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.     

Stress intensity factor solution for a semi-elliptical crack in an arbitrarily distributed stress field

An equation for the stress intensity factor (SIF) for semi-elliptical crack has been developed. It is based on the Newman-Raju's solution for the crack in a plate under bending or tension. The equation can be applied when a stress distribution is described by a power function. Using the approach outlined, the SIF for a surface crack in a T-butt welded connection has been estimated. The results obtained can be used in a fracture-mechanics-based fatigue analysis.     

Developments in the analysis of interacting cracks

This paper presents an overview of the finite element alternating technique for the analysis of interacting cracks. To illustrate the ease and accuracy of this method the technique is used to analyse several problems associated with both widespread fatigue and multi-site damage, a problem which is attracting worldwide attention. Whilst this paper presents an overview of the technique for both two- and three-dimensional problems attention is focused on three-dimensional problems. In particular, the interaction effects between two fully embedded elliptical flaws and between two semi-elliptical surface flaws, and the effects of crack proximity and crack aspect ratio on the stress intensity factors are presented. For semi-elliptical surface flaws these results indicate that as the cracks approach each other the position of the point on the crack front with the highest stress intensity factor shifts. This subsequently suggests that surface cracks will tend to grow preferentially towards each other. The same trend is evidenced for fully embedded cracks. However, in this case there is no shift in the position of the maximum stress intensity factor. A discussion of the results in terms of stress intensity magnification factors is also presented.     

Fatigue crack growth simulation in a first stage of compressor blade

A first-stage rotary compressor blade of a Model GE-F6 gas turbine failed due to vibration in early March 2008. Initial investigations showed that pitting on the pressure side of the blade caused micro cracks, leading to larger cracks due to high cycle fatigue. To assess this failure, a series of experimental, numerical, and analytical analyses were conducted. Fractography of the fractured surface of the blade indicated that two semi-elliptical cracks incorporated and formed a single crack. In this study, static and dynamic stress analyses were performed in Abaqus software. Moreover, fracture mechanics criterion was accomplished to simulate fatigue crack growth. This was carried out using a fracture analysis code for 3-dimensional problems (Franc3D) in two states. Firstly, stress intensity factors (SIFs) for one semi-elliptical surface crack and then SIFs for two semi-elliptical surface cracks were taken into account. Finally, the Paris and Forman–Newman–De Koning models were used to predict fatigue life. Since stress level and crack shape in both conditions are the same and the SIF at the crack tip reaches the fracture toughness of the blade, SIFs results indicate that insertion of a second crack has no effect on the final SIF, however, the second crack facilitates the process of reaching the critical length. So, fatigue life in two-crack condition is less than in the one-crack state.     

Simplified analysis of fracture behaviour of a Francis hydraulic turbine runner blade

Each welded connection between blades and band or crown of a Francis hydraulic turbine runner can be considered as a T‐joint subjected to pure bending induced by the water action. A semi‐elliptical crack is assumed to exist at the surface of one of the aforementioned welded connections. The actual geometry of the T‐joint can be simplified, that is, only the cracked plate (representing the blade) under a given stress distribution acting on the defect faces is examined. A numerical procedure already proposed by the authors to compute the stress‐intensity factor (SIF) along the crack front is here applied by introducing some changes to simplify such computations. The obtained values of SIF are compared with some results available in the literature.     

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.     

Stress intensity factor analysis for part-elliptical cracks in structures

A method based on the generalized weight function theory is used for solving three-dimensional linear elastic fracture mechanics problems. A complete system of equations of the weight function method (WFM) has been obtained for the calculation of stress intensity factors (SIF) for part-elliptical cracks subjected to arbitrary normal loading.A procedure of the WFM is described to analyze structural components containing surface (semi-elliptical), corner (quarter-elliptical) and embedded (elliptical) flaws. The efficiency of the proposed method is illustrated by solving a number of methodical problems     

Stress intensity factor calculation of steel-lined hoop-wrapped cylinders with internal semi-elliptical circumferential crack

The purpose of this paper is to present mode I stress intensity factor for a circumferential semi-elliptical crack on the inner surface of a hoop-wrapped steel-lined CNG cylinder. The stress intensity factors along the crack front are directly computed by 3D finite element method for a wide range of variations of the crack geometry. Also influence of many parameters such as cylinder internal pressure, composite layer thickness, composite material properties and undertaking Auto-Frettage pressure are studied on the stress intensity factor of the crack and some conclusive results are drawn. For the sake of validation of the results and because of lack of the results for a circumferential semi-elliptical crack in the literature, a semi-elliptical axial crack in a composite hoop-wrapped cylinder has been modeled and the results have been compared with those in the literature showing a good agreement.     

Growth of three-dimensional cracks in finite-thickness plates

The stress field in a finite-thickness plate weakened by a three-dimensional crack and subjected to tension in a direction perpendicular to the crack plane is studied. The cases of an embedded elliptical, semi-elliptical and quarter-elliptical surface crack are considered. The stress analysis takes place by a finite element computer program which uses twenty-node isoparametric and fifteen-node enriched elements. The stress intensity factor variations along the periphery of the elliptical cracks are given for various plate thicknesses. The results of the stress analysis are used in conjunction with the strain energy density theory to study the growth characteristics of the cracks. The history of non-self-similar crack growth from initiation to final instability through the intermediate stage of stable growth is analyzed. The increments of crack growth from each point of its front are determined on the basis that the critical element in the direction of crack growth absorbs a critical amount of strain energy density. Crack growth becomes unstable when the last increment from the critical point of the crack front takes a limiting value. Results for the crack growth characteristics are given for the three types of cracks considered and various plate thicknesses.     

Calculation of Stress Intensity Factors for semi-elliptical surface cracks in a turbine rotor

Stress intensity factors based on linear elastic behaviour were calculated for semi-elliptical surface cracks in the front face of a cylindrical disk. The small semi-axis of the cracks coincides with the axis of rotation of the disk. The disk represents a part of a turbine rotor and is used to simplify the calculations. The uncracked rotor is loaded by a radial varying hoop stress distribution, caused by rotation, thermal gradients and the influence of the turbine blades. Following a procedure proposed by Mattheck at al. [1] the stress points of the cracks were calculated by means of the weight functions method.     

Estimation of three‐dimensional stress intensity factor for structural ‘T’ details

Three‐dimensional (3D) opening mode stress intensity factors (SIFs) for structural steel‐welded ‘T’ details were investigated by the finite element method. A 3D shape‐dependent correction factor is proposed for semi‐elliptical surface cracks. The aspect ratio (a/c) of a semi‐elliptical crack plays a key role in the approximation of 3D‐SIF values, and in the present study, it was estimated for a 3D crack analysis. The estimated 3D‐SIF was determined through a correlation between the a/c ratio and the two‐dimensional SIF for semi‐elliptical cracks in the thickness direction adjacent to the web‐flange junction of a welded ‘T’. The resulting equation can be used to estimate the 3D‐SIF values from the two‐dimensional SIF without much ambiguity.     

弯扭组合载荷下圆管半椭圆表面裂纹应力强度因子的有限元分析

对表面裂纹复合型应力强度因子的研究一直是线弹性断裂力学中的重要课题,例如弯扭组合载荷下圆管半椭圆表面裂纹应力强度因子的计算,到现在也没有一个正确的分析解。考虑到裂尖的应力奇异性,在裂纹前沿手动设置三维奇异单元,用三维有限元法中的1/4点位移法计算弯扭组合载荷下圆管表面椭圆裂纹前沿的Ⅰ型、Ⅱ型和Ⅲ型应力强度因子,并分析其随裂纹深度增加时的变化规律。运用该方法计算了有关模型的应力强度因子,并与该模型的实验值进行了比较,计算结果和实验结果吻合良好。     

Subcritical Crack Growth of semi-elliptical surface cracks in a turbine rotor

The subcritial growth of semi-elliptical surface cracks in the front face of a cylindrical disk was calculated. The disk represents a part of a turbine rotor. The stress intensity factors, which are the basis for the calculations presented here, were given in a former paper [1]. The loads steps considered consist of an inhomogeneous hoop stress distribution, which occurs at a distinct time-step of the start – up procedure of the turbine, and of the stress free state, respectively. By use of a Paris-law the growth of different initial cracks was evaluated under the assumption that the crack geometry remains semi-elliptic.     

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.     

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.     

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.     

Transient thermal stress intensity factors for an internal longitudinal semi-elliptical crack in a thick-walled cylinder

In this paper, the stress intensity factors are derived for an internal semi-elliptical crack in a thick-walled cylinder subjected to transient thermal stresses. First, the problem of transient thermal stresses in a thick-walled cylinder is solved analytically. Thermal and mechanical boundary conditions are assumed to act on the inner and outer surfaces of the cylinder. The quasi-static solution of the thermoelasticity problem is derived analytically using the finite Hankel transform and then, the stress intensity factors are extracted for the deepest point and the surface points of the semi-elliptical crack using the weight function method. The results show to be in accordance with those cited in the literature in the special case of steady-state problem. Using the closed-form relations extracted for the transient thermal stress intensity factors, some conclusive results are drawn.     

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.     

Stress intensity factors for a wide range of long-deep semi-elliptical surface cracks, partly through-wall cracks and fully through-wall cracks in tubular members

To calculate the rate of fatigue crack growth in tubular members, one approach is to make use of the fracture mechanics based Paris law. Stress intensity factors (SIF) of the cracked tubular members are prerequisite for such calculations. In this paper, stress intensity factors for circumferential deep semi-elliptical surface crack (a/t > 0.8), semi-elliptical partly through-wall crack and fully through-wall crack cracks in tubular members subjected to axial tension are presented. The work has produced a comprehensive set of equations for stress intensity factors as a function of a/T, c/πR and R/T for deep surface cracks. For the partly through-wall cracks and fully through-wall cracks, two sets of bounding stress intensity factor equations were produced based on which all stress intensity factors within the range of parameters can be obtained by interpolation.     

Calculating stress intensity factors for a semi-elliptical surface crack in a heterogeneous stress field

The author proposes an equation for calculating the stress intensity factor (SIF) for a semi-elliptical surface crack for uniform, linear, and quadratic laws of variation of the load applied to its edges. The derivation of the equation is based on the well-known Newman—Raju solution for a bent plate. The distribution of the values of SIF along the crack front, obtained using the empirical equations, coincides with the results of calculations carried out using the finite element method (FEM).Translated from Problemy Prochnosti, No. 7, pp. 38–41, July, 1990.