Weibull Stress Research Articles

Fracture mechanics approach to TRISO fuel particle failure analysis

Weibull stress-based methods for failure probability assessment have been developed and analyzed to assess the integrity of tristructural isotropic (TRISO) fuel particles during fuel life cycles and accident operating conditions. While simple, these methods entail a number of drawbacks when stress concentrates near crack tips, including finite element mesh size dependency when the Weibull stress is averaged over the finite element domain. Fracture mechanics approaches involving the use of interaction integrals eliminate this lack of mesh convergence and produce consistent fracture predictions. In this work, we use an interaction integral approach to computing stress intensity factors for a crack in the inner pyrolytic carbon layer perpendicular to the silicon carbide layer, which is simplified representation of a failure mode in TRISO particles. The interface between these two TRISO layers has been shown to become porous, which we simulate by considering a transition of mechanical properties over such porous length, i.e. the layers are modeled as a functionally graded material. Aspects such as porosity and thermal and irradiation eigenstrains are considered in computing the stress intensity factor from a fracture mechanics approach and compared with the known Weibull stress failure approach. The methodology introduced in this paper enables a more general fracture probability assessment in TRISO particles and eliminates the need to identify best suited parameters when using local or averaged stress-based failure criterion. Finally, the numerical sensitivity studies show how parameters such as the porous transition zone length, the material stiffness, and creep affect the probability of TRISO fuel particle failure.

Bayesian Estimation of Exponentiated Exponential Strength and Exponentiated Weibull Stress Reliability Model for Real Data

Stress-strength reliability is an important concept in reliability analysis, quantifies the probability that the strength of a system surpasses its applied stress. This paper focuses on the reliability analysis for exponentiated exponential distribution strength variable and the exponentiated Weibull distribution stress variable. The study explores the estimation of the parameters in stress-strength reliability model using maximum likelihood estimation and Bayesian estimation. In particular, the Bayesian estimator of stress-strength reliability is obtained by utilizing Lindley’s approximation by considering both linear exponential loss function and squared error loss function for informative and non-informative priors. A comprehensive simulation study is conducted and the performances of estimators are compared using mean squared errors. The stress-strength reliability for real datasets is also investigated for real-time data sets.

A comprehensive approach for quantifying the reliability of ship hulls under propagating fatigue cracks

Traditional approaches for reliability assessment of ship hulls under propagating cracks mostly rely on comparing the value of a certain fracture mechanics parameter calculated at the crack tip to the fracture toughness of the material. This assumption may not accurately represent the failure event since the fracture resistance is not only affected by the material properties but also by the component geometry. Furthermore, this approach may not properly account for the possibility of elastic-plastic fracture in the ship hull. This paper presents a novel probabilistic approach for quantifying the reliability of ship hulls under propagating fatigue cracks. The approach utilizes the failure assessment diagram (FAD) to define the performance function of the hull and can account for the occurrence of various possible failure mechanisms ranging from brittle fracture to full plastic collapse. The Weibull stress criterion is utilized to quantify the critical crack tip opening displacement which is adopted herein to represent the fracture resistance of the cracked hull. Monte Carlo simulation is used to quantify the failure probability and the reliability index of the ship hull under encountered wave loads and propagating cracks. The approach is illustrated on a tanker operating in the Atlantic Ocean.

Weibull stress method to determine the minimum required strength to improve the actual reliability

AbstractThe method to determine the minimum strength that a product must present to improve the product's reliability from its actual R(t) to a desired R(t2) index are given. The method can be performed in both the normal and accelerated scenarios. It can be done by using either complete failure time data or based only on the applied stress range. The addressed minimum strength that corresponds to the R(t2) index is then used in the stress/strength reliability function to determine the corresponding reliability R(t/S,s) index. From the comparison between the R(t2) and R(t/S,s) indices, we found they are similar only for high reliability indices (say R(t) > 0.90). For lower indices their difference is significant, implying more research must be undertaken. In the application a zero accelerated test plan (ZATP) is performed to experimentally demonstrate the improved product presents the desired R(t2) index. Finally, because by applying the proposed method we always can reproduce the product's average, then the formulation to determine the capability Cp and ability Cpk indices of the R(t2) index is also given.

Brittle fracture prediction method considering mixed modes for column-to-beam connections

This study entailed an improvement in fracture assessment based on mixed-mode loading. Brittle fractures can be evaluated using the Weibull stress regardless of the differences in plastic constraints. However, the Weibull stress is proposed for cracks subjected to open loading (mode I), and it does not consider the effects of in-plane (mode II) or out-plane shear loadings (mode III). Structural connections under large deformations are subjected to modes I, II, and III at the crack tip. Fracture tests on beam-to-diaphragm connections in buildings have indicated that failure occurs when ductile cracks propagate from the strain concentration region of a weld access hole, where mode II dominates. Additionally, they cannot be appropriately evaluated using the conventional Weibull stress approach. This study investigated the relationship between mode differences and fracture initiation for several specimens modeled as beam-to-diaphragm connections. A mixed-mode ratio with stress intensity factor was used as an index to evaluate the mode differences. The fracture prediction considers the differences in modes by correcting the evaluated values on the basis of the mixed-mode ratio. The correction results shows more reasonable fracture prediction than conventional methods for beam-to-column connections under different modes.

Weibull strength distribution and reliability S-N percentiles for tensile tests

Based on the true stress, the ultimate material’s strength, and the fatigue slope b values, the probabilistic percentiles of the S-N curve of ductile materials are formulated. The Weibull β and η parameters used to determine the product’s reliability are determined directly from the material’s strength values corresponding to 103 and 106 cycles. And since in Table corresponding to the properties of this A538 A (b) steel and collected by table 23-A of Shigley Mechanical Engineering Design book; authors present the σt, Sut, and b values of several materials, then the Weibull parameters for each one of these materials as well as the 95% and 5% reliability percentiles of their S-N curves are given. A step-by-step application to the steel A538 A (b) material is presented. And based on the maximum and minimum applied stress values, the corresponding Weibull stress distribution was fitted and used with the Weibull strength distribution, in the stress/strength reliability function to determine the element’s reliability.

Inference of reliability in a multicomponent stress–strength model under generalized progressive hybrid censoring

This paper considers the reliability inference of a multicomponent stress–strength model by assuming the dependent Weibull stress and strength variables based on Gumbel copula under generalized progressive hybrid censoring (GPHC). The point estimates of unknown parameters and reliability are obtained by using frequentist and Bayesian approaches, and their interval estimates are constructed utilizing the asymptotic normality property of maximum likelihood estimates (MLE) and Markov Chain Monte Carlo (MCMC) samples. Extensive simulations are performed to compare the proposed methods. Finally, one real data set is analyzed to demonstrate the applicability of our study.

Scaled Fracture Toughness Based on the Weibull Stress for the Ferritic Steel Used in Nuclear Power Plants

As an important material property in structural integrity assessment of nuclear power components, fracture toughness can be measured by single edge-notched bend (SENB) specimens or compact tension (CT) specimens. However, the tested values may be inhomogeneous with the crack size and specimen thickness. Some toughness scaling models (TSMs) were proposed to transfer the tested value to fracture toughness under small-scale yielding. Combined with the tested data by SENB specimens with different crack sizes and CT specimens with different thicknesses, the scaled fracture toughness is investigated on the global and local approaches to fracture. Using the scaled toughness values, the Weibull scale parameter K0 is estimated and compared for the ferritic steel DIN 22NiMoCr37 widely used in nuclear power plants. The results show that the estimated scale parameter K0 using the scaled toughness obtained by the global approach on the J-Q theory is reasonable in agreement with that of the local approach on the Weibull stress with a relative error of about 10%.

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.

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.

Damage Assessment for Steel Structures Subjected to cyclic pre-strain

Plastic deformation provoked by unexpected events, such as earthquakes, can deteriorate the steel fracture toughness, and it can also induce premature failure of a structure and accidents. This study tries to formulate new equations to quantify the fracture toughness degradation of steel that was cyclic pre-strained. Four point bending tests were used to carry out the cyclic plastic pre-strain and Crack Tip Opening Displacement (CTOD) tests were conducted to compare pre-strained specimens with the steel without plastic deformation (as-received). The cumulative damage law was employed to account for the difference between each specimen, associated with Weibull stresses and critical CTOD values. Furthermore, additional Finite Element Method (FEM) simulations were conducted, using the new approach for the establishment an accurate Weibull Stress criterion, which can represent different cleavage fracture resistance according to the last pre-strain cycle. In the end, a new CTOD equation was also formulated using macro parameters, and it can account for the fracture toughness degradation of material that was subjected to cyclic plastic deformation.

Brittle fracture assessment of narrow weld part in steel pipe joint structure by using Weibull stress concept

Brittle fracture assessment of narrow weld part in steel pipe joint structure by using Weibull stress concept

Brittle fracture estimation at ends of groove welded joints under cyclic loading

To prevent brittle fracture in structural components during earthquakes, their cumulative ductility (plastic deformation capacity) should be determined beforehand and appropriate countermeasures applied accordingly. In this paper, a method for estimating the cumulative ductility of welded joints with defects based on the σW–Nf relationship and Miner's rule is proposed for steel structure components to facilitate brittle fracture estimation. To verify its effectiveness, it was applied to the testing of welded joints with defects and compared to experimental results. The experimental tests were conducted under cyclic loading conditions to reproduce brittle fractures starting from the ends of complete joint penetration groove welded joints. The joint models used in this study were part of beam-to-column connections between beam flanges and diaphragms. Defects were artificially introduced into the welded joints before the experimental tests; brittle fractures occurred at these preinstalled defects in the welded joint specimens. The calculated cumulative ductility was obtained from the fracture toughness and Weibull stress of the joint model measured using finite element analyses. The calculated cumulative ductility values corresponded well with the experimental results, albeit with some explicable errors. The results indicate that the proposed method can estimate the cumulative ductility for structural components with defects under cyclic loading independent of their materials, notch types, and connection details. Thus, the findings could contribute to the assessment of defects in welded joints in steel structures subjected to earthquake loadings.

Loading mode effect on brittle fracture resistance of cracked component under large-scale yielding

This study assesses the applicability of the Weibull stress for the assessment of the crack-tip plastic constraint effect as well as the mixed mode I and II loading effect on brittle fracture resistance of a ferritic steel. Brittle fracture toughness under different crack-tip plastic constraint conditions and different loading mode conditions was respectively obtained by conducting 3-point bend (3PB) test for single edge cracked specimens with different crack depth subjected to mode I load, and 4-point shear (4PS) test for a single edge cracked specimen with deep crack depth subjected to mixed mode I and II load. By using the results of 3PB tests, the critical Weibull stresses distribution independent of crack-tip plastic constraint was identified, and the critical Weibull stresses obtained by 4PS tests provided significantly smaller distribution than that for 3PB specimen. The mixed mode loading provides the different combined stress field around the crack-tip from that under mode I loading, where the 4PS specimen showed the lower distribution of σ2/σ1 and σ3/σ1 than the 3PB specimens. The lower principal stress ratio to be enlarge the fracture driving force was found to have no or much less influence to linear energy release rate for micro-crack, that was employed in the conventional derivation of the Weibull stress. Consequently, the reason why the conventional Weibull stress could not evaluate the effect of loading mode on fracture resistance could be that the linear energy release rate could not exactly take into account the local fracture driving force enlarged due to the combined stress field under mixed mode loading.

Study on Establish a Brittle Fracture Prediction Considering Different Crack Opening Modes Using Mixed-Mode Ratio

In this study, we propose a method for predicting the occurrence of brittle fractures in the beam-to-column joints of steel structures, considering different crack opening modes. We conducted experiments on beam-to-diaphragm joint specimens with varying plastically constrained cracks to reproduce brittle fractures. The experiments’ results demonstrated the effectiveness of the toughness scale model and the Weibull stress approach. In addition, we propose the mixed-mode ratio, which is a quantitative index of the mode difference, and we applied it to the finite element models of the specimens. In this study, we evaluate the validity of the mixed-mode ratio and explore the differences in crack opening modes, as they pertain to the occurrence of brittle fractures.

Bonding States of In Vitro Class 2 Direct Resin Composite Restoration Applied by Various Incremental Techniques.

Incremental techniques are always required for clinical cases of deep and/or large cavities restored with resin composite materials. The purpose of this study was to examine the bonding states of class 2 direct resin composite restoration applied by various incremental techniques after cyclic loading to simulate the intra-oral environment to define the appropriate technique. Three types of resin composites, namely, bulk-fill (B), flowable (F), and conventional resin composite (C), were applied to standardized class 2 cavities by incremental techniques with single- or bi-resin restoratives. After cyclic loading, the micro-tensile bond strength (μ-TBS) of the dentin cavity floor was measured. The Weibull modulus and Weibull stress values at 10%/90% probability of failure were analyzed. Single-resin incremental restorations with B or F and bi-resin incremental restorations with F + B and F + C demonstrated superior μ-TBS (quantitative ability), bonding reliability, and durability (qualitative ability) compared with the single-resin restoration with C (as control). Furthermore, F + B and F + C restoration yielded an excellent performance compared with the single-resin restorations with B, F, and C. In particular, the F + C restoration, which indicates not only the maximum mean µ-TBS, but also the highest values of the Weibull parameters, may be the optimal restoration method, including the esthetic benefits.

Comparing Weibull Stress – Strength Reliability Bayesian Estimators for Singly Type II Censored Data under Different loss Functions

The stress(Y) – strength(X) model reliability Bayesian estimation which defines life of a component with strength X and stress Y (the component fails if and only if at any time the applied stress is greater than its strength) has been studied, then the reliability; R=P(Y<X), can be considered as a measure of the component performance. In this paper, a Bayesian analysis has been considered for R when the two variables X and Y are independent Weibull random variables with common parameter α in order to study the effect of each of the two different scale parameters β and λ; respectively, using three different [weighted, quadratic and entropy] loss functions under two different prior functions [Gamma and extension of Jeffery] and also an empirical Bayes estimator Using Gamma Prior, for singly type II censored sample. An empirical study has been used to make a comparison between the three estimators of the reliability for stress – strength Weibull model, by mean squared error MSE criteria, taking different sample sizes (small, moderate and large) for the two random variables in eight experiments of different values of their parameters. It has been found that the weighted loss function was the best for small sample size, and the entropy and Quadratic were the best for moderate and large sample sizes under the two prior distributions and for empirical Bayes estimation.

Weibull stress solutions and applications under mixed mode I-II loading for 2-D cracks in elastic and elastic–plastic materials

Since the fracture toughness of materials in the ductile and brittle transition region presents obvious scatter, Weibull stress based on the local approach to fracture is widely used to clarify the randomness. A large number of detailed finite element analyses are necessary to be carried out for the calculation of Weibull stress, especially for the complex structures and loads. In order to simplify the calculation of Weibull stress, this paper derives Weibull stress solutions as functions of stress intensity factor or J integral for elastic and elastic–plastic materials under mode I-II loading. These solutions are verified by using modified boundary layer model and applied in the analysis of fracture toughness test results by four-point bending specimens. The relationship between failure probability Pf and SIF or J integral under mode I-II loading is derived and verified with experimental data. The result shows that the failure probability can predict the experimental data well.

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.

Incorporation of Obstacle Hardening into Local Approach to Cleavage Fracture to Predict Temperature Effects in the Ductile to Brittle Transition Regime.

Ductile-to-brittle-transition refers to observable change in fracture mode with decreasing temperature—from slow ductile crack growth to rapid cleavage. It is exhibited by body-centred cubic metals and presents a challenge for integrity assessment of structural components made of such metals. Local approaches to cleavage fracture, based on Weibull stress as a cleavage crack-driving force, have been shown to predict fracture toughness at very low temperatures. However, they are ineffective in the transition regime without the recalibration of Weibull stress parameters, which requires further testing and thus diminishes their predictive capability. We propose new Weibull stress formulation with thinning function based on obstacle hardening model, which modifies the number of cleavage-initiating features with temperature. Our model is implemented as a post-processor of finite element analysis results. It is applied to analyses of standard compact tension specimens of typical reactor pressure vessel steel, for which deformation and fracture toughness properties in the transition regime are available. It is shown that the new Weibull stress is independent of temperature, and of Weibull shape parameter, within the experimental error. It accurately predicts the fracture toughness at any temperature in the transition regime without relying upon empirical fits for the first time.