含圆形孔口和水平直裂纹的平面问题的应力分析

利用复变函数方法和积分方程理论研究了既含有圆形孔口又含有水平裂纹的无限大平面的平面弹性问题,将复杂的解析函数的边值问题化成了求解只在裂纹上的奇异积分方程的问题.此外,还给出了裂纹尖端附近的应力场和应力强度因子的公式.     

Approximate solution of a class of singular integral equations of second kind

A simple method based on polynomial approximation of a function is employed to obtain approximate solution of a class of singular integral equations of the second kind. For a hypersingular integral equation of the second kind, this method avoids the complex function-theoretic method and produces the known exact solution to Prandtl's integral equation as a special case. For a particular singular integro-differential equation of the second kind, this also produces an approximate solution which compares favourably with numerical results obtained by various Galerkin methods. The convergence of the method for both the equations is also established.     

热载荷作用下平面裂纹问题的奇异积分方程与边界无法

从边界积分方程出发,导出了二维裂纹体热传导问题及热弹性问题的积分方程组,继而使用奇异积分方程与边界元相结合的方法,为其建立了相应的数值求解方法。此外,利用奇异积分方程的主部分析法,严格地证明了裂纹尖端温度梯度场的1/√r 奇异性,并且给出了奇性温度梯度场的精确解。最后。对一些典型例子,做了数值计算。     

Singular integral operators in a weighted L2-space

Cauchy singular integral operators are characterized as operators in a weighted L²-space. The integral operator arises from a singular integral equation with variable coefficients. An appropriate weight function associated with the singular integral operator is constructed, and the set of polynomials orthogonal with respect to this weight function is defined. The action of the operator on polynomial sets is studied, and the definition of the operator is extended to a weighted L²-space. In this space, the operator is shown to be bounded, and, in some cases, isometric. Formulas are developed for the composition of the singular integral operator and its one sided inverse.     

An equilibrium penny-shaped crack in an inhomogeneous elastic medium

The problem of a penny-shaped tensile crack in a continuously-inhomogeneous space is considered. The problem reduces to a dual integral equation for which an approximate analytical solution is constructed. It is proved that the approximate solution of the integral equation is asymptotically exact for both small and large values of the dimensionless geometric parameter of the problem. The accuracy of the solution obtained is investigated. Expressions are presented for the stress intensity factor, the energy of the opening of the crack, the displacements of its sides and the normal components of the stress tensor in the neighbourhood of its contour. In the numerical analysis of the solution of the problem, special attention is paid to analysing of the problem when the first derivative of the change in the elastic properties of the material changes sign.     

功能梯度材料有限宽板的反平面断裂问题研究

研究了功能梯度材料有限宽板中与板边平行的III型裂纹问题．假设材料的剪切模量沿板宽度方向呈指数规律变化,利用Fourier变换将问题描述为奇异积分方程,并进一步将未知的位错密度函数表示为Chebyshev多项式的级数式,从而将奇异积分方程化为线性代数方程组进行配点数值求解．基于数值结果,讨论了材料非均匀性参数、板和裂纹的几何参数等对应力强度因子（SIF）的影响．研究表明,SIF随裂纹长度的增大而增大,随裂纹所在区域材料刚度的增大而减小;板越窄,SIF对非均匀性参数的变化越敏感,且变化规律也越复杂．随着非均匀性参数的增大,SIF既可能增大也可能减小还可能基本保持不变,这主要取决于板的相对宽度和裂纹的相对位置．当裂纹位于板的中央或当板较宽时,SIF对非均匀性参数的变化都不太敏感．     

Approximate solution of one singular integro-differential equation

In this paper we construct and theoretically justify a computational scheme for solving the Cauchy problem for a singular integro-differential equation of the first-order, where the integral over a segment of the real axis is understood in the sense of the Cauchy principal value. In addition, we study and solve approximately the integral equation with a special logarithmic kernel. We obtain uniform estimates for errors of approximate formulas. Orders of errors of approximate solutions are proved to be proportional to the order of the approximation error for the derivative of the density of the singular integral in the integro-differential equation.     

非对称载荷作用的外部圆形裂纹问题

使用边界积分方程方法,研究了三维无限弹性体中受非对称载荷作用的外部圆形裂纹问题。通过使用Fourier级数和超几何函数,将问题的二维边界奇异积分方程简化为Abel型方程,获得了一般非对称载荷作用的外部圆形裂纹问题的应力强度因子精确解,比用Hankel变换法得到的结果更为一般。结果表明:边界积分方程法在解析分析方面还有很大的潜力。     

裂纹与弹性夹杂的相互影响*

本文利用无限域上单根弹性夹杂和单根裂纹产生的位移和应力,将裂纹与弹性夹杂的相互影响问题归为解一组柯西型奇异积分方程,然后用此对夹杂分枝裂纹解答的奇性性态作了理论分析,并求得了振荡奇性界面应力场,对于不相交的夹杂裂纹问题,具体计算了端点的应力强度因子及夹杂上的界面应力,结果令人满意。     

Singular integral equation with Cauchy kernel on the real axis

We find closed-form formulas for the solution of the simplest singular integral equation with Cauchy kernel on the real axis and use them to reduce the full singular integral equation considered in the paper to a Fredholm equation. We construct numerical schemes for the above-mentioned equations and estimate the accuracy order of the approximate solution.     

有一条裂纹的圆形焊接问题

本文讨论了在带圆孔的无限平面中焊接一个不同材料的带裂纹的近似圆板的问题.该问题化为求解解析函数边值问题然后又转化为求解沿裂纹的奇异积分方程.后者的数字解法也已给出.文末并对Ⅰ型、Ⅱ型情况得出了应力强度因子的公式以及数字结果.     

Two problems with mixed boundary conditions for an elastic orthotropic strip

Two problems are considered for an elastic orthotropic strip: the contact problem and the crack problem. Both problems are reduced to integral equations of the first kind with different kernels, containing a singularity: logarithmic for the first problem and singular for the second problem. Regular and singular asymptotic methods are employed to construct approximate solutions of these integral equations. Numerical results are presented.     

Doubly singular boundary value problems with derivative dependent source function: A fast‐converging iterative approach

In this paper, a fast‐converging recursive scheme is presented to approximate the solution of a class of derivative dependent doubly singular boundary value problems (DDSBVP). First, the original problem is reformulated as an equivalent integral equation. The resulting integral equation is then efficiently tackled by an improved homotopy analysis method (IHAM). This method contains a parameter, which greatly accelerates the convergence of the series solution. The convergence of the method is carried out. To illustrate the efficiency and accuracy of the proposed recursive approach, we consider three nonlinear examples, including one physical model problem, which describes stress distribution on a rotationally shallow membrane cap. Results show that our method excels over the existing methods.     

Uniform convergence of the rectangle method for singular integral equation with the Hölder density

We study an approximate method for solving singular integral equations. It implies an approximation of a singular operator by means of a compound quadrature formula similar to the rectangle one. The corresponding systems of linear algebraic equations are solvable if so is the integral equation, while its coefficients satisfy the strong ellipticity condition. Under these restrictions we obtain a bound for the rate of convergence of solutions of systems of linear equations to the solution of the considered integral equation in the uniform vector norm.     

An interface crack moving between magnetoelectroelastic and functionally graded elastic layers

A constant crack moving along the interface of magnetoelectroelastic and functionally graded elastic layers under anti-plane shear and in-plane electric and magnetic loading is investigated by the integral transform method. Fourier transforms are applied to reduce the mixed boundary value problem of the crack to dual integral equations, which are expressed in terms of Fredholm integral equations of the second kind. The singular stress, electric displacement and magnetic induction near the crack tip are obtained asymptotically and the corresponding field intensity factors are defined. Numerical results show that the stress intensity factors are influenced by the crack moving velocity, the material properties, the functionally graded parameter and the geometric size ratios. The propagation of the moving crack may bring about crack kinking, depending on the crack moving velocity and the material properties across the interface.     

A CRACK PROBLEM WITH A BROKEN LINE INTERFACE

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A simple Taylor-series expansion method for a class of second kind integral equations

A simple yet effective Taylor-series expansion method is presented for a class of Fredholm integral equations of the second kind with smooth and weakly singular kernels. The equations studied in this paper arise in a number of applications, e.g., potential theory, radiative equilibrium, radiative heat transfer, and electrostatics. The approach leads to an approximate solution of the integral equation which can be expressed explicitly in a simple, closed form. The approximate solution is of sufficient accuracy as illustrated by the numerical examples arising from radiative heat transfer and electrostatics.     

A dynamic contact problem which reduces to a singular integral equation with two fixed singularities

The problem of the harmonic sheat oscillations of an elastic strip, coupled to an elastic half-space is considered. Using the method of integral transformations, the problem is reduced to a singular integral equation in the contact stresses in the region where the strip and the half-space are coupled when there are two fixed singularities at points bounding the integration intervals. One of the main results of this paper is the method of solving this equation numerically, taking into account the true singularity of the solution and based on the use of special quadrature formulae for singular integrals. The approximate solution obtained provides the possibility of numerically investigating the effect of the oscillation frequency and the ratio of the elastic constants of the strip and the half-space on the stress distribution in the contact area.     

Regularization and Fourier Series for Fredholm Integral Equations of the Second Kind with a Weakly Singular Kernel

To compute an approximate solution, of a weakly singular integral equation, we first regularize the kernel, and then truncate the associated Fourier series. An application to an astrophysical transfer equation is presented.