Beremin Model Research Articles

Modelling the Charpy Impact Ductile-Brittle Transition of a Ship Plate Steel with CAFE Modelling

In the present work, a cellular automata finite element model (CAFE) was developed to model the ductile-brittle transition of a Grade A ship plate steel. Therefore, ductile and brittle cellular automata (CA) arrays of cells were created in the model to integrate material data at microstructural level, along with the ductile and brittle fracture processes. Microstructural data was analysed with Weibull distributions and incorporated in CAFE model using random number generators, along with ductile and brittle fracture parameters. Ductile fracture was modelled with Rousselier damage model; hence damage model parameters were calibrated with experimental data. Brittle fracture was modelled with Beremin model, and four different cleavage particles, found in a Grade A ship plate steel, were incorporated in CAFE model in order to model a competition of particles nucleating microcracks of critical size in the damage regions of Impact Charpy tests and four-point double-notch bend tests performed at low temperature. The mechanical properties the plate steel was measured in the transition region and incorporated in CAFE model, along with ductile-brittle transition rules. The present CAFE model was able to simulate distributions of microcracks in the notch region of four-point double-notch bend models (in the transition region), which correlated with experimental data. CAFE model was also able to simulate microvoids in the notch region of Charpy specimens along with the load-displacement Charpy curve for room test temperature, with very good agreement with experimental data. Once CAFE model was validated at micro and structural level, it was applied to model the typical scatter of impact Charpy energy values in the transition region of Grade A ship plate steel with good agreement with the measured ductile-brittle transition curved of the plate steel. Keywords: cellular automata, finite element modelling, ductile-brittle transition, damage modelling.

Influence of heterogeneity due to toughness variations on weakest-link modeling for brittle failure

The effect of heterogeneous microstructures on the macroscopic probability of failure is studied by use of weakest-link modeling. Heterogeneity is here associated with a local variation of toughness, where a size scale characteristic of this variation defines a length parameter. The ratio between this length parameter and the size of the active fracture process zone, defined as the heterogeneity ratio, is key to evaluating the impact of a heterogeneous microstructure. Two extremes are identified; small-scale heterogeneity (SSH) and large-scale heterogeneity (LSH). For these cases, it is possible to formulate analytical expressions based on the weakest-link concept, and references are made to existing models in the literature. Typically, heterogeneity along the crack front, where gradients of the mechanical fields are small, falls under the category of SSH. On the other hand, the effect of heterogeneity in a plane perpendicular to the crack front depends strongly on the heterogeneity ratio. Cases that can neither be identified with SSH nor LSH must be addressed with care. How this can be done is discussed, and examples are given for four different microstructure configurations of interest. The investigation is carried out by numerical analysis of a modified boundary layer model. The cumulative probability of failure by cleavage fracture is evaluated in a post-processing step, where two different statistical models are examined; the Beremin model and the Kroon–Faleskog model. Both models render the same conclusion about the alteration of the overall failure probability distributions caused by heterogeneity.

An Improved Method for Calibration of Beremin Model Parameters

An Improved Method for Calibration of Beremin Model Parameters

Prediction of Fracture Toughness Scatter Based on Weibull Stress Using Crystal Plasticity Finite Element Method

A multi-scale prediction method was proposed to investigate the scatter of fracture toughness by combining the local approach (LA) to cleavage fracture and the crystal plasticity finite element method (CPFEM). The parameters in the crystal plasticity constitutive model were firstly determined by comparing the simulated stress-strain curves with tested curves for SA508-III steel. Then CT samples were modeled using the CPFEM to calculate Weibull stress. Using the calibration process of local approach, the relevant parameters of the Beremin model were obtained with m = 30 and σu = 2590 MPa. The fracture toughness was analyzed including the scatter for a given temperature, the master curve in a temperature range. The distribution of predicted fracture toughness shows good agreement with the test results. All of the tested fracture toughness value are fall in the range of 5% to 95% that precited using the proposed combined approach.

Probabilistic analysis of cleavage fracture in commercial polycrystalline tungsten

Polycrystalline tungsten is a promising candidate as plasma-facing armour material in future fusion reactors. Below the DBTT tungsten shows brittle or semi-brittle behaviour with no or little plastic deformation before cleavage fracture. Moreover, tungsten has an inherent low fracture toughness as well as a large scatter within the strength properties. This requires a statistical treatment of the strength values. Therefore, four-point bending tests in the temperature range of room temperature (RT) to 400 °C were performed to allow the Weibull analysis in brittle failure regime and/or the Beremin model analysis in case of onset of plastic behaviour while approaching the DBTT temperature. The analyses are supported by FE simulations. Weibull and Beremin parameter values are determined from the simulation results using post-processing codes.

Study of Weibull parameters at different loading rates for 20MnMoNi55 steel using Beremin model

This work considers the influence of loading rate on the Weibull stress parameters in the Beremin model for 20MnMoNi55 RPV steel using finite element modelling of compact tension specimen. For failure analysis of cleavage fracture at component level, the Beremin model is very useful. To implement the Beremin model, the model parameters needed to be calibrated for this particular material with experimental observation. In the first portion of this study, some fracture experiments are conducted at -100°C for four different loading rates ranging from quasi-static to dynamic rates (0.625–255.8 MPa√m/s) to obtain master curves and reference temperatures. Then calibration of the Weibull stress parameters is done in a specific loading rate by linear regression analysis for different a/W ratios. For the other loading rates, Monte Carlo simulation technique along with respective master curve is used to calibrate the model parameters. Then variation of the Weibull stress parameters with loading rate is studied. This study shows that loading rate has a significant effect on the Weibull stress parameters. Reference temperatures are also determined for different a/W ratios and loading rate by calibrated Weibull stress parameters from experiments as well as Monte-Carlo simulation. Obtained reference temperatures are compared with experimental values for respective loading rates. With the increasing loading rate reference temperature, obtained from the Beremin model is found to be increased and it matches well with the experimental reference temperature values.

Size Effect on Fracture Toughness of A508-3 Steel Predicted by Using Beremin Model

Size Effect on Fracture Toughness of A508-3 Steel Predicted by Using Beremin Model

A Study on Fracture Toughness Based on a Modified Beremin Model

A Study on Fracture Toughness Based on a Modified Beremin Model

Validation of Monte Carlo Simulation Technique for Calibration of Cleavage Fracture Model Parameters, With the Calibrated Values From Experimental Results for Reactor Pressure Vessel Material 20MnMoNi55 Steel in the Lower Self of Ductile-to-Brittle Transition Region

Abstract In the earlier work done by the author for the reactor pressure vessel (RPV) material 20MnMoNi55 steel, Monte Carlo simulation technique is shown as an effective statistical technique to calibrate Weibull modulus “m” and Weibull scale parameter “σu” for temperatures −100 °C, −110 °C, −120 °C, −130 °C, and −140 °C by performing only six fracture toughness tests instead of (30 × 5 = 150) number of tests. But it lacks validation with the experimental work. So, in this paper, huge numbers of fracture toughness tests (minimum 30 for each temperature) are performed experimentally at −100 °C, −110 °C, and −130 °C in the lower self of the ductile-to-brittle transition (DBT) region for the RPV material 20MnMoNi55 steel. Then, linear regression analysis of the experimental data is performed to calibrate Weibull modulus m and Weibull scale parameter σu of Beremin (a cleavage fracture model) for the above mentioned temperatures. The calibrate values show a matching trend with that of experimental calibrated values and justify the prediction capability of Monte Carlo simulation to predict the Beremin model parameters with the help of only six fracture toughness tests. This statistical simulation procedure can be considered as an effective technique for the proper calibration of Beremin model parameters for any material.

Experimental and numerical investigations of tensile specimens containing multiple flaws in the ductile-to-brittle transition region

In thick-walled ferritic steel forgings, a large number of flaws can arise due to an unfavorable chemical composition or an erroneous heat treatment. Thus, a comprehensive understanding of the interaction of multiple flaws and their failure mechanisms is necessary.In this paper, experimental and numerical results from tensile specimens containing multiple flaws nearly parallel to the loading direction are presented. The material used for these investigations is the ferritic pressure vessel steel 22NiMoCr3-7, which is similar to an SA 508 Cl.2 steel. The experiments were conducted in the upper and in the lower shelf of the ductile-to-brittle transition region of fracture toughness. In addition, several specimens were tested to determine the material characteristics and to examine the fracture mechanisms for different levels of stress triaxiality.For numerical evaluations, damage mechanics models were used to investigate flaw interaction and the fracture mechanisms. In the upper shelf, an extended non-local Rousselier model was used. The numerical investigations in the lower ductile-to-brittle transition region were carried out with a coupled damage mechanics model. This coupled model consists of the extended non-local Rousselier model and an extended Beremin model, which was newly developed. The modifications of the models are accounting for low stress triaxiality and mixed fracture (dimple and cleavage fracture).A comparison between the numerical und experimental results shows that the defined damage mechanics models are able to describe the deformation and failure behavior of the tested specimens with a high accuracy. The investigations show that the flaw interaction behavior is strongly dependent on the temperature and the arrangement of the cracks. In the upper shelf, the crack growth is shear-stress dominated, whereas in the lower ductile-to-brittle transition region the crack growth is preferably normal to the loading direction.

Investigation on Beremin parameters based on the toughness scaling model

The toughness scaling model (TSM for short) is one of the methods to correct the constraint loss of small specimens. The principle of this method is the fracture toughness conversion at equal Weibull stress. However, related research shows that this fracture toughness scaling model can only accurately estimate the characteristic toughness J0. The scaled toughness distribution of small specimens always deviates from that of the standard specimens. In response to this widespread deviation, this paper studies the factors that significantly affect Weibull stress, such as the crack tip radius and parameter calibration methods. This research aims to determine the requirements for accurate fracture toughness conversion and minimize the deviation under the existing conditions. After studying the data of three different specimens, it is found that the TSM based on equal σw/σu and Minami's calibration method can accurately convert the toughness distribution of small specimens. However, the calibrated Weibull modulus of the two specimens must be close enough. Since it is difficult to guarantee that the calibrated parameters of the Beremin model are constant, the requirements for this accurate toughness conversion are relatively harsh. However, appropriately relaxing the Weibull modulus requirements can make the toughness scaling model more comfortable to build. Although the deviation cannot be avoided, it is still significantly reduced compared to the original toughness scaling model. On the other hand, this paper found that as long as the scale J0 of small specimens is equal to the J0 of standard specimens, an accurate reference temperature estimation can be easily obtained.

Comparison between Loading Rate and Local Stress Rate When Applying ASTM E1921 at Elevated Loading Rates

Abstract ASTM E1921-2019b, Standard Test Method for Determination of Reference Temperature, T0, for Ferritic Steels in the Transition Range, Annex 1, describes the determination of the reference temperature T0,X for a loading rate X for ferritic steels in the transition range, where X corresponds to the logarithm of the average loading rate for the tests performed, rounded to the nearest integer. Within the framework of three research projects on dynamic crack initiation and crack arrest, funded by the German government, fracture mechanics tests using specimens of 22NiMoCr3-7 steel (A 508 Grade 2 Cl.1) were performed in a range from 102 MPa√m s−1 to 106 MPa√m s−1 at different temperatures in the ductile to brittle transition region. The Weibull stress rate was calculated from the results of a numerical analysis of tests with 1-inch compact tension (1T C(T)) specimens. The modified three-parameter Beremin model of Petti and Dodds was used, since this approach is congruent with the Master Curve method. In addition to the stress parameters corresponding to Kmin and K0, the Weibull exponent can be calibrated using the experimental results. The analysis shows the effect of the Weibull exponent on the time-dependent Weibull stress rate for a dynamic 1T C(T) test. Contrary to the loading rate calculated from KJc divided by the time to cleavage according to ASTM E1921, the Weibull stress rate decreases with increasing plastic behavior. This has to be considered when the dynamic reference temperature T0,X is determined using results from tests with large plastic contribution.

Probability of brittle failure in different geometries using a simplified constraint based local criterion method

This paper introduces a novel calibration method for the original Beremin model to predict cleavage fracture in various fracture mechanics geometries with differing crack tip constraints. Calibration of Weibull parameters were conducted with respect to the goodness of fitness and the probability of the fracture toughness distributions for low and high constraint specimens. By introducing a reference diagram for a specific material based on the T-stress as a crack tip constraint parameter, the applicability of calibrated parameters was investigated for the experimental data of specimens with different geometry. The findings indicated that the ratio of Weibull stresses to associated calibrated scale parameters (here denoted as σw,R) at fracture loads for specimens with different shapes were in a same range. Therefore, by calculating T-stress as crack tip constraint for a typical geometry, a suitable prediction of probability of failure would be possible with respect to the reference diagram.

Fracture toughness evaluation for dissimilar steel joints by Charpy impact test

The influence of strength mismatch in a dissimilar steel joint on the Charpy absorbed energy was numerically analyzed on the basis of the Beremin model. The toughness correction ratio γCVN was defined as the ratio of the Charpy absorbed energy of a dissimilar steel joint specimen to that of a homogeneous specimen under the same Weibull stress. The Charpy test results for a clad joint of dissimilar steels were characterized using γCVN. A fracture toughness evaluation method was developed from the Charpy data for dissimilar steel joints considering γCVN. The fracture toughness at temperature T was predicted according to ASTM E1921 and BS 7910:2015 standards using the 27 J transition temperature T27J determined from the estimated Charpy transition curve. The predicted KJc transition curve showed reasonable agreement with the experimental results.

Variation of Beremin Model Parameters With Temperature by Monte Carlo Simulation

In this work, variation of the Beremin parameters with temperature for reactor pressure vessel material 20MnMoNi55 steel is studied. Beremin model is used, including the effect of plastic strain as originally formulated in the Beremin model. A set of six tests are performed at a temperature of −110 °C in order to determine reference temperature (T0) and master curve for the entire ductile-to-brittle transition (DBT) region as per the ASTM Standard E1921. Monte Carlo simulation is employed to produce a large number of 1 T three-point bending specimen (TPB) fracture toughness data randomly drawn from the scatter band obtained from the master curve, at different temperatures of interest in the brittle dominated portion of DBT region to determine Beremin model parameters variation with temperatures.

A simplified method for parameters calibration of the new local approach model for cleavage fracture in a ferritic steel

This work presents a simplified method for parameters calibration of the new local approach model (NLAM) for cleavage fracture in a ferritic steel. Based on the Beremin model, a NLAM was recently proposed by Lei (2016) to calculate the Weibull parameters. Although the above NLAM proves to be mathematically and physically self-consistent, yet there are still no reliable and convenient methods to calibrate the model parameters. Considering the plastic volume is determined by the threshold coefficient of fracture process zone λ, we adopt four different values of λ in this work to calculate the plastic volume so as to obtain four groups of calibrated Weibull parameters. The material used in this study is the pressure vessel steel 20MnMoNi55 with some published strength data under two subzero temperatures. And the finite element analysis FEA) software ABAQUS is employed to obtain the stress distributions, then the calibration results for four different values of λ are compared and analyzed. These efforts show that taking λ=1.5 is a more reasonable and reliable choice.

Calibration of Beremin Parameters for 20MnMoNi55 Steel and Prediction of Reference Temperature (T0) for Different Thicknesses and a/W Ratios

Master curve and reference temperature (T0) from three-point bending specimens of 20MnMoNi55 steel for different thicknesses and a/W ratios are determined using Kim Wallin’s master curve methodology (ASTM E1921-02) to study the effect of variation in thickness and a/W ratio at reference temperature (T0). Weibull stress at the crack tip is calculated from FE analysis of each fracture test using FE software ABAQUS. Calibration of Beremin parameters, like Weibull modulus (m) and scaling parameter (σu), and Cm,n is done using linear regression analysis of a large number of fracture test data at single test temperature. T0 for different thicknesses and a/W ratios are also evaluated from corresponding Weibull stress based on Beremin model using calibrated m, σu and Cm,n which are compared with experimental results showing case-specific good matching. The same calibrated values of Beremin parameters and Cm,n are also used to evaluate T0 for CT specimen of the same material using Beremin model, and an excellent matching with the experimental result is found.

Weibull triaxiality: A novel constraint assessment parameter for cleavage fracture in ductile to brittle transition region

AbstractThe cleavage fracture probability of ferritic and ferritic/martensitic steels is generally computed using Beremin's model. In this work, a novel approach is developed for Beremin's model to be applicable to crack tip conditions under nonself similar stress field. The modification of Beremin's model corrects the inaccurate integration of local cleavage fracture probabilities of reference volume elements due to the incorrect assumption of self similar stress field under large‐scale yielding. Under small‐scale yielding condition, where stresses remain self similar, integration of local cleavage failure probability is accurate. However, when the stresses differ along the thickness due to constraint loss, the integration of local cleavage fracture probability is inaccurate due to the fact that the stress levels are not same in all the reference volume elements. The modification developed in this work in the form of a parameter named Weibull Triaxiality, which measures the deviation of a crack tip volume from one under self similar stress field.

Determination of global and local cleavage fracture characteristics of high strength bolt steels

Abstract High strength bolts with large diameters are widely used in steel structures from the infrastructure and energy conversion sectors. The brittle fracture behavior of these connections has to be taken into account, especially for structures operating at cold environment, such as the offshore wind energy plant. Generally, two groups of methods are employed for cleavage fracture assessment, i. e., global approaches developed in the fracture mechanics frame and local criteria based on the damage mechanics approach proposed by Beremin. This work aims to determine characteristics of high strength bolt steels in the strength class 10.9 for cleavage fracture assessment. Fracture mechanics tests are conducted to determine the experimental master curves. In addition, numerical approaches are employed to derive the parameters of Beremin model from the master curves.

The dependence of Weibull parameters on preloads and its implication on brittle fracture probability prediction using a local criterion

A linear applicable relationship is proposed to improve the cleavage failure probability prediction of preloaded specimens using the modified Beremin model. Maximum stress triaxility factor shows fracture load independency for enough high loads and a good sensitivity to crack tip stress changes due to residual stresses and preloads. Hence, it is selected as representative of crack tip stress state. Using recalibration for six experimental datasets of CT including different levels of pretension and precompression shows that only reference stress changes with changing the preload level and type. A linear relationship is derived to modify the reference stress for the preloaded cases as a function of maximum stress triaxiality factor which can be easily computed from the FE analysis. Using the proposed relationship simply leads to a considerable improvement for failure predictions in all cases with different geometries, materials and preloading conditions.