A rectangular crack in an infinite solid,a semi-infinite solid and a finite-thickness plate subjected to tension

This paper is concerned with the tension problems of a semi-infinite solid with an embedded rectangular crack and a rectangular surface crack as well as an infinite solid with a rectangular crack as a special case. The analyses are based on the body force method, and are performed by generalizing the previous analyses of an embedded elliptical crack and a semi-elliptical surface crack. Furthermore, approximate results are given for a finite-thickness plate with an embedded rectangular crack, by superposing the effects of the free surfaces obtained in the above analysis for a semi-infinite solid.     

Weight Function Method for Stress Intensity Factor of Internal Surface Semi-elliptical Crack in Elliptical Holes

The hatches for inspecting are usually designed with elliptical holes in airplane structures, so computation of the stress intensity factor of three dimensional crack at elliptical holes is pivotal for damage tolerance analysis of these structures. In this paper, weight function is derived for a two dimensional through cracks at elliptical holes by applying a compounding method. Stress intensity factor formulas for an internal surface semi-elliptical crack in elliptical holes are obtained wing the three dimensional weight function method. Stress intensity factors for an internal surface semi-elliptical crack in elliptical holes under remote tension are computed. At the same time, research on how radius of curvature for elliptical holes affect stress intensity factors was conducted. Stress intensity factors decrease when radius of curvature increases. Some results and conclusions which are of practical value are given.     

Determination of stress intensity factor for embedded elliptical crack in turbine rotor

The stress intensity factor (SIF) for an embedded elliptical crack in a turbine rotor and the thermal shock stress intensity factor for a semi-elliptical surface crack in a finite plate are determined by means of Vainshtok's weight function method. The solution for the semi-elliptical surface crack is in good agreement with the previous one. The value of the SIF for the embedded elliptical crack in the turbine rotor under centrifugal and thermal loading is larger at the crack contour near the inner radius surface and almost constant at the opposite crack contour. The SIF decreases by increasing the crack ratio, and the distance between the inner radius surface and the crack center.     

Approximate Closed from Weight Function for an Elliptical Crack in an Infinite Body

It have been reported that in literature there exists some number of approximate closed form weight functions for elliptical crack in an infinite body. The general procedure for refining them by Rice's integral formula is suggested. A tentative formula obtained in this way has been numerically verified by a carefully elaborated procedure. For uniform loading the accuracy of suggested formula lies within 6 percent of the strict solution. The practical usage of the formula in the point weight function method for semi-elliptical crack is demonstrated. This revised version was published online in July 2006 with corrections to the Cover Date.     

Interacting dissimilar semi-elliptical surface flaws under tension and bending

Stress intensity factors for two dissimilar interacting semi-elliptical coplanar surface flaws (cracks) in a semi-infinite elastic body are obtained under overall tension and bending. First the basic equations for a general planar crack normal to the free surface are established, using the method of equivalent eigen- or transformation strains (the body force method). Then the results are specialized for application to elliptical cracks. Numerical values are obtained for various configurations and crack shapes. Results are compared with those of two-dimensional collinear cracks. Finally, an approximate procedure for estimating the stress intensity factors for a general three-dimensional crack is suggested.     

Determination of approximate point load weight functions for embedded elliptical cracks

This paper presents the application of weight function method for the calculation of stress intensity factors in embedded elliptical cracks under complex two-dimensional loading conditions. A new general mathematical form of point load weight function is proposed based on the properties of weight functions and the available weight functions for two-dimensional cracks. The existence of this general weight function form has simplified the determination of point load weight functions significantly. For an embedded elliptical crack of any aspect ratio, the unknown parameters in the general form can be determined from one reference stress intensity factor solution. This method was used to derive the weight functions for embedded elliptical cracks in an infinite body and in a semi-infinite body. The derived weight functions are then validated against available stress intensity factor solutions for several linear and non-linear stress distributions. The derived weight functions are particularly useful for the fatigue crack growth analysis of planer embedded cracks subjected to fluctuating non-linear stress fields resulting from surface treatment (shot peening), stress concentration or welding (residual stress).     

The combined weight function method application for a hole emanated crack

An approximate weight function construction for a hole emanated crack as a sum of an asymptotic term and a correction one is considered. Based on the known approximate fundamental solution for a half-plane with an edge crack the new weight function asymptotic term which satisfies both known solutions for an edge crack in a half-plane and for an inner crack in an infinite plane is proposed. The practical usefulness of the method is demonstrated for a hole emanated crack in an infinite plate and for a thick cylinder with an edge crack at the inner surface.     

Variation of stress intensity factor and crack opening displacement of semi-elliptical surface crack

In this paper a singular integral equation method is applied to calculate the stress intensity factor along crack front of a 3D surface crack. Stress field induced by body force doublet in a semi infinite body is used as a fundamental solution. Then the problem is formulated as an integral equation with a singularity of the form of r ^-3. In solving the integral equations, the unknown functions of body force densities are approximated by the product of a polynomial and a fundamental density function; that is, the exact density distribution to make an elliptical crack in an infinite body. The calculation shows that the present method gives the smooth variation of stress intensity factors along the crack front and crack opening displacement along the crack surface for various aspect ratios and Poisson's ratio. The present method gives rapidly converging numerical results and highly satisfactory boundary conditions throughout the crack boundary.     

Stress intensity factors of an elliptical crack or a semi-elliptical crack subject to tension

This paper is concerned with the analysis of stress intensity factors of a semi-infinite body with an elliptical or a semi-elliptical crack subject to tension. Analysis is based on the body force method [1] which has been applied to the various plane stress problems. In this paper the method is extended to three-dimensional problems. The numerical calculations are performed for various shapes and configurations of ellipses and the results are in agreement with the two-dimensional cases by M. Isida asb/a→0. The stress intensity factor of a semi-elliptical crack in a plate of finite width is also discussed.     

Variations of stress intensity factor of a semi-elliptical surface crack subjected to mixed mode loading

Maximum stress intensity factors of a surface crack usually appear at the deepest point of the crack, or a certain point along crack front near the free surface depending on the aspect ratio of the crack. However, generally it has been difficult to obtain smooth distributions of stress intensity factors along the crack front accurately due to the effect of corner point singularity. It is known that the stress singularity at a corner point where the front of 3 D cracks intersect free surface is depend on Poisson's ratio and different from the one of ordinary crack. In this paper, a singular integral equation method is applied to calculate the stress intensity factor along crack front of a 3-D semi-elliptical surface crack in a semi-infinite body under mixed mode loading. The body force method is used to formulate the problem as a system of singular integral equations with singularities of the form r ⁻³ using the stress field induced by a force doublet in a semi-infinite body as fundamental solution. In the numerical calculation, unknown body force densities are approximated by using fundamental density functions and polynomials. The results show that the present method yields smooth variations of mixed modes stress intensity factors along the crack front accurately. Distributions of stress intensity factors are indicated in tables and figures with varying the elliptical shape and Poisson's ratio.     

Computation of stress intensity factor in cracked plates under bending in static and fatigue by a hybrid method

A hybrid weight-function technique is presented. It consists of dividing an elliptical crack into two zones, then using the appropriate weight function in the area where it is more efficient. The proportion between zones is determined by optimizing two crack parameters (axis ratio and curvature radius). Stress intensity factors for plates containing elliptical and semi-elliptical cracks are hence computed by a self developed computer code. Static and fatigue loadings of bending are considered. The results found by the present approach are in good correlation with the analytical solutions (when available) as well as with those of other researchers.     

Weight functions and Stress Intensity Magnification factors for elliptical and semi-elliptical cracks under variable normal stress – part I

Surface cracks under peak stresses are investigated. The calculational procedure is based on the general form of the weight function for an elliptical crack embedded in an infinite solid. Two points on the contour of the ellipse are investigated. The superposition method is used for transfer from the embedded crack to surface crack configurations. Weight functions for both points have been found with the crack aspect ratio a/c as parameter. For the point at the end of the minor axis all weight functions are describable by one equation only (Heuman's lambda function). For various a/c ratios of the surface crack under different stress distributions the stress intensity magnification factors are given.     

A new method to calculate k1 for semi-elliptical surface cracks in finite plates

Irwin's solution of the stress intensity factor K_I for an embedded elliptical cracks was extended to solve for K_I for semi-elliptical surface cracks in finite plates. A double series was set up to express the displacement of the crack surface, and the unknown coefficients of the series were determined by the crack surface displacements of two dimensional edge cracks and center cracks. The maximum displacement was determined with an energy method. The results reflected the influence of both the relative crack depth a/t and the relative crack length c/W. The cases in which elliptical axis ratio a/c > 1 were also included.     

Oblique semi-elliptical surface crack in semi-infinite solid subjected to tension

This note is concerned with a semi-elliptical surface crack arbitrarily inclined to the free surface of a semi-infinite solid subjected to tension. The analysis is performed by the body force method, and the stress intensity factor at the maximum depth point on the crack front are given for various shapes and inclination angles of the crack. The numerical results are fitted to a reliable polynomial for convenience in engineering applications.     

Variation of mixed modes stress intensity factors of an inclined semi-elliptical surface crack

In this paper, a singular integral equation method is applied to calculate the stress intensity factor along crack front of a 3D inclined semi-elliptical surface crack in a semi-infinite body under tension. The stress field induced by displacement discontinuities in a semi-infinite body is used as the fundamental solution. Then, the problem is formulated as a system of integral equations with singularities of the form r ^–3. In the numerical calculation, the unknown body force doublets are approximated by the product of fundamental density functions and polynomials. The results show that the present method yields smooth variations of mixed modes stress intensity factors along the crack front accurately for various geometrical conditions. The effects of inclination angle, elliptical shape, and Poisson's ratio are considered in the analysis. Crack mouth opening displacements are shown in figures to predict the crack depth and inclination angle. When the inclination angle is 60 degree, the mode I stress intensity factor F _I has negative value in the limited region near free surface. Therefore, the actual crack surface seems to contact each other near the surface.     

Stress intensity factors for an elliptical crack approaching the surface of a semi-infinite solid

Stress intensity factors for an embedded elliptical crack approaching the free surface of the semi-infinite solid that is subjected to uniform tension perpendicular to the plane of crack are presented in a nondimensional form for various crack aspect ratios and crack distances from the free surface. Stress intensity factors are determined numerically using an alternating technique with two solutions. The first solution involves an elliptical crack in a solid and subjected to normal loading expressible in a polynomial of x and y. The second solution involves stresses in the half space due to prescribed normal and shear stresses on the surface. Effect of the Poisson's ratio on these stress intensity factors is also investigated. Stress intensity factors for a semi-elliptical surface crack in a tinite thickness plate are then estimated in a nondimensional form for various crack aspect ratios and crack depth to plate thickness ratios.Specialist Engineer, Aerospace Group, The Boeing Company, Seattle, Washington.Professor, Department of Mechanical Engineering, University of Washington, Seattle, Washington, and also Aerospace Group, The Boeing Company, Seattle.     

Characterization of the Behaviour of Part-Through Cracks under stable crack extension

The behaviour of part-through cracks, modelled by semi-elliptical surface cracks, under variable normal stresses (linearized) is investigated. The stable crack contour, given as crack aspect ratio, is determined for an infinite body as a circle; in a finite body semi-ellipses are “quasi”- stable. The effects of crack growth law types, crack location, geometry and stress redistribution together with plasticity effects are discussed. The final relative crack lengths for wall penetration are determined. The data are discussed with respect to the conditions for leak-before-break, sudden rupture and no-leak-before-break/no-sudden-rupture.     

Calculation of stress intensity factors for a longitudinal semi-elliptical crack in a finite-length thick-walled cylinder

The purpose of this paper is to present the effect of finite boundary on the stress intensity factor of an internal semi-elliptical crack in a pressurized finite-length thick-walled cylinder  ( R _i/ t = 4)  . The three-dimensional finite element method, in conjunction with the weight function method, is used for computing the stress intensity factor at the deepest and surface points of an axial semi-elliptical crack in a cylinder. The transition aspect ratios, the aspect ratios in which the maximum stress intensity factor translates from the deepest to the surface points of the crack, are calculated for different relative depths and cylinder lengths. The results show that the stress intensity factor increases as the cylinder length decreases, especially at the corner point of the crack compared with the deepest point. The major advantage of this paper is that a closed-form expression is extracted for the stress intensity factor at the surface point of a semi-elliptical crack, which experiences higher changes due to the effect of the finite boundary of the cylinder.     

椭圆裂纹的权函数和应力强度因子

本文推导了承受作用在同心椭圆周上法向均匀分布力拉伸的无限大体椭圆裂纹的权函数,并用它计算了一些应力强度因子K_1。对于椭圆裂纹的特殊情况——圆盘裂纹,则相应的应力强度因子K_1与文献[1]完全相同。本文所提出的方法具有计算简便的优点。     

Weight functions and Stress Intensity Magnification factors for elliptical and semi-elliptical cracks under variable normal stress – Part II

Surface cracks under peak stresses are investigated. The calculational procedure is based on the general form of the weight function for an elliptical crack embedded in an infinite solid. Two points on the contour of the ellipse are investigated. A new correction procedure for transfer from the embedded crack to surface crack configurations is presented, which is valid for all a/t-values. Weight functions for both points have been found with the crack aspect ratio a/c as parameter. For the point at the end of the minor axis all weight functions for embedded cracks are describable by one equation only (using Heuman's lambda function). For various a/c-ratios of the surface crack under different stress distributions the stress intensity magnification factors are given.