Beremin Model Research Articles

Some Modifications to Temperature Dependence of Fracture Toughness in Beremin Model

Some Modifications to Temperature Dependence of Fracture Toughness in Beremin Model

A simple method for estimation of preloading effects on Beremin model

A simple method for estimation of preloading effects on Beremin model

An analysis of fracture under biaxial loading using the non‐singular T‐stress

Finite element analysis using a two‐dimensional modified‐boundary‐layer approach wasused to model the effects of biaxial loading on crack tip stress fields. Loadings wereapplied corresponding to an elastic KI field, non‐singular T‐stress and a biaxial stress.For through‐thickness cracks the T‐stress inherent in the specimen geometry isaugmented by the external biaxial stress. For surface‐notched specimens the biaxialstress acts out of the crack plane. This effect was modelled with generalized plane strainelements. Results were analysed using the Anderson‐Dodds approach for cleavage andthe Beremin model in the ductile regime. Biaxial loading is predicted to have a largeeffect on the toughness of a through‐thickness crack but little effect on a surface crack.Experimental results from a previous series of large‐scale biaxial fracture tests aregenerally consistent with these predictions.

Quantification of the warm prestressing effect in a shape welded 10 MnMoNi 5-5 material

This work describes investigations of the warm prestressing (WPS) effect in a 10 MnMoNi 5-5 shape welded material. Three point bend tests of precracked specimens were carried out for the following load cycles: cooling and loading to fracture without prestressing (CF), warm prestressing with loading, cooling, fracturing (LCF), and loading, unloading, cooling, fracturing (LUCF). The specimens were prestressed at room temperature (RT) and at a load level corresponding to K WPS of 70–80 MPa m 1/2, and fractured in the temperature range of −130°C to −158°C. Fracture surfaces of a number of specimens were observed in detail to measure the stretch zone width, the size and distance of cleavage origins from the blunted crack tip and to determine the character of the origins. For various loading cycles the stress distribution ahead of the crack tip and the crack tip opening displacement were numerically calculated. An increase in the apparent fracture toughness after WPS can be correlated with the geometry of the specimens, the loading cycle and, in particular, with the blunting of the original crack during prestressing. Based on the numerical results, the local fracture stress, σ f , was estimated. The local fracture stress turns out to be almost constant (2400–2500 MPa) for different specimens and load cycles. This suggests the use of σ f as a criterion to predict cleavage fracture after WPS. A statistical model is developed to predict failure loads from the calculated stress and strain fields. It is based on the Beremin concept and evaluates the incremental fracture probability, which depends not only on the maximum principal stress, but also on the equivalent plastic strain and its increment. This incremental probabilistic failure model predicts realistic fracture loads after WPS compared to the experiments, whereas the original Beremin model yields values which are too low.

Experimental and numerical investigations of the warm-prestressing (WPS) effect considering different load paths

The effect of warm prestressing has been investigated representative for the core weld metal of the RPV Stade. Model experiments on CT specimens show a significant rise of effective fracture toughness K eff after warm prestressing and the conservative WPS hypothesis, ‘no failure, if ∂ K I/∂ t≤0’, is verified. Partial unloading and reheating show no influence on the effective fracture toughness K eff. The magnitude of the WPS effect as a function of warm prestress level and temperature, path of unloading and cooling can be predicted using a modified Beremin model with temperature dependent parameters. It is shown that the Weibull stress is an appropriate crack tip loading parameter for decreasing load paths.

Three-dimensional interpretation of cleavage fracture tests of cladded specimens with local approach to cleavage fracture

Electricité de France has conducted during these last years an experimental and numerical research programme in order to evaluate fracture mechanics analyses used in nuclear reactor pressure vessels integrity assessment, regarding the risk of brittle fracture. Two cladded specimens made of ferritic steel A508 Cl3 with stainless steel cladding, and containing shallow subclad flaws, have been tested in four point bending at very low temperature to obtain cleavage failure. The crack instability was obtained in base metal by cleavage fracture, without crack arrest. The tests have been interpreted by local approach to cleavage fracture (Beremin model) using three-dimensional finite element computations. After the elastic–plastic computation of stress intensity factor K J along the crack front, the probability of cleavage failure of each specimen is evaluated using m, σ u Beremin model parameters identified on the same material. The failure of two specimens is conservatively predicted by both analyses. The elastic–plastic stress intensity factor K J in base metal is always greater than base metal fracture toughness K 1c. The calculated probabilities of cleavage failure are in agreement with experimental results. The sensitivity of Beremin model to numerical aspects is finally exposed.

Modifications of the Beremin model for cleavage fracture in the transition region of a ferritic steel

A numerical model based on the ‘local approach’ of Beremin for predicting cleavage fracture in the transition temperature region of a ferritic steel is presented. For that purpose, two modifications of the Beremin model are introduced. The first modification accounts for the decreasing number of cleavage initiation sites due to preceding ductile void formation. The second modification concerns the parameter determination. The Weibull exponent is fixed a priori to avoid numerical problems. Compared to the original Beremin model, the new model is better suited for a material characterisation which is, to a great extent, independent of geometry, temperature, and strain rate.

Application of coupled brittle–ductile model to study correlation between Charpy energy and fracture toughness values

A coupled brittle–ductile fracture local approach model has been developed to predict either Charpy energy or fracture toughness and to investigate conditions for correlations between them. A modification to the Beremin model has been introduced to assess failure in individual elements and make it compatible with the Lemaitre model for implementation as a user subroutine for the program ABAQUS. A probabilistic based fracture criterion was introduced to monitor the damage within individual elements by either cleavage or ductile tearing. The fracture toughness and Charpy energy for a nuclear industry steel BPL A508 were determined for the transition temperature range using finite element analyses with three-dimensional solid elements for both the standard 25 mm SE(B) three-point bend specimen and the Charpy 45° V-notch specimen. Prescribed failure probabilities of 0.05, 0.5 and 0.95 were imposed as critical values to predict the scatter of fracture toughness experimental data. The evaluation of constraint, equivalent plastic strain and maximum principal stress amongst other parameters gave the basis for comparison of the response of the two geometries. Critical values of fracture stress in the modified Beremin model for cleavage fracture and of the Lemaitre damage parameter for ductile fracture were calibrated from experimental fracture toughness data. These same critical values were then used in the analysis of the Charpy specimens, including allowance for strain rate effects, and good agreement found between predicted and experimental energy absorption values. The only conditions where the maximum constraint and equivalent plastic strain in the two specimen geometries were found to match were at the temperature for 28 J energy absorption in the Charpy impact energy and that for fracture toughness, KIC, of 100 MPa-m1/2 for the BPL A508 steel. This may be an important factor in the correlation between Charpy and fracture toughness tests proposed by Wallin.

Scatter of a ferritic steel in the transition region analyzed by Charpy tests and dynamic tensile tests

Several series of static and dynamic experiments with a German reactor pressure vessel steel have been performed at different temperatures. Besides results of standardized compact tension tests and static and dynamic tensile tests, results of instrumented Charpy-V tests and of dynamically loaded notched round bars are presented. In addition to the evaluation of J values these tests were analyzed numerically by means of a micromechanical damage model in order to finally determine the Weibull parameters of the Beremin model. However, not all of the results of the different series of tests are consistent, suggesting the need for modifications of the evaluation or the model to enable the application to arbitrary temperatures and loading conditions.

Global and local approaches to fracture normal to interfaces

The problem of a crack perpendicularly approaching a bimaterial interface is examined using both global and localapproaches to fracture. The global approach is based on the J-integral with a second parameter, Q, which scales the stress triaxiality ahead of the crack. The local approach is based on either brittle fracture(Beremin model ) or ductile fracture (Rice and Tracey model ). In the first case, the Weibull stress over the plasticzone is calculated. In the second case, the void growth rate is calculated at the tip of the crack over a representativevolume (generally associated with a characteristic length of the material ). After a brief summary of each approach,the results for a crack near an elastically homogeneous, plastically mismatched interface are presented. Thebehaviour of the bimaterial is expressed in relation to the behavior of the homogeneous material. It is shown thatthere is an effect on the crack behavior which depends on the direction of crack propagation, i.e. from the hardermaterial to the softer material or vice versa. This effect is examined as a function of change in yield strength ratioand hardening exponent, n. For the case of brittle fracture, the effect of changing the Weibull modulus, m, is also examined. The models based on the local approach show that both stress- and strain-controlledfracture mechanisms must be accounted for. This implies the necessity of using the two parameters J and Q in the global approach. This is due to the fact that the stress–strain fields ahead of the crack tip areaffected by the nature of the second material.

STATISTICAL MODELLING OF INTERGRANULAR BRITTLE FRACTURE IN A LOW ALLOY STEEL

The intergranular brittle fracture (IBF) behaviour of a low alloy steel 16MND5 (A508 Cl. 3) was investigated. A temper embrittlement heat treatment was applied to the material to simulate the effect of local brittle zones (ghost lines) which can be found in the as‐received material condition. An increase in the Charpy V toughness transition temperature and a significant decrease in the fracture toughness measured on CT‐type specimens were observed in the embrittled material, as compared to the reference material which was submitted to the same austenitizing and tempering heat treatment, but which was not subjected to the temper embrittlement treatment. Tensile tests on notched specimens were carried out to measure the Weibull stress and scatter in the results. A statistical model, the Beremin model, originally proposed for brittle cleavage fracture was applied to IBF. It is shown that this model is not able to fully account for the results, in particular for the existence of two slopes in a Weibull plot. Systematic fractographic observations showed that the low slope regime in this representation was associated with the existence of MnS inclusions initiating brittle fracture, while the larger slope was related to microstructural defects. Initiation of IBF from MnS inclusions can occur when the material is still elastically deformed while the second population of microstructural defects is active in the plastic regime. A modified statistical model based on the Beremin model and taking into account these specific aspects is proposed in the framework of the weakest link theory. The parameters of this model are identified from test results on notched specimens. It is shown that this model is able to predict the temperature dependence of fracture toughness and the scatter in the experimental results.

New approaches to improve the RPV materials data base

The models for the local approach to cleavage fracture and to ductile tearing include the effects of specimen geometry and loading situation on the damage development. Thus, these concepts can be used to predict fracture toughness and tearing resistance curves for different sizes of specimens or structures, based on an analysis of tests with small, simple specimens. This advantage is especially valuable if only small pieces of material are available for testing which do not allow for standard fracture specimens. The paper outlines the basic ideas of the Beremin concept for the fracture process in brittle-to-ductile transition regime and of the modified Gurson model for ductile fracture in the upper shelf. A systematic study for the brittle-to-ductile transition regime showed that the Beremin model has to be modified to take into account the effect plastic strain has on the fracture process. For ductile fracture, it was demonstrated that by testing and modeling a smooth bar and a fracture mechanics specimen, one set of material parameters may be found which can be subsequently applied to other structures and, hence, used to extend the material data base.

Micromechanical Modeling of Ductile Fracture Initiation to Predict Fracture Toughness of Reactor Pressure Vessel Steels

Two micromechanical models of ductile fracture are investigated and have been applied to two reactor pressure vessel steels, 18MND5 and A508 CI.3: the Beremin model, based on the Rice et Tracey void growth model, and the damage work model that combines the plastic strain work to the work spent in void growth. Due to the local nature of these models, finite element analysis needs to be performed to derive stress and strain history in order to obtain the damage kinetics of the material and geometry under consideration. Tensile tests were performed on various geometries (notched and precracked tensile) and sizes. All specimens are in large scale yielding condition. It is found that while the critical void growth ratio decreases with the triaxiality ratio, the critical damage work is not affected. Geometry, size and orientation effects are also investigated. These effects are well described by these micromechanical models. Similar concepts are applied to sharp notch and crack tip situations. An additional parameter, the so-called characteristic distance, characterizing the process zone size, is introduced. The fracture toughness, derived from the notched bars, is within experimental uncertainties in reasonable agreement with the results obtained for cracked geometries.

Local Approach to Fracture Applied to Reactor Pressure Vessel : Synthesis of a Cooperative Programme Between EDF, CEA, Framatome and AEA

A cooperative research programme in the field of the local approach to cleavage fracture applied to reactor pressure vessels is conducted between EDF, CEA, Framatome and AEA Technology. The purpose of such a programme is to have a better knowledge of this approach for RPV fracture mechanics integrity assessments regarding the risk of brittle fracture. This programme includes both an experimental part conducted by several laboratories in order to identify m and σ u parameters of Beremin model on a A508 C13 steel, and numerous numerical computations of specimens and structures. Finite element programs are compared on some reference calculations in order to validate local approach to fracture postprocessors. The capability of Beremin model to explain the shallow flaw effect in cleavage fracture is finally discussed.

Ductile - Brittle Transition of Ferritic Steels Modelled by the Local Approach to Fracture

Tensile tests on axisymmetrically notched specimens were performed at various temperatures in the ductile-brittle transition region to study the initiation of fracture. Fracture toughness tests on CT specimens were also carried out in the same region to investigate the critical values of stable crack growth occurring before unstable cleavage fracture. Probabilistic modelling is made by coupling the Beremin model for cleavage crack initiation and a model derived from the Gurson potential. In an elementary volume the probability for fracture depends on the probability of both modes of failure. In a larger volume the probability to is expressed in terms of elementary probabilities of fracture through the weakest link theory. Modelling is applied to axisymmetrically notched bars and to CT specimens. The analysis of results shows that the ductile-brittle transition of ferritic steels can be interpreted quantitatively as a competition between both types of damage.

Prediction of Fracture in the Transition Regime : Application to an A533B Pressure Vessel Steel

In order to model the fracture behaviour of pressure vessel steels in the transition regime, a new model has been developped in 1991 at EDF R&D Division in the framework of local approach to fracture This approach couples the damage mechanics model developped by Rousselier which is accounting for ductile propagation of a crack, and the Beremin's model based on Weibull's statistics which stands for cleavage. It is possible to predict, by this coupled approach, safe lower bound transition curves for a temperature range up to RTNDT+50°C. It has been shown that the predictions of the model agree well with the experimental data, both in terms of fracture toughness at cleavage instability and the amount of pre-cleavage tearing. Those predicted curves have also been compared to the ASME design curve, and substantial safety margins have been exhibited.

A CRITERION FOR INTERGRANULAR BRITTLE FRACTURE OF A LOW ALLOY STEEL

Abstract— The intergranular brittle fracture behaviour of a A533 B Cl 1 steel was investigated by applying a temper embrittlement heat treatment to simulate the presence of local brittle zones (ghost lines) in the as‐received material. This heat treatment produced an embrittled material that failed by an intergranular mode at low temperature. An increase in the Charpy transition temperature and a significant decrease in fracture toughness were observed in the embrittled steel as compared to the as‐received material. Tensile tests on smooth specimens and on notched bars were carried out to determine the critical stress corresponding to brittle intergranular fracture and to investigate the scatter in the results. It is shown that the mean value of the critical stress seems to be an increasing function of temperature. A statistical model, the Beremin model, initially proposed to describe brittle cleavage fracture was applied to intergranular fracture. Modifications are introduced in this model to take into account the temperature dependence of the critical stress. It is shown that this modified model is able to predict the temperature dependence of fracture toughness and the scatter in the experimental results.

IRRADIATION EMBRITTLEMENT OF A LOW ALLOY STEEL INTERPRETED IN TERMS OF A LOCAL APPROACH OF CLEAVAGE FRACTURE

Abstract— One heat of A 508 steel is investigated in both the unirradiated and the irradiated condition to determine the variation of the fracture toughness with temperature and specimen thickness. CT type specimens with three thicknesses B (12.5, 25 and 50 mm) are used. Two fluence levels (3·1019 and 8·19 n/cm2)(E < 1 MeV) are investigated. It is shown that the fracture toughness is a decreasing function of both specimen thickness and temperature. A model developed previously by Beremin is used to interpret the results. Axisymmetric notched specimens are tested to determine the factors used in the statistical approach of cleavage fracture. It is confirmed that the Beremin model is able to account for the large scatter in fracture toughness observed at a given temperature on the unirradiated material. The specimen thickness effect is also reasonably well interpreted by the model. The irradiation embrittlement can be explained by assuming that the cleavage fracture resistance is not modified by irradiation and by taking into account only the variations of yield strength with irradiation and test temperature.