Hypersingular integral equation for multiple curved cracks problem in plane elasticity

The complex variable function method is used to formulate the multiple curved crack problems into hypersingular integral equations. These hypersingular integral equations are solved numerically for the unknown function, which are later used to find the stress intensity factor, SIF, for the problem considered. Numerical examples for double circular arc cracks are presented.     

磁电热弹耦合材料三维多裂纹超奇异积分法

用超奇异积分方程法将多场耦合载荷作用下磁电热弹耦合材料内含任意形状和位置三维多裂纹问题转化为求解一以广义位移间断为未知函数的超奇异积分方程组问题,退化得到内含任意形状平行三维多裂纹问题的超奇异积分方程组;推导出平行三维多裂纹问题的裂纹前沿广义奇异应力场解析表达式、定义了广义(应力、应变能)强度因子和广义能量释放率;应用有限部积分概念及体积力法,为超奇异积分方程组建立了数值求解方法,编制了FORTRAN程序,以平行双裂纹为例,通过典型算例,研究了广义(应力、应变能)强度因子随裂纹位置、裂纹形状及材料参数变化规律,得到裂纹断裂评定准则. 最后,分析了裂纹间干扰、屏蔽作用及其在工程实际中的应用.      

横观各向同性材料的三维断裂力学问题

从三维横观各向同性材料弹性力学理论出发, 使用Hadamard有限部积分概念, 导出了三维状态下单位位移间断(位错)集度的基 本解. 在此基础上, 进一步运用极限理论, 将任意载荷作用下, 三维无限大横观各向 同性材料弹性体中, 含有一个位于弹性对称面内的任意形状的片状裂纹问题, 归结为求 解一组超奇异积分方程的问题. 通过二维超奇异积分的主部分析方法, 精确地求得了裂纹前沿光滑点附近的应力奇异指数和奇异应力场, 从而找到了以裂纹表面位移间断表示的应力强度因子表达式及裂纹局部扩展所提供 的能量释放率. 作为以上理论的实际应用,最后给出了一个圆形片状裂纹问题 的精确解例和一个正方形片状裂纹问题的数值解例. 对受轴对称法向均布载荷作用下圆形片状裂纹问题, 讨论了超奇异积分方程的精确求解方法, 并获得了位移间断和应力强度因子的封闭解, 此结果与现有理论解完全一致.     

Two coplanar Griffith cracks in an infinitely long elastic strip

We consider the problem of determining the stress intensity factors and the crack energy in an infinitely long isotropic, homogeneous elastic strip containing two coplanar Griffith cracks. We assume that the cracks are opened by an internal pressure and the edges of the strip are rigidly fixed. By the use of Fourier transforms we reduce the problem to solving a set of triple integral equations with cosine kernel and a weight function. The triple integral equations are solved exactly. Exact analytical expressions are derived for the stress intensity factors, shape of the deformed crack and the crack energy. Solutions to some particular problems are derived as limiting cases.This work was supported by the National Research Council of Canada through NRC-Grant No. A4177.     

三维断裂力学的超奇异积分方程方法

本文利用有限部积分的概念和方法,严格地证明了三维弹性体中受任意载荷作用的平片裂纹问题的超奇异积分方程组,并对未知解的性态作了理论分析,得到了性态指数,在此基础上通过主部分析,精确地求得了裂纹前沿光滑点附近的奇性应力场,从而找到了以裂纹面位移间断(位错)表示的应力强度因子表达式,最后对所得的超奇异积分方程组建立了数值法,并用此计算了若干典型的平片裂纹问题,数值结果令人满意。     

三维无限体中两平行平片裂纹相互干扰的超奇异积分方程解法

本文利用三维断裂力学的超奇异积分方程求解理论，对三维无限体中两平行平片裂纹在任意载荷作用下的相互干扰问题作了研究。首先导出了以裂纹面移间断（位借）为未知函数的超奇异积分方程组，然后为其建立了有限积分边界元法；在此基础上，讨论用了裂纹面位移间断计算应力强度因子的方法，最后用此计算了两平行平片裂纹的相对位置对裂前沿应力强度因子的影响，其数值结果令人满意。     

Isolated crack in three-dimensional piezoelectric solid. Part II: Stress intensity factors for circular crack

This is part II of the work concerned with finding the stress intensity factors for a circular crack in a solid with piezoelectric behavior. The method of solution involves reducing the problem to a system of hypersingular integral equations by application of the unit concentrated displacement discontinuity and the unit concentrated electric potential discontinuity derived in part I [1]. The near crack border elastic displacement, electric potential, stress and electric displacement are obtained. Stress and electric displacement intensity factors can be expressed in terms of the displacement and the potential discontinuity on the crack surface. Analogy is established between the boundary integral equations for arbitrary shaped cracks in a piezoelectric and elastic medium such that once the stress intensity factors in the piezoelectric medium can be determined directly from that of the elastic medium. Results for the penny-shaped crack are obtained as an example.     

Closed-form solution for two collinear mode-III cracks in an orthotropic elastic strip of finite width

The problem of an orthotropic strip containing two collinear cracks normal to the strip boundaries is considered. The Fourier series method is used to reduce the associated boundary value problem to triple series equations, then to a singular integral equation, which can be solved analytically. Under remote uniform antiplane shear loading, the stress field and the crack sliding displacement are determined analytically and stress intensity factors are also given in a closed form.     

Application of hypersingular integral equation method to three-dimensional crack in electromagnetothermoelastic multiphase composites

A three-dimensional crack problem in electromagnetothermoelastic multiphase composites (EMTE-MCs) under extended loads is investigated in this paper. Using Green’s functions, the extended general displacement solutions are obtained by the boundary element method. This crack problem is reduced to solving a set of hypersingular integral equations coupled with boundary integral equations, in which the unknown functions are the extended displacement discontinuities. Then, the behavior of the extended displacement discontinuities around the crack front terminating at the interface is analyzed by the main-part analysis method of hypersingular integral equations. Analytical solutions for the extended singular stresses, the extended stress intensity factors (SIFs) and the extended energy release rate near the crack front in EMTE-MCs are provided. Also, a numerical method of the hypersingular integral equations for a rectangular crack subjected to extended loads is put forward with the extended displacement discontinuities approximated by the product of basic density functions and polynomials. In addition, distributions of extended SIFs varying with the shape of the crack are presented. The results show that the present method accurately yields smooth variations of extended SIFs along the crack front.     

Finite-part integral and boundary element method for interaction between two parallel planar cracks in three-dimensional finite bodies

By using the Somigliana representation and the concepts of finite-part integrals, a set of hypersingular integral equations of the interaction between two parallel planar cracks in a three-dimensional finite body subjected to arbitrary loads is derived, and then its numerical method is proposed by the finite-part integral method combined with the boundary element method. According to the analytic theory of hypersingular integral equations, the square root models of displacement discontinuities in the elements near the crack front are applied, and thus the computational precision is raised. Based on this, the stress intensity factors can be directly calculated. Finally, the stress intensity factors of several typical interaction problems are calculated.     

APPLICATION OF THE HYPERSINGULAR INTEGRAL EQUATIONS METHOD TO THE INTERACTION BETWEEN TWO PARALLEL PLANAR CRACKS IN 3-D ELASTICITY

By using the analytic theory of hypersingular integral equations in three-dimensional fracture mechanics, the interactions between two parallel planar cracksunder arbitrary loads are investigated. According to the concepts and method of finite-part integrals, a set of hypersingular integral equations is derived, in which theunknown functions are the displacement discontinuities of the crack surfaces. Then itsnumerical method is proposed by combining the finite-part integral method with theboundary element method. Based on the above results, the method for calculating thestress intensity factors with the displacement discontinuities of the crack surfaces ispresented. Finally, several typical examples are calculated and the numerical resultsare satisfactory.     

Dynamic interaction of multiple arbitrarily oriented planar cracks in a piezoelectric space: A semi-analytic solution

The problem of determining the electro-elastic fields around arbitrarily oriented planar cracks in an infinite piezoelectric space is considered. The cracks which are acted upon by a transient load are either electrically impermeable or permeable. A semi-analytic method based on the theory of exponential Fourier transformation is proposed for solving the problem in the Laplace transform domain. The Laplace transforms of the jumps in the displacements and electric potential across opposite crack faces are determined by solving a system of hypersingular integral equations. Once these displacement and electric potential jumps are obtained, the displacements and electric potential and other physical quantities of interest, such as the crack tip stress and electric displacement intensity factors, can be computed with the help of a suitable algorithm for inverting Laplace transforms. The stress and electric displacement intensity factors are computed for some specific cases of the problem.     

三维有限体中两平行裂纹干扰问题的有限部积分与边界元法

利用Ｓｏｍｉｇｌｉａｎａ公式及有限部积分的概念，导出含两平行平片裂纹三维有限体裂纹干扰问题的超奇异积分方程组，联合使用有限部积分与边界元法，建立了数值求解方法，为提高数值计算结果的精度，在裂纹前疝附近单元，采用平方根位移模型，并在此基础雌出直接计算应力强度因子的公式，最后计算若干典型例子裂纹前沿的应力强度因子。     

横观各向同性压电材料中裂纹问题的边界元法

采用Somigiliana公式给出了三维横观各向同性压电材料中的非渗漏裂纹问题的一般解和超奇异积分方程,其中未知函数为裂纹面上的位移间断和电势间断．在此基础上,使用有限部积分和边界元结合的方法,建立了超奇异积分方程的数值求解方法,并给出了一些典型数值算例的应力强度因子和电位移强度因子的数值结果,结果令人满意．     

三维界面裂纹的奇性应力场和应力强度因子分析

使用有限部积分概念和极限方法得到了三维平片界面裂纹的超奇异积分－微分方程组后,进一步利用二维超奇异积分主部分析方法,对裂纹前沿的应力场作了理论分析,并获得了其奇性应力场和裂纹面位移间断表示复位应力强度因子的精确表达式,为三维平片界面裂纹的超奇异积分－微分方程组的求解建立了数值方法,并分析了界面椭圆平片裂纹问题,和现有解比较,所得数值结果令人满意。     

垂直磁压电材料界面三维裂纹的超奇异积分法

应用有限部积分概念和广义位移基本解,垂直于磁压电双材料界面三维复合型裂纹问题被转 化为求解一组以裂纹表面广义位移间断为未知函数的超奇异积分方程问题. 进而,通过主部 分析法精确地求得裂纹尖端光滑点附近的奇性应力场解析表达式. 然后,通过将裂纹表面 位移间断未知函数表达为位移间断基本密度函数与多项式之积,使用有限部积分法对超奇异 积分方程组建立了数值方法. 最后,通过典型算例计算,讨论了广义应力强度因子的变化规 律.     

Finite-part integral and boundary element method to solve three-dimensional crack problems in piezoelectric materials

Using the hypersingular integral equation method based on body force method, a planar crack in a three-dimensional transversely isotropic piezoelectric solid under mechanical and electrical loads is analyzed. This crack problem is reduced to solve a set of hypersingular integral equations. Compare with the crack problems in elastic isotropic materials, it is shown that for the impermeable crack, the intensity factors for piezoelectric materials can be obtained from those for elastic isotropic materials. Based on the exact analytical solution of the singular stresses and electrical displacements near the crack front, the numerical method of the hypersingular integral equation is proposed by the finite-part integral method and boundary element method, which the square root models of the displacement and electric potential discontinuities in elements near the crack front are applied. Finally, the numerical solutions of the stress and electric field intensity factors of some examples are given.     

三维间断位移法及强奇异和超奇异积分的处理方法

从积分方程Somigliana等式出发,导出三维状态下单位位错集度的基本解.在此基础上,建立了边界积分方程,并给出了其离散形式.对强奇异和超奇异积分,采用了Hadamard定义的有限部分积分来处理.最后,给出了计算裂纹应力强度因子的算例,并与解析解进行了比较,证实了该方法的有效性.     

Periodic system of collinear cracks in an elastic porous medium

We consider a problem for cracks in a specific porous elastic material described by the Cowin-Goodman-Nunziato model. For an arbitrary set of collinear cracks, the problem is reduced to an integral equation over the crack surface. In the special case of a periodic crack arrangement, the kernel of the integral equation is written in the form of a Fourier series. By explicitly summing the leading part of the kernel, we can show that it exhibits hypersingular behavior as the argument tends to zero. Further, we solve the main integral equation by using the direct numerical method proposed in our preceding paper. In conclusion, we study the stress concentration at the crack tips and compare this case with the case of a single isolated crack.     

Dynamic behavior of two parallel symmetric permeable cracks in a piezoelectric strip

The dynamic behavior of two parallel symmetric cracks in a piezoelectric strip under harmonic anti-plane shear waves is studied using the Schmidt method for permeable crack surface conditions. The cracks are parallel to the edge of the strip. By means of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. These equations are solved using the schmidt method. The results show that the stress and the electric displacement intensity factors depend on the geometry of the cracks, the frequency of incident waves, the distance between cracks and the thickness of the strip. It is also found that the electric displacement intensity factors for the permeable crack surface conditions are much smaller than those for the impermeable crack surface conditions. Project supported by the Post Doctoral Science Foundation of Heilongjiang Province, the Natural Science Foundation of Heilongjiang Province, the National Science Foundation with the Excellent Young Investigator Award (No. 19725209) and the Scientific Research Foundation of Harbin Institute of Technology (HIT.2000.30).