacement solution of an elastic half space with a movable rigid half-cylindrical inclusion impacted by out-of-plane harmonic line source loaded at any point of its horizontal surface. By using the Green's function, a series of Fredholm integral equations of the first kind which determine the scattering far field can be set up. Then the paper gives the expressions on the far field including the displacement mode of scattering wave and the solution of scattering cross-section. Finally, some examples and numerical results are discussed to analyze the influence of the combination of different media parameters on the answer of far field.     

Analysis of transient wave scattering from an inclusion with an unilateral smoothly contact interface by BEM

Summary A 2D time-domain Boundary Element Method (BEM) is applied to solve the problem of transient scattering of plane waves by an inclusion with a unilateral smooth contact interface. The incident wave is assumed strong enough so that localized separations take place along the interface. The present problem is indeed a nonlinear boundary value problem since the mixed boundary conditions involve unknown intervals (separation and contact regions). In order to determine the unknown intervals, an iterative technique is developed. As an example, we consider the scattering of plane waves by the cross section of a circular cylinder embedded in an infinite solid. Numerical results for the near field solutions are presented. The distortion of the response waves and the variation of the interface states are discussed. The financial support by the China National Natural Science Foundation under Grant No. 19872001 and No. 59878004 is gratefully acknowledged. The second author is also grateful to the support of the National Science Fund for Distinguished Young Scholars under Grant No. 10025211.     

On the Solution of an Elliptical Inhomogeneity in Plane Elasticity by the Equivalent Inclusion Method

Stress analysis of an elliptical inhomogeneity in an infinite isotropic elastic plane is a classical elasticity problem, which is usually solved by means of the complex variable formulation. In this work, we demonstrate that an alternative method of solution for such a problem, via the equivalent inclusion method, may be more convenient and straightforward without recourse to complex potentials or curvilinear coordinates. The explicit analytical solution can be derived through simple algebraic manipulation, although the longitudinal eigenstrain component should be handled with care in the case of plane strain. Since the exterior Eshelby tensor for an elliptical inclusion is available in closed-form, the present study provides a full field stress solution expressed in Cartesian coordinates. Furthermore, the in-plane stress components are represented in terms of Dundurs’ parameters. The solution methodology and the convenient formulae of the stress concentration may be of practical use to the engineers in developing benchmarks for design evaluation.     

界面脱胶圆夹杂对SH波散射的远场解

采用Green函数方法和复变函数法研究了SH波对界面脱胶圆夹杂的散射问题,并给出 了远场解答. 首先,沿双质材料界面将整个空间分成上下两部分, 在下半空间,给出了在水 平表面上任意一点承受时间谐和的出平面线源载荷作用时的位移函数,取该位移函数作为 Green函数. 其次,在下半空间,利用相关文献给出的Green函数,在上下空间连接时在双质材料界面处满足 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,建立积分方 程组, 给出了散射波远场位移模式和散射截面的解答, 分析了在不同参数变化时SH 波散射的远场特性. 结果表明,脱胶结构的存在对位移和散射截面有较大的放大作用.     

Scattering of plane SH-waves on cylindrical canyon of arbitrary shape in anisotropic media

The complex function method is used to solve problems of scattering of plane SH-waves on cylindrical canyon topography of arbitrary shape in anisotropic media. This paper gives the complete function series and general expressions with boundary condition to approach the solution of steady state scattering of plane SH-waves on two-dimensional canyon topography in anisotropic media. The problem to be solved can be reduced to a solution of infinite algebraic equation series by using Hermite function and it's orthogonal conditions. The solution can be obtained directly by using computers. Finally, as an example, computational results of scattering of plane SH-waves on a semi-cylindrical canyon topography are presented. The projects sponsored by The Joint Seismological Science Foundation.     

Propagation of axial shear magneto–electro-elastic waves in piezoelectric–piezomagnetic composites with randomly distributed cylindrical inhomogeneities

The integral equations of the scattering problem for piezoelectric–piezomagnetic composites with an inhomogeneity are derived. In the long-wave limit, the solutions of these integral equations for the composites containing a single inhomogeneous fiber are solved in close forms. The total scattering cross-section for the one-fiber composites is also obtained. By the so-called effective field method, the multi-fiber scattering problem is simplified to the one-fiber scattering problem, and the analytical expressions of magneto–electro-elastic fields for the multi-fiber composites are obtained in the long-wave limit. These solved magneto–electro-elastic fields are then used to solve the expressions of the static effective moduli, effective wave velocity and attenuation factor of piezoelectric–piezomagnetic composites with randomly distributed cylindrical inhomogeneities. Through numerical examples, it concludes that, if the random set of fiber cross-sections is homogeneous and isotropic, the effective field method is coincident with the Mori–Tanaka mean field method when the static effective moduli of piezoelectric–piezomagnetic composites are looked for. Moreover, the rules of the effective wave velocity versus the volume fraction of fibers are investigated for specific materials.     

二维直角平面内固定圆形夹杂对稳态入射反平面剪切波的散射

利用复变函数法、多极坐标及傅立叶级数展开技术求解了二维直角平面内固定圆形夹杂对稳态入射反平面剪切（shearing horizontal, SH）波的散射问题。首先构造出介质内不存在夹杂时的入射波场和反射波场,然后建立介质内存在夹杂时由夹杂边界产生的能够自动满足直角边应力自由条件的散射波解，从而利用叠加原理写出介质内的总波场。利用夹杂边界处位移条件和傅立叶级数展开方法列出求解散射波中未知系数的无穷代数方程组，在满足计算精度的前提下通过有限项截断，得到相应有限代数方程组的解，最后通过算例具体讨论了二维直角平面水平边界点的位移幅度比和相位随量纲一波数、入射波入射角及夹杂位置的不同而变化的情况，结果表明了算法的有效实用性。     

The rigid movable inclusion in elastostatics and elastodynamics

The problem of a single rigid movable inclusion is solved both for elastostatics and elastodynamic, using the null field approach. It is also shown how the solution can be obtained as the limit of the solution for an elastic inclusion. Numerical results for scattering by a superspheroidal inclusion are given.     

弹性椭圆夹杂的线性温变问题

研究了线性温变作用下椭圆夹杂的热弹性问题。通过构造辅助函数，将复变函数的分区全纯函数理论，Riemann边值问题和Cauchy型积分相结合，求得各分区之间的解析关系，从而获得了无穷远均匀加载和线性温变共同作用下椭圆夹杂平面热弹性场的封闭形式解。从本文解答的特殊情况可直接得到已有的若干结果，并可得到一些具有实际意义的新结果。本文发展的分析方法，为求解复杂多连通域的平面热弹性问题提供了一条有效途径。     

Exact solution of a semi-infinite crack in an infinite piezoelectric body

Summary The paper presents an exact and complete solution of the problem of a semi-infinite plane crack in an infinite transversely isotropic piezoelectric body. The upper and lower crack faces are assumed to be loaded symmetrically by a couple of normal point forces in opposite directions and a couple of point charges. The solution is derived through a limiting procedure from the one of a penny-shaped crack. The expressions for the elastoelectric field are given in terms of elementary functions. Received 10 August 1998; accepted for publication 18 November 1998     

直角平面内弹性圆夹杂对入射平面SH波的散射

利用复变函数法、多极坐标移动技术及傅立叶级数展开求解二维直角平面内圆形弹性夹杂对稳态入射平面SH波的散射问题。首先写出直角平面内不含夹杂时的入射波场和反射波场;其次建立直角平面内含夹杂时夹杂外的散射波解和夹杂内的驻波解,并利用叠加原理写出问题的总波场,借助夹杂边界处应力和位移的连续条件建立求解散射波解和驻波解中未知系数的无穷代数方程组并求解,通过算例具体讨论了直角平面水平边界点的位移幅度比和夹杂边界处径向应力集中系数随不同无量纲波数、入射角及圆孔位置的变化情况,结果表明了算法的有效实用性。     

压电材料反平面应变状态的任意形状夹杂问题

应用复函数的Ｆａｂｅｒ级数展开方法，分析了含任意形状夹杂的压电材料反平面应变问题，给出了问题的复势函数解。利用这个解，具体讨论了椭圆形夹杂及其极限（几何方面与物理方面）问题。并给出了三角形、正方形夹杂的近似结果。其特例结果与早期工作一致     

Strain gradient solution for a finite-domain Eshelby-type plane strain inclusion problem and Eshelby’s tensor for a cylindrical inclusion in a finite elastic matrix

A solution for the finite-domain Eshelby-type inclusion problem of a finite elastic body containing a plane strain inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived in a general form using a simplified strain gradient elasticity theory (SSGET). The formulation is facilitated by an extended Betti’s reciprocal theorem and an extended Somigliana’s identity based on the SSGET and suitable for plane strain problems. The disturbed displacement field is obtained in terms of the SSGET-based Green’s function for an infinite plane strain elastic body, which differs from that in earlier studies using the three-dimensional Green’s function. The solution reduces to that of the infinite-domain inclusion problem when the boundary effect is suppressed. The problem of a cylindrical inclusion embedded concentrically in a finite plane strain cylindrical elastic matrix of an enhanced continuum is analytically solved for the first time by applying the general solution, with the Eshelby tensor and its average over the circular cross section of the inclusion obtained in closed forms. This Eshelby tensor, being dependent on the position, inclusion size, matrix size, and a material length scale parameter, captures the inclusion size and boundary effects, unlike existing ones. It reduces to the classical elasticity-based Eshelby tensor for the cylindrical inclusion in an infinite matrix if both the strain gradient and boundary effects are not considered. Numerical results quantitatively show that the inclusion size effect can be quite large when the inclusion is very small and that the boundary effect can dominate when the inclusion volume fraction is very high. However, the inclusion size effect is diminishing with the increase of the inclusion size, and the boundary effect is vanishing as the inclusion volume fraction becomes sufficiently low.     

Wiener-Hopf analysis of an acoustic plane wave in a trifurcated waveguide

In this paper, we have made Wiener-Hopf analysis of an acoustic plane wave by a semi-infinite hard duct that is placed symmetrically inside an infinite soft/hard duct. The method of solution is integral transform and Wiener-Hopf technique. The imposition of boundary conditions result in a 2 × 2 matrix Wiener-Hopf equation associated with a new canonical scattering problem which is solved by using the pole removal technique. In the solution, two infinite sets of unknown coefficients are involved that satisfy two infinite systems of linear algebraic equations. These systems of linear algebraic equations are solved numerically. The graphs are plotted for sundry parameters of interest. Kernel functions are also factorized.     

Elasticity-theoretic plane problem for an anisotropic body subject to an electrical effect

The elastic properties of the body are assumed inseparable from its electrical properties (piezoelectric effect). The static problem is considered for an infinitely long homogeneous rod under the action of a plane system of forces or a lengthwise uniform distribution of electric field potential. The general equations of theory of elasticity and of electrical field theory, together with the piezoelectric relationships, are used. Conditions on the material parameters are obtained, under which a plane electric field and a plane deformation state appear in the body.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 2, pp. 96–103. March–April, 1970.