Cracks Of Arbitrary Shape Research Articles

Diffraction of elastic waves by three-dimensional cracks of arbitrary shape in a plane

A variational-difference method, proposed for solving two-dimensional integral equations of the convolution type in arbitrary regions [1], and highly recommended for solving dynamic contact problems [2,3], is modified for the case of three-dimensional cracks. A general scheme of the method is given and ways of overcoming the difficulties that arise due to the singularity of the kernel, the increase in its symbol at infinity and taking into account the behaviour of the solution at the boundary of the region, are indicated. Calculations are carried out for rectangular and L-shaped cracks which show the effects of the shape of the crack, the angle of incidence, the type of incident wave and the frequency on the reflection coefficient, the radiation pattern and the redistribution of the energy in the reflected field.

An analysis of a semi-infinite solid with three-dimensional cracks

A method of analysis is proposed for a semi-infinite solid with a surface or embedded plane crack of arbitrary shape. The analysis is fundamentally based on the body force method, but proper expressions for the body force densities are introduced and the stress boundary conditions of the crack surface are replaced by the resultant force conditions in order to improve the accuracy and validity of the method. Numerical results for typical crack problems, based on coarse meshes, are shown to be in remarkable agreement with known solutions. Moreover, new numerical results are obtained in many problems, which are difficult to solve by other methods when considering these crack shapes and complex loading conditions.

Elastic Analyses of Planar Cracks of Arbitrary Shape

A numerical method is presented for planar cracks of arbitrary shape. The fundamental solution for a dislocation segment is obtained from the point force solution and used to derive three coupled surface integral equations in which the crack-face tractions are expressed in terms of the gradients of the relative crack-surface displacements. Because the singularity of the kernel in the integral equations is one order less for fundamental solutions based on dislocation segments than for those based on dislocation loops or the body force method, no special numerical techniques are required. Most of the integrations over elements are evaluated analytically. The integral equations are solved numerically by covering the crack surface with triangular elements, and taking the relative displacements to vary linearly over the elements. The mesh is generated by optimizing the local aspect ratio, which is related to the difference in the principal stretches of the mapping of a square reference mesh onto the fracture surface. This mesh generator allows cracks of a wide variety of shapes to be analyzed with good accuracy. Comparison with known solutions indicate that accurate numerical solutions are obtained with a relatively coarse mesh.

An integral representation of the stress intensity factors for three-dimensional static problems

For finding suitable expressions for the stress intensity factors (SIFs) under a general three-dimensional condition, the first stress invariant and the displacement tangent to a crack edge are analyzed. By using Green's theorem, the SIFs are expressed by integrals for the most general situations. K I and K II are expressed by integrals of the first stress invariant and its partial derivative. K III is expressed by an integral of the displacement tangent to the crack edge and its partial derivative. The integrals include a surface integral on a smooth surface of arbitrary shape, and a line integral along part of the surface's boundaries. The expressions are valid for an arbitrarily shaped elastic medium with stationary cracks of arbitrary shape. The expressions provide a new approach for the determination of the SIFs.

AN ANALYSIS FOR THREE DIMENSIONAL CRACKS

Abstract— A new method of analysis is proposed for an infinite solid containing an embedded plane crack of arbitrary shape. The analysis is fundamentally based on the body force method, but proper expressions of the body force densities are introduced and the stress conditions of the crack surface are replaced by the resultant force conditions in order to improve the accuracy and validity of the method. Numerical results for typical crack problems, based on coarse mesh patterns, are shown to be in remarkable agreement with known solutions. The method is also applied to the bending of circular and rectangular cracks generated from compressive surface contacts for which reliable solutions have not yet been proposed by any other methods. The results are found to be very different from conventional free‐surface solutions.

Representation of stress intensity factors by path integrals of the first stress invariant or anti-plane displacement for general two-dimensional static problems

Determining stress intensity factors (SIFs) is a difficult task either analytically or experimentally. The difficulty arises from the fact that there is no simple and accurate expression for the SIFs under general circumstances. As a result, the determination of the SIFs is usually a complex process. For finding a suitable expression for the SIFs, the first stress invariant and anti-plane displacement are analyzed, and Green's theorem is used. It is found that the stress intensity factors can be represented by path integrals involving only the first stress invariant or anti-plane displacement for general two-dimensional static problems. KI and KII are represented by path integrals of the first stress invariant and its partial derivative. KIII is represented by a path integral of the anti-plane displacement as well as its partial derivative. The integrals are path-independent and valid for an arbitrarily shaped elastic medium with stationary cracks of arbitrary shape. They are also valid for a body containing isolated inhomogeneities such as holes and inclusions. If a crack is straight near its tip, and if the straight portion of the crack can be treated as a cut along the radius of a simply connected circular disk, there exists another kind of integrals representation that does not include the partial derivative terms in the representation for KI. The representation by these integrals provides a new approach to determine the SIFs experimentally, which is simpler and more accurate. This is because the integrals are exact expressions for the SIFs and involve only the first stress invariant or anti-plane displacement.

Eddy-current interaction with an ideal crack. I. The forward problem

The impedance of an eddy-current probe changes when the current it induces in an electrical conductor is perturbed by a flaw such as a crack. In predicting the probe signals, it is expedient to introduce idealizations about the nature of the flaw. Eddy-current interaction is considered with an ideal crack having a negligible opening and acting as a impenetrable barrier to electric current. The barrier gives rise to a discontinuity in the electromagnetic field that has been calculated by finding an equivalent electrical source distribution that produces the same effect. The choice of source is between a current dipole layer or a magnetic dipole layer; either will give the required jump in the electric field at the crack. Here a current dipole layer is used. The strength of the equivalent source distribution has been found by solving a boundary integral equation with a singular kernel. From the solution, the probe impedance due to the crack has been evaluated. Although analytical solutions are possible for special cases, numerical approximations are needed for cracks of arbitrary shape. Following a moment method scheme, numerical predictions have been made for both rectangular and semielliptical ideal cracks. These predictions have been compared with experiments performed on narrow slots used to simulate ideal cracks. Good agreement has been found between the calculations and the measurements.

A new boundary integral equation method of three-dimensional crack analysis

Introducing the mode II and mode III dislocation densities W2(y) and W3(y) of two variables, a new boundary integral equation method is proposed for the problem of a plane crack of arbitrary shape in a three-dimensional infinite elastic body under arbitrary unsymmetric loads. The fundamental stress solutions for three-dimensional crack analysis and the limiting formulas of stress intensity factors are derived. The problem is reduced to solving three two-dimensional singular boundary integral equations. The analytic solution of the axisymmetric problem of a circular crack under the unsymmetric loads is obtained. Some numerical examples of an elliptical crack or a semielliptical crack are given. The present formulations are of basic significance for further analytic or numerical analysis of three-dimensional crack problems.

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

AbstractThe 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.

Application of the weight function method to solving multiparametric three-dimensional fracture mechanics problems

A system of equations of the weight function method is obtained for the calculation of the stress intensity factors (SIF) along the crack front under arbitrary normal loading, as well as the crack opening displacement field (CODF) for cracks of arbitrary shape. The equations, in particular, make it possible to estimate the error of the known empirical formulas for the SIF, as well as to calculate the variation of the SIF values with the crack shape. It is shown that the SIF values for the problems with the mixed boundary conditions can be calculated making use of the weight functions obtained for the problems with the boundary conditions in stresses.

Three-Dimensional Analysis of Cracking Through the Boundary of a Two-Phase Material

The penetration through a two-phase boundary by a biplanar (kinked) crack of arbitrary shape is considered in this paper. The two-phase boundary is modeled as the interface between two perfectly-bonded elastic, isotropic, homogeneous half spaces with different elastic constants. The planar crack on either side of the interface may be arbitrarily orientated with respect to the interface boundary. The body-force method is used to derive a set of coupled two-dimensional singular integral equations which are solved numerically. The solution yields the three crack opening displacements as well as the three modes of stress intensity factors along the crack contour. Numerical results are given for a penny-shaped crack symmetrically oriented with respect to the interface. Mode I stress intensity factors are given for the biplanar crack that experiences a kink when passing through the interface.

Three‐dimensional crack analysis using singular boundary elements

AbstractA multi‐domain method of solving three‐dimensional elastic crack problems in an infinite elastic body using the boundary element method is proposed. The displacement and traction behaviours near a crack front are incorporated in special crack elements. The elimination of singularities arising from the term combined with Kelvin's kernel for displacement in the integrals is discussed in detail. Stress intensity factors of modes I, II and III are obtained directly from crack‐front nodal values, without any extrapolation as in some other methods. No differentiation of conventional boundary integral equations (with Kelvin's tensor kernels) is necessary in the current approach. This method is applicable to cracks of arbitrary shape. Infinite bodies are modelled precisely as such, not approximated as large finite bodies. Numerical solutions of stress intensity factors are given for several problems involving a penny‐shaped crack.

Three-dimensional elastic wave scattering by surface-breaking cracks

Scattering of time-harmonic elastic waves by a surface-breaking crack of arbitrary shape, which is normal to the traction-free surface of a half-space, is investigated by the boundary integral equation method. The use of full-space Green’s functions requires a discretization of the free surface near the crack mouth. Convergence of the crack-opening displacements with successively larger regions of discretization of the free surface is demonstrated. The crack-opening displacements are used to calculate the farfield amplitude spectra of scattered longitudinal and transverse body waves. The required representation integral makes use of the farfield expansions of the half-space Green’s functions. Solutions of both symmetric and antisymmetric problems are obtained by considering two special angles of incidence of a transverse wave field. Numerical results, in the form of crack-opening displacements and scattered farfield amplitude distributions, are presented for scattering by a semicircular normal-edge crack of radius a. A breakdown of contributions to the farfield amplitudes, by ray cases, provides insight as to the relative strengths of the signals radiating from the crack along the various ray paths.

A New Boundary Integral Equation Formulation for Elastodynamic and Elastostatic Crack Analysis

An elastodynamic conservation integral, the J˜k integral, is employed to derive boundary integral equations for crack configurations in a direct and natural way. These equations immediately have lower-order singularities than the ones obtained in the conventional manner by the use of the Betti-Rayleigh reciprocity relation. This is an important advantage for the development of numerical procedures for solving the BIE’s, and for an accurate calculation of the strains and stresses at internal points close to the crack faces. For curved cracks of arbitrary shape the BIE’s presented here have simple forms, and they do not require integration by parts, as in the conventional formulation. For the dynamic case the unknown quantities are the crack opening displacements and their derivatives (dislocation densities), while for the static case only the dislocation densities appear in the formulation. For plane cracks the boundary integral equations reduce to the ones obtained by other investigators.

Stress analysis of cracks of arbitrary shape in finite plate subjected to uniform tension

Stress analysis of cracks of arbitrary shape in finite plate subjected to uniform tension

Crack characterization by an inverse scattering method

Elastic wave scattering data is utilized to characterize a crack of arbitrary shape by an equivalent elliptical crack contained in an unbounded solid. The inverse method is based on an integral representation for the scattered field in the frequency domain. The proposed method is valid for both small and intermediate wavelengths as compared with the size of the crack. In particular for intermediate wavelengths and normal incidence a simple method is discussed. Its solution gives the location of the geometrical center of the crack, the crack size, and the crack-opening displacement.

Flat crack of arbitrary shape in an elastic body: Analytical approach

Abstract A new method is proposed for the analysis of an elastic space weakened by a flat crack of arbitrary shape under the action of a uniform normal pressure. The method is based on an integral representation for the reciprocal distance between two points obtained earlier by the author. A simple but yet accurate relationship is established between the crack face displacements and the applied pressure for an arbitrary flat crack. Specific formulae are derived for a crack in the shape of a polygon, a rectangle, a rhombus, a cross, a circular sector and a circular segment. All the formulae are checked against the solutions known in the literature, and their accuracy is confirmed. A similar approach can be used for the analysis of a crack under a general polynomial loading.

Stress intensity factors for cracks of arbitrary shape near an interfacial boundary

Modes I, II and III stress intensity factors for a crack of arbitrary planar shape near a bimaterial interface are calculated. The solution utilizes the body-force method and requires Green's functions for perfectly bonded elastic half-spaces. The formulation leads to a system of two-dimensional singular integral equations whose solutions represent the three modes of crack opening displacement. Numerical examples of a semicircular or semielliptical crack terminating at the interface and circular or elliptical cracks contained in one material are given for both internal pressure and farfield tension.

On the solution of the two-dimensional problem of a plane crack of arbitrary shape in an anisotropic material

In the present report we investigate the formulae of the stress field in the neighbourhood of a plane crack of arbitrary shape in an anisotropic material for the two-dimensional case. Moreover we consider the expression of the stress field in the neighbourhood of a plane crack in a transversely isotropic solid for the two-dimensional case. The construction of the solution for the anisotropic problem is presented as is the derivation of the expression for the surface tractions necessary to maintain the fundamental solution in a bounded region.

An Integral Technique to Evaluate Opening Mode Stress Intensity Factors for Embedded Planar Cracks of Arbitrary Shape

In this paper, techniques are developed to handle the integrable singularities of the integral proposed by Burns and Oore for the estimation of opening mode stress intensity factors for embedded planar defects of arbitrary shape. The hybrid numerical-analytical integration techniques developed consider separately two crack front zones and one interior zone of the crack surface. Parameters are established for the sizing of the integration elements within each zone. Studies of elliptical defects with aspect ratios between 1 and 10 demonstrate the accuracy and efficiency of this procedure for computing opening mode stress intensity factors. A simple method which compensates for the quadrature error associated with computationally inexpensive, coarse grids is outlined.