A practical method for computation of ductile crack growth by means of finite elements and parametric 3D-modelling

The present paper originates from a contribution to the safety assessment of a reactor pressure vessel (RPV). Investigations evaluating the safety against brittle fracture (exclosure of crack initiation and arrest assessments) are completed by calculations concerning ductile crack extension. Crack geometries including the expected crack extension are generated parametrically by a computer code and are used for further calculations with finite element programs. J-integrals of ductile growing cracks located between two comparative contours are determined by interpolation. The comparative contours are loaded by instationary temperature and pressure fields and are evaluated in advance. Taking the stability condition into consideration, the ductile crack extension is determined by pursuing the equilibrium between loading and crack resistance. The automatic modelling and a mathematical program processing the finite element results evaluate the crack growth of the finite element results very effectively.     

Investigation on constraint effect of reactor pressure vessel under pressurized thermal shock

In recent years, the integrity of reactor pressure vessels (RPVs) under pressurized thermal shock (PTS) accident has been treated as one of the most critical issues. Under PTS condition, the combination of thermal stress due to a steep temperature gradient and mechanical stress due to internal pressure causes considerable tensile stress inside the RPV wall. As a result, cracks on the inner surface of RPVs can experience elastic-plastic behavior that can be explained using the J-integral. In such a case, however, the J-integral may possibly lose its validity due to the constraint effect. The degree of constraint effect is influenced by the loading mode, the crack geometry and the material properties. In this paper, three-dimensional finite element analyses are performed for various surface cracks to investigate the effect of clad thickness and crack geometry on the constraint effect. A total of 36 crack geometries are analyzed and results are presented by the two-parameter characterization based on the J-integral and the Q-stress.     

Fracture mechanical evaluation of an in-vessel melt retention scenario

This paper presents methods to compute J-integral values for cracks in two- and three-dimensional thermo-mechanical loaded structures using the finite element code ANSYS. The developed methods are used to evaluate the behavior of a crack on the outside of an emergency cooled reactor pressure vessel (RPV) during a severe core melt down accident. It will be shown, that water cooling of the outer surface of a RPV during a core melt down accident can prevent vessel failure due to creep and ductile rupture. Further on, we present J-integral values for an assumed crack at the outside of the lower plenum of the RPV, at its most stressed location for an emergency cooling (thermal shock) scenario.     

The J integral evaluation of a nozzle corner crack under thermal transient loading condition

This paper presents the results of the evaluation of the three dimensional J integral of a nozzle corner crack which is initiated by fatigue under thermal transient loading conditions of a BWR type reactor vessel. Analyses are carried out by using the finite element method in the following two cases. One case consideres the effect of the stainless steel cladding deposited over the inside surface of the reactor vessel, and the other neglects it. In both cases, the extended J integral concept, called integral, is used to obtain the path independent J value in the thermal stress fields.By changing the shapes and the dimensions of the crack, some elastic analyses are carried out in the two dimensional space. The effects of the cladding are studied qualitatively, and the integrals are compared with the critical J value, and discussed.Three dimensional values along the three dimensional crack front are evaluated for the embedded crack. The results are compared with those for two dimensional analysis.The nozzle corner crack is treated under thermal transient condition and distributions of values, and their change with time is obtained. The shapes and dimensions are changed by the assumption that the crack growth occurs at the point where the vector has its maximum, and the three dimensional shape of the propagating crack is estimated.     

3D analyses of surface flaws in thick-walled reactor pressure-vessels using displacement-hybrid finite element method

Solutions of stress intensity factors for external and internal unpressurized and pressurized surface cracks in internally pressurized thick-walled reactor pressure vessels are determined directly by a three-dimensional displacement-hybrid finite element method. The finite element method is based on a rigorous modified variational principle of the total potential energy, with arbitrary element interior displacements, interelement boundary displacements and element boundary tractions as variables. Special crack front elements, developed using the hybrid displacement model, which contain the proper square root and inverse square root variations of displacements and stresses, are used in this analysis and the three stress intensity factors, K_I, K_II and K_III are solved directly along with the unknown nodal displacements. Stress intensity factor variations for pressurized and unpressurized semi-elliptical inner surface cracks in pressurized cylinders with crack aspect ratios of 0.2 and 1.0, crack depth to cylinder wall thickness ratios of 0.5 and 0.8 and outer to inner diameter ratios of 1.5 and 2.0, are presented. Also, for unpressurized outer surface cracks in pressurized cylinders, the solutions are presented for crack aspect ratios of 0.6 and 1.0, crack depth to cylinder wall thickness ratios of 0.4, 0.6 and 0.8, and outer to inner diameter ratio of 1.5.     

A parametric study on pressure–temperature limit curve using 3-D finite element analyses

In order to operate a reactor pressure vessel (RPV) safely, it is necessary to keep the pressure–temperature (P–T) limit during the heatup and cooldown process. While the ASME Code provides the P–T limit curve for safe operation, this limit curve has been prepared under conservative assumptions. In this paper, the effects of conservative assumptions involved in the P–T limit curve specified in the ASME Code Sec. XI were investigated. Three different parameters, the crack depth, the cladding thickness and the cooling rate, were reviewed based on 3-D finite element analyses. Also, the constraint effect on P–T limit curve generation was investigated based on J–T approach. It was shown that the crack depth and constraint effect change the safety region in P–T limit curve dramatically, and thus it is recommended to prepare a more precise P–T limit curve based on finite element analysis to obtain P–T limit for safe operation of a RPV.     

A multibarrier safety concept for the reactor pressure vessel of the Stade Nuclear Power Plant

Two specific problems within the safety case of Stade RPV have been analysed: brittle fracture initiation and arrest under strip type emergency core cooling conditions and safety margins against ductile failure from deep cracks as postulated by ASME- and German KTA-rules. For EOL material conditions exclusion of initiation is shown for cracks of more than twice the size which is safely detectable by NDE; for arbitrarily postulated large cracks it is demonstrated that they are arrested well within the allowed depth of of the wall thickness; therefore no critical crack size exists for Stade RPV under strip cooling. Growth in depth of an assumed circumferential flaw in the girth weld embrittled at EOL could occur only at upper shelf temperatures and by loads higher than about twice the service pressure; leak before break was demonstrated in a constraint-modified J − R-curve crack-growth analysis. But neither a transient nor the plant itself would be able to provide the necessary high loads. The LEFM and EPFM proofs are summarized in a multibarrier safety scheme.     

Effect of cladding on the stress intensity factors in the reactor pressure vessel

In general, reactor pressure vessels (RPV) are cladded with stainless steel to prevent corrosion and radiation embrittlement. The ASME Sec. XI specifies that a subclad crack which may be found during the in-service inspection must be considered as a semi-elliptical surface crack when the thickness of cladding is less than 40% of the crack depth. In order to refine the fracture assessment procedures for such subclad cracks, three-dimensional finite element analyses were applied for various subclad cracks embedded in the base metal. A total of 18 crack geometries were analyzed, and the results were compared with those for idealized semi-elliptical surface cracks for two different loading conditions, i.e. internal pressure and pressurized thermal shock. The resulting stress intensity factors for subclad cracks were 50–70% less than those for idealized surface cracks. It has been proven that the condition specified on the ASME Sec. XI is overly conservative for subclad cracks which are assumed to be surface cracks.     

Significance of warm prestress to crack initiation during thermal shock

During a loss of coolant accident (LOCA) followed by operation of the emergency core cooling system, the inside wall of a nuclear pressure vessel is subjected to high thermal stresses that can cause extension of a pre-existing flaw. During this event the crack-tip stress intensity factor, K_I, may achieve its maximum value early in the transient, but the critical level for crack initiation, K_Ic, may not be reached until minutes later at which time the loading has decreased from its peak. It is shown that this phenomenon, termed warm prestress (WPS), can preclude crack extension when K_I equals or exceeds K_Ic.NRL has conducted an experimental study, employing three-point bend specimens, to investigate the potential for elevation in K_Ic by WPS, and to translate the significance of this behavior into structural terms in the sense of minimizing crack extension in a nuclear vessel during a LOCA. From the experiments it is concluded that the mechanisms associated with WPS act to elevate the K_Ic of the material at the crack tip and that this fact can greatly minimize crack extension that would have been predicted theoretically without consideration of WPS. The experiments demonstrated that failure never occurs during the unloading portion of the simulated LOCA path. This finding is of major significance to the integrity of a vessel. For example, with relatively deep cracks there is a combination of conditions wherein initiation ordinarily would be predicted as K_Ic is reached along a decreasing K_I path. For this set of conditions the present research studies have shown that the WPS phenomenon will preclude all such crack initiation.In terms of margin of safety against fracture it is shown that the elevation in K_Ic caused by WPS is not uniform but depends upon the WPS level, the degree of unloading, and the increment between the temperature of WPS and the failure temperature. For LOCA conditions, however, it is concluded that WPS can result in an effective elevation in K_Ic up to the WPS level assuming, of course, that metallurgically the material is capable of exhibiting this level of toughness.In terms of structural significance it is clear that WPS by itself cannot prevent the initiation of shallow cracks. Specifically, for a reference calculational vessel under the worst combination of conditions including a long, axial flaw and severe radiation embrittlement it is shown that a shallow crack can extend to a relative depth of 0.34 of the wall thickness; further crack extension is prevented by WPS. However, cracks having initial depths greater than 0.2 of the wall are prevented, by WPS, from extending any amount. Finally, it was observed that an elastic analysis of crack extension during a LOCA has predicted nearly complete penetration of the wall without consideration of WPS. Factoring WPS into the same analysis results in predicted crack extension of greatly reduced proportions such that complete penetration of the wall does not occur. Thus, WPS may form a key element upon which to base the assurance of vessel integrity during a LOCA.     

LOCA下具有表面裂纹的反应堆压力容器承压热冲击分析

失水事故（LOCA）瞬态下,具有半椭圆形表面裂纹的反应堆压力容器（RPV）承压热冲击（PTS）问题被研究。采用有限元方法计算瞬态过程的热-应力响应;采用影响函数法计算应力强度因子,分别对母材和堆焊层内的应力进行分解,从而解决了由于堆焊层存在造成的应力拟合困难带来的计算偏差。编制了相应的断裂分析程序,对LOCA下RPV的结构完整性进行了分析。结果表明,在研究的LOCA下,整个瞬态过程中RPV应力强度因子均未超过材料断裂韧性,压力容器结构安全。本文研究为RPV在PTS下的结构完整性评估提供理论指导。     

Numerical investigation of J-characterization of growing crack tips

Two different geometries, a centrally cracked panel and a three-point bend bar, are modelled with aid of the finite element program ABAQUS. Elastic-plastic behaviour with a realistic linear hardening modulus is assumed. By simulation of the growth with the aid of nodal relaxation, the J-value for a remote path around the growing tip is obtained for some different load-crack growth histories. This J_F-value is compared to the J_D-value that results if the crack tip is assumed to be stationary at the current length. It is found that the J_D- and J_F-values agree well for crack growth histories satisfying the criteria for J-characterization. However, after examination of the crack surface displacements it was found that the results for the bend geometry and the tension geometry, respectively, did not coincide for corresponding J-values, except at low load levels. This raises doubts about the abilities of J to characterize the state at a growing tip.     

Stable crack growth of axial flaws in pressure vessels

The ductile crack growth of axial through and part-through cracks in a vessel under internal pressure has been studied experimentally to contribute to the fundamental problem whether or not and under which conditions resistance curves obtained from specimens can be transferred to large scale components. The experiments and numerical analyses are part of a research program on fracture mechanics failure concepts for the safety assessment of nuclear components.Whereas only an averaged crack extension is determined in specimen tests, the local propagation of cracks may be of main importance for surface cracks in vessels and pipes. In the present experiments, the surface cracks revealed the well known canoe shape, i.e. a larger crack extension has occurred in the axial direction than in the wall thickness direction. Two of these tests have been analysed by finite element calculations to obtain the variation of the J-integral along the crack front and the stress and strain state in the vicinity of the crack. The local crack resistance appeared to depend on the local stress state. To Predict ductile crack extension correctly, J_R-curves have to account for the varying triaxiality of the stress state along the crack front.     

Analysis of stable crack growth of a semi-elliptical surface crack by numerical simulation

In the present paper, the canoeing effect during stable crack growth of a semi-elliptical surface crack in a side-grooved panel under tension was investigated by means of a three-dimensional elastic-plastic finite element analysis. The numerical crack growth simulation was performed by using crack mouth opening displacement resistance curves obtained from experiment on panels by the multi-specimen procedure. The influence of the crack tip constraint and the stress triaxiality on the ductile crack resistance property was studied. It is shown that the tearing modulus T_JR increases proportionally with decreasing stress triaxiality implying constraint dependent J_R-curves.     

Assessment of cracks existing in the strain concentrated region from a viewpoint of brittle fracture

The significance of cracks existing in the strain concentrated region is discussed from a viewpoint of brittle fracture.In this study, the J-integral, of which the mathematical treatment is easier and stricter than that of COD (crack opening displacement) is considered as the fracture parameter. Also, averaged local strain, , in the strain concentrated region without a crack is considered as the mechanical parameter which characterizes the state of the elastic-plastic region generated near the stress raiser. The relation between the J-integral of the cracked body and the local strain is investigated experimentally and theoretically with the aid of the elastic-plastic finite element analysis.Brittle fracture tests on the strain concentration models of the four kinds of the structural steel (SM41, SM50, HT80 and A508 C1.3) show that a newly proposed J design curve provides a good estimation for the strength of brittle fracture from a crack existing in the stress/strain concentrated region.     

Elastic-plastic analysis of small cracks in tubes under internal pressure and bending

Elastic-plastic finite element analyses were conducted to generate new solutions of J-integral and crack-opening displacement (COD) for short through-wall cracks in pipes subjected to combined bending and tension loads. The results are presented in terms of the well-known GE/EPRI influence functions to allow comparisons with some limited results in the literature. Two different pipe pressures with values of 7.24 MPa (1050 psi) and 15.51 MPa (2250 psi) simulating BWR and PWR operating conditions, respectively, were used to evaluate the effects of pressure on J and COD. Pipes with various radius-to-thickness ratios, crack sizes, and material parameters were analyzed. Limited analyses were also performed to evaluate the effects of hoop stresses in pipes under pure pressure loads. The results suggest that the fracture response parameters can be significantly increased by pressure-induced axial tension for larger crack size, material hardening constant, and radius-to-thickness ratio of the pipe. The presence of pressure-induced hoop stresses also increases the fracture response, but in low-hardening materials their effects are insignificant due to small plastic-zone size that was expected for the intensity of pipe pressure and crack size considered in this study. However, for high-hardening materials when the plastic-zone size is not negligible, the hoop stresses can moderately increase J and COD.     

The importance of the triaxiality modified J concept for the assessment of pressurized components with surface flaws

With the progress of stable crack growth of surface flaws observed in panels or pressure vessels a canoe-shaped crack front is formed. The crack propagation in the longitudinal direction is more pronounced that in the wall thickness direction. Therefore, the canoe effect is important with respect to a leak-before-break assessment because the actual through crack length is influenced by this effect. Based on the J integral concept crack initiation and crack propagation in ductile materials are described by J resistance curves which were found to be dependent on the constraint effect of the specimen geometry. Prediction of local crack growth by taking a conservative (flat) J_R-curve into account results in a nonconservative estimate of the axial extension of the surface crack [W. Brocks, H. Veith and K. Wobst, in K. Kussmaul (ed.), Fracture Mechanics Verification by Large Scale Testing, Mech. Eng. Publication Limited, London, 1991]. This means that the influence of local constraint effects on crack resistance has to be considered.Ductile crack growth of semi-elliptical surface cracks in side-grooved specimens F(SCTsg) under tension made from German standard steel StE 460 will be reported on. The development of the canoe effect of an SCTsg specimen was also analysed by a finite element simulation of ductile crack growth which was modelled by using the node shift and node release technique and controlled by crack mouth opening displacement versus crack growth curves from the experiment. The simulation allows the determination of local J_R-curves in dependence on the local multiaxility of the stress state to verify the constraint modified J concept. It is demonstrate that the slope of the J_R-curves decreases with increasing multiaxiality of the stress state near by the crack front.     

Finite element analysis of crack propagation in three-dimensional solids under cyclic loading

This paper is concerned with an efficient computational procedure for analyzing crack propagation in solids. The method is general; however, its application to semi-elliptical surface cracks in thick plates is discussed in particular. The strain energy release rate G for a crack in mode I is a function of the crack geometry, the direction of crack propagation and the state of loading. When G is known, the stress intensity factor K_I can easily be obtained. In this paper the strain energy of the plate is computed numerically for a wide range of crack geometries using the finite element method. A 20-node isoparametric solid element is employed in modelling the structure. Certain special techniques for increasing the computational efficiency of the method, such as multilevel subdivision of the structure (substructuring) and condensation of degrees of freedom that are not needed in the crack propagation analysis, are emphasized. In fact, analysis of a large number of crack geometries requires only insignificantly more computational efforts than treating a single crack. Certain other aspects of the finite element modelling are also discussed.Two methods for replacing the computed discrete values of strain energy by continuous functions are presented. These functions are expressed in terms of the two half-axes defining the geometry of the elliptical crack and they are determined using a least square technique. G and K_I are easily deduced from these functions. As an example, a semi-elliptical, part-through, surface crack in a thick nickel steel plate is analyzed. The crack is subjected to a combination of axial and bending loading, applied cyclically. From the finite element calculations of the strain energy and the stress intensity factors which are computed accordingly, crack propagation along the two half-axes of the ellipse is calculated by utilization of a formula suggested by Paris. The results are checked against laboratory fatigue tests. The method has proved to be very efficient and accurate, and due to its generality it can also be applied to complicated geometries and complex states of loading.     

Elastic-plastic FEM-analysis of a nozzle corner crack and discussion of the results by some fracture mechanics concepts

This paper presents the results of an elastic-plastic threedimensional finite element analysis for a nozzle corner crack in a pressurized reactor test vessel. The calculations were performed by the finite element program ADINA incorporating von Mises' yield condition and isotropic hardening. The crack plane was taken parallel to the axis of the vessel and the crack front straight and perpendicular to the symmetry line of the nozzle corner in order to obtain the worst position for a nozzle corner crack. The calculations were performed up to that pressure level where general yield of the ligament in the nozzle corner section takes place. The results of the finite element analysis are compared with figures obtained from analytical procedures of elastic-plastic fracture mechanics.     

反应堆压力容器接管嘴内隅角应力强度因子计算研究

本文针对反应堆压力容器接管嘴内隅角,采用含真实裂纹的三维有限元法对温度与压力作用下应力强度因子的计算进行了研究。以某工程压力容器接管嘴内隅角为例,用含真实裂纹的三维有限元法和目前使用的简化工程算法对压力与热载荷作用下的接管嘴内隅角应力强度因子进行了计算,并对两种方法的计算结果进行对比分析。结果表明:当简化工程算法得到的应力强度因子接近规范限值时,应对热载荷引起的应力强度因子进行详细有限元计算,以规避简化工程算法的不保守性给压力容器带来的快速断裂风险。