Anti-plane elastodynamic analysis of orthotropic planes weakened by several cracks

Stress analysis is carried out in an orthotropic plane containing a Volterra-type dislocation, the distributed dislocation technique is employed to obtain integral equations for an orthotropic plane weakened by cracks under time-harmonic anti-plane traction. The integral equations are of Cauchy singular type at the location of dislocation which are solved numerically. Several examples are solved and the stress intensity factors for multiple cracks with different configuration are obtained.     

Anti-plane analysis of an orthotropic strip weakened by several moving cracks

In this paper several finite cracks with constant length (Yoffe-type crack) propagating in an orthotropic strip were studied. The distributed dislocation technique is used to carry out stress analysis in an orthotropic strip containing moving cracks under anti-plane loading. The solution of a moving screw dislocation is obtained in an orthotropic strip by means of Fourier transform method. The stress components reveal the familiar Cauchy singularity at the location of dislocation. The solution is employed to derive integral equations for a strip weakened by moving cracks. Finally several examples are solved and the numerical results for the stress intensity factor are obtained. The influences of the geometric parameters, the thickness of the orthotropic strip, the crack size and speed have significant effects on the stress intensity factors of crack tips which are displayed graphically.     

Stress analysis of orthotropic planes weakened by cracks

The stress fields in an orthotropic infinite plane containing Volterra type climb and glide edge dislocations are derived. The dislocation solutions are utilized to formulate integral equations for dislocation density functions on the surfaces of smooth cracks. The integral equations are of Cauchy singular type and are solved for several different cases of crack configurations and arrangements. The results are used to evaluate modes I and II stress intensity factors for multiple smooth cracks.     

Anti-plane elastodynamic analysis of cracked graded orthotropic layers with viscous damping

Stress analysis is carried out in a graded orthotropic layer containing a screw dislocation undergoing time-harmonic deformation. Energy dissipation in the layer is modeled by viscous damping. The stress fields are Cauchy singular at the location of dislocation. The dislocation solution is utilized to derive integral equations for multiple interacting cracks with any location and orientation in the layer. These equations are solved numerically thereby obtaining the dislocation density function on the crack surfaces and stress intensity factors of cracks. The dependencies of stress intensity factors of cracks on the excitation frequency of applied traction and material properties of the layer are investigated. The analysis allows the determination of natural frequencies of a cracked layer. Furthermore, the interactions of two cracks having various configurations are studied.     

Stress analysis in a sheet with multiple cracks

The stress field due to the presence of a Volterra dislocation in an isotropic elastic sheet is obtained. The stress components exhibit the familiar Cauchy type singularity at dislocation location. The solution is utilized to construct integral equations for elastic sheets weakened by multiple embedded or edge cracks. The cracks are perpendicular to the sheet boundary and applied traction is such that crack closing may not occur. The integral equations are solved numerically and stress intensity factors (SIFs) are determined on a crack edges.     

Mixed mode axisymmetric cracks in transversely isotropic infinite solid cylinders

The present study examined mixed mode cracking in a transversely isotropic infinite cylinder. The solutions to axisymmetric Volterra climb and glide dislocations in an infinite circular cylinder of the transversely isotropic material are first obtained. The solutions are represented in terms of the biharmonic stress function. Next, the problem of a transversely isotropic infinite cylinder with a set of concentric axisymmetric penny-shaped, annular, and circumferential cracks is formulated using the distributed dislocation technique. Two types of loadings are considered: (i) the lateral cylinder is loaded by two self-equilibrating distributed shear stresses; (ii) the curved surface of the cylinder is under the action of a distributed normal stress. The resulting integral equations are solved by using a numerical scheme to compute the dislocation density on the borders of the cracks. The dislocation densities are employed to determine stress intensity factors for axisymmetric interacting cracks. Finally, a good amount of examples are solved to depict the effect of crack type and location on the stress intensity factors at crack tips and interaction between cracks. Numerical solutions for practical materials are presented and the effect of transverse isotropy on stress intensity factors is discussed.     

Stress intensity factors for three cracks at the interfaces of a graded layer bonding two different materials

We consider two dissimilar elastic half-planes bonded by a nonhomogeneous elastic layer in which there is one crack at the lower interface between the elastic layer and the lower half-plane and two cracks at the upper interface between the elastic layer and the upper half-plane. The stress intensity factors for these three cracks are solved for when tension is applied perpendicular to the interface cracks. The material properties of the bonding layer vary continuously between those of the lower half-plane and those of the upper half-plane. The differences in the crack surface displacements are expanded in a series of functions that are zero outside the cracks. The unknown coefficients in the series are solved by the Schmidt method so as to satisfy the conditions inside the cracks. The stress intensity factors are calculated numerically for selected crack configurations.     

Anti-plane elastodynamic analysis of planes with multiple defects

The solution of a screw dislocation under time-harmonic condition is obtained in an infinite isotropic plane by means of the Fourier transform method. The stress components reveal the familiar Cauchy singularity at the location of dislocation. The solution is employed to derive integral equations for a plane weakened by cracks and cavities. Cavities are considered as closed curved cracks without singularity. Several examples are solved and the stress intensity factor of cracks and hoop stress on cavities are obtained.     

Interaction between Griffith cracks in a sandwiched orthotropic layer

A study is made of the interaction between three coplanar Griffith cracks which are located symmetrically in the midplane of an orthotropic layer of finite thickness 2h sandwiched between two identical orthotropic half planes. The Fourier transform technique is used to reduce the elastostatic problem to a set of integral equations which have been solved by using the finite Hilbert transform and Cooke's results. Analytical expressions for the stress intensity factors at the tips of cracks are obtained for large values of h. Numerical results concerning the interaction effects are presented with physical significance. It is shown that interaction effects are either shielding or amplification depending on the location of cracks, spacing of crack-tips, and the thickness of the layer. The stress magnification factors at the crack-tips are also calculated.     

Exact solution of four cracks originating from an elliptical hole in one-dimensional hexagonal quasicrystals

Based on the Stroh-type formalism for anti-plane deformation, the fracture mechanics of four cracks originating from an elliptical hole in a one-dimensional hexagonal quasicrystal are investigated under remotely uniform anti-plane shear loadings. The boundary value problem is reduced to Cauchy integral equations by a new mapping function, which is further solved analytically. The exact solutions in closed-form of the stress intensity factors for mode III crack problem are obtained. In the limiting cases, the well known results can be obtained from the present solutions. Moreover, new exact solutions for some complicated defects including three edge cracks originating from an elliptical hole, a half-plane with an edge crack originating from a half-elliptical hole, a half-plane with an edge crack originating from a half-circular hole are derived. In the absence of the phason field, the obtainable results in this paper match with the classical ones.     

位于两不同正交各向异性半平面间张开型界面裂纹的性能分析

利用Schmidt方法分析了位于正交各向异性材料中的张开型界面裂纹问题．经富立叶变换使问题的求解转换为求解两对对偶积分方程,其中对偶积分方程的变量为裂纹面张开位移．最终获得了应力强度因子的数值解．与以前有关界面裂纹问题的解相比,没遇到数学上难以处理的应力振荡奇异性,裂纹尖端应力场的奇异性与均匀材料中裂纹尖端应力场的奇异性相同．同时当上下半平面材料相同时,可以得到其精确解．     

Stress concentration in an anisotropic half-plane with holes and cracks

On the basis of general representations of the generalized complex potentials for a multiconnected half-plane, which the authors have obtained, we solve problems for a multiconnected half-plane with holes and cracks when external forces or dies act on the boundary of the half-plane. Using conformal mapping for an ellipse and the method of least squares, we reduce these problems to solving a system of linear algebraic equations. For different anisotropic materials we give the results of studies of the stress distributions and the variation of the stress intensity factors for a half-plane with a crack in the case of tension at infinity, internal pressure on the edges of the crack, and the action of normal forces on the rectilinear boundary. Two figures, 2 tables. Bibliography: 2 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 63–72.     

Multiple moving cracks in a functionally graded strip

This paper considers several finite moving cracks in a functionally graded material subjected to anti-plane deformation. The distributed dislocation technique is used to carry out stress analysis in a functionally graded strip containing moving cracks under anti-plane loading. The Galilean transformation is employed to express the wave equations in terms of coordinates that are attached to the moving crack. By utilizing the Fourier sine transformation technique the stress fields are obtained for a functionally graded strip containing a screw dislocation. The stress components reveal the familiar Cauchy singularity at the location of dislocation. The solution is employed to derive integral equations for a strip weakened by several moving cracks. Numerical examples are provided to show the effects of material properties, the crack length and the speed of the crack propagating upon the stress intensity factor.     

The interaction between a screw dislocation and a rigid line in a confocal elliptical inhomogeneity

The interaction between a screw dislocation and an elastic elliptical inhomogeneity which contains a confocal rigid line is investigated. The screw dislocation is located inside either the elliptical inhomogeneity or the infinite matrix. By using the complex potential method, explicit series solutions of complex potentials are obtained. The image force acting on the screw dislocation and the stress intensity factor at the tip of the rigid line are derived. As a result, the analysis and discussion show that the influence of the rigid line on the interaction effects between a screw dislocation and an elliptical inhomogeneity is significant. The rigid line enhances the repulsive force exerted on the dislocation produced by the stiff inhomogeneity and abates the attractive force produced by the soft inhomogeneity. For the soft inhomogeneity, there is an unstable equilibrium position when the dislocation is inside the matrix and there is a stable equilibrium position when the dislocation is inside the inhomogeneity. The stress intensity factor contour around the rigid line tip shows that when a dislocation with positive burgers vector is in the upper half-plane, stress intensity factor will be positive; while in the lower half-plane, stress intensity factor will be negative; and in the x-axis, it will be zero. The absolute value of the stress intensity factor will increase when the dislocation approaches the tip of the rigid line. The stress intensity factor at the rigid line tip is enhanced by a harder matrix and abated by a softer matrix.     

Singular integral equation method for 2D fracture analysis of orthotropic solids containing doubly periodic strip-like cracks on rectangular lattice arrays under longitudinal shear loading

The doubly periodic arrays of cracks represent an important mesoscopic model for analysis of the damage and fracture mechanics behaviors of materials. Here, in the framework of a continuously distributed dislocation model and singular integral equation approach, a highly accurate solution is proposed to describe the fracture behavior of orthotropic solids weakened by doubly periodic strip-like cracks on rectangular lattice arrays under a far-field longitudinal shear load. By fully comparing the current numerical results with known analytical and boundary element solutions, the high precision of the proposed solution is verified. Furthermore, the effects of periodic parameters and orthotropic parameter ratio on the stress intensity factor, crack tearing displacement, and effective shear modulus are studied, and an analytically polynomial estimation for the equivalent shear modulus is proposed in a certain range. The interaction distances among the vertical and horizontal periodic cracks are quite different, and their effects vary with the orthotropic parameter ratio. In addition, the dynamic problem is discussed briefly in the case where the material is subjected to harmonic longitudinal shear stress waves. Further work will continue the in-depth study of the dynamics problem of the doubly periodic arrays of cracks.     

HYPERSINGULAR INTEGRAL EQUATIONS FOR PERIODIC ARRAYS OF PLANAR CRACKS IN A PERIODICALLY-LAYERED ANISOTROPIC ELASTIC SPACE UNDER ANTIPLANE SHEAR STRESS

1991MRSubjectClassification75M25,45E991IntroductionDuringthelasttenyearsorsojmanyresearchersinappliedmathematicsandmechanicshaveshownasurginginterestinformulatinglinearcrackproblemsillterlllsofsystel-alsofHadamardfillite-part(hypersingular)integralequations,e.g.Ioakimidis['],Lin'kovandMogilevskaya[']andAnal'].Anadvantageofsuchaformulationisthatthe11nkllowllfllnctionsaredirectlyrelatedtothejlllxlpillthedisplacementsacrossoppositeera(:kfaces.Oncetheyaredeterlttillied,crackparaliietersofinter…     

The stress distribution in an anisotropic half-plane with two elliptic holes or cracks

By use of the method of complex potentials, conformal mappings and least squares this problem is reduced to solving a system of linear algebraic equations with respect to the unknown constants that occur in the required functions. We describe the results of numerical studies of the variation of the stress intensity factors for cracks in an anisotropic half-plane under tension of the half-plane and force on its boundary. Two figures, two tables. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 57–61.     

Antiplane elastodynamic analysis of a strip weakened by multiple defects

The method of images is utilized to derive the solution of a screw dislocation under time-harmonic conditions for an elastic strip from the solution of infinite planes. The displacement and stress components are obtained for a strip under concentrated antiplane, time-harmonic traction. The dislocation solution is employed to formulate integral equation for a strip weakened by cracks and cavities. The effects of load frequency and crack orientation on the stress intensity factors are studied.     

Anti-plane stress analysis of dissimilar sectors with multiple defects

In this article, the anti-plane deformation of a typical dissimilar sector consists of two sub-sectors attached to each other on one circular edge is studied. The solution of a Volterra type screw dislocation problem in the sector is obtained through finite Fourier cosine transform. Exact closed-form solutions for the displacement and stress fields are also presented. Next, using a distributed dislocation method, integral equations of the sectors weakened by cracks and cavities under an anti-plane traction are obtained. The defects are assumed to be located only in one of the sub-sector regions. The obtained equations for the latter problem are of the Cauchy singular type and have been solved numerically. Several examples are presented to demonstrate the efficiency and applicability of the proposed solution procedure. The geometric and force singularities of the stress field are studied and compared to those reported in the literature.     

On an integral equation of the problem for an elastic strip with a slit

A new singular integral equation is obtained that describes the elastic equilibrium of a strip with both an inner and an edge slit (crack) and has a considerable advantage over existing equations /1–9/, etc.) from the viewpoint of a numerical realization and clarification of the analytical relationship with an analogous equation for a half-plane. Numerical results are given of a computation of the stress intensity coefficients at the tips of the inner and edge cracks that refine data in the literature.