Approximate expressions of maximum value of stress intensity factor of nozzle corner cracks

Based on the flat plate-nozzle model loaded by uniform uniaxial tension, through some suitable reduction and approximation, the maximum value of stress intensity factor (K_I)_max along a crack front can be expressed by the weight function method for the flat plate-nozzle that is loaded by the uniform uniaxial tension and cylinder-nozzle loaded by internal pressure separately. This approach can be valuable for the prediction of fatigue life and failure analysis of pressure vessels. The results so calculated are compared with those reported in the literature and from experiments here reported; and they show good agreement, the discrepancies being less than 15%. This shows that the approximate expressions are effective and can be used in practice.     

Computation of SIF (stress intensity factor) of corner crack in interior wall of nozzle of nuclear vessel

The computation of the stress intensity factor (SIF) of a corner crack in the interior wall of a nozzle for a steel nuclear vessel is a complicated 3-D case. The finite element method is a powerful tool for this case. An improved 3-D collapsed isoparametric singular element with quarter-points is presented in this paper and the required inverse square root singularity in these vertical planes of a crack is derived. The method of construction for the corresponding transitional element that was extensively used in the 3-D case is also presented. The Williams formula as well as the least-squares linear fitting method was used to compute K_I according to the displacements near the crack front.

The SIF of a corner crack in the interior wall of a nozzle of a Reactor Pressure Vessel (RPV) of a typical 300-MW nuclear power plant was computed and verified by the 3-D photo-elastic test and the diffusion of light test. To analyse the effect of various factors on the SIF, the computations were carried out on 12 models.     

Methods of calculating stress intensity factors for nozzle corner cracks

The various methods of determining stress intensity factors for transverse cracks in pressure vessel nozzles are reviewed. An estimate of the accuracy of experimental and finite element analyses is made by comparing the results of different studies. Methods are proposed for calculating approximate stress intensity factors for part-circular and straight fronted cracks using the weight function method. Approximate procedures are described for taking into account the two-dimensional nature of the nozzle corner stress distribution, and the curvature of the inner wall local to the nozzle corner. Where possible, stress intensity factors derived using these procedures are compared with published results. In general, agreement to within 20 per cent is obtained and, in most cases, the agreement is significantly better than this.     

An approximate expression of KImax for sphere/nozzle corner cracks

Based on the flat-nozzle model loaded by uniform biaxial tension, using the O-integral of the weight function method and introducing suitable free-surface correcting factors M₁, M₂, and a curvature correcting factor M_c, an approximate expression of K_Imax of sphere/nozzle circular corner cracks has been presented in this paper. The results so calculated are compared with those reported in the literatures and they show good agreement, the discrepancies being less than 10%. This shows that the approximate expression is effective and can be used in practice.     

Stress intensity factor equations for mixed-mode surface and corner cracks in finite-thickness plates subjected to tension loads

Normalized mixed-mode stress intensity factor equations are presented for deflected and inclined circular surface and corner cracks in finite-thickness plates under uniform remote tensile loading. The equations are obtained by performing non-linear regression analyses on the data from previous numerical solutions based on three-dimensional enriched finite elements. In the equations, the effects of deflection/inclination angles and plate thickness on mixed-mode stress intensity factors are included. The comparisons of normalized stress intensity factors from the equations with those of the finite element analyses show good agreement. Thus, it is concluded that, as a reasonable approximation, the presented equations can be used to assess stress intensity factors and fracture conditions of mixed-mode circular surface and corner cracks in finite-thickness plates.     

Effects of location,thermal stress and residual stress on corner cracks in nozzles with cladding

The stress intensity factors (K_I) for corner cracks in a boiling water reactor feedwater nozzle with stainless steel cladding are obtained for loading by internal pressure and a fluid quench in the nozzle. Conditions both with and without residual stress in the component are considered. The residual stress is simulated by means of a reference temperature change. The stress distribution for the uncracked structure is obtained from a three-dimensional finite element model.A three-dimensional influence function (IF) method, in conjunction with the boundary-integral equation method for structural analysis, is employed to compute K_I values from the uncracked stress distribution. For each type of loading K_I values are given for cracks at 15 nozzle locations and for six crack depths. Reasonable agreement is noted between calculated and previously published pressure-induced K_I values. Comparisons are made to determine the effect on K_I of crack location, thermal stress and residual stress, as compared with pressure stress. For the thermal transient it is shown that K_I for small crack depths is maximised early in the transient, while K_I for large cracks is maximised later under steady state conditions. Computations should, therefore, be made for several transient time points and the maximum K_I for a given crack depth should be used for design analysis. It is concluded that the effects on K_I of location, thermal stresses and residual stresses are significant and generally too complex to evaluate without advanced numerical procedures. The utilised combination of finite element analysis of the uncracked structure and three-dimensional influence function analysis of the cracked structure is demonstrated and endorsed.     

Calculation of stress intensity factors for semielliptical cracks in a thick-wall cylinder

Weight functions for the surface and the deepest point of an internal semielliptical crack in a thick-wall cylinder were derived from a general weight function and two reference stress intensity factors. For several linear and nonlinear crack face stress fields, the weight functions were validated against finite element data. Stress intensity factors were also calculated for the Lamé through the thickness stress distribution induced by internal pressure. The weight functions appear to be particularly suitable for fatigue and fracture analysis of surface semielliptical cracks in complex stress fields. All stress intensity factor expressions given in the paper are valid for cylinders with an inner radius to wall thickness ratio, R_i/t = 4.     

Finite element based evaluation of stress intensity factors for interactive semi-elliptic surface cracks

A finite thickness plate with two coplanar self-same shallow and deep semi-elliptical surface cracks subjected to remote tensile surface traction is considered for fracture analysis. Based on three-dimensional (3D) finite element solutions, stress intensity factors (SIFs) are evaluated along the entire crack front using a force method. The line spring model has also been used to evaluate crack depth point SIFs using shell finite element analysis. A wide range of geometric dimensions and crack configurations viz. crack shape aspect ratio (0.3≤a/c≤1.2), crack depth ratio (1.25≤t/a≤6), relative crack location (0.33≤2c/d≤0.9) and normalized location on the crack front (0≤2φ/π≤2) are considered for numerical estimation of crack interaction factors. SIFs evaluated at the depth point using the force method from the 3D finite element results are compared with SIFs evaluated using the line spring model. Finally, using finite element results, an empirical relation is proposed for the evaluation of crack interaction factors. For the ranges considered, the proposed empirical relation predicts crack interaction factors at critical locations within ±2% of the 3D finite element solutions.     

Assessment of the fatigue life in relation to cylinder-nozzle corner cracks

Based on former studies this paper deals with both the front shape variation of nozzle corner cracks and the calculation of crack propagation rates. An approximate method describing fatigue crack growth at nozzle corners under cyclic internal pressure is proposed here. The results calculated are compared with those reported in the literature. The good agreement shows that the views proposed here are effective and may be valuable for the researchers in this field.     

An explicit integral expression for the stress intensity factor of a semi-elliptic surface crack subjected to thermal transient loading

An explicit integral expression for the stress intensity factor of a semi-elliptic surface crack in a plate subjected to thermal transient loading was developed. The stress intensity factor of a semi-elliptic surface crack in a plate, which is exposed to a step change of fluid temperature, was calculated on the basis of the weight function method. The change of the stress intensity factor for a semi-elliptic surface crack subjected to an arbitrary change of the boundary fluid temperature was obtained by Duhamel integration for the product of the step function result and the time varying fluid temperature. The result obtained by the present method has shown good agreement with those obtained by the influence function method. As a practical application, a parametric analysis was performed for the crack behavior during the emergency cool down of reactor coolant in the reactor pressure vessel. Also, the present expression can be effectively applied to the simulation of fatigue crack growth of a semi-elliptic surface crack subjected to various thermal transient loading.     

Experimental determination of the stress intensity factor for a part-through thickness crack

Based on the relationship between the stress intensity factor and the plastic zone size at the crack tip, an experimental method for the evaluation of the stress intensity factor for a part-through thickness crack is proposed. Approximating the geometrical shape of the crack to a semi-elliptical surface crack, an empirical K-expression for the surface crack in a plate under bend loading configuration is derived and verified experimentally where the new K-expression contains both crack depth and length as a crack tip singularity parameter. The experimental work makes use of newly developed etching techniques to reveal the plastic zone size in deformed A533B pressure vessel steel.     

A simplified method for calculating the stress intensity factor at crack arrest

This paper outlines a new, simplified method for approximating K_I at arrest during fast fracture events. The analysis method used in this report was developed as a simplified tool to approximate arrest conditions in a series of experiments performed by Combustion Engineering for EPRI. This simplified analysis method is applied to the CE/EPRI, HSST wide-plate, and ORNL pressurized-thermal-shock crack arrest experiments. The results presented in this paper are the first which use a single analysis technique, and thereby allow a comparison of these diverse experiments. This consistent evaluation places all the data from these experiments on an equal footing and provides a comparison with small specimen test data from the HSST program.     

A simple procedure to evaluate the stress intensity factor in internally pressurized cylinders

The evaluation of stress intensity factors in internally pressurized cylinders, with both surface and sub-surface flaws, is examined. The method of analysis is based on the equivalent linear representation of the circumferential stress distribution in accordance with ASME rules, the non-linear hoop stress distribution then being conservatively approximated by the membrane and bending stresses. The stress intensity factor for an elliptical crack embedded in an elastic solid and subjected to internal pressure is considered for two conditions of load (tension and bending) and the effects are added.The results are presented in non-dimensional form to evaluate the effect on stress intensity factor of the various parameters (outside and inside radius, crack position, cylinder thickness, form of ellipse).     

A comparison of methods of calculating stress intensity factors for cracks in pipes and thin walled cylinders

This paper presents results of a study undertaken to compare stress intensity factor solutions for various crack geometries in pipes and thin walled cylinders against the equivalent flat plate K solutions. The exercise was restricted to cylinders and pipes with wall thickness to radius ratios (t/R) of 0·1.

The results of the exercise indicate that structural integrity assessments of pipes and thin walled cylinders which contain flaws should ideally incorporate representative stress intensity factor solutions. Nevertheless there are a number of crack geometries for which flat plate K solutions can provide reasonable estimates of the stress intensity factor.     

Experimental calibration of stress intensity factor for piping with circumferential through-wall crack

A survey of the literature shows that the existing stress intensity factor solutions for circumferential through-walled cracks in piping may be classified into three categories. One category is based on Sanders' analytical results for long pipe cracks, with various corrections in the short crack range and different curve-fitting formulae to give convenient closed form expressions. The second category consists of various independent finite element solutions. Each of these solutions is for a discrete pipe geometry and crack length and so is not practical to be used in fracture mechanics analysis. Lastly there is Kanninen & Zahoor's solution, derived independently of Sanders' results. Comparison showed that the results from the first two categories roughly agreed but were vastly different from Kanninen & Zahoor's results. Experimental calibration using the strain gauge method and the fatigue crack growth rate back-tracking method has been carried out. The experimental results agreed with the family of solutions derived from Sanders' work. Details about this experimental calibration are presented.     

The effect of spatial incoherence in surface temperature fluctuations on the stress intensity factor in thermal striping

When incompletely mixed hot and cold fluid streams pass adjacent to the surface of a component of a structure, they can cause that component to suffer thermal fatigue damage. Spatial incoherence in the thermal fluctuations will affect the stress intensity factor (SIF), which is frequently calculated assuming perfect spatial coherence. A model is developed to assess this effect for a circumferential line crack in a thin cylinder. The dimensionless SIF can be increased or decreased by reduced spatial coherence, depending on a combination of factors such as Fourier number, Poisson ratio, scale of coherence and crack depth.     

Weight functions for the determination of stress intensity factor and T-stress for edge-cracked plates with built-in ends

This paper presents the weight functions for the determination of the stress intensity factor and T-stress solutions for edge-cracked plates with built-in ends under complex stress distributions. First, a compliance analysis approach is used to calculate stress intensity factor and T-stress for edge cracks in finite width plates with built-in ends with uniform or linear stress distributions acting on the crack face. The results serve as the reference solutions for the next step in which the approaches of deriving weight functions from reference stress intensity factor and T-stress solutions developed for stress boundary conditions are extended to obtain the corresponding weight functions for edge-cracked plates with built-in ends. Finite element analysis is conducted to validate the derived solutions. The weight functions derived are suitable for obtaining stress intensity factors and T-stress solutions under any complex stress field.     

Stress intensity factor analyses of surface cracks in three-dimensional structures—Comparison of the finite element solutions with the results obtained by the simplified estimation methods

The stress intensity factor analyses of surface cracks in various three-dimensional structures were performed using the finite element computer program EPAS-J1. The results obtained by EPAS-J1 were compared with other finite element solutions or the results obtained by the simplified estimation methods. Among the simplified estimation methods, the equations proposed by Newman and Raju give the distributions of the stress intensity factors along a crack front, which were compared with the results obtained by EPAS-J1. It was confirmed by comparing the results that the finite element program EPAS-J1 gives the reasonable stress intensity factors of surface cracks in three-dimensional structures.     

The shape of a surface crack in a plate based on a given stress intensity factor distribution

A new method is developed for the evaluation of a crack shape based on a given stress intensity factor (SIF) distribution for a surface crack under Mode-I loading conditions. The SIF distribution along the crack front is investigated using a direct simulation technique, in which the effect of crack closure on fatigue crack growth is considered. Then a SIF distribution function is chosen based on the numerical results. Crack shape (and the SIF) is achieved based on the given SIF distribution function using a numerical iterative procedure. Empirical SIF equations for surface cracks in plates subjected to tension and pure bending fatigue load are determined by systematic curve fitting of the numerical results. The depth ratio and the aspect ratio are considered in the ranges of 0.1–0.9 and 0.2–1.2, respectively. The aspect-ratio variation of surface cracks under fatigue loading is predicted. The application of the new method to predict the shape of a surface crack in plates subjected to tension and pure bending and comparisons of the results obtained with the predictions of the empirical equations proposed by Newman and Raju are presented.     

Theory model combined with XFEM of threshold stress intensity factor and critical hydride length for delay hydride cracking

In order to determine the threshold stress intensity factor and critical hydride length for delayed hydride cracking in Zr-2.5Nb pressure tube alloy, the distribution of normal stress in the plastic zone of crack tip by the developed method that combines theory calculation with extended finite element method (XFEM) was improved. The fracture process of two-phase composites containing Zr-2.5Nb and hydride precipitate was simulated by XFEM. Based on that, critical hydride length $L_C$ corresponding to the theoretical model for K_1H was estimated. Meanwhile, to illustrate the validity of theoretical and numerical methods, recent theoretical models and experimental measurements were utilized to verify the results of this paper. The theoretical model of DHC was improved to estimate the critical hydride length corresponding to threshold stress intensity factor. The predicted value of critical hydride length is close to the experimental values.