Complex potentials and singular integral equation for curve crack problem in antiplane elasticity

Four types of the fundamental complex potential in antiplane elasticity are introduced: (a) a point dislocation, (b) a concentrated force, (c) a dislocation doublet and (d) a concentrated force doublet. It is proven that if the axis of the concentrated force doublet is perpendicular to the direction of the dislocation doublet, the relevant complex potentials are equivalent. Using the obtained complex potentials, a singular integral equation for the curve crack problem is introduced. Some particular features of the obtained singular integral equation are discussed, and numerical solutions and examples are given.

Cauchy singular integral equation method for transient antiplane dynamic problems

In this paper, by use of the boundary integral equation method and the techniques of Green basic solution and singularity analysis, the dynamic problem of antiplane is investigated. The problem is reduced to solving a Cauchy singular integral equation in Laplace transform space. This equation is strictly proved to be equivalent to the dual integral equations obtained by Sih [Mechanics of Fracture, Vol. 4. Noordhoff, Leyden (1977)]. On this basis, the dynamic influence between two parallel cracks is also investigated. By use of the high precision numerical method for the singular integral equation and Laplace numerical inversion, the dynamic stress intensity factors of several typical problems are calculated in this paper. The related numerical results are compared to be consistent with those of Sih. It shows that the method of this paper is successful and can be used to solve more complicated problems. Copyright (C) 1996 Elsevier Science Ltd

The mixed-type integral equations for transient elastodynamic plane crack problems

In the present paper, by use of the boundary integral equation method and the techniques of Green fundamental solution and singularity analysis, the dynamic infinite plane crack problem is investigated. For the first time, the problem is reduced to solving a system of mixed-typed integral equations in Laplace transform domain. The equations consist of ordinary boundary integral equations along the outer boundary and Cauchy singular integral equations along the crack line. The equations obtained are strictly proved to be equivalent with the dual integral equations obtained by Sih in the special case of dynamic Griffith crack problem. The mixed-type integral equations can be solved by combining the numerical method of singular integral equation with the ordinary boundary element method. Further use the numerical method for Laplace transform, several typical examples are calculated and their dynamic stress intensity factors are obtained. The results show that the method proposed is successful and can be used to solve more complicated problems.

Dynamic behavior of a cylindrical crack in a functionally graded interlayer under torsional loading

In this paper the problem of a cylindrical crack located in a functionally graded material (FGM) interlayer between two coaxial elastic dissimilar homogeneous cylinders and subjected to a torsional impact loading is considered. The shear modulus and the mass density of the FGM interlayer are assumed to vary continuously between those of the two coaxial cylinders. This mixed boundary value problem is first reduced to a singular integral equation with a Cauchy type kernel in the Laplace domain by applying Laplace and Fourier integral transforms. The singular integral equation is then solved numerically and the dynamic stress intensity factor (DSIF) is also obtained by a numerical Laplace inversion technique. The DSIF is found to rise rapidly to a peak and then reduce and tend to the static value almost without oscillation. The influences of the crack location, the FGM interlayer thickness and the relative magnitudes of the adjoining material properties are examined. It is found among others that, by increasing the FGM gradient, the DSIF can be greatly reduced.

Dynamic SIF of interface crack between two dissimilar viscoelastic bodies under impact loading

In this paper, the transient dynamic stress intensity factor (SIF) is determined for an interface crack between two dissimilar half-infinite isotropic viscoelastic bodies under impact loading. An anti-plane step loading is assumed to act suddenly on the surface of interface crack of finite length. The stress field incurred near the crack tip is analyzed. The integral transformation method and singular integral equation approach are used to get the solution. By virtue of the integral transformation method, the viscoelastic mixed boundary problem is reduced to a set of dual integral equations of crack open displacement function in the transformation domain. The dual integral equations can be further transformed into the first kind of Cauchy-type singular integral equation (SIE) by introduction of crack dislocation density function. A piecewise continuous function approach is adopted to get the numerical solution of SIE. Finally, numerical inverse integral transformation is performed and the dynamic SIF in transformation domain is recovered to that in time domain. The dynamic SIF during a small time-interval is evaluated, and the effects of the viscoelastic material parameters on dynamic SIF are analyzed.

The scattering of general SH plane wave by interface crack between two dissimilar viscoelastic bodies

The scattering of general SH plane wave by an interface crack between two dissimilar viscoelastic bodies is studied and the dynamic stress,intensity factor at the crack-tip is computed. The scattering problem can be decomposed into two problems: one is the reflection and refraction problem of general SH plane waves at perfect interface (with no crack); another is the scattering problem due to the existence of crack. For the first problem, the viscoelastic wave equation, displacement and stress continuity conditions across the interface are used to obtain the shear stress distribution at the interface. For the second problem, the integral transformation method is used to reduce the scattering problem into dual integral equations. Then, the dual integral equations are transformed into the Cauchy singular integral equation of first kind by introduction of the crack dislocation density function. Finally, the singular integral equation is solved by Kurtz's piecewise continuous function method. As a consequence, the crack opening displacement and dynamic stress intensity factor are obtained. At the end of the paper, a numerical example is given. The effects of incident angle, incident frequency and viscoelastic material parameters are analyzed. It is found that there is a frequency region for viscoelastic material within which the viscoelastic effects cannot be ignored.

Torsional impact response of a penny-shaped interface crack in bonded materials with a graded material interlayer

In this paper, the dynamic response of a penny-shaped interface crack in bonded dissimilar homogeneous half-spaces is studied. It is assumed that the two materials are bonded together with such a inhomogeneous interlayer that makes the elastic modulus in the direction perpendicular to the crack surface is continuous throughout the space. The crack surfaces art assumed to be subjected to torsional impact loading. Laplace and Hankel integral transforms are applied combining with a dislocation density,function to reduce the mixed boundary value problem into a singular integral equation with a generalized Cauchy kernel in Laplace domain. By solving the singular integral equation numerically, and using a numerical Laplace inversion technique, the dynamic stress intensity factors art obtained. The influences of material properties and interlayer thickness on the dynamic stress intensity factor are investigated.

Singularity of dynamic stress fields around an interface crack between viscoelastic bodies

The singular nature of the dynamic stress fields around an interface crack located between two dissimilar isotropic linearly viscoelastic bodies is studied. A harmonic load is imposed on the surfaces of the interface crack. The dynamic stress fields around the crack are obtained by solving a set of simultaneous singular integral equations in terms of the normal and tangent crack dislocation densities. The singularity of the dynamic stress fields near the crack tips is embodied in the fundamental solutions of the singular integral equations. The investigation of the fundamental solutions indicates that the singularity and oscillation indices of the stress fields are both dependent upon the material constants and the frequency of the harmonic load. This observation is different from the well-known -1/2 oscillating singularity for elastic bi-materials. The explanation for the differences between viscoelastic and elastic bi-materials can be given by the additional viscosity mismatch in the case of viscoelastic bi-materials. As an example, the standard linear solid model of a viscoelastic material is used. The effects of the frequency and the material constants (short-term modulus, long-term modulus and relaxation time) on the singularity and the oscillation indices are studied numerically.

The annular crack in a nonhomogeneous matrix surrounding a fiber under torsional loading .2. The crack going through the bimaterial interface into the fiber

The axisymmetric problem of an elastic fiber perfectly bonded to a nonhomogeneous elastic matrix which contains an annular crack going through the interface into the fiber under axially symmetric shear stress is considered. The nature of the stress singularity is studied. It is shown that at the irregular point on the interface, whether the shear modulus is continuous or discontinuous the stresses are bounded. The problem is formulated in terms of a singular integral equation and can be solved by a regular method. The stress intensity factors and crack surface displacement are given.

Kinking of an interface crack between 2 dissimilar anisotropic elastic solids

A detailed analysis of kinking of an interface crack between two dissimilar anisotropic elastic solids is presented in this paper. The branched crack is considered as a distributed dislocation. A set of the singular integral equations for the distribution function of the dislocation density is developed. Explicit formulas of the stress intensity factors and the energy release rates for the branched crack are given for orthotropic bimaterials and misoriented orthotropic bicrystals. The role of the stress parallel to the interface, sigma0 is taken into account in these formulas. The interface crack can advance either by continued extension along the interface or by kinking out of the interface into one of the adjoining materials. This competition depends on the ratio of the energy release rates for interface cracking and for kinking out of the interface and the ratio of interface toughness to substrate toughness. Throughout the paper, the influences of the inplane stress sigma0 on the stress intensity factors and the energy release rates for the branched crack, which can significantly alter the conditions for interface cracking, are emphasized.

粘弹性材料界面裂纹的瞬态响应和散射研究

本文研究粘弹性材料界面裂纹对冲击载荷的瞬态响应和对广义平面波的稳态散射。相对于已有广泛研究的弹性材料裂纹瞬态响应和稳态散射问题，本文的研究有三个突出特点：1）粘弹性材料；2）界面裂纹；3）广义平面波入射。粘弹性材料界面裂纹对冲击载荷的瞬态响应和对广义平面波的散射尚无开展研究，本文在弹性材料相应问题的研究基础上，首先开展了这一问题的研究。对于冲击载荷下粘弹性界面裂纹的瞬态响应问题，利用Laplace积分变换方法，将粘弹性材料卷积型本构方程转化为Laplace变换域内的代数型本构方程，从而可以在Laplace变换域内象处理弹性材料的冲击响应一样，将相应的混合边值问题归结为关于裂纹张开位移COD的对偶积分方程，并进一步引入裂纹位错密度函数CDD （Crack Dislocation Density），将对偶积分方程化成关于CDD的奇异积分方程（SIE）。用数值方法求解奇异积分方程得到变换域内的动应力强度因子数值解，最后利用Laplace积分逆变换数值方法得到时间域内的动应力强度因子的时间响应。理论分析考虑了两种裂纹模型，即Griffith界面裂纹和柱面圆弧型界面裂纹。考虑的载荷包括反平面冲击载荷和平面冲击载荷。对于平面冲击载荷，通过对裂尖应力场的奇性分析，首次发现粘弹性界面裂纹裂尖动应力场奇性指数不是常数0.5，而是与震荡指数一样依赖材料参数。针对反平面冲击载荷给出了一个算例，计算了裂尖动应力强度因子的时间响应，并与弹性材料的结果作了比较，发现粘弹性效应的影响不仅使过冲击峰值降低，而且使峰值点后移。粘性效应较大时，过冲击现象甚至不出现。关于粘弹性界面裂纹对广东省义平面波的散射问题，首先研究广义平面波在无裂纹存在的理想界面的反射和透射，再研究由于界面裂纹的存在而产生的附加散射场。利用粘弹性材料的复模量理论，可将粘弹性材料的卷积型相构方程化成频率域内的代数型本构方程。类似弹性平面波的处理，在频率域内将问题最终归结为关于裂纹位错密度CDD的奇异积分方程。数值方法求解奇异积分方程即可得到频率域内的散射场，并进而得到裂尖动应力强度因子和远场位移型函数和散射截面。理论分析考虑了两种裂纹模型：Griffith界面裂纹和柱面圆弧型界面裂纹。研究的入射波有广义的SH波和P波。对于广义平面P波入射的情况，通过对裂尖应力场的奇性分析，同样发现粘弹性界面裂纹裂尖动应力场奇性指数不地常数0.5，而是与震荡指数一样依赖于材料参数。对柱面裂纹散射远场的渐近分析，发现远场位移和应力除含有几何衰减因子外，都含有一个材料衰减速因子。散射截面由于材料衰减因子的存在也成为依赖散射半径的量。为了使散射截面仍有意义，文中提出一种修正办法。对Griffith界面裂纹，给出了一个广义平面SH波入射的算例；对柱面界面裂纹，给出了一个广义平面P波入射的算例。计算了不同入射角和入射频率下裂纹的张开位移和动就应力强度因子，并分析了其依赖关系。求解奇异积分方程的数值方法和Laplace积分逆变换数值方法是本文的基本数值方法。本文对这两种方法作了大量的调研和系统的研究。在对比分析的基础上，对现有的各种方法从原理，适用范围，计数效率，优势及特点进行了归纳总结。并尝试了奇异积分方程的最新数值方法--分片连续函数法，证实了其适用性和方便性．

基于广义Lippmann-Schwinger方程的地震波场模拟算法与应用

In the last several decades, due to the fast development of computer, numerical simulation has been an indispensable tool in scientific research. Numerical simulation methods which based on partial difference operators such as Finite Difference Method (FDM) and Finite Element Method (FEM) have been widely used. However, in the realm of seismology and seismic prospecting, one usually meets with geological models which have piece-wise heterogeneous structures as well as volume heterogeneities between layers, the continuity of displacement and stress across the irregular layers and seismic wave scattering induced by the perturbation of the volume usually bring in error when using conventional methods based on difference operators. The method discussed in this paper is based on elastic theory and integral theory. Seismic wave equation in the frequency domain is transformed into a generalized Lippmann-Schwinger equation, in which the seismic wavefield contributed by the background is expressed by the boundary integral equation and the scattering by the volume heterogeneities is considered. Boundary element-volume integral method based on this equation has advantages of Boundary Element Method (BEM), such as reducing one dimension of the model, explicit use the displacement and stress continuity across irregular interfaces, high precision, satisfying the boundary at infinite, etc. Also, this method could accurately simulate the seismic scattering by the volume heterogeneities. In this paper, the concrete Lippmann-Schwinger equation is specifically given according to the real geological models. Also, the complete coefficients of the non-smooth point for the integral equation are introduced. Because Boundary Element-Volume integral equation method uses fundamental solutions which are singular when the source point and the field are very close，both in the two dimensional and the three dimensional case, the treatment of the singular kernel affects the precision of this method. The method based on integral transform and integration by parts could treat the points on the boundary and inside the domain. It could transform the singular integral into an analytical one both in two dimensional and in three dimensional cases and thus it could eliminate the singularity. In order to analyze the elastic seismic wave scattering due to regional irregular topographies, the analytical solution for problems of this type is discussed and the analytical solution of P waves by multiple canyons is given. For the boundary reflection, the method used here is infinite boundary element absorbing boundary developed by a pervious researcher. The comparison between the analytical solutions and concrete numerical examples validate the efficiency of this method. We thoroughly discussed the sampling frequency in elastic wave simulation and find that, for a general case, three elements per wavelength is sufficient, however, when the problem is too complex, more elements per wavelength are necessary. Also, the seismic response in the frequency domain of the canyons with different types of random heterogeneities is illustrated. We analyzed the model of the random media, the horizontal and vertical correlation length, the standard deviation, and the dimensionless frequency how to affect the seismic wave amplification on the ground, and thus provide a basis for the choice of the parameter of random media during numerical simulation.

Crack Tip Field and J-Integral with Strain Gradient Effect

The mode I plane strain crack tip field with strain gradient effects is presented in this paper based on a simplified strain gradient theory within the framework proposed by Acharya and Bassani. The theory retains the essential structure of the incremental version of the conventional J_2 deformation theory No higher-order stress is introduced and no extra boundary value conditions beyond the conventional ones are required. The strain gradient effects are considered in the constitutive relation only through the instantaneous tangent modulus. The strain gradient measures are included into the tangent modulus as internal parameters. Therefore the boundary value problem is the same as that in the conventional theory Two typical crack Problems are studied: (a) the crack tip field under the small scale yielding condition induced by a linear elastic mode-I K-field and (b) the complete field for a compact tension specimen. The calculated results clearly show that the stress level near the crack tip with strain gradient effects is considerable higher than that in the classical theory The singularity of the strain field near the crack tip is nearly equal to the square-root singularity and the singularity of the stress field is slightly greater than it. Consequently, the J-integral is no longer path independent and increases monotonically as the radius of the calculated circular contour decreases.

Using wavelet transform to study the Lipschitz local singular exponent in wall turbulence

In this paper, wavelet,transform is introduced to study the Lipschitz local singular exponent for characterising the local singularity behavior of fluctuating velocity in wall turbulence. I, is found that the local singular exponent is negative when the ejections and sweeps of coherent structures occur in a turbulent boundary layer.

Stress intensity factors calculation in anti-plane fracture problem by orthogonal integral extraction method based on femol

For an anti-plane problem, the differential operator is self-adjoint and the corresponding eigenfunctions belong to the Hilbert space. The orthogonal property between eigenfunctions (or between the derivatives of eigenfunctions) of anti-plane problem is exploited. We developed for the first time two sets of radius-independent orthogonal integrals for extraction of stress intensity factors (SIFs), so any order SIF can be extracted based on a certain known solution of displacement (an analytic result or a numerical result). Many numerical examples based on the finite element method of lines (FEMOL) show that the present method is very powerful and efficient.