A general method for multiple crack problems in a finite plate

A novel method for the multiple crack problems in a finite plate is proposed in this paper. The basic stress functions of the solution consist of two parts. One is the Fredholm integral equation solution for the crack problem in an infinite plate, and the other is that of the weighted residual method for general plane problems. The combined stress functions are used in the analysis and the boundary conditions on the crack surfaces and the boundary are considered. After the coefficients of the functions have been determined, the stress intensity factors (SIF) at the crack tips can be calculated. Some numerical examples are given and it was observed that when the cracks are very short, the results compare very favorably with the existing results for an infinite plate. Furthermore, the influence of the boundary can be considered. This method can be used for arbitrary multiple crack problems in a finite plate.     

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

Introducing the mode II and mode III dislocation densities W ₂(y) and W ₃(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.     

Stress intensity factors for cracks emanating from a triangular or square hole in an infinite plate by boundary elements

This paper concerns stress intensity factors of cracks emanating from a triangular or square hole in an infinite plate subjected to internal pressure calculated by means of a boundary element method, which consists of constant displacement discontinuity element presented by Crouch and Starfied and crack tip displacement discontinuity elements proposed by the author. In the boundary element implementation the left or the right crack tip displacement discontinuity element is placed locally at corresponding left or right crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. Numerical examples are included to show that the method is very efficient and accurate for calculating stress intensity factors of plane elasticity crack problems. Specifically, the numerical results of stress intensity factors of cracks emanating from a triangular or square hole in an infinite plate subjected to internal pressure are given.     

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.     

A generalised solution for the crack surface displacement of mode I two-dimensional part-elliptical crack

A generalised approximate crack surface displacement solution for the two-dimensional part-elliptical mode I crack was developed. This solution includes the surface crack, corner crack and embedded crack, which is subjected to the arbitrary crack surface pressure. The crack surface displacement is derived from stress intensity factor solution and corresponding crack surface pressure distribution. Comparisons of the solution with accurate solutions showed that rather high accuracy has been achieved with the developed solution for various surface, embedded and corner crack problems. This solution can be used to derive three-dimensional weight functions as long as the stress intensity factor and the corresponding crack surface pressure are available for arbitrary mode I problems.     

Vibrations of elastic infinite and semi-infinite media containing cracks

The problems of stress distribution in an infinite medium and in an elastic half-plane containing line cracks, when the pressure which opens the crack is periodic in time, are considered. These are (1) a cruciform crack in an elastic infinite medium, (2) an edge crack perpendicular to the surface of an elastic half-plane, and (3) their corresponding “exterior” problems. The integral equations corresponding to these problems are obtained. Expressions for the stress intensity factor and the crack energy are derived and numerical results are presented. The equivalence of the stress intensity factor and the crack energy for “exterior” and “interior” problems as established by Stallybrass for the static case is obtained from the dynamic results by letting the frequency tend to zero.     

A MODEL FOR THE BEHAVIOUR OF MATERIALS WITH CRACKS UNDER HYDROGEN EMBRITTLEMENT CONDITIONS

Abstract— We consider the slow growth of normal tension cracks as quasi-brittle behaviour under hydrogen embrittlement conditions. Experiments show that the cracking resistance of a material in such cases is not a constant of the material, but is characterized by some function that relates the rate of crack growth to the stress intensity factor. We propose a numerical method for the calculation of opening mode crack growth when the kinetics are controlled by the gas diffusion into the material. The problems under consideration model the fracture phenomena inherent to structures (e.g. pressure vessels, pipelines) that operate in an aggressive medium and in particular a hydrogen environment.
In such problems it is necessary to calculate the pressure variation inside a crack as a result of gas diffusion and crack growth under the action of this pressure. Hence it is necessary to solve problems of diffusion theory and elasticity theory for a cracked medium together with some additional conditions that provide the link between these two fundamental problems.
We study the case of an infinite medium containing a crack which occupies a plane domain of arbitrary shape. To avoid difficulties related to the three-dimensionality of the problems, we reduce them to two-dimensional integro-differential equations for the crack domain. The integro-differential equation of the elasticity problem of the crack is solved on the basis of the Boundary Element Method (BEM). The crack kinetics are calculated using a scheme previously introduced by one of the authors and then the BEM is used to solve the integral equation for the diffusion-into-the-crack problem similar to the analogous problem of filtration of the fluid into a crack.     

Boundary-integral equation analysis of cracked anisotropic plates

A numerical procedure based on the boundary-integral equation method, is formulated using the fundamental solution (Green's function) for an infinite anisotropic plate containing an exact crack. The boundary-integral equation developed can be solved numerically for the mode 1 and mode 2 stress intensity factors by approximating boundary data on the surface of an arbitrary body, excluding the crack surface. Thus the efficiency and generality of the boundary-integral equation method and the precision of exact crack model analyses are combined in a direct manner. The numerical results reported herein are as accurate as previously published isotropic results. The effects of material anisotropy are reported for center and double-edge cracked geometries. A path independent integral for obtaining mode 1 and mode 2 stress intensity factors directly for arbitrary loading is reported.     

An Analysis of a Quarter-Infinite Solid with a Corner Crack of Arbitrary Shape

In this paper, a versatile body force method for a quarter-infinite solid with a corner crack of arbitrary shape is proposed under two types of pressure: constant and linear. New numerical results are obtained for different corner crack cases. Fatigue crack growth from a corner crack has been analysed successively with the present method. Moreover, the stress intensity factor of a corner crack is proposed in a simple form for an arbitrary shape.     

Point weight function method application for semi-elliptical mode I cracks

The method of the approximate weight function construction for a semi-elliptical crack was suggested. The weight function sought was written as the sum of asymptotic (weight function for an elliptical crack in an infinite body) and correction components. To take into account the influence of a body free surface on the asymptotic component behavior, fictitious forces symmetric with respect to the body free surface were introduced.As an example of the efficiency of the proposed method semi-elliptical axial cracks in pressure vessels were considered. The results of the stress intensity factor prediction are in good agreement with the corresponding results obtained by Raju and Newman. The only exception are the results for the points located near the major ellipse axis. This may be explained by the shortcomings of the employed empirical weight function expression for an elliptical crack in an infinite body.     

Tension of a cylindrical bar having an infinite row of circumferential cracks

In this paper, the crack problems in the case of a cylindrical bar having a circumferential crack and a cylindrical bar having an infinite row of circumferential cracks under tension are analyzed by the body force method. The stress field for a periodic array of ring forces in an infinite body is used to solve the problems. The solution is obtained by superposing the stress fields of ring forces in order to satisfy a given boundary condition. The stress intensity factors are calculated for various geometrical conditions. The obtained values of stress intensity factor of a single circumferential crack are considered to be more reliable than the results of other paper's. As the crack becomes very shallow, the stress intensity factor of a row of circumferential cracks approaches the value corresponding to that of a row of edge cracks in a semi-infinite plate under tension. As the crack becomes very deep, it approaches the values corresponding to that of a single deep circumferential crack.     

AN IMPROVED NUMERICAL TECHNIQUE FOR SIMULATING THE GROWTH OF PLANAR FATIGUE CRACKS

Abstract— This paper describes a versatile technique for simulating the fatigue growth of a wide range of planar cracks of practical significance. Crack growth is predicted on a step-by-step basis from the Paris law using stress intensity factors calculated by the finite element method. The crack front is defined by a cubic spline curve from a set of nodes. Both the 1/4-node crack opening displacement and the three-dimensional J -integral (energy release rate) methods are used to calculate the stress intensity factors. Automatic remeshing of the finite element model to a new position which defines the new crack front enables the crack propagation to be followed. The accuracy and capability of this finite element simulation technique are demonstrated in this paper by the investigation of various problems of both theoretical and practical interest. These include the shape growth trend of an embedded initially penny-shaped defect and an embedded initially elliptical defect in an infinite body, the growth of a semi-elliptical surface crack in a finite thickness plate under tension and bending, the propagation of an internal crack in a round bar and the shape change of an external surface crack in a pressure vessel.     

Weight function for an elliptic crack in an infinite medium. I. Normal loading

A recently developed integral equation method has been used to derive the crack opening displacement of an elliptic crack in an infinite elastic medium subjected to a concentrated pair of point force loading at an arbitrary location on the crack faces. These results have been used to obtain the stress intensity factor along the elliptic crack front which corresponds to the weight function for an elliptic crack under normal loading. Analytical expression of the weight function can be used to derive the stress intensity factor for both polynomial loading as well as non-polynomial loading.     

Crack problems in magnetoelectroelastic solids. Part II: general solution of collinear cracks

In this paper, we study mechanically traction-free and electromagnetically permeable crack problems in infinite magnetoelectroelastic solids with linear coupling between the elastic and electromagnetic fields. Using the Stroh-formalism, we first obtain the general solution for collinear cracks in a magnetoelectroelastic medium subjected to arbitrary loads. Then, we give specific solutions for several examples: finite or infinite number of collinear crack subjected to arbitrary remote loads, and a single crack subjected to a line load at an arbitrary point. It is found that in the most general cases, the singularity of electric-magnetic field is always dependent on that of stress. Especially when the medium is only loaded by the remote uniform field, the intensity factor of stress is the same as that of isotropic materials, and the electric-magnetic field inside any crack is uniform.     

A natural approach to the introduction of finite-part integrals into crack problems of three-dimensional elasticity

The classical singular integral equation for the problem of a plane crack inside an infinite isotropic elastic medium and under an arbitrary normal pressure distribution was recently modified and written without the use of the Laplace operator Δ or the derivatives of the unknown function, but with the use of a finite-part integral. In this paper, a second complete derivation of the same equation is made (not based on previous forms of this equation) by using a limiting procedure, which makes it clear why the finite-part integral results in this equation. It is believed that the present results will be used in future for the introduction of finite-part integrals into a lot of crack problems in the theory of three-dimensional elasticity.     

The infinite sheet with cracked cylindrical hole under inclined tension or in-plane shear

The stress intensity factors are determined at the root of a radial crack emanating from a circular hole in an infinite sheet, under uniform tension in the direction at an arbitrary inclination with angle β and uniform in-plane shear, respectively. The stress analysis is carried out using the Muskhelishvili formulation and the conformal mapping. Numerical results of the stress intensity factors are obtained for varying crack length-to-hole radius ratio, L/R.     

Determination of thermal stress intensity factors for an interface crack under vertical uniform heat flow

In the case where an interface crack exists in an infinite two-dimensional elastic bimaterial, the crack surface is insulated under traction-free conditions and the uniform heat flow vertical to the crack from an infinite boundary is given, temperature and stress potentials are obtained by using the complex variable approach to solve Hubert problems, and the results are used to obtain thermal stress intensity factors. The mode II thermal stress intensity factor only occurs if both the shear moduli, as well as the Poisson's ratios in the upper and lower material, are the same. Otherwise, mode I and II thermal stress intensity factors exist but the value of the mode I thermal stress intensity factor is much smaller than that of mode II.     

The approximate analytical solution for both surface and embedded cracks in cylinder with finite size

The exact solution for a penny-shaped crack in an infinite elastic body under the action of an arbitrary normal load and the trigonometric series solution for a cylinder with finite size under an arbitrary lateral surface load are taken as basic solutions. From the alternate action of the basic solutions, we have obtained the approximate analytical expressions of K_I for both the surface and embedded cracks in cylinder with finite size. The first approximate expressions of K_I for a cylinder under tensile load and pure bending respectively and some numerical results of K_I are given as an example of practical calculation.     

A solution for SIFs of part-elliptical cracks based on approximate crack surface displacement and energy method

A procedure has been developed to derive stress intensity factors (SIFs) for part-elliptical cracks based on an approximate crack surface displacement mode assumption for general configurations. The crack surface displacement mode is composed of available 2D crack surface displacement modes at intersections of the crack surface and boundaries, or in symmetry planes. Along with the obtained crack surface displacement mode, SIFs are determined by the magnitude of the crack surface displacement derived from energy release rate for virtual crack increments. The procedure was analytically verified with the exact solution for an embedded crack in an infinite body subjected to uniform crack surface pressure. Several examples show the obtained results in acceptable agreements with available solutions.     

Analytical solution for embedded elliptical cracks,and finite element alternating method for elliptical surface cracks,subjected to arbitrary loadings

The complete solution for an embedded elliptical crack in an infinite solid and subjected to arbitrary tractions on the crack surface is rederived from Vijayakumar and Atluri's general solution procedure. The general procedure for evaluating the necessary elliptic integrals in the generalized solution for elliptical crack is also derived in this paper. The generalized solution is employed in the Schwartz alternating technique in conjunction with the finite element method. This finite element-alternating method gives an inexpensive way to evaluate accurate stress intensity factors for embedded or elliptical cracks in engineering structural components.