Ductile crack growth simulation from near crack tip dissipated energy

A method to calculate ductile tearing in both small scale fracture mechanics specimens and cracked components is presented. This method is based on an estimation of the dissipated energy calculated near the crack tip. Firstly, the method is presented. It is shown that a characteristic parameter G_fr can be obtained, relevant to the dissipated energy in the fracture process. The application of the method to the calculation of side grooved crack tip (CT) specimens of different sizes is examined. The value of G_fr is identified by comparing the calculated and experimental load line displacement versus crack extension curve for the smallest CT specimen. With this identified value, it is possible to calculate the global behaviour of the largest specimen. The method is then applied to the calculation of a pipe containing a through-wall thickness crack subjected to a bending moment. This pipe is made of the same material as the CT specimens. It is shown that it is possible to simulate the global behaviour of the structure including the prediction of up to 90-mm crack extension. Local terms such as the equivalent stress or the crack tip opening angle are found to be constant during the crack extension process. This supports the view that G_fr controls the fields in the vicinity near the crack tip.     

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.     

Elastic-plastic fracture mechanics analysis of a pressure vessel with an axial outer surface flaw

The stress and strain state in pressure vessel containing an axial semi-elliptical surface flaw is analyzed by elastic-plastic finite element (FE) calculations. The variation of J along the crack front is presented. Stresses and strains in the vicinity of the surface flaw are compared with those of a compact specimen of the same material at a similar J level. The FE results are taken to examine the ductile crack growth obtained in a vessel test and to discuss the validity of J-controlled crack growth. It is shown that the local constraint of the component affects the crack resistance significantly and that, therefore, J_R-curves have to account for the varying triaxiality of the stress state. This improved two parameter approach yields a much better prediction of the stable crack growth and, especially, is able to describe the canoe shape of the surface crack.     

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.     

Results and conclusions from the first pressurized-thermal-shock experiment

The first pressurized-thermal-shock test of a 148 mm thick steel pressure vessel with a 1 m long flaw was performed to investigate fracture behavior of a vessel under conditions relevant to a flawed nuclear reactor pressure vessel during an overcooling accident. The objectives were to observe crack arrest and stability on the ductile upper shelf and the effects of warm prestressing on crack initiation. Three coordinated pressure and thermal transients were imposed on the vessel, which was preheated to 290°C. Two episodes of crack propagation and arrest occurred. The thermal transients were induced by coolant at −29 to 15°C. Pressure transients were as high as 94.4 MPa. The experimental objectives were attained. The inhibiting effects of warm prestressing were definitely demonstrated. Crack propagation was nearly pure cleavage, and arrest at 30 K above the onset of the Charpy upper-shelf was experienced in a positive K₁ gradient and with K₁ = 300 MPa√m. Fracture-mechanics analysis of brittle fracture based on small-specimen toughness measurements was reasonably accurate. Flaw evaluation by procedures of the ASME Boiler and Pressure Vessel Code conservatively predicted vessel failure, which did not occur. No ductile tearing occurred after each crack arrest, although some stable tearing had been predicted on the basis of tearing resistance data.     

Elastic-plastic fracture toughness behavior of heavy section steels for nuclear pressure vessels

Fracture toughness tests were performed in the transition region for ASTM A508 Class 3 steel using about 160 specimens. The K_J-values which are converted from J_c of the smaller specimens indicated a wide scatter ranging from below the K_Ic-value to much higher toughness. The fast brittle fracture behavior in the transition regime can be divided into two regions: (1) the region where fracture occurs on a blunting line (Region I) and (2) the region where fracture occurs on an R-curve (Region II). The scatter of the K_J-values in each region is caused by the amount of crack extension contained in the specimens. The methods to obtain the fracture toughness equivalent to the K_Ic from the K_J values were also presented.In the upper shelf region, the ductile fracture behavior of A508 Class 3 base metal and weldments was investigated. The 25% side grooved specimen was recommended for measuring the resistance against ductile crack growth. The weld heat affected zone (HAZ) has comparatively higher tearing modulus, whereas the weld metal shows the lowest one.     

Evaluation of SCC crack behavior in zirconium and zircaloy-2 using nonlinear fracture mechanics parameters

Applicability of nonlinear fracture mechanics parameters, i.e. J-integral, crack tip opening displacement (CTOD), and crack tip opening angle (CTOA), to evaluation of stress corrosion crack (SCC) propagation rate was investigated using fully annealed zirconium plates and Zircaloy-2 tubing, both of which produce SCC with comparatively large plastic strain in an iodine environment at high temperatures.Tensile SCC tests were carried out at 300°C for center-notched zirconium plates and internal gas pressurization SCC tests at 350°C, for Zircaloy-2 tubing, to measure the SCC crack propagation rate. The J-integral around semi-elliptical SCC cracks produced in Zircaloy-2 tubing was calculated by a three-dimensional finite element method (FEM) code.The test results revealed that the SCC crack propagation rate dc/dt could be expressed as a function of the J-integral, which is the most frequently used parameter in nonlinear fracture mechanics, by the equation dc/dt = C · Jⁿ, where C and n were experimental constants.Among the other parameters, CTOD and CTOA, the latter appeared to be useful for assessing the crack propagation rate, because it had a tendency to hold a constant value at various crack depths.     

A critical survey on the application of plastic fracture mechanics to nuclear pressure vessels and piping

Plastic fracture mechanics techniques have been developed to treat the regime where extensive plastic deformation and stable crack growth occur prior to fracture instability in the tough ductile materials used in nuclear systems. As described in this paper, a large number of crack tip parameters can be used in a plastic fracture resistance curve approach. However, applications using the J-integral currently predominate. This parameter has significant advantages. It offers computational ease and can provide a lower bound estimate of the fracture condition. But, J also has a disadvantage in that only a limited amount of stable crack growth can be accommodated. The crack tip opening angle parameter, in contrast, can be valid for extensive stable crack growth. But, with it and most other realistic alternatives, the computational convenience associated with the J-integral is lost and finite element or other numerical methods must be employed. Other possibilities such as the two-criterion approach and the critical net section stress are also described in the paper. In addition, current research work focussed upon improving the theoretical basis for the subject is reviewed together with related areas such as dynamic plastic analyses for unstable crack propagation/arrest and creep crack growth at high temperatures. Finally, an application of plastic fracture mechanics to stress corrosion cracking of nuclear piping is made which indicates the possible anti-conservative nature of the current linear elastic assessments.     

Fracture initiation and propagation in a cylindrical shell containing an initial surface flaw

The paper deals with the problem of fracture initiation, propagation, and arrest in a pressurized cylindrical vessel which contains an initial surface flaw. It is assumed that the flaw has the most unfavorable geometry and orientation, namely, it is a relatively long part-through axial crack.First we consider the problem of a crack which is sufficiently ‘shallow’ so that the plastic deformations are confined to the neighborhood of the crack border and part of the net section near the inner wall is still elastic. The plasticity-corrected stress intensity factor obtained from this analysis is the controlling load factor in failure considerations related to fatigue crack propagation, stress corrosion cracking, and static fracture (with the use of fracture toughness, COD, or a K_R curvetype failure criterion).The problem of relatively deep crack with fully-yielded net ligament is then considered. Plastic deformations are also assumed to spread around the crack ends through the entire wall thickness. A perfectly plastic strip model (with an eight order shell theory) is used to calculate the plastic zone size and the crack opening displacement along the crack border. Previous studies indicate that for the analysis of the type of stable and subsequent unstable crack propagation problems under consideration, the crack opening displacement δ is a more suitable load factor than the stress intensity factor K, or the crack extension force G. Thus, in this paper a ‘crack opening stretch’ type material characterization will be used.After the rupture of the net ligament under the crack, the axial crack propagation is accompanied by the depressurization of the vessel caused by leakage. From this point on the fracture problem is coupled with the related fluid mechanics or gas dynamics problem where the primary unknowns are the pressure and the crack length as functions of time. In the present study it is assumed that the volume of the vessel is finite and the crack propagation is quasi-static (this assumption, which is necessary to keep the problem within manageable proportions, is justified by the relatively low crack velocities, i.e. v_c < 0.25 c₂, c₂ being the shear wave velocity).     

Micromechanisms and toughness for cleavage fracture of steel

A complete understanding of the fracture mechanisms of steel in the ductile/brittle transition region requires analysis not only of crack initiation, but also of crack propagation. This paper reviews micrographic and fractographic experiments that give insight into both phenomena, and suggests a frame-work through which both may be related.Unstable cleavage crack initiation can occur after some blunting of the original fatigue precrack or after some stable crack growth. In either event, instability appears to be triggered by the fracture of a brittle micro-constituent ahead of the precrack. The large scatter in reported K_Ic values within the transition region reflects the size distribution and relative scarcity of these “trigger” particles.While a large number of models have attempted to correlate toughness in the ductile/brittle transition regime to events occurring ahead of the crack tip, surprisingly little attention has been paid to events occurring behind the crack front. Fractographic evidence as well as metallographic sectioning of arrested cracks show that the mechanism of rapid crack propagation by cleavage is affected strongly by partial crack-plane deflection which leaves unbroken ligaments in its wake. The tearing of these ligaments by dimple-rupture is the dominant energy-absorbing mechanism. Etch-pit experiments using an Fe-Si alloy show that the crack-tip stress intensity based on plastic zone size is extremely low. It is suggested that the mechanism of crack arrest should be modeled using a sharp crack which is restrained by a distribution of discrete pinching forces along its faces. The same model is applied to crack initiation.     

The application of elastic-plastic fracture mechanics parameters in fracture safe design

The report summarizes some of the methods which are currently used for assessing the fracture toughness of materials under elastic and elastic-plastic conditions. The main parameters which are considered are (1) plane strain fracture toughness (K_Ic), (2) equivalent energy (K_Icd), (3) contour integral (J) and (4) crack opening displacement (COD). Gross strain crack tolerance and stress concentration methods are also discussed.It is concluded that of these parameters, the contour integral and the crack opening displacement have most potential for future development. These two parameters are shown to be equivalent, however, at the present stage of development the COD concept has several advantages over the J concept. Firstly, the COD concept is able to take into account, secondary stresses, such as welding residual stresses. Because these stresses are in equilibrium, they do not appear in energy measurements to evaluate J. Secondly, the COD value is a physical measure of the crack tip conditions which includes the effect of stress state and thickness. It is, therefore, possible to measure and calculate COD levels for cracks in real structures. It is not possible to evaluate J for real structures since J methods are appropriate only to in-plane problems. This also means that partial wall (thumbnail) flaws are better characterized by the COD concept.The COD concept has been developed to a stage where it is possible to estimate the significance of flaws in welded structures provided the toughness of the material and the acting stresses or strains are known. This development is described and the method used to analyze tests on model pressure vessels with 6″ thick walls. A comparison is made with other methods, and it is concluded that although the COD analysis gives conservative estimates of the flaw size to cause failure, further work is necessary to be able to predict vessel burst conditions when failure is preceded by extensive plasticity and stable ductile tearing. A simple nomogram to determine COD levels to ensure leak before break conditions is also developed.     

Ductile fracture behavior of circumferentially cracked type 304 stainless steel piping under bending load

A ductile pipe fracture test program has been conducted in Japan Atomic Energy Research Institute (JAERI) to investigate the ductile fracture behavior of circumferentially cracked pipes and to demonstrate the validity of the leak before break concept in LWR pipings.In the paper are described the scope of the pipe test program and current test results for 6-inch diameter type 304 stainless steel pipes. Test pipes with a through-wall or a part-through crack in the circumferential direction were bent under low or high compliance condition, and stable or unstable pipe fracture behavior was investigated. J based tearing instability criterion and the net section collapse criterion are compared with the pipe test results, and the validity of these fracture criteria is discussed. Furthermore, geometries of acceptable flaws in pipes are evaluated considering the leak before break condition.     

Probabilistic elastic-plastic fracture analysis of circumferentially cracked pipes with finite-length surface flaws

A probabilistic model was developed for predicting elastic-plastic fracture response and reliability of circumferentially cracked pipes with finite-length, constant-depth, internal surface cracks subject to remote bending loads. It involves engineering estimation of energy release rate, J-tearing theory for characterizing ductile fracture, and standard methods of structural reliability theory. The underlying J-estimation model is based on the deformation theory of plasticity, a constitutive law characterized by power law model for stress-strain curve, and an equivalence criterion incorporating reduced thickness analogy for simulating system compliance due to the presence of a crack. New analytical equations were developed to predict J-integral and are shown to be fairly accurate when compared with generally more accurate elastic-plastic finite-element results. Using this J-estimation method, fast probability integrators and simulation methods were formulated to determine the probabilistic characteristics of J. The same methods were used later to predict the probability of crack initiation and net-section collapse as a function of the applied load. Numerical examples are provided to illustrate the proposed methodology. The results show that probabilistic analysis based on net-section collapse (without any margin) may significantly overpredict the reliability of surface-cracked pipes.     

Probabilistic prediction of multiple fracture under service conditions

The probability of fracture initiation, propagation, and arrest is one of the important problems facing designers, analysts, and operators of modern structures including nuclear reactors. The question of fracture requires special considerations which include the random nature of the loading and the statistical nature of the material's response to fracture under the imposed service loads. In most structural applications it is essential to know not only if and when fracture will occur under normal and off-normal operating conditions, but also to have some knowledge of the fracture propagation configurations and their resulting influence on the integrity of the structure. In this paper we address the problem of multiple fracture propagation configurations in structures under service conditions. This is accomplished by introducing a generalized energy criterion for multiple brittle fracture in nonhomogeneous and anisotropic materials.The fracture criterion is expressed in terms of the so-called Hartz function H which measures the difference between the total instantaneous strain energy release rate G_T and the rate at which energy is required for the formation of new fracture surfaces R_T. Strain energy release rates are computed for a variety of symmetric as well as asymmetric fracture propagation configurations from finite element solutions of incrementally related boundary value problems. These solutions yield a deterministic influence parameter α which is used to relate the applied loading to the probabilistic expression for the strain energy release rate. A similar treatment is given to the function R_T for which the influence parameter β must be determined experimentally. The parameters, α and β, depend upon the relative locations, sizes and orientations of the primary and secondary cracks as well as flaws, and secondary imperfections present in the material. In addition the parameter β depends upon the relative values of the specific surface energy associated with the possible primary and secondary fracture paths. This is important for anisotropic materials such as ceramics mixed-oxide fuels and concrete, and for materials experiencing stress-corrosion fracture, where the energies associated with intergranular and transgranular fracture, for example, may differ significantly.From the probabilistic expressions for G_T and R_T, the Hartz function is determined as a random variable h which describes the multiple fracture process. The formulations developed in this presentation are applied to a single edge notched panel experiencing multiple fracture under assumed random loadings. Some interesting symmetric as well as asymmetric fracture configurations are studied and the results are related to reported experimental observations. The approach presented here may have applications in the areas of fatigue crack propagation, stress corrosion cracking, and fail-safe design optimization. Current studies are aimed at simulating the influence of grain structure anisotropy, intergranular corrosive attack, and creep deformation on multiple fracture in polycrystalline materials.     

First spinning cylinder test analysis using a local approach to fracture

In recent years, several experimental programmes on large-scale specimens have been organized to evaluate the capabilities of the fracture mechanics concepts employed in structural integrity assessment of pressurized water reactor pressure vessels. During the first spinning cylinder test, a geometry effect was revealed experimentally showing the difficulties of transferring toughness data from small-scale to large-scale specimens. An original analysis of this test, by means of a local approach to fracture, is presented in this paper. Both compact tension specimen and spinning cylinder fracture behaviour were computed using a continuum damage mechanics model developed at EDF. We confirmed by numerical analysis that the cylinder's resistance to ductile tearing was considerably larger than in small-scale fracture mechanics specimen tests, about 50%. The final crack growth predicted by the model was close to the experimental value. Discrepancies in J R curves seemed to be due to an effect of stress triaxiality and plastic zone evolution. The geometry effect inducing differences in resistance to ductile tearing of the material involved in the specimens can be investigated and explained using a local approach to fracture methodology.     

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.     

Large stable crack growth in fracture mechanics specimens

According to the J concept, information is reported about the crack resistance behaviour up to 8 mm crack growth of side-grooved CT-25 as well as CCT-25 specimens made from German standard steel StE 460. Numerical simulations controlled by J_R curves make the calculation of J from the stresses and strains of specimen models during large crack growth feasible. These data allow a comparison to standards and rules describing the evaluation of J from experiments. Using stress, strain and displacement fields from a plane-strain finite-element analysis, the extended J concept is discussed concerning larger ductile crack growth. Additionally, the distribution of other fracture mechanics parameters such as the crack tip opening displacement (CTOD) and the crack tip opening angle (CTOA) are presented for larger crack growth.     

The use of acoustic emission to determine characteristic dynamic strength and toughness properties of steel

The dynamic J integral at crack initiation (J_id) and dynamic yield stress (σ_yd) are useful parameters to characterize elastic-plastic material behaviour under rapid loading rates. The critical step for evaluating J_id and σ_yd under the condition of the three point impact bending test is the detection of the crack initiation and of the yield point in the impact load–deflection curve, respectively. This paper presents an acoustic emission (AE) based method to determine the ductile crack initiation and additionally the beginning of yield. The experimental techniques used to evaluate σ_yd and J_id include both instrumented pendulum impact tests with the AE transducer within the striker (tup) and medium rate three point bend (TPB) tests with additional AE transducers on the specimen surface. Results obtained from the tests indicate that the AE method is capable of detecting general yielding and the onset of ductile crack growth (initiation). Different types of pulse shaped AE signals can be observed. They were connected with characteristic features during the loading process.     

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.     

A comparison of the use of the modified J-integral and the deformation J-integral in procedures for estimating the critical crack length for CANDU pressure tubes

The paper presents the results of a theoretical investigation whose objective has been to see whether there are advantages to be gained from using the modified J-integral in procedures for estimating the critical crack length for CANDU pressure tubes. For typical operation conditions, and with irradiated tubes having critical crack lengths over a wide range, it is shown that the slope of the modified J-integral J_M-Δa crack growth resistance curve for a pressure tube crack is only marginally greater than the slope of the corresponding deformation J-integral J_D-Δa curve; the results are expressed in terms of the parameter Z_*, which is dJ_M/da − dJ_D/da and the parameter Q, which is the fractional difference between dJ_M/da and dJ_D/da. In the light of these findings, there would appear to be little advantage to be gained in using J_M, rather than J_D, as a characterizing parameter for crack growth in a CANDU pressure tube.