Modelling the fissuring flat-fracture mode of crack growth in Zr-2.5Nb Candu pressure-tube material

Modelling of the fissuring mode of fracture in Candu pressure-tube material and, in particular, Stage 1 crack growth (essentially flat J _r curve) as observed in some irradiated compact toughness specimens, is reported. A preliminary attempt has been made at modelling the characteristics of the fracture process zone associated with a crack that tunnels at the specimen mid-section. Based on the experimentally determined J _r curve, both the fracture-process zone size and the crack opening displacement at the trailing edge of the zone have been predicted, and their values are seemingly not unreasonable. The initial considerations enable specific issues to be highlighted which need to be addressed before a complete picture is obtained of the fissuring mode of crack growth. A theoretical analysis has also shown that the non-tunnelling material at the specimen side surfaces exerts very little restraint on the cumulative (tunnelling) mode of crack propagation, a prediction that is consistent with the experimental finding that Stage 1 crack growth in irradiated material is associated with an essentially flat J _r curve.     

A review of the J and I integrals and their implications for crack growth resistance and toughness in ductile fracture

The application of the J and the I-integrals to ductile fracture are discussed. It is shown that, because of the finite size of the fracture process zone (FPZ), the initiation value of the J-integral is specimen dependent even if the plastic constraint conditions are constant. The paradox that the I-integral for steady state elasto-plastic crack growth is apparently zero is examined. It is shown that, if the FPZ at the crack tip is modelled, the I-integral is equal to the work performed in its fracture. Thus it is essential to model the fracture process zone in ductile fracture. The I-integral is then used to demonstrate that the breakdown in applicability of the J-integral to crack growth in ductile fracture is as much due to the inclusion in the J-integral of progressively more work performed in the plastic zone as it is to non-proportional deformation during unloading behind the crack tip. Thus J _R -curves combine the essential work of fracture performed in the FPZ with the plastic work performed outside of the FPZ. These two work terms scale differently and produce size and geometry dependence. It is suggested that the future direction of modelling in ductile fracture should be to include the FPZ. Strides have already been made in this direction.     

Constraint effects during stable transient crack growth

The effect of plastic-flow constraint on the field of a crack, during stable propagation within the area, covered by the plastic zone at initiation of growth, has been examined. Plane strain, mode I and contained yielding have been considered. The material is homogeneous with elastic-plastic behavior, described by Hook's law, J ₂-flow theory and isotropic hardening. The numerical investigation has been performed within the framework of a boundary layer formulation, whereby the remote loading is fully specified by the first two terms in Williams' expansion, characterized by K and T. It is shown, that a self-similar state is reached, after growth of the order of the fracture process zone size at initiation. The characteristic length of the self-similar field is the fracture process zone size. Under contained yielding, the self-similar field depends on an appropriately normalized T/K ratio, which is a measure of the deviation of the stress field from small scale yielding distribution at distances of the order of the damage process zone size. According to the analysis, the effect of constraint on near-tip triaxiality, during transient growth is moderate. Also the effect of constraint on crack growth resistance is weak at initiation and increases with crack propagation.     

J resistance behavior in functionally graded materials using cohesive zone and modified boundary layer models

This paper describes elastic–plastic crack growth resistance simulation in a ceramic/metal functionally graded material (FGM) under mode I loading conditions using cohesive zone and modified boundary layer (MBL) models. For this purpose, we first explore the applicability of two existing, phenomenological cohesive zone models for FGMs. Based on these investigations, we propose a new cohesive zone model. Then, we perform crack growth simulations for TiB/Ti FGM SE(B) and SE(T) specimens using the three cohesive zone models mentioned above. The crack growth resistance of the FGM is characterized by the J-integral. These results show that the two existing cohesive zone models overestimate the actual J value, whereas the model proposed in the present study closely captures the actual fracture and crack growth behaviors of the FGM. Finally, the cohesive zone models are employed in conjunction with the MBL model. The two existing cohesive zone models fail to produce the desired K–T stress field for the MBL model. On the other hand, the proposed cohesive zone model yields the desired K–T stress field for the MBL model, and thus yields J _R curves that match the ones obtained from the SE(B) and SE(T) specimens. These results verify the application of the MBL model to simulate crack growth resistance in FGMs.     

Prediction of mode I crack growth resistance based on a comparative investigation of J-integral and energy dissipation rate concept

An energy dissipation rate concept is employed in conjunction with the J-integral to calculate crack growth resistance of elastic-plastic fracture. Different from Rice’s J-integral, the free energy density is employed in place of the stress working density to define an energy-momentum tensor, which yields that the slightly changed J-integral is path dependent regardless of incremental plasticity and deformational plasticity. The J-integral over the remote contour is split into the plastic influence term and the J _FPZ-integral over the fracture process zone which is an appropriate estimate of the separation work of fracture. Finite element simulations are carried out to predict the plane strain mode I crack growth behavior by an embedded fracture process zone. It can be concluded that J-integral characterization is in essence a stress intensity-based fracture resistance similar to the K criterion of linear elastic fracture, and energy dissipation rate fracture resistance can be taken as an extension of the Griffith criterion to the elastic-plastic fracture.     

Experimental and numerical investigation of cracked chevron notched Brazilian disc specimen for fracture toughness testing of rock

The cracked chevron notched Brazilian disc (CCNBD) specimen has been suggested by the International Society for Rock Mechanics to quantify mode I fracture toughness (K_Ic) of rock, and it has also been applied to mode II fracture toughness (K_IIc) testing in some research on the basis of some assumptions about the crack growth process in the specimen. However, the K_Ic value measured using the CCNBD specimen is usually conservative, and the assumptions made in the mode II test are rarely assessed. In this study, both laboratory experiments and numerical modeling are performed to study the modes I and II CCNBD tests, and an acoustic emission technique is used to monitor the fracture processes of the specimens. A large fracture process zone and a length of subcritical crack growth are found to be key factors affecting the K_Ic measurement using the CCNBD specimen. For the mode II CCNBD test, the crack growth process is actually quite different from the assumptions often made for determining the fracture toughness. The experimental and numerical results call for more attention on the realistic crack growth processes in rock fracture toughness specimens.     

The effect of T-stress on plane strain dynamic crack growth in elastic–plastic materials

In this paper, the effects of T‐stress on steady, dynamic crack growth in an elastic–plastic material are examined using a modified boundary layer formulation. The analyses are carried out under mode I, plane strain conditions by employing a special finite element procedure based on moving crack tip coordinates. The material is assumed to obey the J₂ flow theory of plasticity with isotropic power law hardening. The results show that the crack opening profile as well as the opening stress at a finite distance from the tip are strongly affected by the magnitude and sign of the T‐stress at any given crack speed. Further, it is found that the fracture toughness predicted by the analyses enhances significantly with negative T‐stress for both ductile and cleavage mode of crack growth.     

Numerical study of geometric constraint and cohesive parameters in steady-state viscoelastic crack growth

This paper presents a finite element study of cohesive crack growth in a thin infinite viscoelastic strip to investigate the effects of viscoelastic properties, strip height, and cohesive model parameters on the crack growth resistance. The results of the study show that the dependence of the fracture energy on the viscoelastic properties for the strip problem is similar to that obtained for the infinite body problem even when the cohesive zone length is large compared to the height of the strip. The fracture energy also depends on the crack speed v through the dimensionless parameter v τ/L _∞ where L _∞ is the characteristic length of the cohesive zone and τ is the characteristic relaxation time of the bulk material. This relationship confirms that at least two properties of the fracture process must be prescribed accurately to model viscoelastic crack growth. In contrast, the fracture energy and crack speed are insensitive to the strip height even in situations where the growth of the dissipation zone is severely constrained by the strip boundaries. We observe that at high speeds, where the fracture energy asymptotically approaches the maximum value, the material surrounding the cohesive zone is in the rubbery (equilibrium) state and not the glassy state.     

Characterization of fracture toughness of high strength low alloy steel weldments using stretch zone width measurements

Most of the industrial applications involving use of high-strength low-alloy steels require good weldability. Thus it is important to characterize the properties of the welded steels, especially the heat affected zone. Attempts have been made to characterize the fracture toughness of the HAZ by the use of theJ-integral and the crack opening displacement. In the present study, the effect of addition of titanium, vanadium and niobium, as well as combinations of these, on the fracture toughness of the heat affected zone of welded steel plates is examined. Six compositions were used in this study. Three-point bending specimens as well as tensile specimens were prepared. The fracture surfaces were examined in a scanning electron microscope to determine the fracture mode as well as the extent of the stretch zone as the crack blunts. Calculation of the fracture toughness parameter,J _lc, is carried out through a quantitative stereofractographic analysis of the stretch zone at the crack tip. The results show that there is a marked increase inJ _lc due to the addition of the various alloying elements. Generally, the addition of niobium and titanium alone produce the highestJ _lc due to the extent of grain refinement that these elements produce.     

A theory of the initiation of creep crack growth

A computer simulation of the time dependent development of the plastic zone ahead of a crack loaded in uniform tension was performed. The material was assumed to deform according to a creep law relating the local strain rate to the local stress. The plastic zone was modelled by an array of edge dislocations coplanar with the crack. For a given time the stress was found to be uniform in a region ahead of the crack. This region increased and the local stress decreased with increasing time. The distribution of dislocations in the zone at a given time was found to be almost the same as that given by the Bilby, Cottrell and Swinden model (1963) if the friction stress in that model was replaced by an apparent friction stress equal to the uniform stress ahead of the crack. This apparent friction stress is dependent on both the applied stress and time. Assuming a critical crack opening displacement (COD) or a critical value of theJ integral,J _c, to be the criteria for the onset of the creep crack growth the initiation time can be calculated using the results of this study. A good agreement between the theory and experiment is obtained for two different CrMoV steels. This comparison with experiments suggests that the COD is an appropriate crack growth initiation parameter for both ductile and brittle materials whilstJ _cdoes not seem to be applicable in creep fracture.

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Résumé Une simulation par calculateur du développement fonction du temps de la zône plastique située devant une fissure soumise à tension uniforme a été effectuée. Le matériau est supposé pouvoir se déformer suivant une loi de fluage mettant en relation la vitesse de déformation locale et la tension locale. La zône plastique est représentée par une série de dislocations-coin coplanaires à la fissure. Pour un temps déterminé, la tension a été trouvée uniforme dans la région située de vant la fissure. Cette région s'étend, et la tension locale décroit, lorsque le temps s'accroit. La distribution des dislocations dans la zône à un moment déterminé est trouvée être sensiblement la même que celle donnée par le modèle de Bilby, Cottrell et Swinden (1963), pour autant que la tension de friction dans ce modèle soit remplacée par une tension de friction apparente égale à la tension uniforme située en avant de la fissure. Cette tension apparente de friction dépend à la fois de la contrainte appliquée et du temps. En supposant qu'une valeur critique du COD ou de l'intégraleJ,J _cconstitue le critère pour le démarrage d'une fissure de fluage, la durée de l'amorçage peut être calculée en utilisant les résultats de cette étude. Un bon accord entre la théorie et l'expérience a été obtenu dans le cas de deux aciers au CrMoV différents. Cette comparaison avec l'expérience suggère que le COD est un paramètre valable pour l'accroissement d'une fissure dans le cas de matériaux ductiles et de matériaux fragiles, tandis queJ _cne paraît pas applicable dans le cas des ruptures dues au fluage.

     

Study of the relationship between fracture toughness (J IC) and bulge ductility

A theoretical model relating fracture toughness expressed as J _IC and bulge ductility {ie71-1} for a material exhibiting linear elastic behavior at low temperature and elastic-plastic behavior at higher temperatures is proposed. This model shows a variation of J _IC with {ie71-2} for linear elastic behavior and J _IC with {ie71-3} for elastic-plastic behavior. The model contains three constants to be determined experimentally for a given material, specimen geometry and testing conditions. A case study on 1045 steel in the temperature range ?60 to 25°C confirms the validity of the model. The experimental results help in determining the size of the fracture zone ahead of the crack as well as the mechanisms for crack blunting and crack growth.     

Quasi-static extension of cohesive crack described by the energy partition technique

Nonlinear differential equation governing mode I fracture under small scale yielding condition has been derived on the basis of the energy partition concept. This technique is associated with a cohesive crack model. The nonlinear zone which precedes a propagating crack has been assumed to have a structured nature. In addition to this microstructural assumption, it has been postulated that the energy dissipated within the process zone (Δ), embedded in a larger nonlinear zone (R), remains invariant to the extent of crack growth. Upper and lower bounds of the tearing modulus have been related to the material ductility via closed form expressions. It has been demonstrated that the energy screening, measured by the ratio of the true fracture energy (Ŵ) to the total work expended in the cohesive zone during the process of irreversible deformation, is a monotonic function of the crack growth increment, resembling a reciprocal of the apparent material resistance to cracking described by anR-curve.     

Mixed mode plane stress ductile fracture

Mixed mode ductile fractures in thin sheets are shown to be possible. The staggered deep edge notch tension specimen enablesJ _p , the plane stress propagation value of theJ integral, and dJ/d, the rate of increase inJ with crack growth to be measured from the specific work of fracture. TheJ integral can also be separated into its two component modesJ ₁ andJ _n.For the particular low alloy steel testedJ _p is virtually independent of the mode of fracture, but for other materialsJ _p may be dependent on the fracture mode.

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Résumé On montre que des ruptures mixtes de mode ductile dans des feuillards minces sont possibles. En utilisant une éprouvette de traction comportant deux entailles latérales, profondes et décalées, on peut mesurerJ _p composante de propagation de l'intégraleJ en état plan de tension et dJ/d _a , le taux d'accroissement deJ en fonction de la croissance de la fissure à partir du travail spécifique de rupture. L'intégraleJ peut également être séparée en ses deux composantes des modesJ ₁ etJ _n.Dans le cas de l'acier faiblement allié particulier qui a été traité,J _p est virtuellement indépendant du mode de rupture; cependant, pour d'autres matériaux,J _p peut dépendre du mode de rupture.

     

Experimental and finite element study on stable crack growth in three point bending

Experimental and finite element results are presented on mode I and mixed mode (involving I and II only) stable crack growth under static loading through an aircraft grade aluminium alloy (D16AT) in three point bending. The results include load-displacement diagrams, J-integrals, plastic zones, tunneling (or crack front curving), etc. During experiment a substantial amount of tunneling is observed, the extent of which increases as the extension progresses in both mode I and mixed mode. The tunneling reduces as a_o/w increases. The crack extends initially almost along a straight line at an angle with the initial crack in a mixed mode. The maximum load is observed to be as high as 1.6 times the initiation load in the whole range examined. From the finite element study it is seen that, in a mixed mode, the J-integral at the onset of extension is the lowest compared with the values at the later stages. The plastic zone size grows as the stable extension progresses; the growth is approximately the maximum along the crack extension line. The direction of initial crack extension in a mixed mode can be predicted through an elastic finite element analysis and using the criterion of maximum tangential principal stress. The study also indicates that the load-displacement diagram associated with a mixed mode stable crack growth can be predicted reasonably accurately using the criterion of crack opening angle.     

Analysis of steady-state ductile crack growth along a laser weld

The fracture toughness in an elastic-plastic material joined by a laser weld is analyzed for steady-state crack growth along the weld. The analysis is performed for laser welds in steel. Laser welding gives high mismatch between the yield stress within the weld and that in the base material, due to the fast thermic cycle that the material undergoes in welding. The material is described by J ₂-flow theory, and the analysis is performed using a special numerical algorithm, in which the finite element mesh remains fixed relative to the tip of the growing crack, so that the material moves through the mesh. Fracture is modelled by using a cohesive zone criterion in front of the crack tip along the fracture zone. It is found that in general a thinner laser weld gives a higher interface toughness. Furthermore, it is shown that the preferred path of the crack is in the base material slightly outside the weld; a phenomenon also observed in experiments.     

A numerical investigation on the geometry dependence of the crack growth resistance in CT specimens

The influence of the specimen thickness B and the ligament length b on the J _R -curves is numerically investigated for CT specimens. The thickness effect is taken into account with 2-D analyses by dividing a plain sided specimen into a plane stress part and a plane strain part. The fracture process is controlled by experimentally determined critical values of the crack tip opening displacement for crack growth initiation (CTOD_i) and the crack tip opening angle for stable crack growth (CTOA_C). It is shown that for the global behaviour of a plain sided specimen, the B/b ratio is essential. The difference between the geometry dependence of the initiation value of the J-integral and the geometry dependence of the slope of the J _R -curves is also shown.     

Evaluation of the ISO <Emphasis Type="Italic">J</Emphasis> initiation procedure using the EURO fracture toughness data set

The multiple specimen J _0.2/BL initiation fracture toughness test procedure from the ISO standard, ISO 12135:2002, is evaluated using the EURO fracture toughness data set. This standard is also compared with the ASTM standard, ASTM E 1820, multiple specimen J _Ic procedure. The EURO round robin data set was generated to evaluate the transition fracture toughness methods for steels. However, many of the tests resulted in ductile fracture behavior giving final J versus ductile crack extension points. This is the information that is measured in a multiple specimen J initiation fracture toughness test. The data set has more than 300 individual points of J versus crack extension with four different specimen sizes. It may be the largest data set of that type produced for one material. Therefore, its use to determine J initiation values can provide an important evaluation of the standard procedures. The results showed that a J _0.2/BL value could be determined from the ISO standard for three of the four specimen sizes, the smallest size did not meet the specimen size requirement on J. The construction line slopes in this method are very steep compared with the ASTM construction line slopes. This resulted in low J initiation values, about a factor of two lower than the one from the ASTM method. Of the various criteria imposed to determine a valid J _0.2/BL value, the one limiting the maximum J value was the most questionable. It had an effect of eliminating small specimen data that was identical to acceptable large specimen data.     

Crack growth on elastic-plastic bimaterial interfaces

Finite element calculation based on finite strain theory is carried out to simulate the crack growth on bimaterial interfaces under the assumption of small scale yielding and plane strain condition. The modified Gurson's constitutive equation and the element vanish technique introduced by Tvergaard et al. are used to model the final formation of an open crack. The crack growths in homogeneous material and in bimaterials are compared. It is found from the calculation that the critical macroscopic fracture toughness for crack growth J _IC is much lower in bimaterials than in homogeneous material. For bimaterial cases, the J _IC of a crack between two elastic-plastic materials which have identical elastic properties with different yield strength is lower than that of a crack between an elastic-plastic material and a rigid substrate. It seems that the difference in yield strength between the dissimilar materials has more significant influence on the void nucleation and crack growth than the difference in hardening exponent.     

Quasi-static extension of cohesive crack described by the energy partition technique

Nonlinear differential equation governing mode I fracture under small scale yielding condition has been derived on the basis of the energy partition concept. This technique is associated with a cohesive crack model. The nonlinear zone which precedes a propagating crack has been assumed to have a structured nature. In addition to this microstructural assumption, it has been postulated that the energy dissipated within the process zone (), embedded in a larger nonlinear zone (R), remains invariant to the extent of crack growth.Upper and lower bounds of the tearing modulus have been related to the material ductility via closed form expressions. It has been demonstrated that the energy screening, measured by the ratio of the true fracture energy () to the total work expended in the cohesive zone during the process of irreversible deformation, is a monotonic function of the crack growth increment, resembling a reciprocal of the apparent material resistance to cracking described by an R-curve.     

The jump-like crack growth model, the estimation of fracture energy and JR curve

In this paper the jump-like crack growth model for monotonic loading is applied to re-examine both the onset of crack growth and process of stable crack growth. In the former case the fracture energy associated with a new surface creation is estimated and the in-plane constraint influence on this quantity is examined using the J-A₂ approach. In the later case the formula to compute the J-resistance curve is re-examined and compared with the one known from the standards. In the analysis the plane strain model of a structural element made of elastic-plastic material is assumed.