Method of effective stress field in three-dimensional problems of interaction of plane cracks

We propose a modification of the method of boundary integral equations capable of the efficient solution of the problems of interaction of large numbers of arbitrarily oriented plane cracks on the engineering level. The approach is based on the determination of the effective stress field formed in the vicinity of a fixed crack by neighboring cracks interacting with this crack. The reliability of the results obtained by the method of effective stress field is checked by comparing with the exact solution of the problem of interaction of two plane circular cracks for different mutual orientations of the cracks. The efficiency of the proposed approach is illustrated by an example of interaction of an aperiodic system of six cracks located in different planes.     

Mixed mode fatigue growth of curved cracks emanating from fastener holes in aircraft lap joints

The finite element alternating method is extended further for analyzing multiple arbitrarily curved cracks in an isotropic plate under plane stress loading. The required analytical solution for an arbitrarily curved crack in an infinite isotropic plate is obtained by solving the integral equations formulated by Cheung and Chen (1987a, b). With the proposed method several example problems are solved in order to check the accuracy and efficiency of the method. Curved cracks emanating from loaded fastener holes, due to mixed mode fatigue crack growth, are also analyzed. Uniform far field plane stress loading on the plate and sinusoidally distributed pin loading on the fastener hole periphery are assumed to be applied. Small cracks emanating from fastener holes are assumed as initial cracks, and the subsequent fatigue crack growth behavior is examined until long arbitrarily curved cracks are formed near the fastener holes under mixed mode loading conditions.     

Solving two-dimensional cases for cracks in piecewise-homogeneous bodies

Summary A survey is made on papers concerned with a general approach to solving planar problems in the theory of elasticity for finite and infinite piecewise-homogeneous bodies containing curvilinear cracks, where singular integral equations are used. Cases are considered for an infinite plane and half-plane containing arbitrary and periodic elastic-inclusion systems together with curvilinear cracks, as well as a curvilinear two-component ring having internal and edge cracks.Translated from Fiziko-Khimicheskaya Mekhanika Materialov, No. 4, pp. 80–89, July–August, 1989.     

Elastostatic interaction of multiple arbitrarily shaped cracks in plane inhomogeneous regions

A numerical technique has been developed for the determination of stress fields associated with multiple arbitrarily shaped cracks in plane inhomogeneous regions. The procedure allows the elastostatic analysis of cracks interacting with one or more straight bimaterial interfaces; of cracks located near, or emanating from, circular inclusions; and of cracks that emanate from single or multiple origins. The cracks may be branched or blunted, and may be subjected to arbitrarily applied stresses. The technique employs an efficient surface integral method, using distributions of edge dislocations to represent the cracks. The resulting singular integral equations are solved using a Gauss-Chebyshev integration formula; appropriate conditions are developed for closing the set of equations governing cracks intersecting inhomogeneity boundaries, based on a consideration of the stresses and displacements at the points of intersection. Crack-tip stress intensity factor results are presented for several crack configurations. The overall scheme provides a more general, direct, and convenient approach than other available schemes. A computer program has been developed to implement the various formulations in a single framework.     

Thermal stresses in an elastic solid weakened by two coplanar circular cracks

Some linear thermoelastic problems are studied for the thermal stress and displacement fields in an infinite elastic medium weakened by cracks occupying the space interior to two coplanar circular regions with equal radii. The thermal stresses are caused by the uniform heating or heat flow disturbed by the presence of the coplanar cracks. The problem is reduced to the determination of the solution of infinite sets of Fredholm integral equations. Attention is given to the case when the plane occupying the space external to the cracks is insulated from uniform heat flow. The sets of integral equations are solved iteratively by assuming the spacing between the center of the cracks is large as compared to the radii. Physical quantity of interest such as crack-opening displacement is investigated.     

A half plane and a strip with an arbitrarily located crack

The paper introduces a technique to deal with the problem of an elastic domain containing an arbitrarily oriented internal crack. The problem is formulated as a system of integral equations for a fictitious layer of body forces imbedded in the plane along a closed smooth curve encircling the original domain. The problems of a half plane with a crack in the neighborhood of its free boundary and of an infinite strip containing a symmetrically located internal crack with an arbitrary orientation are considered as examples. In each case the stress intensity factors are computed and are given as functions of the crack angle.     

Two coplanar Griffith cracks in an infinite elastic layer under torsion

Summary We consider the problem of determining the stress distribution in an infinitely long isotropic homogeneous elastic layer containing two coplanar Griffith cracks which are opened by internal shear stress acting along the lengths of the cracks. The faces of the layer are assumed to be stress free. The cracks are located in the middle plane of the layer parallel to its faces. By using Fourier transforms, we reduce the problem to the solution of a set of triple integral equations with a cosine kernel and a weight function. These equations are solved exactly by using finite Hilbert transform techniques. Finally we derive the closed form expressions for the stress intensity factors and the crack energy. Solutions to the following problems are derived as particular cases: (i) a single crack in an infinite layer under torsion, (ii) two coplanar cracks in an infinite space under torsion, (iii) a single crack in an infinite space under torsion.     

Numerical analysis of doubly periodic array of cracks/rigid-line inclusions in an infinite isotropic medium using the boundary integral equation method

In this paper, the boundary integral equation approaches are used to study the doubly periodic array of cracks/rigid-line inclusions in an infinite isotropic plane medium. For the doubly periodic rigid-line inclusion problems, the special integral equation containing the axial and shear forces within the rigid-line inclusion is used. The doubly periodic crack problems are dealt with using the displacement discontinuous integral equation approach. Stress intensity factors, effective elastic properties for doubly periodic array of cracks/rigid-line inclusions are calculated and compared with the available numerical solutions.     

The gauss-hermite numerical integration method for the solution of the plane elastic problem of semi-infinite periodic cracks

The problem of parallel semi-infinite periodic cracks subjected to a transversely directed load in an infinite isotropic and elastic medium under conditions of plane stress or plane strain can be reduced to the solution of a Cauchy-type singular integral equation along one of the cracks. This equation can be transformed into a system of linear equations by means of an approximation of the integrals through the Gauss-Hermite procedure and application of the equation to distinct points along the faces of the crack. Stress intensity factors thus determined for the crack tips under constant load along the cracks are in satisfactory agreement with corresponding values derived previously.     

The elastostatic problem of an infinite strip containing a periodic row of line cracks

The two dimensional problem of an infinite strip containing a periodic row of line cracks, each subjected to arbitrary but identical pressure distribution, is solved by superposing two solutions: the problem of a row of line cracks in an infinite medium and the solution of an infinite strip loaded at the edges. This procedure leads to a system of simultaneous integral equations. The solution is obtained by reduction of these equations into algebraic equations with the help of Fourier expansion of involved functions. An analytic expression for the stress-intensity factor is also derived. Boundary conditions are later modified to get the solution of the problem of an infinite sheet containing doubly periodic array of line cracks.     

Identification of multiple cracks based on optimization methods using harmonic elastic waves

A theoretical study of the identification of multiple cracks in an elastic medium based on optimization methods using harmonic elastic waves is presented in the paper. A general interacting crack model is first used to determine the dynamic interaction between arbitrarily located and oriented cracks subjected to plane harmonic waves. The solution of this problem is then implemented into an optimization process for the identification of unknown cracks from known strain components at discrete locations. An optimization scheme based on sensitivity analysis is used to determine the length, the orientation and the position of cracks. Numerical simulation indicates that the sensitivity analysis and the optimization method used are effective in identifying multiple cracks. It is observed that convergent results can be achieved from a set of arbitrarily determined initial values of the crack parameters. Numerical examples are presented to illustrate the determination of the length, orientation, and position of different interacting cracks.     

The influence of a finite stringer on the stress intensities around cracks in plates

The influence of a stringer embedded in a plate, containing a number of cracks of arbitrary shape and arbitrarily oriented relatively to the stringer, on the stress field and the stress intensity factors at the tips of the cracks was studied by using the method of singular integral equations. An exact expression for the complex stress function was given for the most general case. The method was applied to an example of an arbitrarily oriented internal crack relatively to the position of the stringer and the influence of its orientation on the stress intensity factor was determined numerically. The method constitutes a potential technique for solving problems of stress concentrations and intensities in reinforced plates.     

Electrostatic interactions of arbitrarily dispersed multicoated elliptic cylinders

We propose a theoretical framework for evaluation of electrostatic potentials in an unbounded isotropic matrix containing a number of arbitrarily dispersed elliptic cylinders subjected to a remotely prescribed potential field. The inclusions could be homogeneous or confocally multicoated, and may have different sizes, aspect ratios and different conductivities. The approach is based on a multipole expansion formalism, together with a construction of consistency conditions and translation operators. This procedure generalizes the approach of the classic work of Rayleigh [1] for a periodic array of circular disks or spheres to an arbitrarily dispersion of elliptic cylinders. We combine the methods of complex potentials with a re-expansion formulae and the generalized Rayleigh’s formualtion to obtain a complete solution of the many-inclusion problem. We show that the coefficients of field expansions can be written in the form of an infinite set of linear algebraic equations. Numerical results are presented for several configuarions. We further apply the obtained field solutions to determine the effective conductivity of the composite.     

The problem of internal cracks in an infinite strip having a circular hole

Summary. The elastostatic plane problem of an infinite strip having a circular hole and containing two symmetrically located internal cracks perpendicular to the boundary is formulated in terms of triply coupled integral equations. The solution of the problem is obtained for various crack geometries and for uniaxial tension applied to the strip away from the crack region. Quantities of physical interest are displayed in graphical forms.     

Anti-plane transient analysis of planes with multiple cracks

In this paper, the transient dynamic stress intensity factor is determined for multiple curved cracks under impact loading. The dislocation method has rarely been applied to the problems involving dynamic loading. The transient response of Volterra-type dislocation in a plane is obtained by means of the Cagniard-de Hoop method. The distributed dislocation technique is used to construct integral equations for an infinite isotropic plane weakened by cracks. These equations are of Cauchy singular type at the location of dislocation which are solved numerically to obtain the dislocation density on the faces of the cracks. The dislocation densities are employed to determine stress intensity factors for multiple smooth cracks. Numerical results are obtained to validate the formulation and illustrate its capabilities.     

The longitudinal shear problem for an array of cracks at the edge of a circular hole in an infinite elastic solid

Summary A Mellin-type transform technique reduces the longitudinal shear problem for a set of cracks at the edge of a circular hole in an infinite elastic solid to that of solving a system of integral equations. The stress intensity factors and crack formation energy are calculated. Three special cases are considered in detail and graphical results given.     

含任意方向裂纹功能梯度材料的应力分析研究

功能梯度材料是在航空航天领域的需求背景下发展起来的,但由于生产技术及工作环境等方面的原因,功能梯度材料内部常常产生各种形式的裂纹并最终导致材料破坏,因此研究含任意方向裂纹功能梯度材料的断裂问题具有重要意义。以含有任意方向裂纹的功能梯度材料为对象,运用积分变换方法,给出了相应材料平面问题的位移场的形式解。通过引入辅助函数并利用相关条件,可将问题转化为求解一组带有Cauchy核的奇异积分方程,继而采用Lobatto-Chebyshev方法对奇异积分方程进行数值求解。最后分析了裂纹方向、材料非均匀指数、载荷条件对混合型应力强度因子的影响。      

Scattering of a gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation: numerical results

We present numerical results concerning the properties of the electromagnetic field scattered by an infinite circular cylinder illuminated by a circular Gaussian beam. The cylinder is arbitrarily located and arbitrarily oriented with respect to the illuminating Gaussian beam. Numerical evaluations are provided within the framework of a rigorous electromagnetic theory, the generalized Lorenz-Mie theory, for infinite cylinders. This theory provides new insights that could not be obtained from older formulations, i.e., geometrical optics and plane-wave scattering. In particular, some emphasis is laid on the waveguiding effect and on the rainbow phenomenon whose fine structure is hardly predictable by use of geometrical optics.     

Method of potentials in three-dimensional static and dynamical problems of the theory of cracks

By the method of potentials, three-dimensional static and dynamical problems of the theory of elasticity for an infinite body with arbitrarily located cracks are reduced to boundary integral equations. We obtain regular representations of these equations and their discrete analogs in the form of systems of linear algebraic equations. In the case of two disk-shaped coplanar cracks subjected to the action of dynamical forces (described by the Heaviside function), we construct time dependences of the stress intensity factors. Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 32, No. 1, pp. 22–32, January–February, 1996.     

Hypersingular integral equations for a thermoelastic problem of multiple planar cracks in an anisotropic medium

The problem of calculating the thermoelastic stress around an arbitrary number of arbitrarily located planar cracks in an infinite anisotropic medium is considered. The cracks open up under the action of suitably prescribed heat flux and traction. With the aid of suitable integral solutions, we reduce the problem to solving a system of Hadamard finite-part (hypersingular) integral equations. The hypersingular integral equations are solved for specific cases of the problem.