A probabilistic model for cleavage fracture with a length scale-influence of material parameters and constraint

A probabilistic model for the cumulative probability of failure by cleavage fracture with a material related length scale is developed in this study. The model aims at describing the random nature of fracture in ferritic steels in the brittle-to-ductile transition temperature region. The model derives from use of an exponential function to describe the distribution of microstructural entities eligible to take part in the fracture initiation process, where also a dependence on effective plastic strain is incorporated. A nonlocal stress measure, calculated as the average stress in a spherical volume, drives the contribution to failure probability of an infinitesimal material volume. The radius of the spherical volume enters as the material length in this model. This length has a significant influence on failure probability predictions in geometries exposed to strong stress gradients as found ahead of cracks. The material length is associated with a fracture toughness threshold value. In a fracture application three model parameters need to be estimated based on testing; a parameter directly related to the mean fracture toughness, a parameter that primarily is related to crack-tip constraint effects and the material length parameter. The model is explored in a parametric study showing model features in concord with typical features found in toughness distributions from fracture mechanics testing in the transition region.     

Residual stress and constraint effects on cleavage fracture in the transition temperature regime

Defects in structural components are often associated with welds that may contain significant levels of residual stress. Whilst the primary load acting on the component may induce low constraint conditions at the crack tip, the presence of residual stresses, e.g. due to welding, can modify this constraint level and consequently influence the observed fracture toughness behaviour. This paper presents the results of a combined experimental and numerical programme aimed at exploring this issue. Cleavage fracture toughness data for high and low constraint specimens are presented with and without residual stresses. The results indicate that under certain conditions, the constraint-induced increase in cleavage fracture toughness may be eroded by the presence of a residual stress in the vicinity of the crack. The results are quantified with respect to two-parameter fracture mechanics in which the  T  and  Q  parameters are appropriately defined. Preliminary guidance is provided for the assessment of defects when residual stresses may influence crack-tip constraint.     

Identification of probabilistic distribution parameters for the mesoscopic stochastic fracture model

As a kind of multiphase composite material, the basic mechanical behaviors of concrete are randomness and nonlinearity. The mesoscopic stochastic fracture model (MSFM) which can reflect the coupling effect of randomness and nonlinearity, has been widely used for the nonlinear analysis of concrete structures. In this paper, we proposed a new stochastic modeling principle to identify the probabilistic distribution parameters of MSFM. In order to reduce the modeling works, a dimension-reduced algorithm is proposed as well. In this paper, an overview of MSFM is firstly presented to introduce the background of the research. Then the stochastic harmonic function (SHF) representation is introduced to express the random field mentioned in the MSFM, and the probability density evolution method (PDEM) is applied to obtain the theoretical probability density function (PDF) of the stress–strain relationships. Furthermore, a stochastic modeling principle is proposed, in which minimizing the Kullback–Leibler divergence (KLD) is taken as the optimization object. Based on the framework of genetic algorithm, a dimension-reduced algorithm is proposed to identify the parameters with reference to the data from tested complete curves of uniaxial compressive and uniaxial tensile stress–strain relationship of concrete. The results indicate that the proposed principle and algorithm can be used to identify the parameters of MSFM accurately and efficiently.     

Constraint effects on the ductile-to-brittle transition temperature of ferritic steels: a Weibull stress model

This study examines crack front length and constraint loss effects on cleavage fracture toughness in ferritic steels at temperatures in the ductile-to-brittle transition region. A local approach for fracture at the micro-scale of the material based on the Weibull stress is coupled with very detailed three-dimensional models of deep-notch bend specimens. A new non-dimensional function g(M) derived from the Weibull stress density describes the overall constraint level in a specimen. This function remains identical for all geometrically similar specimens regardless of their absolute sizes, and thus provides a computationally simple approach to construct (three-dimensional) fracture driving force curves _w vs. J, for each absolute size of interest. Proposed modifications of the conventional, two-parameter Weibull stress expression for cumulative failure probability introduce a new threshold parameter _w–min. This parameter has a simple calibration procedure requiring no additional experimental data. The use of a toughness scaling model including _w–min>0 increases the deformation level at which the CVN size specimen loses constraint compared to a 1TSE(B) specimen, which improves the agreement of computational predictions and experimental estimations. Finally the effects of specimen size and constraint loss on the cleavage fracture reference temperature T ₀ as determined using the new standard ASTM E1921 are investigated using Monte Carlo simulation together with the new toughness scaling model.     

Effect of specimen geometry on impact strength and ductile-to-brittle transition temperature

Data on the effect of specimen geometry on the ductile-to-brittle transition temperature of low-alloy steel are reported. The influence of the cross-sectional area, specimen width and thickness, notch angle, and root radius were studied.     

An enhanced probabilistic model for cleavage fracture assessment accounting for local constraint effects

Objective of the present study is the development of an enhanced model for the probabilistic cleavage fracture assessment of ferritic materials considering the conditions for both, nucleation and propagation of micro defects. In a first step, the local load and deformation history at the cleavage initiation spot is analysed numerically for a variety of fracture mechanics specimens. The experimental data base includes experiments on standard deep and shallow crack specimens with different geometries as well as novel small scale cruciform bending specimens. These specimens enable the application of an additional stress component along the crack front. Based on the results, a two-criteria concept for cleavage initiation is proposed, assuming that the propagation of existing micro defects is controlled by the maximum principal stress whereas the nucleation of potentially critical micro defects is governed by a combination of the local plastic strain and the local stress triaxiality at the respective material point. Based on these assumption, a probabilistic cleavage fracture model is formulated and validated against the experimental data base.     

A probabilistic model for cleavage fracture with a length scale--parameter estimation and predictions of stationary crack experiments

This study presents a large experimental investigation in the transition temperature region on a modified A508 steel. Tests were carried out on single-edge-notch-bend specimens with three different crack depth over specimen width ratios to capture the strong constraint effect on fracture toughness. Three test temperatures were considered, covering a range of 85 °C. All specimens failed by cleavage fracture prior to ductile tearing. A recently proposed probabilistic model for the cumulative failure by cleavage was applied to the comprehensive sets of experimental data. This modified weakest link model incorporates a length scale, which together with a threshold stress reduce the scatter in predicted toughness distributions as well as introduces a fracture toughness threshold value. Model parameters were estimated by a robust procedure, which is crucial in applications of probabilistic models to real structures. The conformity between predicted and experimental toughness distributions, respectively, were notable at all the test temperatures.     

A model to predict the fracture distance of cleavage fracture as a function of temperature

We propose a model which computes the fracture distance of materials (reactor pressure vessel steel, C–Mn steel, and four heat-treated HSLA steels) as a function of temperature. The fracture distance, an important length scale for predicting cleavage fracture, is determined using the parameters such as mean grain size, fracture toughness and yield stress of the material in the range of −250 to 0 °C. The fracture distances computed from the proposed model agreed with measurements for the materials. Some differences between the predictions and the measurements were observed for HSLA steel with fine grain size (30 μm) and coarse grain size (55 μm).     

A micromechanical model for the prediction of the temperature fracture behaviour dependence in metallic alloys

In the present paper, a micro-mechanical model based on energetic considerations is developed to simulate the effect of environmental temperature on the fracture toughness of metallic alloys. By considering a reference elementary volume (REV) with the same composition of the real material, the stress-strain field inside such a volume and the corresponding strain energy due to a temperature variation is determined. The energy balance to determine the material fracture toughness is generalised in order to take into account the temperature effects. The proposed micro-mechanical model is governed by few parameters which can be simply estimated, and allows us to determine the fracture toughness for any temperature below the room temperature. Such a model is applied to three metallic alloys which show a ductile-brittle transition temperature: ASTM A471, Carbon Steel D6ac, Steel S275 J2. From the comparison of theoretical results with experimental data, it can be concluded that the model seems to be able to correctly predict the fracture toughness at low temperatures.     

Competing risks models for fracture in the ductile to brittle transition temperature region

When fracture toughness testing is carried out over the ductile to brittle transition temperature region cleavage instability may be observed at the initiation of cracking or after some prior ductile crack growth. The amount of precleavage ductile crack growth increases with increasing temperature. At the lower test temperatures, it is possible to assume that all tests will result in cleavage instability. However, as the test temperature increases, at some limiting temperature, the failure mode during the final instability changes from cleavage to ductile. These two different types of behaviour can be accommodated in a statistical analysis which is based on the method of competing risks. A statistical approach is presented for the analysis of data by competing risks and a procedure is given for the estimation of the probability of cleavage failure.     

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

The 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.     

A probabilistic analysis on thickness effect in fracture toughness

Fracture toughness values are often influenced by specimen thickness and they indicate generally decreasing toughness with increasing thickness. In the present paper, a probabilistic analysis has been carried out by using various kinds of toughness data in order to clarify the applicability of the weakest link model to thickness effect in fracture toughness. Moreover, a new statistical method is proposed for determining fracture toughness distribution, which is necessary for the above analysis, with taking the temperature dependency of fracture toughness into account. Thickness effect in fracture toughness is brought about by its statistical nature and the weakest link model can be applied to evaluate the thickness effect for both steel plate and its welds with heterogeneity in toughness. This thickness effect is considerably affected by Weibull shape parameter and the probability of cleavage fracture for the material. The statistical method proposed newly in this paper is sufficiently applicable and superior to the conventional method. By using this new method, Weibull parameters at a temperature of interest can be determined with much the same reliability as in the conventional method, and also Weibull parameters at lower and higher temperatures can be obtained with a certain confidence depending on the number of specimens tested.     

A statistical model of cleavage fracture

The aim of the paper is to provide a sound theoretical basis to the statistics of cleavage fracture in three-dimensional cracked structures. The probability of critically sized carbide being present in a Fracture Initiation Zone ahead of the crack tip has been derived, and shown to have a two-parameter Weibull distribution, with a shape parameter that is proportional to the strain-hardening exponent of the material. In a three-dimensional structure the cracking of such critically sized, intergranular carbide is necessary, but may not be sufficient to precipitate brittle fracture; this is because intergranular carbide is randomly orientated within the crack-opening stress field, so its orientation must also be unfavourable. It has been hypothesised that in three-dimensional structures the actual probability of fracture will be an extreme from the necessary distribution, in which case a sample of fracture toughness observations will be described by a Gumbel distribution, called here the LED model. After discussing the minimum number of fracture toughness observations needed to fit the model, its strength of evidence is compared with those of other candidate models, including the Master Curve model, and the LED model is shown to be the best.     

A probabilistic model for cleavage fracture with a length scale - Parameter estimation and predictions of growing crack experiments

A probabilistic model for the cumulative probability of failure by cleavage fracture was applied to experimental results where cleavage fracture was preceded by ductile crack growth. The model, introduced by Kroon and Faleskog [Kroon M, Faleskog J. A probabilistic model for cleavage fracture with a length scale - influence of material parameters and constraint. Int J Fract 2002;118:99-118], includes a non-local stress with an associated material related length scale, and it also includes a strain measure to account for the number of nucleated cleavage initiation sites. The experiments were performed on single edge cracked bend test specimens with three different crack lengths at the temperature 85 °C, which is in the upper transition region for the steel in question. The ductile rupture process is modelled using the cell model for nonlinear fracture mechanics. The original cleavage fracture model had to be modified in order to account for the substantial number of cleavage initiators being consumed by the ductile process. With this modification, the model was able to accurately capture the experimental failure probability distribution.     

Influence of threshold parameters on cleavage fracture predictions using the Weibull stress model

This study explores applications of three-parameter Weibull stress models to predict cleavage fracture behavior in ferritic structural steels tested in the transition region. The work emphasizes the role of the threshold parameters (_th and _{w – min}) in cleavage fracture predictions of a surface crack specimen loaded predominantly in tension for an A515-70 pressure vessel steel. A recently proposed procedure based upon a toughness scaling methodology using a modified Weibull stress (^* _w) extends the calibration scheme for the Weibull modulus, m, to include the threshold parameters. The methodology is applied to calibrate the Weibull stress parameter for the tested material and then to predict the toughness distribution for the surface crack specimen. While the functional relationship between ^* _w and m suggests a strong effect of the threshold stress, _th, on the calibrated m-parameter, the results show a remarkably weak dependence of fracture predictions on _th as does the dependence of fracture predictions on _w–min for this specimen.     

The effect of aluminum alloying on ductile-to-brittle transition in Hadfield steel single crystal

The ductile-to-brittle transition (DBT) in Fe-13Mn-1.3C (Hadfield steel, I) and Fe-13Mn-2.7 Al-1.3C (Hadfield steel, II) (wt.%) single crystals oriented along [011], [[`1]44]{[011], [{\bar{{1}}}44]}, and [[`1]11{\bar{{1}}11}] directions was investigated under tension in the temperature interval of 77 to 673 K. The DBT temperature interval was found to be independent of single crystal orientation. The DBT temperatures were estimated (1) as the mean value between the temperature corresponding to the minimum crystal ductility and the one coinciding with the onset of the plateau of the e{\varepsilon}(T)-dependence (T_DBT1); and (2) as the temperature where the volume fraction of brittle failure on the fracture surfaces was 50% (T_DBT2). The DBT temperatures estimated this way, do not coincide for both steels. Mechanical twinning has been reported as the primary reason for the occurrence of the DBT in austenitic high-carbon Hadfield steel and appears to account for the difference in DBT temperatures as well. Alloying with aluminum partially suppresses twinning in steel (II). Twinning sets in only after a certain amount of dislocation slip, but still influences the fracture mechanism of steel (II).     

A probabilistic model for the pearlite-induced cleavage of a plain carbon structural steel

The cleavage fracture of a carbon structural steel is shown to deviate from what is typical of a ferritic matrix. This occurs at temperatures ranging from −150 to 20 °C and affects the scatter of fracture toughness, and fractographic features. To explain the observed discrepancies a probabilistic model is developed by linking physical mechanisms of cleavage with continuum mechanics analysis applied to fracture. The model assumes that cleavage is nucleated and triggered in pearlite within the blunting zone of the crack tip by the mechanism of Miller and Smith. Once the model is calibrated for the steel, its predictions are shown to agree with the experimental results.     

Influence of statistical and constraint loss size effects on cleavage fracture toughness in the transition - A model based analysis

A database derived from tests on specimens with a large range of ligament (b) and thickness (B) dimensions was systematically analyzed to evaluate constraint loss and statistical size effects on cleavage fracture toughness. The objectives were to: (1) decouple size effects related to constraint loss, mediated by b and B, from those arising from statistical effects, primarily associated with B; and, (2) develop procedures to transfer toughness data to different conditions of constraint and B. The toughness database for a Shoreham pressure vessel steel plate, tested at a common set of conditions, was described in a companion paper. Quantification of constraint loss was based on an independently calibrated 3D finite-element critical stress-area, σ^∗-[K_Jm/K_Jc], model. The measured toughness data, K_Jm, were first adjusted using computed [K_Jm/K_Jc] constraint loss factors to the corresponding values for small scale yielding conditions, K_Jc=K_Jm/[K_Jm/K_Jc]. The K_Jc were then statistically adjusted to a K_Jr for a reference B_r = 25.4 mm. The B adjustment was based on a critically stressed volume criterion, modified to account for a minimum toughness, K_min, consistent with modest modifications of the ASTM E 1921 Standard procedure. The combined σ^∗-[K_Jm/K_Jc]-K_min adjustment procedure was applied to the Shoreham b − B database, producing a homogeneous population of K_Jr data, generally within the expected scatter. The analysis suggests that: (1) there may be a maximum B beyond which statistical size effects diminish, and (2) constraint loss in the three-point bend specimens begins at a relatively low deformation level. A corresponding analysis, based on a Weibull stress, σ_w-[K_Jm/K_Jc]-K_min, adjustment procedure, yielded similar, but somewhat less satisfactory, results. The optimized adjustment procedure was also applied to other K_Jm data for the Shoreham plate from this study, as well as a large database taken from the literature. The population of 489K_Jr data points, covering an enormous range of specimen sizes, geometries and test temperatures, was found to be consistent with the same master curve T₀ = −84 °C derived from the b − B database. Thus, calibrated micromechanical models can be used to treat size and geometry effects on K_Jm, facilitating using small specimens and data transfer to predict the fracture limits of structures.     

方钢管混凝土柱-外环板式组合梁节点在地震损伤后的耐火性能分析

为研究FRP增强混凝土梁失稳前断裂过程,该文基于混凝土断裂力学理论和非线性FRP-混凝土界面粘结滑移规律,建立了一个跨中裂缝导致界面脱粘的粘聚区模型,采用解析方法推导了FRP增强混凝土梁界面剪切应力、FRP拉应力以及失稳前断裂韧度的公式,为分析FRP-混凝土界面脱粘提供了一种有效的方法,并开展了动态荷载下4种不同初始缝高比(0.2、0.3、0.4和0.5)的FRP增强混凝土梁三点弯曲试验。结果表明,FRP增强混凝土梁的起裂荷载和阻裂荷载随着初始缝高比的增大而逐渐减小,但初始缝高比为0.4时,试件起裂最晚;起裂韧度和阻裂韧度不随初始缝高比的变化而变化,表现出与其他文献类似的规律,验证了断裂韧度解析解的正确性。     

Dependence of ductile-to-brittle transition temperature on phosphorus grain boundary segregation for a 2.25Cr1Mo steel

Equilibrium grain boundary segregation of phosphorus in a 2.25Cr1Mo steel was studied by means of Auger electron spectroscopy (AES). After quenching from 980 °C and tempering at 650 °C, the steel samples were aged for different periods of time at 560, 520 and 480 °C, respectively, followed by AES measurements. Based on the experimental values of the equilibrium segregation levels at different temperatures along with equilibrium segregation theories, the thermodynamic parameters of segregation were evaluated. With the use of Charpy impact testing along with the AES results on phosphorus grain boundary concentration, a relationship between ductile-to-brittle transition temperature (DBTT) and phosphorus boundary concentration was established. On the basis of this relationship, predictions of the DBTT were made through predicting the grain boundary segregation of phosphorus.