带裂缝的半平面弹性基本问题

本文用复变方法讨论了半平面内含若干条任意形状裂缝时的弹性基本问题,包括各向同性和各向异性两种情况,把寻求复应力函数的问题归结为求解某种带若干待定常数的正则型奇异积分方程,证明了若适当且唯一地选择这些常数的值,该方程的解存在且唯一。     

Elasticity problems involving coupled half-planes

A general method is proposed for reducing problems concerning cracks, cuts, inclusions and interacting blocks in coupled half-planes to complex integral equations, both singular and hyper-singular. The method is based on the fact that if the Kolosov-Muskhelishvili functions are known for a whole plane, then the corresponding functions for coupled half-planes are obtained from them by simple transformations. Boundary integral equations (BIE) are presented, as well as fundamental solutions for isolated forces and periodic systems of forces, which may be used to construct new complex BIEs.     

Dynamic crack analysis in 2D elastic solids with the singular edge-based smoothed finite element method

The singular edge-based smoothed finite element method (sES-FEM) is developed for stationary dynamic crack analysis in two-dimensional (2D) elastic solids. The paper aims at providing a better understanding of the dynamic fracture behaviors in linear elastic solids by means of the strain smoothing technique. The strains are smoothed and the system stiffness matrix is performed using the strain smoothing over the smoothing domains associated with the element edges. A two-layer singular five-node crack-tip element is employed while the standard implicit time integration scheme is used for solving the discrete sES-FEM equation system. Dynamic stress intensity factors (DSIFs) are extracted using the domain-form of interaction integrals in terms of the smoothing technique. The normalized DSIFs are compared with reference solutions showing a high accuracy of the sES-FEM. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical Solution for the Helmholtz Equation with Mixed Boundary Condition

We consider the numerical solution for the Helmholtz equation in R~2 with mixed boundary conditions.The solvability of this mixed boundary value problem is estab- lished by the boundary integral equation method.Based on the Green formula,we express the solution in terms of the boundary data.The key to the numerical real- ization of this method is the computation of weakly singular integrals.Numerical performances show the validity and feasibility of our method.The numerical schemes proposed in this paper have been applied in the realization of probe method for inverse scattering problems.     

Stressed state of an anisotropic body with elliptical holes, elastic inclusions, and cracks

A method is proposed for studying the two-dimensional stressed state of a multiply connected anisotropic body with cavities and elastic and rigid inclusions, as well as planar cracks and rigid laminar inclusions. Generalized complex potentials, conformal mapping, and the method of least squares are used. The problem is reduced to solving a system of linear algebraic equations. Formulas are given for finding the stress intensity factors in the case of cracks and laminar inclusions. For an anisotropic plate with a single elliptical hole or a crack and an elastic (rigid) inclusion, some numerical results are presented from a study of the effect of the rigidity of the inclusion and the closeness of the contours to one another on the distribution of stresses and the stress intensity factor. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 175–187, 1999.     

Simulation of microstructural damage evolution during very high cycle fatigue (VHCF) using the boundary element method

A simulation model for damage evolution in slip bands under VHCF condition is presented. By use of a numerical method it is applied to a real simulated microstructure of AISI304. It considers orientations of slip systems as well as individual anisotropic elastic properties in each grain. The numerical method is the two-dimensional boundary element method which is based on two integral equations and implies fundamental solutions for anisotropic elastic solids. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the determination of stress concentration in a stretched plate with two holes

We present a short survey of studies of the elastic interaction of two holes in a stretched plate. Special attention is paid to the study of the concentration of stresses on the contours of closely positioned holes. For two identical elliptic holes, numerical results are obtained by the method of singular integral equations. With the help of the limit transition, we determined the stress intensity factors at the vertices of semi-infinite parabolic notches. A comparison of the numerical data with known analytic solutions for two circular holes and collinear cracks is performed.     

A Novel Time‐Domain BIEM for Wave Propagation Analysis in Anisotropic Solids With Cracks

Analysis of elastic wave propagation in anisotropic solids with cracks is of particular interest to quantitative non‐destructive testing and fracture mechanics. For this purpose, a novel time‐domain boundary integral equation method (BIEM) is presented in this paper. A finite crack in an unbounded elastic solid of general anisotropy subjected to transient elastic wave loading is considered. Two‐dimensional plane strain or plane stress condition is assumed. The initial‐boundary value problem is formulated as a set of hypersingular time‐domain traction boundary integral equations (BIEs) with the crack‐opening‐displacements (CODs) as unknown quantities. A time‐stepping scheme is developed for solving the hypersingular time‐domain BIEs. The scheme uses the convolution quadrature formula of Lubich [1] for temporal convolution and a Galerkin method for spatial discretization of the BIEs. An important feature of the present time‐domain BIEM is that it uses the Laplace‐domain instead of the more complicated time‐domain Green's functions. Fourier integral representations of Laplace‐domain Green's functions are applied. No special technique is needed in the present time‐domain BIEM for evaluating hypersingular integrals.     

ON THE FUNDAMENTAL PROBLEM FOR AN INFINITE ELASTIC PLANE BONDED BY DIFFERENT ANISOTROPIC MATERIALS WITH CRACKS

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实Clifford分析中六类拟Bochner-Martinelli型高阶奇异积分的几个问题

本文借助于Ｈａｄａｍａｒｄ关于高阶奇异积分有限部分的思想,研究关于实 Ｃｌｉｆｆｏｒｄ分析中六个类型(含一个奇点或二个奇点的)拟Ｂｏｃｈｎｅｒ－Ｍａｒｔｉｎｅｌｌｉ型高阶奇异积分的归纳定义、Ｈａｄａｍａｒｄ主值的存在性、递推公式、计算公式、微分公式、Ｐｏｉｎｃａｒｅ－Ｂｅｒｔｒａｎｄ置换公式以及拟Ｂ－Ｍ型高阶奇异积分的Ｈｏｌｄｅｒ连续性等问题．这些问题是研究单、多元复分析的学者们在研究奇异积分时,通常要涉及到的几个问题．     

Reconstruction of cracks in an inhomogeneous anisotropic elastic medium

In this paper we give in two and three dimensions a reconstruction formula for determining cracks buried in an inhomogeneous anisotropic elastic body by making elastic displacement and traction measurements at the boundary. The information is encoded in the local Neumann-to-Dirichlet map. With the help of the Runge property, the local Neumann-to-Dirichlet map is connected to the so-called indicator function. This function can be expressed as an energy integral involving some special solutions, called reflected solutions. The heart of our method lies in analyzing the blow-up behavior at the crack of the indicator function, which is by no means an easy task for the inhomogeneous anisotropic elasticity system. To overcome the difficulties, we construct suitable approximations of the reflected solutions that capture their singularities. The indicator function is then analyzed by the Plancherel formula.     

Computational modeling of elastic and plastic multiple cracks by the fundamental solutions

The fundamental solutions of elasticity are used to establish a numerical method for elastic and plastic multiple crack problems in two dimensions. The continuous distributions of the point forces, dislocations, and the plastic sources are used systematically to model the crack, non-crack boundary, and the plastic deformation. Use of these singularities are guided strictly by the physical interpretation of the problem. We adopt Muskhelishvili's complex variable formalism that facilitate the analytical evaluation of the integrals representing the continuous distributions of the singularities. The resulting numerical method is concise and accurate enough to be used for elastic and plastic multiple crack problems.     

Hybrid boundary element formulation for acoustic wave propagation

The so‐called hybrid stress boundary element method (HSBEM) is introduced in a frequency domain formulation for the computation of acoustic radiation and scattering in closed and in finite domains. Different from other boundary element formulations, the HSBEM is based on an extended Hellinger‐Reissner variational principle and leads to a Hermitian, frequency‐dependent stiffness equation. Due to this, the method is very well suited for treating fluid structure interaction problems since the effort for the coupling the structure, discretized by a finite elements, and the fluid, discretized by the HSBEM is strongly reduced. To arrive at a boundary integral formulation, the field variables are separated into boundary variables, which are approximated by piecewise polynomial functions, and domain variables, which are approximated by a superposition of singular fundamental solutions weighed by source strength. This approximation cancels the domain integral over the equation of motion in the hybrid principle and leads to a boundary integral formulation, incorporating singular integrals. Comparing to previous results published by the authors, new considerations concerning the interpretation of singular contributions in the stiffness matrix for exterior domain problems are communicated here.     

Trapped-mode and gap-band effects in elastic waveguides with obstacles

Traveling wave propagation in elastic waveguides with obstacles in the form of cracks, voids, inclusions or surface irregularities is considered. The investigation is focused on the trapped–mode phenomena featured by the time–averaged harmonic wave energy localization near the obstacles in the form of energy vortices. The latter results, in particular, in narrow gap bands in the frequency plots of transmission coefficients. The study is carried out using analytically based computer models relying on wave expressions in terms of path Fourier integrals, Green's matrices for the laminate structures, and asymptotics for body and traveling waves derived from those integrals. The connection between the resonance effects and natural frequencies (spectral points of the related boundary value problems) in the complex frequency plane is analyzed as well. Examples of spectral points touching the real axis in the course of varying crack size are presented. The eigenforms associated with such discrete spectral points lying in a continuous spectrum depict strong wave energy localization. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Construction of fundamental solutions of two-dimensional problems of the nonstationary theory of elasticity

We propose a method of constructing the images of the fundamental solutions in the space of the Laplace transform with respect to time, leading to simple formulas. The method is illustrated using three dynamical problems: planar deformation for an anisotropic body; flexural vibrations of an anisotropic plate; and vibrations of a shallow isotropic shell of arbitrary Gaussian curvature. Quadrature formulas are given for computing the values of the fundamental solutions. We give a new interpretation and a new method of computing the values of the special functions used in the construction of singular solutions in problems of the static theory of shells. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 86–92.     

SINGULAR INTEGRALS IN SEVERAL COMPLEX VARIABLES(Ⅱ)——HADAMARD PRINCIPAL VALUE ON A SPHERE

Hadamard introduced the concept of finite parts of divergent integrals.i.e.Hadamardprincipal value,when he researched the Cauehy problems of the hyperbolic type partialdifferential equations.In this paper,the authors try to generalize this concept to the singularintegrals on a sphere of several complex variables space C~n.The Hadamard principal valueof higher order singular integralis defined and the corresponding Plemelj formula is obtained.     

Asymptotics of negative eigenvalues of the Dirichlet problem with the density changing sign

In a three-dimensional anisotropic elastic space with either a bounded foreign inclusion or a void, we derive asymptotic formulas for the increment of the polarization tensor of a defect caused by a smooth variation of the defect boundary. The formulas involve weighted integrals of jumps of the surface enthalpy evaluated for solutions to the problem about deformation of an unperturbed composite space by constant stress at infinity. The study of the positiveness/negativeness of the polarization matrix increment leads to inferences with a clear physical interpretation, in particular, for elastic solids admitting phase transitions. For homogeneous ellipsoid shaped inclusions we derive a relation between the polarization tensor and the Eshelby tensor and obtain miscellaneous consequences of this relation as well. In particular, we introduce the notion of the link tensor which is symmetric and positive definite for any elastic properties of homogeneous materials of the composite space. Bibliography: 60 titles. Illustrations: 5 figures. Dedicated to Nina Nikolaevna Uraltseva Translated from Problemy Matematicheskogo Analiza, 41, May 2009, pp. 3–36.     

复合材料焊接线出现裂缝的平面弹性基本问题

本文用复变方法讨论了复合材料任意形状焊接线上出现若干条裂缝时的平面弹性第一和第二基本问题，把寻求复应力函数的问题分别归结为求解某种正则型奇异积分方程和正则型奇异积分方程组，并证明了其解存在且唯一。     

A semi-analytic solution for multiple curved cracks emanating from circular hole using singular integral equation

This paper provides an elastic solution for an infinite plate containing multiple curved edge cracks emanating from a circular hole. A fundamental solution is suggested, which represents a particular solution for a concentrated dislocation in an infinite plate with the traction free hole. The generalized image method and the concept of the modified complex potentials are used in the derivation of the fundamental solution. After using the fundamental solution and placing the distributed dislocations at the prospective sites of cracks, a singular integral equation is formulated. The singular integral equation is solved by using the curve length method in conjunction with the semi-opening quadrature rule. By taking an additional point dislocation at the hole center, the number of the unknowns is equal to the number of the resulting algebraic equations. This is a particular advantage of the suggested method. Finally, several numerical examples are given to illustrate the efficiency of the method presented. Numerical examinations are carried out and sufficient accurate results have been found.     

Global approaches for accurate loading analyses and crack path predictions at single and two-cracks systems

Based on the work by Eshelby, the path-independent J_k-, M-, L- and interaction- or I_k-integrals were introduced and applied to cracks for the accurate calculation of crack tip loading quantities. Applying the FE-method to solve boundary value problems with cracks, numerically inaccurate values are observed within the crack tip region affecting the accuracy of local approaches. Simulating crack paths, local approaches face further problems as cracks are running towards interfaces, internal boundaries or other crack faces. Within global approaches, path-independent integrals are calculated along remote contours far from the crack tip, essentially exploiting numerically reliable data requiring special treatment only for the near-tip crack faces. To provide path-independence, additional integrals along interfaces, internal boundaries and crack faces are necessary. In this paper, new global approaches of path-independent integrals are presented and applied to the calculation of crack paths at two-cracks systems. A second focus is directed to the accurate loading analysis and crack path prediction considering anisotropic properties and material interfaces. The numerical model provides crack paths which are in good agreement with those obtained from crack growth experiments. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)