J integral solution for semi-elliptical surface crack in high density poly-ethylene pipe under bending

The goal of this work is to analyse the severity of semi-elliptical crack defects and to study the degree of damage in the poly-ethylene pipe in bending during the crack propagation. The semi-elliptical cracks are considered in this work located in different position in the wall of the pipe. The three finite element method based on the computation of the J integral was used to analyse the fracture behaviour of these structures. The effect of the position, shape and size of the crack on the J integral values was highlighted. The effects of strain rate and the temperature on the J integral values were also examined. The obtained results show that the strain rates have a strong influence on the J integral values especially for circumferential crack at higher bending moment. However, the energy for circumferential crack is more important compared to axial crack. The effect of the depth of the crack becomes important when the ratio (a/t) reaches a critical value of 0.6 (a/t = 0.6), especially when the ratio a/c is weak (semi-elliptical crack, a/c = 0.2) where the J integral values becomes independently of the crack depth, this conclusion is opposite to the above for the poly-ethylene pipe subjected to internal pressure. We recall finally, that the temperature effect on circumferential cracks behaviour is more important compared to the axial cracks at critical crack size (a/c = 0.2 and a/t = 0.6). It is also shown that in the wall of pipe, the internal cracks are more dangerous than the external cracks.     

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.     

T-Stress solutions for a semi-elliptical axial surface crack in a cylinder subjected to mode-I non-uniform stress distributions

This paper presents the T-stress solutions (T₁₁ and T₃₃) for semi-elliptical axial surface cracks in a cylinder subjected to mode-I non-uniform stress on the crack surface. Two cylindrical geometries with inner radius (R_i) to wall thickness (t) ratios R_i/t = 5 and 10 were considered. The T-stresses were applied along the crack front for normalized crack depth values a/t of 0.2, 0.4 and 0.5 and aspect ratios a/c of 0.2, 0.4, 0.6 and 1.0. Three stress distribution; uniform, linear and parabolic were applied to the crack face. In addition to these solutions, concrete formulation of the superposition principle is given for the T₃₃-stress, which is known as an elastic parameter that describes the out-of-plane crack tip constraint effect. Then, the validity of the formulation was shown through application of our T-stress solutions to the problem of an axial semi-elliptical surface crack in a cylinder subjected to internal pressure, and checking that the principle of superposition holds for the problem.     

Elastic T-stress solutions for semi-elliptical surface cracks in finite thickness plates

Three-dimensional finite element analyses have been conducted to calculate the elastic T-stress for semi-elliptical surface cracks in finite thickness plates. Far-field tension and bending loads were considered. The analysis procedures and results were verified using both exact solutions and approximate solutions. The T-stress solutions are presented along the crack front for cracks with a/t values of 0.2, 0.4, 0.6 or 0.8 and a/c values of 0.2, 0.4, 0.6 or 1.0. Based on the present finite element calculations for T-stress, empirical equations for the T-stress at three locations: the deepest, the surface and the middle points of the crack front under tension or bending are presented. The numerical results are approximated by empirical formulae fitted with an accuracy of 1% or better. They are valid for 0.2?a/c?1 and 0?a/t?0.8. These T-stress results together with the corresponding K or J values for surface cracks are suitable for the analysis of constraint effects for surface cracked components.     

3-D stress intensity factors for arrays of inner radial lunular or crescentic cracks in a typical spherical pressure vessel

Many spherical pressure vessels are manufactured by methods such as the integrated hydro-bulge forming (IHBF) method, where the sphere is composed of a series of double curved petals welded along their meridional lines. Such vessels are susceptible to multiple radial cracking along the welds. For fatigue life assessment and fracture endurance of such vessels one needs to evaluate the stress intensity factors (SIFs) distribution along the fronts of these cracks. However, to date, only two 3-D solutions for the SIF for one inner semi-elliptical crack in thin or thick spheres are available, as well as 2-D SIFs for one through-the-thickness crack in thin spherical shells. In the present paper, mode I SIF distributions for a wide range of lunular and crescentic cracks are evaluated. The 3-D analysis is performed, via the FE method employing singular elements along the crack front, for a typical spherical pressure vessel with outer to inner radius ratios of η = R_o/R_i = 1.1. SIFs are evaluated for arrays containing n = 1-20 cracks; for a wide range of crack depth to wall thickness ratio, a/t, from 0.025 to 0.95; and for various ellipticities of the crack, i.e., the ratio of crack depth to semi crack length, a/c, from 0.2 to 1.5. The obtained results clearly indicate that the SIFs are considerably affected by the three-dimensionality of the problem, and the following parameters: the number of cracks in the array-n, the relative crack depth a/t, and the crack ellipticity a/c.     

The T-stress solutions for through-wall circumferential cracks in cylinders subjected to general loading conditions

The elastic T-stress is a parameter used to define the level of constraint at a crack tip. It is important to provide T-stress solutions for practical geometries to apply the constraint-based fracture mechanics methodology. In the present work, T-stress solutions are provided for circumferential through-wall cracks in thin-walled cylinders. First, cylinders with a circumferential through-wall crack were analyzed using the finite element method. Three cylinder geometries were considered; defined by the mean radius of the cylinder (R) to wall thickness (t) ratios: R/t = 5, 10, and 20. The T-stress was obtained at eight crack lengths (θ/π = 0.0625, 0.1250, 0.1875, 0.2500, 0.3125, 0.3750, 0.4375, and 0.5000, θ is the crack half angle). Both crack face loading and remote loading conditions were considered including constant, linear, parabolic and cubic crack face pressures and remote tension and bending. The results for constant and linear crack face pressure were used to derive weight functions for T-stress for the corresponding cracked geometries. The weight functions were validated against several linear and non-linear stress distributions. The derived weight functions are suitable for T-stress calculations for circumferential cracks in cylinders under complex stress fields.     

Two-parameter characterization of elastic-plastic crack front fields: Surface cracked plates under tensile loading

In this paper the J-Q two-parameter characterization of elastic-plastic crack front fields is examined for surface cracked plates under uniaxial and biaxial tensile loadings. Extensive three-dimensional elastic-plastic finite element analyses were performed for semi-elliptical surface cracks in a finite thickness plate, under remote uniaxial and biaxial tension loading conditions. Surface cracks with aspect ratios a/c = 0.2, 1.0 and relative depths a/t = 0.2, 0.6 were investigated. The loading levels cover from small-scale to large-scale yielding. In topological planes perpendicular to the crack fronts, the crack stress fields were obtained. In order to facilitate the determination of Q-factors, modified boundary layer analyses were also conducted. The J-Q two-parameter approach was then used in characterizing the elastic-plastic crack front stress fields along these 3D crack fronts. Complete distributions of the J-integral and Q-factors for a wide range of loading conditions were obtained. It is found that the J-Q characterization provides good estimate for the constraint loss for crack front stress fields. It is also shown that for medium load levels, reasonable agreements are achieved between the T-stress based Q-factors and the Q-factors obtained from finite element analysis. These results are suitable for elastic-plastic fracture mechanics analysis of surface cracked plates.     

Fully plastic J-integral solutions for surface cracked plates under biaxial loading

In this paper, surface cracked plates under biaxial tension are studied. Three-dimensional elastic-plastic finite element analyses have been carried out to calculate the J-integral for surface cracked plate for a wide range of geometry, biaxiality and material properties. Fully plastic J-integral solutions along the front of the surface cracks are presented for Ramberg-Osgood power law hardening material of n = 3, 5, 10 and 15. Geometries considered are a/c = 0.2, 1.0 and a/t = 0.2, 0.4, 0.6 and 0.8 and the biaxial ratios of 0, 0.5 and 1. Based on these results, the J-integral along the crack front for general elastic-plastic loading conditions can be estimated using the EPRI scheme. These solutions are suitable for fracture analyses for surface cracked plates under biaxial loading.     

Comparison of semi-elliptical cracks in cylinders with a thickness transition and in a straight cylinders - Elastic-plastic behaviour

This paper presents a study of a circumferential external semi-elliptical crack in a cylinder with a thickness transition. Crack harmfulness is compared to the harmfulness of a similar crack in a straight cylinder, on the basis of J-integral calculations. Loads considered are pure tensile stress and bending moment. A numerical analysis is performed considering elastic plastic behaviour of material. A crack mesh was designed for 3D models. The results show that the cylinder with a thickness transition is more vulnerable to the defect. A simple method for J calculation at the deepest point of the crack in the transition zone is also proposed.     

The geometry dependence of the JR curve for circumferential growth of through-wall cracks in cylindrical pipes subject to bending loads. II: Effect on the instability criterion

Proceeding from the experimentally vindicated basis that circumferential growth of a through-wall crack in a circular cylindrical Type 304 stainless steel pipe subject to bending loads is associated with a constant crack tip opening angle, an earlier paper has predicted the shape of the J_R curve in terms of geometrical and material parameters. The predictions compare favourably with experimental results for 4 in. (～ 10 cm.) diameter pipes containing circumferential through-wall cracks with different sizes. In this paper, the implications of the geometry dependence of the J_R curve are considered in relation to the criterion for the instability of circumferential crack growth. Limitations of the tearing modulus approach are accordingly highlighted, particularly if it is used for a material with a low crack growth resistance.     

Elastic-plastic fracture mechanics analyses of circumferential through-wall cracks between elbows and pipes

The present work proposes a method for elastic-plastic fracture mechanics analysis of the circumferential through-wall crack in weldment joining elbows and attached straight pipes, subject to in-plane bending. Heterogeneous nature of weldment is not explicitly considered and thus, the proposed method assumes cracks in homogeneous materials. Based on small strain finite element limit analyses using elastic-perfectly plastic materials, closed-form limit loads for circumferential through-wall cracks between elbows and straight pipes under bending are given. Then applicability of the reference stress-based method to approximately estimate J and crack opening displacement (COD) is evaluated. It was found that the limit moments for circumferential cracks between elbows and attached straight pipes can be much lower than those for cracks in straight pipes, particularly for a crack length of less than 30% of the circumference; this result is of great interest in practical cases. This result implies that, if one assumes that the crack locates in the straight pipe, limit moments could be overestimated significantly, and accordingly, reference stress-based J and COD could be significantly overestimated. For the leak-before-break analysis, accurate J and COD estimation equations based on the reference stress approach are proposed.     

Stress Intensity Factors of partly circumferential surface cracks at the outer wall of a pipe

By means of the finite element method stress intensity factors were calculated for partly circumferential surface cracks at the outer wall of a pipe. The crack shape considered can be described as curved rectangular shape. The cracks considered have crack depths between 20 and 80 percent of the wall thickness of the pipe and crack lengths (defined by the angle of circumference φ) between φ = 10° and φ = 60°. The pipe is loaded by a constant axial tensile stress σ₀ (equal to 136 Nmm^?2 in the numerial calculations), and the wall thickness to inner radius ratio of the pipe was chosen to 0.1. A wall thickness of 20 mm was used for the numerical calculations.     

Plastic collapse moment equations of throughwall axially cracked elbows subjected to in-plane bending moment

A throughwall axial crack may develop in an elbow or pipe bend due to service related degradation mechanism. It is very important to know the plastic collapse moment (PCM) of an elbow in the presence of a throughwall axial crack. The existing PCM equations of throughwall axially cracked (TAC) elbows are based on very few test data points of Griffiths without detailed analyses and also the range of applicability of their proposed equations are limited. Further, they do not differentiate between closing and opening modes of bending although deformation characteristics under these two modes are completely different. Therefore, the present study has been undertaken to investigate through 3D elastic-plastic finite element analysis. A total of 84 elbows with various sizes of axial cracks (a/D_m = 0-1), different wall thickness (R/t = 5-20), different elbow bend radii (R_b/R = 2, 3) and two different bending modes, namely closing and opening have been considered in the analysis. Elastic-perfectly plastic stress-strain response of material has been assumed. Both geometric and material non-linearity are considered in the analysis. Crack closing is observed in most of the cases. To capture the crack closure effect, contact analysis has been performed. Plastic collapse moments have been evaluated from moment-end rotation curves by twice-elastic slope method. From these results, closed-form equations are proposed to evaluate plastic collapse moments of elbows under closing and opening mode of bending moment. The predictions of these proposed equations are compared with the test data available in the literature. Matching between predictions and experimental results is found to be satisfactory.     

Effect of biaxial loads on elastic-plastic <Emphasis Type="Italic">J</Emphasis> and crack tip constraint for cracked plates: finite element study

Based on detailed two-dimensional (2-D) and three-dimensional (3-D) finite element (FE) analyses, this paper attempts to quantify in-plane and out-of-plane constraint effects on elastic-plastic J and crack tip stresses for a plate with a through-thickness crack and semi-elliptical surface crack under positive biaxial loading. For the plate with a through-thickness crack, plate thickness and relative crack length are systematically varied, whereas for the plate with a semi-elliptical surface crack, the relative crack depth and aspect ratio of the semi-elliptical crack are systematically varied. It is found that the reference stress based approach for uniaxial loading can be applied to estimate J under biaxial loading, provided that the limit load specific to biaxial loading is used, implying that quantification of the biaxiality effect on the limit load is important. Investigation on the effect of biaxiality on the limit load suggests that for relatively thin plates with small cracks, in particular with semi-elliptical surface cracks, the effect of biaxiality on the limit load can be neglected for positive biaxial loading, and thus elastic-plastic J for a biaxially loaded plate could be estimated, assuming that such plate is subject to uniaxial load. Regarding the effect of biaxiality on crack tip stress triaxiality, it is found that such effect is more pronounced for a thicker plate. For plates with semi-elliptical surface cracks, the crack aspect ratio is found to be more important than the relative crack depth, and the effect of biaxiality on crack tip stress triaxiality is found to be more pronounced near the surface points along the crack front.     

Mixed Mode Stress Intensity Factor Determination of Multiple Cracks in Hollow Circular Pipe

Failure of pressure vessels and piping due to high temperature applications occurs due to the formation of fatigue cracks caused by cyclic load. It is well known that, the consequences of collapses of pipes causing enormous disruption of daily life. Thus there is a need to design and manufacture the pipes with precision and care. The major cause of crack nucleation in pipes is due to corrosion and internal fluid pressure. The crack-tip stresses are determined using stress intensity factor (SIF). In the present work an attempt has been made to determine the SIF for multiple cracks in a circular pipe subjected to internal fluid pressure. Two surface cracks of same size were introduced at the inner wall of the tube. The crack depth ratio (a/t) ranging between 0.1 and 0.5 and crack aspect ratio (a/c) of 0.6 and 1.0 was considered. Internal fluid pressure of 100 MPa was applied at the inner surface of the pipe and the corresponding SIF was measured. SIF values were calculated with consideration of mode-II and mode-III fracture in order to predict the exact SIF. As available SIF solutions of cracked pipes are limited to mode-I fracture, present work presents the influence of additional influence of mode-II and mode-III fracture. It is observed that, as crack depth ratio increases, SIF also increases considerably for semi-circular cracks. Higher SIF values were observed at the crack surface region [S/S ₀ = ±1] compared to crack middle [S/S ₀ = 0] region. A crossover in SIF was noted at a crack depth ratio of 0.3. At higher crack depths, SIF values decrease at the crack surface region due to additional influence of mode-II and mode-III fracture. In contrast to semi-circular cracks, SIF values are higher at the crack surface region for semi-elliptic cracks irrespective of the crack depths.     

Approximate elastic-plastic J estimates of cylinders with off-centred circumferential through-wall cracks

This paper provides approximate J estimates for off-centred, circumferential through-wall cracks in cylinders under bending and under combined tension and bending. The proposed method is based on the reference stress approach, where the dependence of elastic and plastic influence functions of J on the cylinder/crack geometry, the off-centred angle and strain hardening is minimised through the use of a proper normalising load. Based on published limited FE results for off-centred, circumferential through-wall cracks under bending, such normalising load is found, based on which the reference stress based J estimates are proposed for more general cases, such as for a different cylinder geometry and for combined loading. Comparison of the estimated J with extensive FE J results shows overall good agreements for different crack/cylinder geometries and for combined tension and bending, which provides sufficient confidence in the use of the proposed method for fracture mechanics analyses of off-centred circumferential cracks. Furthermore, the proposed method is simple to use, giving significant merits in practice.     

Creep crack growth behavior in 316 stainless steel at 594°C (1100°F)

The creep crack growth behavior of a type 316 stainless steel was characterized at 594°C (1100°F) using precracked single edge notch specimens loaded in displacement rate control. The steady-state crack growth rate, da/dt, correlated with J-integral and did not correlate with C ^*. The creep crack growth behavior in this material and temperature is compared with our previous creep crack growth rate data on a Cr-Mo-V steel at 538°C (1000°F) and on type 304 stainless steel at 594°C in which da/dt correlated with C ^*. A detailed discussion is included on why in some materials creep crack growth rate correlates with J integral and in others it correlates with C ^*.     

Stress intensity factors for circumferential semielliptical surface cracks in a pipe under thermal loading

Stress intensity factors are calculated at the deepest point and at the surface points of circumferential semielliptical surface cracks in a thermally shocked pipe. The method of calculation is based on weight functions following a proposal by Munz et al. Numerical values of the stress intensity factors are given for a wide range of crack depths and crack lengths considering a pipe with a wall thickness to inner radius ratio of $\text{1}\text{10}$.     

CALCULATION OF THE STRESS INTENSITY FACTOR FOR A PARTIAL CIRCUMFERENTIALLY CRACKED TUBE LOADED IN BENDING BY USING THE SHELL LINE-SPRING MODEL

Abstract— In this paper the line-spring model (LSM) developed by Rice and Levy is used to obtain an approximate solution of the stress intensity factor for a partial circumferential, externally cracked tube under axial tension and four point bending. The calculation is based on the work done by Delale and Erdogan for cylindrical shells containing a circumferential or an axial semi-elliptical, part-through crack. The range of utility of their analysis is enlarged to thicker wall tubes with nonelliptic and longer part-through circumferential cracks. Values of K ₁ calculated by the LSM are compared with those from a finite element analysis for remote tensile loading and bending cases, which shows fairly good agreement. The calculations are also applied to a fatigue crack growth test in a tube in four point bending to correlate the d a /d N vs λ K ₁ data.     

The behaviour of cracks in elastic-plastic materials under plane normal and shear loadings

Cracks in structures are often subjected to complex loading conditions. The direction of the crack extension depends on the normal and the shear components of the load. This paper is based on the kinking behaviour of cracks taking elastic-plastic behaviour of materials into account. The J-integral and the mixed-mode components J _I and J _II were determined after having performed several finite element analyses for different loading conditions. The path independence of J, J _I and J II is investigated for both, the line integral proposed by Rice and the volume integral proposed by deLorenzi. For correctly determined crack deflection angles the J _II-component vanishes when a FE-model with a kinked crack is considered. Hence, cracks propagate perpendicularly to the local mode I load.