Three finite element analysis of semi-elliptical crack in high density poly-ethylene pipe subjected to internal pressure

The principal objective of this work is to analyze the severity of semi-elliptical crack defects and to study the degree of damage in the equipment under internal pressure during the crack propagation. The semi-elliptical cracks are considered in this work located in different position in the wall of poly-ethylene pipe. The tree finite element method based on the computation of the J integral was used to analyze the fracture behaviour of these structures. The effect of the position shape and size of the crack on the J integral was highlighted. The effects of strain rate and the temperature on the J integral values were also examined. The obtained results show that, whatever the material (strain rate) for a semi-elliptical crack, the J integral value has not an important variation with respect to the crack size. However, the energy for axial crack is more important compared to circumferential 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). We recall finally, that the temperature effect on circumferential cracks behaviour is more important compared to the axial cracks. 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.     

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.     

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.     

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.     

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.     

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.     

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.     

High temperature fatigue deformation behaviors of thermally sprayed steel measured with electronic speckle pattern interferometry method

High temperature fatigue (R=0) damage and deformation behaviors of SUS304 steel thermally sprayed with Al₂O₃/NiCr coating were investigated using an electronic speckle pattern interferometry (ESPI) method. Surface cracks and delamination occurred after 1×10⁵ cycles test when σ_max was 202 MPa at 873 K. The lengths and number of cracks and delamination largely decreased when σ_max or temperature decreased to 115 MPa or 573 K, respectively. Strain values along cracks measured with the ESPI method were much larger than other areas due to crack opening under the tensile load. The positions of strain concentration zones on strain distribution figures by ESPI method were well corresponded to those of cracks on sprayed coatings. Strain values decreased largely where local delamination occurred.     

Edge cracks in a transversely isotropic hollow cylinder

The analytical solution for the linear elastic, axisymmetric problem of inner and outer edge cracks in a transversely isotropic infinitely long hollow cylinder is considered. The z = 0 plane on which the crack lies is a plane of symmetry. The loading is uniform crack surface pressure. The mixed boundary value problem is reduced to a singular integral equation where the unknown is the derivative of the crack surface displacement. An asymptotic analysis is done to derive the generalized Cauchy kernel associated with edge cracks. It is shown that the stress intensity factor is a function of three material parameters. The singular integral equation is solved numerically. Stress intensity factors are presented for various values of material and geometric parameters.     

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.     

High-rate J-testing of toughened polyamide 6/6: Applicability of the load separation criterion and the normalization method

This paper examines the applicability of the load separation criterion and the normalization method in determining J_R curve of a toughened polyamide 6/6 at high loading rates (1 m/s). The analysis of procedure problems associated to this high experimental rate is performed. The results obtained using the normalization method are then compared with those measured via multi-specimen testing procedures proposed by ESIS (Technical Committee 4). The results show that, unlike low loading rate tests, the presence of the oscillations in the load vs displacement traces, due to the inertial effects produced during the impact, complicate considerably the elaboration of the data, with particular reference to the identification of the separable blunting region. The comparison of J_Ic values obtained according to the different procedures examined indicates that the values of J_Ic = J_0.2 (taken at 0.2 mm crack growth) are in good agreement, whereas consistent differences among the values of J_Ic = J-blunting (taken at the blunting line) are observed.     

Characterization of a fracture specimen for crack growth in epoxy due to thermal fatigue

Finite element simulations are carried out to characterize a new fracture specimen, consisting of an outer circular epoxy ring bonded to an inner circular invar plate for accelerated thermal fatigue testing. Radial cracks are introduced in the epoxy ring. The growth of these radial cracks is correlated to the applied energy release rate G. We studied the dependence of G on the crack length, the specimen geometry and the elastic modulus. For short cracks, G is obtained in closed form. Analysis is carried out to determine the critical thermal buckling load the specimen can withstand. Experimental results show that the fatigue crack growth rate per thermal cycle da/dN is given by da/dN = 0.51(ΔG)^0.38 for cycling between 4 and 100 °C but by da/dN = 0.25(ΔG)^0.24 for cycling between 20 and 85 °C, where ΔG is the difference of the energy release rate between the highest and lowest temperatures during a thermal cycle. More severe thermal cycles produce considerably larger fatigue crack growth rates than less severe ones at the same ΔG. This result also implies that isothermal fatigue tests will probably be inadequate to predict thermal fatigue crack growth in epoxies.     

Three-dimensional stress state around quarter-elliptical corner cracks in elastic plates subjected to uniform tension loading

Detailed full-field three-dimensional (3D) finite element analyses have been conducted to study the out-of-plane stress constraint factor T_z around a quarter-elliptical corner crack embedded in an isotropic elastic plate subjected to uniform tension loading. The distributions of T_z are studied in the forward section (0° ? θ ? 90°) of the corner cracks with aspect ratios a/c of 0.2, 0.4, 0.5, 0.6, 0.8 and 1.0. In the normal plane of the crack front line, T_z drops radially from Poisson’s ratio at the crack tip to zero beyond certain radial distances. Strong 3D zones (T_z > 0) exist within a radial distance r/a of about 4.6-0.7 for a/c = 0.2-1.0 along the crack front, despite the stress-free boundary conditions far away. At the same radial distance along the crack front in the 3D zones, T_z increases from zero on one free surface to a peak value in the interior, and then decreases to zero on another free surface. The distributions of T_z near the corner points are also discussed. Empirical formulae describing the 3D distributions of T_z are obtained by fitting the numerical results, which prevail with a sufficient accuracy in the valid range of 0.2 ? a/c ? 1.0 and 0° ? θ ? 90° except very near the free surfaces where T_z is extremely low. Combined with the K-T solution, the transition of approximate plane-stress state near the surfaces to plane-strain state in the interior can be characterized more accurately.     

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.     

Lead substitutions for promoted critical current density Jc and mechanical properties of Mg1−xPbxB2 regime

Lead was used as softener dopant element to replace on magnesium sites in Mg_1−xPb_xB₂ regime (where x = 0.0, 0.05, 0.1 and 0.2 mole). Samples were prepared via high temperature solid-state reaction technique depending upon diffusion mechanism of Mg/Pb-vapour ions through boron-matrix. The maximum solubility limit of lead n was found to be ∼0.2 mole, which emphasises that substitution on MgB₂ system is stilling too limited and needs many precautions to be successful.The crystalline lattice constants were evaluated and exhibit noticeable length elongation in case of c-axis as x increases while a, b-axes were nearly constant. Furthermore, effect of Pb-doping was investigated carefully on microstructure and superconducting properties of MgB₂ system. SE-microscopic analysis indicated that lead ions diffuse regularly through surface and bulk in case of (x = 0.05 and 0.1 mole) and grain size was estimated and found in between 0.43 and 1.6 μm. Critical current densities (J_c's values) recorded an reasonable increase as Pb-content increase (from x = 0 to 0.1 mole) while recorded a slight decrease for sample with maximum lead content (x = 0.2 mole). The mechanical tensile strength of the samples was clearly improved linearly as Pb-content increase recording maximum tensile 31.7 MPa for sample with x = 0.2 mole.     

A modified damage potential to predict crack initiation: theory and experimental verification

Mesh dependency of cavity growth model due to Rice and Tracey has been overcome by integrating it over a process zone surrounding the crack tip. This integral represents a modified damage potential. The critical value of the integral for crack initiation in SA333Gr.6 material has been determined analysing a CT specimen and comparing the computed J with the experimentally measured J-initiation value. The critical value of the integral was then used to compute J-initiation in other fracture specimens having different crack-tip constraints. The critical value was also used to predict crack initiation loads in three 8 in. straight pipes and three 8 in. elbows having different measure of through-wall circumferential flaws. The computed values have been compared with the experimentally measured values. A close agreement between the computed crack initiation loads with the experimentally measured values justified the usefulness of the present modified damage potential.     

Boron substitution in ternary metal phosphide superconductors

Partial substitution of boron for phosphorus in the ternary phosphide superconductors ZrRuP and LaRu₂P₂ induces highly anisotropic structural changes and suppresses the superconducting transition temperatures. Undoped ZrRuP [hexagonal, a=6.445(4) Å, c=3.765(2) Å] systematically expands for ZrRuP_1−xB_x (x=0.05, 0.10, 0.15, 0.20, 0.40) in the a direction to 6.949(4) Å and shrinks in the c direction to 3.601(3) Å. Superconductivity is still present in the boron-substituted samples, although T_c drops from 10.1 K in ZrRuP to 5.6 K in ZrRuP_0.6B_0.4. LaRu₂P₂ [tetragonal, a=4.032(2) Å, c=10.632(7) Å] remains essentially unchanged along the a axis [a=4.037(1) Å] upon substitution at a starting composition of LaRu₂P_1.6B_0.4, but shrinks along the c axis to 10.477(5) Å. Accordingly, the superconducting T_c drops from 4.0 K for LaRu₂P₂ to 2.0 K for LaRu₂P_1.6B_0.4.