Stress concentration of a cylindrical bar with a V-shaped circumferential groove under torsion, tension or bending

The stress concentration of a cylindrical bar with a V-shaped circumferential groove is analyzed by the body force method. The stress field due to a ring force in an infinite body is used to solve this problem. The solution is obtained by superposing the stress fields of ring forces in order to satisfy the given boundary conditions. The present results for semi-circular notches are in close agreement with Hasegawa's results. As a result of the systematic calculation of a 60° V-shaped notch, it is found that the stress concentration factors obtained by Neuber's trigonometric rule used currently have non-conservative errors of about 10% for a wide range of notch depths. The stress concentration factors are illustrated in charts so they can be used easily in design or research.     

KI of a circumferential crack emanating from an ellipsoidal cavity obtained by the crack tip stress method in FEM

In high hardness steel, fatigue crack appears from a microcavity in most cases. Therefore, it is important to know the stress intensity factor of a crack emanating from the cavity. Recently, a method for calculating the highly accurate values of two-dimensional stress intensity factors was proposed by Nisitani, based on the usefulness of the stress values at a crack tip calculated by FEM. This method is called the crack tip stress method. In this paper, the crack tip stress method is extended to the problems of an infinite solid having an ellipsoidal cavity with a circumferential crack emanating from the cavity subjected to tension.     

Practical use of the ‘equivalent’ measured stress intensity factor to control fatigue crack propagation rates in aircraft full-scale fatigue tests—First assessment of the method in testing of a pressurized aircraft fuselage

In the case of circumferential cracks in a cylindrical fuselage, the comparison of some analysis and test results shows that the theoretical stress intensity factor is a suitable correlation parameter of fatigue crack propagation rates, both in aircraft fuselages and in plane panels. Values of the ‘equivalent’ stress intensity factor, computed by applying the Barrois-Bhandari method to slot-opening measurements performed under decreasing loading levels, agree well with the values computed from two dimensional Theory of Elasticity, using the method of finite elements.

In the case of longitudinal cracks, the experimental values of the ‘equivalent’ stress intensity factor, i.e. the stress intensity factor of the infinite plane sheet containing a centre crack with the same elastic strain and stress distributions near the boundary of the plastically strained region around the crack tip, yield a good correlation of fatigue crack propagation rates of the cracked fuselage and of cracked plane structures. The values of the ‘equivalent’ stress intensity factor are lower than those of the theoretical stress intensity factor, the interest of which disappears, but are also far higher than the bidimensionally computed values, which are no longer to be considered.

Some meant of safety provided to limit crack openings will make it possible, in the near future, to investigate test conditions reaching the ultimate residual static strength of cracked structures, while avoiding, however, catastrophic failures.     

Sickle-shaped surface crack in a notched round bar under cyclic tension and bending

A sickle-shaped surface crack, also called crescent-moon (or crescent) crack, is assumed to exist at the root of a circular-arc circumferential notch in a round bar under tension and bending. For different notch sizes (i.e. different values of the stress concentration factor), the stress intensity factor along the crack front is computed through a three-dimensional finite-element analysis. The effect of the stress concentration factor on the stress intensity factor values is examined for several crack configurations. Finally, the surface crack growth under cyclic loading is analysed through a numerical procedure that employs the stress intensity factor values obtained. Some results of the present study are compared with those by other authors.     

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.     

Thermal stresses in a laminate composite with infinite row of parallel cracks normal to the interfaces

Under the assumption of plane strain, some linear thermoelastic problems concerning a laminate composite containing an infinite row of parallel cracks situated normal to the bond lines are solved by the use of integral transform technique. The thermal stresses are caused by a uniform heat flow disturbed by the presence of the cracks and the interfaces. The problem is reduced to that of solving a Fredholm integral equation of the second kind which is solved numerically by the use of Gaussian quadrature formulas. The effect of various quantities of physical interest on the stress intensity factor is shown graphically.Calculations are also carried out for the stress intensity factor for a laminate composite with a crack normal to the interfaces. This is accomplished by taking the limiting case of a laminate composite with an infinite row of parallel cracks normal to the interfaces.     

A study of interference of three parallel cracks

This paper describes the priority of propagation among three parallel cracks. The treated object is an infinite elastic body with three parallel cracks subjected to uniform tension or antiplane shear at infinity. The results make it clear that the longest crack does not always have the biggest value relating to fracture, e.g. the biggest tangential stress at crack tip or the biggest stress intensity factor, because of the interaction among the cracks.     

An interaction energy integral method for computation of mixed-mode stress intensity factors along non-planar crack fronts in three dimensions

A new interaction energy integral method for extracting mixed-mode stress intensity factors along the fronts of non-planar, three-dimensional cracks is described. In the method, interaction energy contour integrals are defined and expressed in domain form. The mixed-mode stress intensity factors are obtained by evaluating the domain integrals as a post-processing step in the finite element method. To assess the accuracy of the method, two benchmark problems are considered. The first problem is that of an arc crack in an infinitely extended solid subjected to equibiaxial tension. The second is that of a lens-shaped crack embedded in an infinite solid subjected to hydrostatic tension. Excellent agreement is obtained between the numerical and corresponding analytical results obtained from the literature.     

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.     

Effect of fiber damage on the overall electrical conductivity of bare carbon fiber strand

The simple method developed by Kachanov (1985) for multiple interacting cracks in homogenous medium is extended to predict complex stress intensity factor for multiple split type interface cracks. Calculations are implemented for two equal cracks and infinite row of periodic cracks at the interface between two dissimilar isotropic materials. Results for infinite row of cracks are compared against the exact analytical solution provided by Sih (1973). The approximate method leads to the results very close the exact solution for crack density up to 0.90 (relative error is less than 3.8% for real part of stress intensity factor) and material dissimilarity does not have a major influence on the error. For crack densities higher than 0.90, the influence of material dissimilarity is more evident and the error increases as material dissimilarity increases. The promising match between the approximate and exact method proves the capability of the approximate method for solving other interacting interface crack problems, such as multiple penny-shaped interface cracks, in which the solution is not obtained in the literature yet.     

A new combined analytical and finite-element solution method for stability analysis of the growth of interacting tension cracks in brittle solids

The analysis of stable and unstable growth regimes of a system of tension cracks in brittle solids, requires accurate estimates of the values of the stress intensity factors at various crack tips, and their derivatives with respect to the crack lengths. A new combined analytical and finite-element solution method is developed which is specially suited to problems of this kind. For illustration the method is applied to estimate the results corresponding to: (1) a single Griffith crack in an infinite space; and (2) possible unstable growth regimes corresponding to a system of edge cracks in a strip of finite width. Comparison of the results with those obtained by other methods, reveals that the present method is indeed very effective and efficient, and therefore provides a useful tool for the stability analysis of interacting cracks.     

Stress intensity factors of cracks emanating from a surface square hole in infinite body in tension

This paper deals with such a kind of surface crack problem with a same depth (called a liked‐plane crack problem for short). Based on the previous investigations on an internal rectangular crack and a surface rectangular crack in an infinite solid in tension and the hybrid displacement discontinuity method, a numerical approach for the liked‐plane crack problem is presented. Numerical examples are given to illustrate the numerical approach is simple, yet accurate for calculating the stress intensity factors (SIFs) of the liked‐plane crack problem. Specifically, SIFs of a pair of cracks emanating from a surface square hole in an infinite body in tension are investigated in detail.     

Stress intensity factors in a circular cylinder containing a pair of radial cracks

The elasticity problem of a circular cylinder having a pair of radial cracks subject to mode I loading is studied in this article. Stress intensity factors of the cracked cylinder under mode I loading are systematically and effectively evaluated with use of an equivalent procedure established in this paper. The equivalent procedure reduces the problem under consideration to that of a simpler geometry—an infinite medium with two similar collinear cracks. Numerical results are obtained for a uniform tension and for a pair of concentrated forces acting on the circumferential boundary of the cylinder. The relative merit of the solution method presented in this paper is also discussed.     

Determination of the stress intensity factor for parallel and symmetric edge cracks under pure bending

An experimental and semi-theoretical investigation concerning thin plates having “Parallel and Symmetric” (P.A.S.)² edge cracks and subjected to pure bending will be discussed in this paper. Experiments were performed extensively on plates with P.A.S. equal edge cracks in order to study the effect of the variation of the length of the cracks and crack spacing seperately.Observations show that when crack spacing becomes larger than crack length, the diameter of the caustic for the equal P.A.S. edge cracks approaches the diameter of the caustic of the double edge cracks. Based on this fact, the equation that expresses the semi-theoretical and experimental stress intensity factor for the double edge cracks was first modified and then utilized in order to express the strese intensity factor of P.A.S. edge cracks. Moreover, an alternative formula for the stress intensity factor was established which is based on a cubic interpolating polynomial. For large crack spacing the cubic interpolating polynomial converges to unity and as a result, the diameter of the caustic and the stress intensity factor approach those of a double edge crack.For the case of unequal P.A.S. edge cracks, a crack closure phenonmenon under the pure bending condition was investigated. Experimental measurements showed that the relative position of the shorter crack with respect to the longer cracks for which the crack closure phenomenon happens is related not only to the crack spacing of the parallel edge cracks in the tension region, but also strongly related to the variation of the existing symmetric edge crack in the compression region.     

On the solution of doubly periodic array of cracks

We present a correct procedure for the determination of the stress intensity factor at the tip of a crack in a doubly periodic array contained in an infinite elastic solid. It is based on the use of weight functions for an infinite row of periodic cracks, thus avoiding the use of divergent double infinite summations.     

Tension of a finite-thickness plate with a pair of semi-elliptical surface cracks

This paper deals with the tension of a finite-thickness plate with a pair of semi-elliptical cracks on both of the free surfaces. The analysis is performed in a similar manner to the previous single crack problem, by using the body force method and the boundary conditions expressed in terms of resultant forces and displacements of the boundary elements. The stress intensity factor at the maximum depth of the crack front is calculated for various values of the parameters and these results are fitted by a reliable polynomial formula for convenience of engineering applications.     

Mode-I stress intensity factor derivation by a suitable Green's function

A suitable Green's function is developed for the infinite elastic solid, containing internal penny-shaped crack and loaded by a singular co-axial tensile and radial ring-shaped source acting outside or on crack faces. The corresponding boundary integral equation (BIE) is solved by the BEM for the calculation of the mode-I stress intensity factor of cracked axisymmetric finite bodies under tension. The proposed technique has three advantages: (a) it does not require discretization of the crack surface, (b) it does not require multiregion modeling and (c) it reduces the 3-D discretization of the solid to 1-D, resulting in substantially reduced effort. Numerical results are derived for the case of a cylindrical bar with a central penny-shaped crack located in a plane normal to its axis, loaded by tensile force. Comparison with results of other methods are included indicating excellent agreement.     

An anisotropic strip weakened by an array of cracks

The paper deals with a 2-dimensional problem of an anisotropic elastic strip having an infinite row of Griffith cracks. By using integral equation approach, the problem is treated analytically. The stress intensity factor, the critical pressure and the energy required to open the crack are studied for two cases—(a) when the edges of the strip are in contact with smooth and rigid planes and (b) when the edges of the strip are free of tractions. Numerical results for the aforementioned quantities are obtained for both the cases for a specific anisotropic material and a comparison is made with the corresponding results for a strip made of an isotropic material.     

Generalized stress intensity factors of V-shaped notch in a round bar under torsion, tension, and bending

In this study, generalized stress intensity factors K_I,λ1, K_II,λ2, and K_III,λ4 are calculated for a V-shaped notched round bar under tension, bending, and torsion using the singular integral equation of the body force method. The body force method is used to formulate the problem as a system of singular integral equations, where the unknown functions are the densities of body forces distributed in an infinite body. In order to analyze the problem accurately, the unknown functions are expressed as piecewise smooth functions using three types of fundamental densities and power series, where the fundamental densities are chosen to represent the symmetric stress singularity and the skew-symmetric stress singularity. Generalized stress intensity factors at the notch tip are systematically calculated for various shapes of V-shaped notches. Normalized stress intensity factors are given by using limiting solutions; they are almost determined by notch depth alone, and almost independent of other geometrical parameters. The accuracy of Benthem-Koiter’s formula proposed for a circumferential crack is also examined through the comparison with the present analysis.     

Cracks located on a single line in an orthotropic elastic medium in which body forces are acting

Plane strain problem of determining the distribution of stress in the neighborhood of cracks located on a single line is examined. The medium containing cracks is infinite, orthotropic and acted upon by an asymmetrical distribution of body forces. The crack surfaces are free from tractions and the crackline coincides with an axis of orthotropy. Fourier transform methods are employed to reduce the problem to that of solving a singular integral equation of Cauchy type.The solution is completed in the case when the infinite orthotropic medium contains a single crack. Particular distributions of concentrated loads and moments are considered in detail. The stress intensity factor and crack shape are determined and shown graphically for oak wood. The results in special and limiting cases are compared with those available in the literature.