A simple solution of the bridged crack problem

A simple, yet effective solution is derived to the problem of a penny-shaped crack, whose faces are bridged by fibers, and subjected to normal loading. The mathematical apparatus developed by the author earlier (Fabrikant, 1989, 1991) is used to reduce the governing integral equation to a sequence of integrals of elementary functions, which can be easily solved by the method of collocations. The case of uniform loading is considered as an example. The accuracy of the obtained solution is rigorously verified by different criteria.     

Stress field around an arbitrary thin inclusion in a transversely isotropic elastic half-space

We consider a thin flat inclusion of arbitrary shape located inside a transversely isotropic elastic half-space in the plane parallel to its boundary z = 0. An arbitrary tangential displacement is prescribed on the inclusion. The boundary of the half-space is stress-free. We need to find the complete field of stresses and displacements in this half-space. A governing integral equation is derived by the generalized method of images, introduced by the author. The case of circular inclusion is considered as an example. Two methods of solution of the governing integral equation are derived. A detailed solution is presented for the particular cases of radial expansion, torsion and lateral displacement of the inclusion. The solution is also valid for the case of isotropy. The governing integral equation for the case of isotropy is derived.     

Dual boundary integral formulation for 2-D crack problems

An efficient integral equation formulation for two-dimensional crack problems is proposed with the displacement equation being used on the outer boundary and the traction equation being used on one of the crack faces. Discontinuous quarter point elements are used to correctly model the displacement in the vicinity of crack tips. Using this formulation a general crack problem can be solved in a single-region formulation, and only one of the crack faces needs to be discretised. Once the relative displacements of the cracks are solved numerically, physical quantities of interest, such as crack tip stress intensity factors can be easily obtained. Numerical examples are provided to demonstrate the accuracy and efficiency of the present formulation.     

Magnetoelectroelastodynamic interaction of multiple arbitrarily oriented planar cracks

The problem of multiple arbitrarily oriented planar cracks in an infinite magnetoelectroelastic space under dynamic loadings is considered. An explicit solution to the problem is given in the Laplace transform domain in terms of suitable exponential Fourier integral representations. The unknown functions in the Fourier integrals are directly related to the Laplace transform of the jumps in the displacements, electric potential and magnetic potential across opposite crack faces and are to be determined by solving a system of hypersingular integral equations. Once the hypersingular integral equations are solved, the displacements, electric potential, magnetic potential and other quantities of interest such as the crack tip intensity factors may be easily computed in the Laplace transform domain and recovered in the physical space with the help of a suitable algorithm for inverting Laplace transforms.     

Analysis of anti-plane interface cracks in one-dimensional hexagonal quasicrystal coating

The displacement discontinuity method is extended to study the fracture behavior of interface cracks in one-dimensional hexagonal quasicrystal coating subjected to anti-plane loading. The Fredholm integral equation of the first kind is established in terms of displacement discontinuities. The fundamental solution for anti-plane displacement discontinuity is derived by the Fourier transform method. The singularity of stress near the crack front is analyzed, and Chebyshev polynomials of the second kind are numerically adopted to solve the integral equations. The displacement discontinuities across crack faces, the stress intensity factors, and the energy release rate are calculated from the coefficients of Chebyshev polynomials. In combination with numerical simulations, a comprehensive study of influencing factors on the fracture behavior is conducted.     

Boundary integral equation method in thermoelasticity: part II crack analysis

The boundary integral equation formulation of thermoelasticity problems from part I is applied to crack problems in both finite and infinite thermoelastic bodies. For a flat crack in an infinite body the normal and tangential crack opening displacement are decoupled. Transient and steady state problems of thermoelasticity, as well as stationary problems, are considered.     

A crack model with delayed embedded discontinuities

In this contribution the combination of a smeared rotating crack model with a crack model based on the strong discontinuity approach and formulated within the framework of elements with embedded discontinuities is presented. This so–called crack model with delayed embedded discontinuities allows considering crack opening in the direction normal to the crack and relative tangential displacements of the crack faces with transfer of shear forces across the crack faces. The advantages of this approach are shown by the numerical simulation of an anchor pull out test. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Continuation of some solutions in potential theory and their applications to elastic contact and crack problems

The following mixed boundary value problem is considered: arbitrary tangential displacements are prescribed inside a circle, while the tangential and normal stresses outside the circle are zero. In this case, a direct and simple formula is derived for the tangential displacements outside the circle in terms of prescribed displacements inside, thus making the tangential displacement known all over the boundary. The original problem is no longer mixed, and the complete solution becomes readily available. Another solution for the case, when arbitrary tangential displacements are prescribed outside a circle, is derived in a similar manner. The reciprocal theorem is used to derive the continuation formulae for the tangential stresses inside and outside a circle. Application of these results to contact and crack problems is demonstrated.     

Non-local theory solution of two collinear mode-I cracks in piezoelectric materials

In this paper, the basic solution of two collinear cracks in a piezoelectric material plane subjected to a uniform tension loading is investigated by means of the non-local theory. Through the Fourier transform, the problem is solved with the help of two pairs of integral equations, in which the unknown variables are the jumps of displacements across the crack surfaces. To solve the integral equations, the jumps of displacements across the crack surfaces are directly expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the interaction of two cracks, the materials constants and the lattice parameter on the stress field and the electric displacement field near crack tips. Unlike the classical elasticity solution, it is found that no stress and electric displacement singularities are present at crack tips. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to using the maximum stress as a fracture criterion in piezoelectric materials.     

Antiplane fracture in a prestressed and prepolarized piezoelectric crystal

The general complex-variable solution to the equations of incrementalantiplane piezoelectricity is used to solve the problem of crackpropagation in a crystal in the presence of an initial stressand electric field. Remote incremental antiplane tractions arespecified and an incremental traction of the same intensityis applied to the crack faces. The initial fields are shownto have a pronounced effect on the near-tip stresses and onthe displacements of the crack faces. Griffith's criterion isused to determine the combination of electromechanical parametersrequired for incipient crack propagation. Received 22 April 2000.     

Boundary integral equations for an interface linear crack under harmonic loading

The present study is devoted to application of boundary integral equations to the problem of a linear crack located on the bimaterial interface under time-harmonic loading. Using the Somigliana dynamic identity the system of boundary integral equations for displacements and tractions at the interface is derived. For the numerical solution the collocation method with piecewise constant approximation on each linear continuous boundary elements is used. The distributions of the displacements are computed for different values of the frequency of the incident tension-compression wave. Results are compared with static ones.     

Solution of problems on the anti-plane deformation of elastic bodies with corner points by the method of integral equations

The problem of the antiplane deformation of an elastic cylinder with a multiconnected finite or infinite section, bounded by a system of closed curves that can have corner points, is examined. Forces or displacements are given on the whole boundary of the body. The problem is reduced to an integral equation whose kernel has strong stationary singularities at the corner points. Results of an investigation of the solvability of this equation and the smoothness of its solution are presented. A procedure for the numerical solution of the integral equation is described. A space with a prismatic hole of rectangular section or a rigid inclusion subjected to a uniform tangential force at infinity is considered as an example. The generalized stress intensity factors are calculated.     

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.     

A normal crack in an elastic half-space with stress-free surface

The problem of an elastic half-space with stress-free surface and a crack of arbitrary shape with prescribed displacements or tractions is reduced to an equivalent system of integral equations on the crack. For a pressurized crack in a plane perpendicular to the free surface, a scalar integral equation is derived. In properly chosen function spaces, unique solvability of the integral equation and regularity of solutions for regular data are proven.     

压电材料中两个非对称平行裂纹的基本解

采用Schmidt方法分析压电材料中非对称平行的双可导通裂纹的断裂性能．利用Fourier变换使问题的求解转换为求解两对以裂纹面位移之差为未知变量的对偶积分方程．为了求解对偶积分方程,直接把裂纹面位移差函数展开成Jacobi多项式形式．最终得到了裂纹的应力强度因子与电位移强度因子之间的关系．数值结果表明,应力强度因子和电位移强度因子与裂纹间的距离、裂纹的几何尺寸有关;与不可导通裂纹有关结果相比,可导通裂纹的电位移强度因子远小于相应问题不可导通裂纹的电位移强度因子．同时可以发现裂纹间的“屏蔽”效应也在压电材料中出现．     

Contact Interaction of Faces of the Plane Crack under Harmonic Loading

The present work is devoted to the solution of the three-dimensional fracture mechanics problem for a linear elastic, homogeneous and isotropic solid with a stationary plane crack under normal time-harmonic loading. The problem has been solved by the method of boundary integral equations with the allowance for the contact interaction of the opposite faces of the crack. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis solution method for 3D planar crack problems of two-dimensional hexagonal quasicrystals with thermal effects

An analysis solution method (ASM) is proposed for analyzing arbitrarily shaped planar cracks in two-dimensional (2D) hexagonal quasicrystal (QC) media. The extended displacement discontinuity (EDD) boundary integral equations governing three-dimensional (3D) crack problems are transferred to simplified integral-differential forms by introducing some complex quantities. The proposed ASM is based on the analogy between these EDD boundary equations for 3D planar cracks problems of 2D hexagonal QCs and those in isotropic thermoelastic materials. Mixed model crack problems under combined normal, tangential and thermal loadings are considered in 2D hexagonal QC media. By virtue of ASM, the solutions to 3D planar crack problems under various types of loadings for 2D hexagonal QCs are formulated through comparison to the corresponding solutions of isotropic thermoelastic materials which have been studied intensively and extensively. As an application, analytical solutions of a penny-shaped crack subjected uniform distributed combined loadings are obtained. Especially, the analytical solutions to a penny-shaped crack subjected to the anti-symmetric uniform thermal loading are first derived for 2D hexagonal QCs. Numerical solutions obtained by EDD boundary element method provide a way to verify the validity of the presented formulation. The influences of phonon-phason coupling effect on fracture parameters of 2D hexagonal QCs are assessed.     

利用Schmidt方法研究压电材料Ⅰ-型界面裂纹问题

在一定的假设条件下,即不考虑界面裂纹尖端处裂纹面的相互叠入现象,研究了压电材料Ⅰ-型界面裂纹问题．利用Fourier变换使问题的求解转换为求解两对对偶积分方程．进而把裂纹表面位移差展开成Jacobi多项式形式来求解对偶积分方程．结果表明裂纹尖端应力场和电位移场的奇异性与均匀材料裂纹问题的奇异性相同．当上下半平面材料相同时,解可以退化而得到其精确解．     

Thermal stress state of a bimaterial with a closed interfacial crack having rough surfaces

We have formulated the problem of thermoelasticity for a bimaterial whose components differ only in their shear moduli, with a closed interfacial crack having rough surfaces. The bimaterial is subjected to the action of compressive loads and heat flow normal to the interfacial surface. We have taken into account the dependence of thermal conductance of the defect on the contact pressure of its faces and heat conductivity of the medium that fills it. The problem is reduced to a Prandtl-type nonlinear singular integro-differential equation for temperature jump between the crack surfaces. An analytical solution of this problem has been constructed for the case of action of the heat flow only. We have analyzed the dependence of contact pressure of the defect faces, temperature jump between them, and the intensity factor of tangential interfacial stresses on the value of given heat flow, roughness of the surfaces, and ratio between the shear moduli of joined materials.     

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…