Indentation of a cantilever beam or plate

In this paper the general plane problem for a semi-infinite strip fixed at its short end, containing a crack perpendicular to its voundaries is considered. The strip is under the effect of a stamp. By extending the crack to the surfaces, one can reduce the problem to that of c cantilever beam or plate. Integral transform technique is used to provide an exact formulation of this problem, in terms of a system of four singular integral equations one of them being second kind. Stress singularities at the corners of the fixed-end, at the crack tips and at the end points of the contact region undermeath the stamp are obtained from the singular integral equations which are then solved numerically.     

Modelling of growth of three-dimensional cracks by a continuous distribution of dislocation loops

This paper is concerned with a numerical simulation of growth of fatigue cracks in a three-dimensional geometry. A continuous distribution of infinitesimal dislocation loops is employed to model the crack faces, so that the crack problem can be formulated as a set of singular integral equations. A numerical procedure based on an analytical treatment of the associated finite part integral is developed to solve the singular integral equations. The Paris law is then used to predict the rate of crack growth, so that the evolution of the crack shape under fatigue can be traced by a step-by-step algorithm. Various crack growth problems, e.g., the growth of a subsurface crack in a surface treated specimen, are analyzed using the technique, providing new data for several cracks of practical interest.     

Axisymmetric elastodynamic response of a flat annular crack to normal impact waves

The axisymmetric response of a flat annular crack in an infinite medium subjected to normal impact load is investigated in this study. A step stress is applied to the crack surface. The singular solution is equivalent to solutions of the problem of diffraction of normally incident tension wave by a flat annular crack, and the problem of the sudden appearance of a flat annular crack in a uniform tensile stress field. Laplace and Hankel transforms are used to reduce the problem to the solution of a set of triple integral equations in the Laplace transform domain. These equations are solved by using a integral transform technique and the result is expressed in terms of a singular integral equation of the first kind with the kernel which is improved by means of a contour integration on the Riemann surface. A numerical Laplace inversion routine is used to recover the time dependence of the solution. Numerical results of the dynamic stress intensity factor are obtained to show the influence of inertia, the ratio of the inner radius to the outer one and Poisson's ratio on the load transmission to the crack tip.     

Fracture of coated plates and shells under thermal shock

The main interest in this study is in the subcritical crack propagation and fracture of coated materials, specifically of cylindrical shells under repeated thermal shock. First it is shown that the circumferential crack problem in a cylindrical shell may be approximated by a plate on an elastic foundation under plane strain conditions. The thermal shock problem for a layered plate supported by an elastic foundation containing a crack in each layer of arbitrary sizes and locations is then considered. An additional factor studied is the influence of the cooling rate of the plate surface on the stress intensity factors at the crack tips. The problem is formulated in terms of a pair of singular integral equations which are solved for a number of typical crack geometries such as an edge crack, a crack terminating at the interface, an undercoat crack, and a crack crossing the interface. The main results of this paper are the stress intensity factors.     

Surface and internal cracks in a residually stressed plate

The problem of a surface or an internal crack in a plate which contains residual stresses is examined. The line spring model, which reduces a three-dimensional elasticity problem into a two-dimensional problem in plate theory, is used to model the crack. The Reissner plate theory, which takes into account transverse shear deformations, is used to model the plate. The formulation is based on Fourier Transforms which lead to a pair of singular integral equations that are solved numerically. The line spring method requires the plane strain solution to both the edge and internally cracked strip with crack surface loads representative of tension, bending, and the given residual stress distribution. For general use, plane strain solutions are presented for polynomial loading through the thickness up to the fifth order. Comparisons are made between the results given by the line spring model for the Reissner plate theory and the finite element method.     

State of stress in a patched plate containing a crack

Summary The state of stress and strain is examined for a plate containing a curvilinear crack reinforced by a finite patch. The elastic patch covers the crack completely and is rigidly connected to the infinite plate only along its edge. It is assumed that the plate and patch are in a general state of planar strain. The boundary-value problem is reduced to a system of three singular integral equations, which is solved by mechanical quadrature. Numerical results are given for a plate containing a crack in the form of an arc of a parabola and reinforced with an elliptical patch for various orientations of the tensile forces at infinity. The stress intensity coefficients at the crack vertices have been calculated along with the contact forces at the junction.Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 27, No. 4, pp. 33–40, July–August, 1991.     

Perturbation method for the solution of a Zener–Stroh crack with a slightly curved configuration

This paper investigates the Zener–Stroh crack with curved configuration in plane elasticity. A singular integral equation is suggested to solve the problem. Formulae for evaluating the SIFs and T-stress at the crack tip are suggested. If the curve configuration is a product of a small parameter and a quadratic function, a perturbation method based on the singular integral equation is suggested. In the method, the singular integral equation can be expanded into a series with respect to the small parameter. Therefore, many singular integral equations can be separated from the same power order for the small parameter. These singular integral equations can be solved successively. The solution of the successive singular integral equations will provide results for stress intensity factors and T-stress at the crack tip. It is found that the behaviors for the solution of SIFs and T-stress in the Zener–Stroh crack and the Griffith crack are quite different. This can be seen from the presented comparison results.     

A method for the numerical solution of singular integral equations with a principal value integral

A method for the numerical solution of singular integral equations with kernels having a singularity of the Cauchy type is presented. The singular behavior of the unknown function is explicitly built into the solution using the index theorem. The integral equation is replaced by integral relations at a discrete set of points. The integrand is then approximated by piecewise linear functions involving the value of the unknown function at a finite set of points. This permits integration in a closed form analytically. Thus the problem is reduced to a system of linear algebraic equations. The results obtained in this way are compared with the more sophisticated procedures based on Gauss-Chebyshev and Lobatto-Chebyshev quadrature formulae. An integral equation arising in a crack problem of the classical theory of elasticity is used for this purpose.     

Estimation of the effects of plasticity and resulting crack closure during small fatigue crack growth

On the basis of a plastic-strip model and the method of singular integral equations, a closed-form analytical solution of the problem of an elastic-plastic plate containing a rectilinear fatigue crack is considered. The solution is used for the prediction of fatigue growth of `mechanically-small' crack by accounting for reverse plastic yielding and plasticity-induced crack closure in the material. The main effects of these factors on the crack-growth rate are analyzed, and the predicted results are compared with experimental data on small fatigue-crack growth in a aluminum-lithium alloy 2091-T351 and Fe-3% Si alloy.     

The effect of a rigid elliptical inclusion on a straight crack

The general problem of a straight crack near a rigid elliptical inclusion is solved. Complex potentials presented in a previous paper (Santare and Keer [6]) for the interaction of an edge dislocation with rigid ellipse are used to formulate the Green's function for this problem. The solution is written as a set of singular integral equations for crack opening displacement which are solved numerically. Stress intensity factors are presented for a variety of crack/inclusion geometries.     

Electroelastic analysis of an interface anti-plane shear crack in a layered piezoelectric plate

The problem of an anti-plane interface crack in a layered piezoelectric plate composed of two bonded dissimilar piezoelectric ceramic layers subjected to applied voltage is considered. It is assumed that the crack is either impermeable or permeable. An integral transform technique is employed to reduce the problem considered to dual integral equations, then to a Fredholm integral equation by introducing an auxiliary function. Field intensity factors and energy release rate are obtained in explicit form in terms of the auxiliary function. In particular, by solving analytically a resulting singular integral equation, they are determined explicitly in terms of given electromechanical loadings for the case of two bonded layers of equal thickness. Some numerical results are presented graphically to show the influence of the geometric parameters on the field intensity factors and the energy release rate.     

Anti-plane problem of periodic interface cracks in a functionally graded coating-substrate structure

In this paper, the interface cracking between a functionally graded material (FGM) and an elastic substrate is analyzed under antiplane shear loads. Two crack configurations are considered, namely a FGM bonded to an elastic substrate containing a single crack and a periodic array of interface cracks, respectively. Standard integral-transform techniques are employed to reduce the single crack problem to the solution of an integral equation with a Cauchy-type singular kernel. However, for the periodic cracks problem, application of finite Fourier transform techniques reduces the solution of the mixed-boundary value problem for a typical strip to triple series equations, then to a singular integral equation with a Hilbert-type singular kernel. The resulting singular integral equation is solved numerically. The results for the cases of single crack and periodic cracks are presented and compared. Effects of crack spacing, material properties and FGM nonhomogeneity on stress intensity factors are investigated in detail.     

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.     

Crack tip opening displacement of a Dugdale crack in a three-phase cylindrical model composite material

The analytical investigation of the plastic zone size of a crack in three-phase cylindrical model composite material was carried out. The physical problem is simulated as a crack near a circular inclusion (a single fiber) in the composite matrix, while the three-phase cylindrical composite model is used to represent the composite matrix. In the solution procedure, the crack is simulated as a continuous distribution of edge dislocations. With the Dugdale model of small scale yielding, a thin strip of yielded plastic zone is introduced at each crack tip. Using the solution for a three-phase model with a single dislocation in the matrix phase as the Green’s function, the physical problem is formulated into a set of singular integral equations. By employing Erdogan and Gupta’s method, as well as iterative numerical procedures, the singular integral equations are solved numerically for the plastic zone sizes and crack tip opening displacements.     

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.     

Kinked crack in a bending trapezoidal plate

The interaction problem of a kinked crack and the edges of a bending trapezoidal plate which takes the effects of transverse shear deformation into account is presented. The research method is based upon the complex potential technique of Muskhelishvili using conformal mapping. Furthermore, for the analysis of the moment intensities at the tips of the kinked crack, the concept of dislocation distribution is applied. The integral equations for the stress disturbance problem along the line that is the presumed location of the kinked crack are then obtained as a system of singular integral equations with simple Cauchy kernels. As a consequence, the variation of moment intensity factors at the crack-tips is also illustrated.     

Various integral equations for a single crack problem of elastic half-plane

Two kinds of the complex potentials used for the crack problem of the elastic half-plane are suggested. First one is based on the distribution of dislocation along a curve, and second one is based on the distribution of crack opening displacement along a curve. Depending on the use of the complex potentials and the right hand term in the integral equation, two types of the singular integral equation for a single crack problem of elastic half-plane are derived. Regularization of the suggested singular integral equations gives three types of the Fredholm integral equation for the relevant problem. The weaker singular integral equation and the hypersingular integral equation are also introduced. Seven types of the integral equation are finally obtainable. The relation between the kernels of the various integral equations is also discussed.     

On the plastic zone size and CTOD study for a Zener�CStroh crack interacting with a circular inclusion

An analytical solution is given for plastic yielding of a Zener?CStroh crack near a circular inclusion embedded in an infinite matrix. The crack is orientated along the radial direction of the inclusion. In the solution procedure, the crack is simulated as a continuous distribution of edge dislocations. Using the Dugdale model of small-scale yielding, plastic zones are introduced at both crack tips. Using the solution of a circular inclusion, interacting with a single dislocation as the Green??s function, the physical problem is formulated into a set of singular integral equations. With the aid of Erdogan??s method and iterative numerical procedures, the singular integral equations are solved numerically for the plastic zone sizes and crack tip opening displacement. The results obtained in the current work are verified by reduction to simpler cases of the Dugdale model. Various parameters such as the distance, shear modulus ratio, Poisson??s ratio, and loading condition are studied.     

Effect of a riveted wide patch on stresses in a cracked plate

We study the stress-strain state and limiting equilibrium of a thin plate with curvilinear cracks reinforced by a wide patch. The patch is arbitrarily located relative to the cracks and attached to the plate with elastic rivets. The boundary-value problem is reduced to a system of singular integral and integro-algebraic equations and this system is solved by the method of mechanical quadratures. Numerical analysis is performed for the case of a plate with one curvilinear or rectilinear crack reinforced by an elliptic patch. The stress intensity factors formed in this reinforced cracked plate and ultimate loads are determined for various geometric and physical parameters of the plate, crack, patch, and rivets. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 34, No. 1, pp. 37–46, January–February, 1998.     

Scattering of antiplane shear wave by a kinked crack

In this paper the scattering of antiplane shear waves by a kinked crack for a linearly elastic medium is considered. In order to solve the proposed problem, at first the broken crack problem is reduced to two coupled single cracks. Fourier integral transform method is employed to calculate the scattered field of a single crack. In order to derive the Cauchy type integral equations of a broken crack and analyze the singular stresses at the breakpoint, the scattered field of a single crack is separated into a singular part and a bounded part. The single crack solution is applied to derive the generalized Cauchy type integral equations of a broken crack. The singular stress and singular stress order are analyzed in the paper and the dynamic stress intensity factor (DSIF) at breakpoint is defined. Numerical solution of the obtained Cauchy type integral equations gives the DSIF at the crack tips and at the breakpoint. Comparison of the present results in some special cases with the known results confirms the proposed method. Some typical numerical results and corresponding analysis are presented at the end of the paper.