Stress singularity at a crack tip for various intermediate zones in bimaterial structures (mode III)

The influence of the geometry of a thin intermediate zone on the stress distribution has been investigated in the vicinity of a crack tip in a bimaterial structure. Corresponding modelling boundary value problems are reduced to functional-difference equations by the Mellin transform technique, and later to singular integral equations with fixed point singularities. It has been observed that the order of the stress singularity is essentially dependent on the model parameters. Numerical results concerning the stress singularity exponents and generalized stress intensity factors are presented.     

The plane solution for bending of joined dissimilar elastic semi-infinite strips

The problem is reduced to a system of two singular integral equations for determining the interface slope and shear stress. The dominant part of the system is analyzed to determine the order of the stress singularity and its dependence on the elastic constants. After removing logarithmic singularities from the right hand sides we solve these equations numerically for several chosen composites and the interface slope and traction are exhibited graphically. The solution should be relevant in studying adhesive joints by means of a bending test.     

Plane steady vibration of an elastic wedge

The problem of plane steady vibration of an elastic wedge of arbitrary angle (less than 180 degrees) subject to harmonic normal and shearing tractions on its faces is reduced to a system of singular integral equations by the superposition of two half-plane solutions. The integral equations have kernels with Cauchy singularities of a non-translation type, except for the 90 degree wedge. The locations of these singularity lines are shown graphically as a function of wedge angle.     

Numerical analysis of singular solutions of two-dimensional problems of asymmetric elasticity

An algorithm for the numerical analysis of singular solutions of two-dimensional problems of asymmetric elasticity is considered. The algorithm is based on separation of a power-law dependence from the finite-element solution in a neighborhood of singular points in the domain under study, where singular solutions are possible. The obtained power-law dependencies allow one to conclude whether the stresses have singularities and what the character of these singularities is. The algorithm was tested for problems of classical elasticity by comparing the stress singularity exponents obtained by the proposed method and from known analytic solutions. Problems with various cases of singular points, namely, body surface points at which either the smoothness of the surface is violated, or the type of boundary conditions is changed, or distinct materials are in contact, are considered as applications. The stress singularity exponents obtained by using the models of classical and asymmetric elasticity are compared. It is shown that, in the case of cracks, the stress singularity exponents are the same for the elasticity models under study, but for other cases of singular points, the stress singularity exponents obtained on the basis of asymmetric elasticity have insignificant quantitative distinctions from the solutions of the classical elasticity.     

Stress singularities for three-dimensional corners using the boundary integral equation method

The boundary integral equation method is developed to study three-dimensional asymptotic singular stress fields at vertices of a pyramidal notch or inclusion in an isotropic elastic space. Two-dimensional boundary integral equations are used for the infinite body with pyramidal notches and inclusions when either stresses or displacements are specified on its surface. Applying the Mellin integral transformation reduces the problem to one-dimensional singular integral equations over a closed, piece-wise smooth line. Using quadrature formulas for regular and singular integrals with Hilbert and logarithmic kernels, these integral equations are reduced to a homogeneous system of linear algebraic equations. Setting its determinant to zero provides a characteristic equation for the determination of the stress singularity power. Numerical results are obtained and compared against known eigenvalues from the literature for an infinite region with a conical notch or inclusion, for a Fichera vertex, and for a half-space with a wedge-shaped notch or inclusion.     

界面下主微裂纹间干涉的研究

涉及两相正交各向异性体界面干涉问题的研究,多裂纹问题被分解为只含单裂纹的子问题,利用位错理论和裂面应力自由条件,列出一组可数值求解位错密度函数的奇异积分方程,从耐 注得应力强度因子。     

Split singularities and the competition between crack penetration and debond at a bimaterial interface

For a crack impinging upon a bimaterial interface at an angle, the singular stress field is a linear superposition of two modes, usually of unequal exponents, either a pair of complex conjugates, or two unequal real numbers. In the latter case, a stronger and a weaker singularity coexist (known as split singularities). We define a dimensionless parameter, called the local mode mixity, to characterize the proportion of the two modes at the length scale where the processes of fracture occur. We show that the weaker singularity can readily affect whether the crack will penetrate, or debond, the interface.     

On the load transfer from elastic inclusions to an infinite elastic orthotropic plane with cuts

The present paper deals with the problem of load transfer from elastic inclusions to an infinite elastic orthotropic plane with cuts located on one of the principal orthotropy directions. The constitutive system of equations of this problem is derived under the assumption that the inclusions are in a uniaxial stress state. The obtained system consists of a singular integro-differential equation and a singular integral equation for the jumps of the tangential stresses acting on the inclusion shores and for the derivative of the the cut opening function. The behavior of solutions of the system of constitutive equations at the endpoints of the inclusions and cuts is studied, and the solution of this system is constructed by the numerical-analytic discrete singularity method.     

基于(α, β)变换和距离变换的三维弱奇异边界积分近奇异性消除方法

通过非线性变换求解三维弱奇异积分时,变换的雅可比消除了被积函数的奇异性。然而,当积分单元形状较差,如顶角过大或者顶角边长比过大时,弱奇异积分中的近奇异性仍然存在,这将导致弱奇异积分计算精度低甚至计算结果完全错误。因此,本文提出了一种基于(α, β)变换和距离变换的弱奇异积分中的近奇异性消除方法,用于精确计算三维弱奇异积分。首先通过(α, β)变换消除弱奇异积分中α方向的奇异性,并分离出β方向的近奇异性;然后针对β方向的积分函数形式,构造对应的距离变换来消除其近奇异性;最后给出具有大顶角和大边长比的弱奇异积分数值算例。结果表明,采用(α, β)变换和β方向距离变换相结合的方案可以精确计算不同单元形状的弱奇异积分。     

Singular loadings in elasticity problems and singular solutions of the corresponding integral equations

The method of singular integral equations is an efficient method for the formulation and numerical solution of plane and antiplane, static and dynamic, isotropic and anisotropic elasticity problems. Here we consider three cases of singular loadings of the elastic medium: by a force, by a moment and by a loading distribution with a simple pole. These loadings cause corresponding singularities in the right-hand side function and in the unknown function of the integral equation. A method for the numerical solution of the singular integral equation under the above singular loadings is proposed and the validity of this equation at the singular points is investigated.     

The Stress State of Three-Dimensional Bodies with Nonsmooth Inclusions

The stress state of a three-dimensional body with inclusions bounded by surfaces with singular lines (sets of corner points) and a conical point is studied. By determining the asymptotics of displacements and stresses at the singularities of interfaces and using the generalized elastic potentials of single and double layers, the problem posed is reduced to a system of singular integral equations. The results obtained are used to analyze the stress state of a body with a circular conical inclusion     

Shrink fit of an elastic layer having a cylindrical cavity

Summary This paper deals with the problem of determining the stress distribution in an elastic layer with a cylindrical cavity when the mixed boundary conditions are prescribed on the curved surface of the cylinder. The problem is simplified to that of finding the solution of dual integral equations arising from the mixed boundary conditions. These dual integral equations are subsequently reduced to a singular integral equation. The solution of this integral equation is obtained numerically, and the quantities of physical interest are calculated.     

Analysis of bonded anisotropic wedges with interface crack under anti-plane shear loading

The antiplane stress analysis of two anisotropic finite wedges with arbitrary radii and apex angles that are bonded together along a common edge is investigated. The wedge radial boundaries can be subjected to displacement-displacement boundary condi- tions, and the circular boundary of the wedge is free from any traction. The new finite complex transforms are employed to solve the problem. These finite complex transforms have complex analogies to both kinds of standard finite Mellin transforms. The traction free condition on the crack faces is expressed as a singular integral equation by using the exact analytical method. The explicit terms for the strength of singularity are extracted, showing the dependence of the order of the stress singularity on the wedge angle, material constants, and boundary conditions. A numerical method is used for solving the resul- tant singular integral equations. The displacement boundary condition may be a general term of the Taylor series expansion for the displacement prescribed on the radial edge of the wedge. Thus, the analysis of every kind of displacement boundary conditions can be obtained by the achieved results from the foregoing general displacement boundary condition. The obtained stress intensity factors （SIFs） at the crack tips are plotted and compared with those obtained by the finite element analysis （FEA）.     

Periodic Traveling Waves for Diffusion Equations with Time Delayed and Non-local Responding Reaction

We develop a singular perturbation technique to study the existence of periodic traveling wave solutions with large wave speed for a class of reaction-diffusion equations with time delay and non-local response. Unlike the classical singular perturbation method, our approach is based on a transformation of the differential equations to integral equations in a Banach space that reduces the singular perturbation problem to a regular perturbation problem. The periodic traveling wave solutions then are obtained by the use of Liapunov-Schmidt method and a generalized implicit function theorem. The general result obtained has been applied to a non-local reaction-diffusion equation derived from an age-structured population model with a logistic type of birth function.     

Singular Casimir Elements of the Euler Equation and Equilibrium Points

The problem of the nonequivalence of the sets of equilibrium points and energy-Casimir extremal points, which occurs in the noncanonical Hamiltonian formulation of equations describing ideal fluid and plasma dynamics, is addressed in the context of the Euler equation for an incompressible inviscid fluid. The problem is traced to a Casimir deficit, where Casimir elements constitute the center of the Poisson algebra underlying the Hamiltonian formulation, and this leads to a study of singularities of the Poisson operator defining the Poisson bracket. The kernel of the Poisson operator, for this typical example of an infinite-dimensional Hamiltonian system for media in terms of Eulerian variables, is analyzed. For two-dimensional flows, a rigorously solvable system is formulated. The nonlinearity of the Euler equation makes the Poisson operator inhomogeneous on phase space (the function space of the state variable), and it is seen that this creates a singularity where the nullity of the Poisson operator (the “dimension” of the center) changes. The problem is an infinite-dimension generalization of the theory of singular differential equations. Singular Casimir elements stemming from this singularity are unearthed using a generalization of the functional derivative that occurs in the Poisson bracket.     

Eccentric crack in a piezoelectric strip bonded to half planes

We consider the problem of determining the singular stresses and electric fields in a piezoelectric ceramic strip containing an eccentric Griffith crack off the centre line bonded to two elastic half planes under anti-plane shear loading using the continuous crack-face condition. Fourier transforms are used to reduce the problem to the solution of two pairs of dual integral equations, which are then expressed to a Fredholm integral equation of the second kind. Numerical values on the stress intensity factor and energy release rate are obtained.     

穿过反平面圆形夹杂界面的曲线型刚性线

利用复变函数方法和叠加原理建立了求解刚性线夹杂问题的弱奇积分方程,利用Ｃａｕｃｈｙ型奇异积分方程主部分方法,研究了穿过反平面圆夹杂界面的曲线型刚性线夹杂在界面交点处点处的奇性应力指数以及交点处角形域内的奇性应力,并定义了交点处的应力奇性因子。利用所得的奇性应力指数,通过对弱奇异积分方程的数值求解,得出了刚性线端点和交点处的应力奇性因子。     

一种处理弹性动力边界元法中奇异积分的方法

利用边界元法求解瞬态弹性动力学问题时,时域基本解函数的分段连续性和奇异性为该问题的求解带来很大的困难。为了解决时域基本解中的奇异性问题,本文依据柯西主值的定义,对经过时间解析积分之后的时域基本解进行奇异值分解,将其分成奇异和正则积分两部分;其中正则部分可通过采用常规高斯积分方法来计算,而奇异部分具有简单的形式,可以利用解析积分计算。经过上述操作之后,就可以达到直接消除时域基本解中奇异积分的目的。和传统方法相比,本文方法并不依赖静力学基本解来消除奇异性,是一种直接求解方法。最后给定两个数值算例来验证本文提出方法的正确性和可行性,结果表明使用本文算法可以解决弹性动力学边界积分方程中的奇异性问题。     

A new variable-order singular boundary element for calculating stress intensity factors in three-dimensional elasticity problems

A new three-dimensional variable-order singular boundary element has been constructed for stress analysis of three-dimensional interface cracks and internal material junctions. The singular fields in the vicinity of crack front or junction have been accurately represented by the singular elements by taking account the variable order of singularities and the angular profiles of field variables. Both the singular stress fields and displacement fields are independently formulated by the element’s shape functions. Different kinds of displacement formulations are investigated. The formulation combining singular and linear terms is found to be the most accurate one. The mixed-mode stress intensity factors are treated as nodal unknowns. The variation of stress intensity factors along the line of singularity can be obtained directly from the final system of equations and thus no post processing, such as three-dimensional J-integral or domain integral, is necessary. Numerical examples involving stress singularity, such as penny-shaped cracks in homogeneous and dissimilar material interface, plates with through-thickness cracks, and a dissimilar inclusion, are investigated. The analysis results are in good agreement with those reported in the literature.