Observations in the statistical analysis of NBG-18 nuclear graphite strength tests

The purpose of this paper is to report on the selection of a statistical distribution chosen to represent the experimental material strength of NBG-18 nuclear graphite. Three large sets of samples were tested during the material characterisation of the Pebble Bed Modular Reactor and Core Structure Ceramics materials. These sets of samples are tensile strength, flexural strength and compressive strength (CS) measurements. A relevant statistical fit is determined and the goodness of fit is also evaluated for each data set. The data sets are also normalised for ease of comparison, and combined into one representative data set. The validity of this approach is demonstrated. A second failure mode distribution is found on the CS test data. Identifying this failure mode supports the similar observations made in the past. The success of fitting the Weibull distribution through the normalised data sets allows us to improve the basis for the estimates of the variability. This could also imply that the variability on the graphite strength for the different strength measures is based on the same flaw distribution and thus a property of the material.     

Design techniques for ceramics in fusion reactors

Unique design techniques are needed for low activity ceramic materials in first wall/blanket regions of fusion reactors. A Weibull probabilistic design approach is used to characterize the scatter in the fracture strength and the size effect. Results indicate that ceramic first wall/blanket structures should be modular and each module should be proof tested. The ceramic materials should have high fracture strength, high Weibull modulus, and minimal strength degradation due to subcritical crack growth. The Weibull statistical analysis is coupled with finite element thermal and stress analysis and the probability of failure of ceramic first wall/blanket design concepts is predicted. The usefulness of the approach is demonstrated by optimizing the geometry of the structure to produce minimum probability of failure.     

Comments on: “Studies on diversion cross-flow between two parallel channels communicating by a parallel slot — Part I” by A. Tapucu

Unique design techniques are needed for low activity ceramic materials in first wall/blanket regions of fusion reactors. A Weibull probabilistic design approach is used to characterize the scatter in the fracture strength and the size effect. Results indicate that ceramic first wall/blanket structures should be modular and each module should be proof tested. The ceramic materials should have high fracture strength, high Weibull modulus, and minimal strength degradation due to subcritical crack growth. The Weibull statistical analysis is coupled with finite element thermal and stress analysis and the probability of failure of ceramic first wall/blanket design concepts is predicted. The usefulness of the approach is demonstrated by optimizing the geometry of the structure to produce minimum probability of failure.     

0Cr18Ni10Ti管道钢的概率机械性能

完成了反应堆管道新材料0Crl8Nil0Ti的概率机械性能试验研究。拓展赵等提出的确定有限疲劳可靠性数据良好假设分布的统一方法到本文机械性能数据分析。结果表明．数据具有明显正偏性，常见正态分布描述不适于该套数据。综合考虑各分布的拟合优度、与失效机制的一致性和尾部预测的安全性．数据的良好统计分布是对数正态分布和极大值分布。从统计分析的方便性角度。提出了给定可靠度和置信度下基于对数正态分布的机械性能参数估计方法。并有效地估计了材料的概率机械性能参数。     

Fracture statistics of brittle materials with intergranular cracks

A statistical theory for intergranular crack frequency as a function of crack size has been developed for polycrystalline materials. On the assumption that the cracks are penny-shaped and that brittle fracture is caused by overstressing the weakest link, the crack frequency was converted into a statistical theory of fracture. The implications of this theory were examined and compared with those of Weibull and of Fisher, Holloman and McClintock. In general, the predicted behavior is in reasonable agreement with experimental observations, although whether agreement is significantly better than Weibull's theory will require detailed examination. Among the attractive features of the new theory are: (1) a closer relationship to fracture mechanics and to material structure and properties compared with the Weibull theory and (2) a capacity for refinement so as to model the actual material structure more closely, which should result in improved agreement with experiment.     

Characterization of tensile strength and fracture toughness of nuclear graphite NBG-18 using subsize specimens

The mechanical properties of NBG-18 nuclear grade graphite were characterized using small specimen test techniques and statistical treatment on the test results. New fracture strength and toughness test techniques were developed to use subsize cylindrical specimens with glued heads and to reuse their broken halves. Three sets of subsize cylindrical specimens of different sizes were tested to obtain tensile fracture strength and fracture toughness. The mean fracture strength decreased as the specimen size increased. The fracture strength data indicate that in the given diameter range the size effect is not significant and much smaller than that predicted by the Weibull moduli estimated for individual specimen groups of the Weibull distribution. Further, no noticeable size effect existed in the fracture toughness data. The mean values of the fracture toughness datasets were in a narrow range of 1.21-1.26 MPa√m.     

Probabilistic failure analysis of a water-cooled tungsten divertor: Impact of embrittlement

Inherent brittleness and neutron embrittlement are critical weaknesses of tungsten for fusion application. Pronounced scattering of the fracture strength of tungsten requires a statistical treatment. Thus, the risk of structural failure of a tungsten component can be estimated only in a probabilistic framework. In this work, we applied a probabilistic failure analysis code STAU to estimate the failure risk of a water-cooled tungsten mono-block divertor component. The STAU code was based on the weakest-link failure theory and linear elastic fracture mechanics. A typical heat flux load being expected for a fusion reactor was considered for the FEM stress analysis. The failure probability was computed considering various mixed-mode fracture criteria. Both the experimentally estimated and hypothetical Weibull parameters were used as material data. In the case of unirradiated tungsten, the failure probability was acceptably small whereas reduced Weibull parameters led to significantly increased failure risk.     

Fracture toughness transition curve estimation from a notched round bar specimen using the local approach method

The local approach method for the brittle or transition region describes the fracture probability of specimens or structures using Weibull statistics. Many papers have discussed the characteristics of the Weibull parameter using notched tensile specimens and the applicability to fracture toughness scatter evaluation using CT specimens. However few papers have made clear whether the Weibull parameter of the Weibull stress is a material property or not. In this paper the distribution of Weibull stress in the brittle fracture region using notched round bar specimens and CT specimens were investigated and it was confirmed that both distributions agreed well. Furthermore the estimation method for the fracture toughness transition curve including its scatter from notched round bar tensile tests was proposed based on the relation between the Weibull stress and the Wallin's fracture toughness transition curve. As a result, the estimated fracture toughness curve in the brittle and lower transition region from the notched round bar specimens coincided with the measured fracture toughness curve from CT specimens. This method will be applicable to fracture toughness curve estimation under plane strain conditions even if there is no possibility of obtaining thick enough CT specimens from a structure because of geometry or some other restrictions.     

Effect of prior crack growth on cleavage fracture of low toughness precracked Charpy specimens

Cleavage fracture of reactor pressure vessel steels in the upper ductile to brittle transition region generally occurs with prior significant ductile crack growth. For low upper shelf materials and using PreCracked Charpy v-notch (PCCv) specimens that can be obtained from conventional surveillance programs, the effect of prior crack growth could be particularly important. In practice, the shape of the Master Curve and the failure distribution could be affected by ductile crack growth. To quantify the effect in practical applications, the effect of prior ductile on cleavage is evaluated on PCCv specimen.The methodology use finite element calculations to grow a ductile crack and infer the brittle failure probability using the local approach to fracture. It is found that for very low upper shelf toughness materials, ductile crack growth enhances the failure probability, induces a steeper failure distribution and affects the shape of the Master Curve. However, for low toughness materials, the enhanced failure probability due to crack growth is compensated by loss of constraint.     

V型凹口对液态熔盐堆用核石墨构件应力分析的影响

本文利用了一个根据球床模块堆(Pebble Bed Modular Reactor,PBMR)用核石墨材料辐照性能数据编写的用户自定义材料模型(User defined Material model,UMAT),按照美国橡树岭国家实验室(Oak Ridge National Laboratory,ORNL)的液态燃料熔盐试验堆(Molten Salt Reactor Experiment,MSRE)用核石墨构件尺寸,为钍基熔盐堆(Thorium-based Molten Salt Reactor,TMSR)设计了一款方型核石墨构件。利用新编UMAT对该核石墨构件进行了初步的应力分析。分析结果表明,在没有预制裂纹的情况下辐照梯度越大核石墨构件中心区域最大主应力值越大,构件的断裂位置可能出现在构件中心位置处;对于有V型凹口预制裂纹的情况,应力集中部位均出现在预制裂纹尖端附近,这将可能导致裂纹尖端附近出现裂纹扩展,从而引起构件断裂失效。     

用当量初始裂纹尺寸法评估石墨的疲劳性能

用当量初始裂纹尺寸法评估某些金属材料的疲劳性能效果良好。本文用这一方法对高温气冷堆用石墨材料进行了评估。结果表明，这种方法应用于存在大量微小缺陷的脆性材料时，需要作适当的修正。     

Weibull统计在硬质合金涂层抗弯强度分析中的应用

首先讨论了ibull 统计用于涂层强度分析的理论基础:如果涂层的局部强度是独立的,随着涂层体积的增大,Weibull 分布可以作为涂层强度分布的渐近分布;然后将其应用于两种硬质合金涂层抗弯强度的分析。结果表明,应用 Weibull 统计可以很好地描述硬质合金涂层的抗弯强度。     

A comparative method for improving the reliability of brittle components

Calculating the absolute reliability built in a product is often an extremely difficult task because of the complexity of the physical processes and physical mechanisms underlying the failure modes, the complex influence of the environment and the operational loads, the variability associated with reliability-critical design parameters and the non-robustness of the prediction models. Predicting the probability of failure of loaded components with complex shape for example is associated with uncertainty related to: the type of existing flaws initiating fracture, the size distributions of the flaws, the locations and the orientations of the flaws and the microstructure and its local properties. Capturing these types of uncertainty, necessary for a correct prediction of the reliability of components is a formidable task which does not need to be addressed if a comparative reliability method is employed, especially if the focus is on reliability improvement. The new comparative method for improving the resistance to failure initiated by flaws proposed here is based on an assumed failure criterion, an equation linking the probability that a flaw will be critical with the probability of failure associated with the component and a finite element solution for the distribution of the principal stresses in the loaded component. The probability that a flaw will be critical is determined directly, after a finite number of steps equal to the number of finite elements into which the component is divided. An advantage of the proposed comparative method for improving the resistance to failure initiated by flaws is that it does not rely on a Monte Carlo simulation and does not depend on knowledge of the size distribution of the flaws and the material properties. This essentially eliminates uncertainty associated with the material properties and the population of flaws.On the basis of a theoretical analysis we also show that, contrary to the common belief, in general, for non-interacting flaws randomly located in a stressed volume, the distribution of the minimum failure stress is not necessarily described by a Weibull distribution. For the simple case of a single group of flaws all of which become critical beyond a particular threshold value for example, the Weibull distribution fails to predict correctly the probability of failure. If in a particular load range, no new critical flaws are created by increasing the applied stress, the Weibull distribution also fails to predict correctly the probability of failure of the component. In these cases however, the probability of failure is correctly predicted by the suggested alternative equation. The suggested equation is the correct mathematical formulation of the weakest-link concept related to random flaws in a stressed volume. The equation does not require any assumption concerning the physical nature of the flaws and the physical mechanism of failure and can be applied in any situation of locally initiated failure by non-interacting entities.     

Rules for design of nuclear graphite core components — Some considerations and approaches

The use of graphite as a structural element presents unusual problems both for the designer and stress analyst. When the structure happens to be a nuclear reactor core, these problems are significantly magnified both by the environment and the attendant safety requirements. In the high temperature gas reactor (HTGR) core a large number of elements are constructed of nuclear graphite. This paper discusses the attendant difficulties, and presents some approaches, for ASME code safety-consistent design and analysis. The statistical scatter of material properties, which complicates even the definitions of allowable stress, as well as the brittle, anisotropic, inhomogeneous nature of the graphite was considered. The study of this subject was undertaken under contract to the U.S. Nuclear Regulatory Commission.     

ASME规范概率评价方法对细颗粒石墨的适用性研究

石墨由于其高中子散射截面和低中子吸收截面特性,被广泛应用于第四代高温气冷堆中作为慢化剂、反射层和堆芯结构,故保证其结构完整性对反应堆的安全运行非常重要。由于石墨材料强度分散,概率论方法评价其失效较常用的确定论评价方法更为合适。目前,美国ASME规范采用的概率方法主要针对NBG-18这种大颗粒石墨,对我国高温气冷堆核电站工程项目采用的细颗粒石墨IG-110的适用性未知。同时,我国成都碳素生产的高温堆备选石墨NG-CT-01颗粒大小与IG-110相似,也为细颗粒石墨。因此,文章研究ASME规范概率方法对细颗粒石墨的适用性,并通过实验数据加以验证。结果表明,对于细颗粒石墨,ASME规范过于保守,低估了材料的强度性能。     

Matrix displacement methods in fracture mechanics analysis of reactor vessels

The technology of fracture mechanics is developing rapidly in response to increased requirements for integrity of engineering structures. It enables structural engineers to evaluate brittle failure resistance of structures within appropriate regimes of temperature, materials and geometry. The evaluation includes the combined effects of material toughness, flaw characteristics, environment and service loadings. Calculations of stress intensity factors associated with the flaws, geometry and applied loading form the basis of fracture analysis and control procedures for reactor vessels.     

基于Weibull分布的电磁脉冲损伤函数Bayesian分析技术

Weibull分布是电磁脉冲易损性分析常常用到的一种分布.论文采用Weibull分析方法运用Gibbs抽样技术求取了基于Weibull分布的固态继电器电磁脉冲损伤函数.     

Radiation damage in silicon carbide and graphite for fusion reactor first wall applications

Silicon carbide and graphite materials were exposed to fast neutron fluences of 2 × 10²³ to 2 × 10²⁴n/m² (E > 1 MeV) and a study was made of changes in fracture strength, Weibull modulus and electrical resistivity. Silicon carbide (Norton NC-430) exhibits a decrease in fracture strength (25%) at the higher fluence if the temperature is kept at 298 K, while at 1473 K the decrease in fracture strength is only 10% indicative of recovery due to thermal annealing. The fracture strength of the graphite (POCO AXF-5Q) tested at 298 K increases rapidly by ~20% after 2 × 10²³n/m² and remains constant at higher fluence. Analyses of the data using the Weibull weakest link model were given, in addition to annealing and swelling results.     

Heat treatment effect of T91 martensitic steel on liquid metal embrittlement

The sensitivity of liquid metal embrittlement of the T91 martensitic steel is investigated with small punch tests at 300 °C in air and in lead bismuth eutectic (LBE). The material was studied in six tempering conditions corresponding to different values of hardness. An effect of LBE has been observed for all the materials excepted for T91 steel tempered at 750 °C, the more ductile material. In high strength materials (T91 steel as quenched, tempered at 600 °C or 500 °C), a ductile to brittle transition is induced by liquid metal, confirmed by the observation of brittle fracture. In relative high strength materials (tempered at 650 °C and 700 °C), LBE promotes a decrease in mechanical properties and a reduction of the ductility of materials, with a mixed ductile and brittle fracture.     

Fracture criteria of reactor graphite under multiaxial stesses

New fracture criteria for graphite under multiaxial stresses are presented for designing core and support materials of a high temperature gas cooled reactor. Different kinds of fracture strength tests are carried out for a near isotropic graphite IG-11. Results show that, under the stress state in which tensile stresses are predominant, the maximum principal stress theory is seen as applicable for brittle fracture. Under the stress state in which compressive stresses are predominant, there may be two fracture modes for brittle fracture, namely, slipping fracture and mode II fracture. For the former fracture mode the maximum shear stress criterion is suitable, but for the latter fracture mode the following mode II fracture criterion including a restraint effect for cracks is verified to be applicable, where δ₁ and δ₃ are the maximum and minimum principal stresses, δ_t and δ_c are the tensile and compressive strengths and K_Ic and K_IIc are the mode I and II fracture toughness values, respectively. The above equation is similar in form to the Coulomb-Mohr criterion.Also a statistical correction for brittle fracture criteria under multiaxial stresses is discussed. By considering the allowable stress values for safe design, the specified minimum ultimate strengths corresponding to a survival probability of 99% at the 950 confidence level are presented.