Weakest Link Model Research Articles

Phase transitions in random circuit sampling

Undesired coupling to the surrounding environment destroys long-range correlations in quantum processors and hinders coherent evolution in the nominally available computational space. This noise is an outstanding challenge when leveraging the computation power of near-term quantum processors1. It has been shown that benchmarking random circuit sampling with cross-entropy benchmarking can provide an estimate of the effective size of the Hilbert space coherently available2–8. Nevertheless, quantum algorithms’ outputs can be trivialized by noise, making them susceptible to classical computation spoofing. Here, by implementing an algorithm for random circuit sampling, we demonstrate experimentally that two phase transitions are observable with cross-entropy benchmarking, which we explain theoretically with a statistical model. The first is a dynamical transition as a function of the number of cycles and is the continuation of the anti-concentration point in the noiseless case. The second is a quantum phase transition controlled by the error per cycle; to identify it analytically and experimentally, we create a weak-link model, which allows us to vary the strength of the noise versus coherent evolution. Furthermore, by presenting a random circuit sampling experiment in the weak-noise phase with 67 qubits at 32 cycles, we demonstrate that the computational cost of our experiment is beyond the capabilities of existing classical supercomputers. Our experimental and theoretical work establishes the existence of transitions to a stable, computationally complex phase that is reachable with current quantum processors.

The study on the wood candidates for the impact limiter of a spent fuel cask

The nuclear fuel transportation package is meticulously designed to enclose radioactive materials, guaranteeing containment even after undergoing a free-drop impact test and considering potential accident situations to prevent radioactive leakage. The impact limiters attached to the transport cask are engineered to absorb impact energy. Wood, renowned for its strong energy absorption capacity, is often specified as the core material within impact limiters. It is crucial to accurately determine the wood material employed in the impact limiter. This study carried out crushing experiments on various common wood materials to obtain stress versus strain data for the selection of the appropriate wood material. Indicators such as density, crushing strength, and densification strain are utilized to assess the suitability of wood as the filler material of the impact limiter. Paulownia wood is chosen based on density, the appropriate energy absorption value per unit volume, and response acceleration. The dispersity of the strengths of Paulownia wood is taken into account with the weakest-link models. The finite element method is adopted to precisely design and verify the impact limiter. The arrangement of the glulam wood is designed to balance the energy absorption in hypothetical accident conditions. Finally, tests on the application of the impact limiter in the transportation cask were conducted to validate the proper design of the impact limiter. This study can offer reference and data support for the research on wood filler materials.

Statistical Scaling in Localization-Induced Failures

Abstract The investigation of statistical scaling in localization-induced failures dates back to da Vinci's speculation on the length effect on rope strength in 1500 s. The early mathematical description of statistical scaling emerged with the birth of the extreme value statistics. The most commonly known mathematical model for statistical scaling is the Weibull size effect, which is a direct consequence of the infinite weakest-link model. However, abundant experimental observations on various localization-induced failures have shown that the Weibull size effect is inadequate. Over the last two decades, two mathematical models were developed to describe the statistical size effect in localization-induced failures. One is the finite weakest-link model, in which the random structural resistance is expressed as the minimum of a set of independent discrete random variables. The other is the level excursion model, a continuum description of the finite weakest-link model, in which the structural failure probability is calculated as the probability of the upcrossing of a random field over a barrier. This paper reviews the mathematical formulation of these two models and their applications to various engineering problems including the strength distributions of quasi-brittle structures, failure statistics of micro-electromechanical systems (MEMS) devices, breakdown statistics of high– k gate dielectrics, and probability distribution of buckling pressure of spherical shells containing random geometric imperfections. In addition, the implications of statistical scaling for the stochastic finite element simulations and the reliability-based structural design are discussed. In particular, the recent development of the size-dependent safety factors is reviewed.

Parameter Estimation of Birnbaum-Saunders Distribution under Competing Risks Using the Quantile Variant of the Expectation-Maximization Algorithm

Competing risks models, also known as weakest-link models, are utilized to analyze diverse strength distributions exhibiting multi-modality, often attributed to various types of defects within the material. The weakest-link theory posits that a material’s fracture is dictated by its most severe defect. However, multimodal problems can become intricate due to potential censoring, a common constraint stemming from time and cost limitations during experiments. Additionally, determining the mode of failure can be challenging due to factors like the absence of suitable diagnostic tools, costly autopsy procedures, and other obstacles, collectively referred to as the masking problem. In this paper, we investigate the distribution of strength for multimodal failures with censored data. We consider both full and partial maskings and present an EM-type parameter estimate for the Birnbaum-Saunders distribution under competing risks. We compare the results with those obtained from other distributions, such as lognormal, Weibull, and Wald (inverse-Gaussian) distributions. The effectiveness of the proposed method is demonstrated through two illustrative examples, as well as an analysis of the sensitivity of parameter estimates to variations in starting values.

A study on the reduction of microstructural non-uniformity in Ni-multilayer ceramic capacitors via repeated highly accelerated life tests and analyses of degraded local areas

The degree of microstructural non-uniformity and the lifetime extension of multilayer ceramic capacitors (MLCCs) were evaluated by repeating the fabrication of “prebreakdown” MLCCs through highly accelerated life tests (HALTs), the removal of the insulation resistance-degraded layer, and the reformation of the terminal electrode. The total mean time to failure (MTTF) tended to extend and converge as HALTs were repeated. Microstructural analysis in the degraded local area revealed a clear correlation between the shorter lifetime and the minimum number of grains per unit dielectric layer. After HALTs were repeated, the number of grains came close to and converged as much as undegraded areas over a longer lifetime. These results imply that degradation occurs in order from the greatest degree of microstructural non-uniformity and that the weakest-link model can be understood by treating the lifetime as MTTF.

Out-of-plane shear strength of cross-laminated timber made of Japanese Larch (Larix kaempferi) with various layups and spans

Cross-laminated timber (CLT) is a promising construction material. When CLT is used for horizontal applications, shear stress occurs in the out-of-plane direction and can fracture the transverse layers owing to the rolling shear. The out-of-plane shear strength of the CLT can be evaluated by an out-of-plane loading test and is affected by the CLT layups and/or span conditions. In this study, we conducted out-of-plane loading tests on 3-layer 4-ply, 5-layer 7-ply, 7-layer 7-ply, and 9-layer 9-ply CLT made of Japanese larch (Larix kaempferi) under various spans and investigated the effect of layups and spans on the out-of-shear strength. The fracture modes of the specimens were classified into three types: shear fracture, shear fracture accompanied by bending fracture, and bending fracture. The out-of-plane shear strength of the specimens except for the 9-layer 9-ply ones decreased as the span increased, and then converged to a constant value (1.0–1.5 kN/mm2). In addition, the shear strength decreased exponentially as the number of laminae in the transverse layers increased and then converged to a constant value (1.0–1.5 kN/mm2). The out-of-plane shear strength of the 9-layer 9-ply specimens decreased as the shear span increased; however, the converged value with a longer span could not be calculated because the tests were conducted under only three-span conditions. The shear strength of 3-layer 4-ply specimens was lower than that of the other layups. The results of the Monte Carlo simulation of the shear strength of the laminae in the transverse layers showed that a model, which assumed that the minimum shear strength of the laminae in the transverse layers determined the shear strength of a specimen, tended to correspond with the decreasing tendency of shear strength with longer spans. The results showed that the weakest link model for the out-of-plane shear fracture of the CLT would relate to a specimen with long span.

The influence of vulnerability on depression among Japanese university athletes.

This study examined the estimated causal relationship between vulnerability and depressive symptoms in Japanese university athletes and how the degree of vulnerability affects depressive symptoms. In Study 1, 248 Japanese university athletes completed a continual survey from Time 1 to Time 3. In Study 2, 562 Japanese university athletes responded to another survey during the same period. Structural equation modeling was performed to estimate the causal relationship using the cross-lagged effects model for the three waves. Next, a binomial logistic regression analysis was performed to examine the influence of vulnerability on depression. Results of the cross-lagged effects model showed that all paths from vulnerability to depressive symptoms were significant, and all paths from depressive symptoms to vulnerability were not significant. Thus, vulnerability was the causative variable and depressive symptoms were the outcome variables within the causal relationship. The logistic regression results showed that those with high vulnerability were 1.7 times more likely to have moderate or higher depressive symptoms than those with low vulnerability. Vulnerable individuals are at a higher risk for developing depressive symptoms. By verifying the causal relationship between vulnerability and depressive symptoms, we can contribute to the enhancement of mental health care in accordance with the weakest link model. Appropriate psychological support for athletes can decrease depression and improve their mental health.

Probabilistic fatigue assessment of notched components under size effect using generalized weakest-link model

Fatigue assessment in the presence of intricate geometries is critical for ensuring reliability and structural integrity of engineering components. This study proposes a generalized weakest-link model for probabilistic life prediction of notched components considering statistical and geometrical size effects, in which the failure probability of materials at different positions is quantified by a weight function of stress field intensity (SFI) approach. In particular, with a new definition of fatigue failure region (FFR), the size effect on notch fatigue behavior is reasonably characterized. Model validation and comparison are performed with experimental data of GH4169 and TC4 notched specimens. Note that the scattering fatigue data points agree well with the model predicted P–S–N curves. Finally, the proposed generalized weakest-link model is applied to fatigue strength assessment of a turbine bladed disk.

Analysis of Roof Damage Distribution Caused by Typhoons and Damage Probability Estimation based on the Weakest Link Model

Analysis of Roof Damage Distribution Caused by Typhoons and Damage Probability Estimation based on the Weakest Link Model

Determination of tensile strength of GFRP bars using flexure tests

This paper presents an experimental and theoretical study on finding tensile strength of glass fibre reinforced polymer (GFRP) reinforcing bars using flexure testing. The tensile strength of these bars is their primary property which must be determined before placing them in concrete members. Direct tensile testing of the bars is tedious and often not practical. In the presented work, the three-point flexure test is proposed as an alternative for direct tensile testing of GFRP bars. The flexure test is quick and easy to setup. The tensile strength in a flexure test is referred to as the modulus of rupture which for brittle materials, is generally higher than tensile strength determined in direct tensile tests. Research presented in this paper shows the correlation of the modulus of rupture to the tensile strength of the GFRP bars. The detail procedures for testing, determination of the cracking load in flexure test, and the subsequent procedures for calculations of tensile strength using the Weibull’s weakest link model are presented and discussed. The experimental part includes flexure tests on bars with diameters ranging from 8 mm to 32 mm, followed by calculations of rupture moduli and correlated tensile strengths. Bars with 8 mm, 13 mm and 16 mm diameters were also tested in direct tensile test and the results were used for comparisons with the correlated tensile strengths from flexure test. The results of this work confirm that flexure testing can provide easy and reliable method for obtaining tensile strength of GFRP reinforcing bars.

Transferability of weakest link model parameters of polysilicon from indentation fracture to failure of MEMS structures

Polysilicon weakest link model parameters obtained from indentation fracture are used to predict failure in MEMS structures. Plain and notched polysilicon micro-cantilevers were fabricated using surface micro-machining and bending tests were carried out till the structure failed catastrophically. To estimate the failure probability of fabricated samples, numerical studies were carried out on models closer to the actual experimental condition, by estimating and incorporating fabrication details such as undercut and roughness. Initially, the undercut inherent to fabrication process was estimated on plain cantilevers by comparing the elastic behavior with experiments. The geometry of open cavities in the sidewall, formed as a consequence of etching, were characterized using SEM micrographs. The stress field obtained from 3D FEM analysis of the notched geometries, including these geometric details, showed great sensitivity to the positioning of cavities in the sidewall relative to the top surface. Applying weakest link theory based local criteria, the probability distribution of failure load of tested structures was generated using weakest link parameters from indentation fracture. Failure loads, thus predicted, were validated with experimental data.

Evaluation of Anti-Adhesion Characteristics of Diamond-Like Carbon Film by Combining Friction and Wear Test with Step Loading and Weibull Analysis.

Anti-adhesion characteristics are important requirements for diamond-like carbon (DLC) films. The failure load corresponding to the anti-adhesion capacity varies greatly on three types of DLC film (hydrogen-free amorphous carbon film (a-C), hydrogenated amorphous carbon film (a-C:H), and tetrahedral hydrogen-free amorphous carbon film (ta-C)) in the friction and wear test with step loading using a high-frequency, linear-oscillation tribometer. Therefore, a new method that estimates a representative value of the failure load was developed in this study by performing a statistical analysis based on the Weibull distribution based on the assumption that the mechanism of delamination of a DLC film obeys the weakest link model. The failure load at the cumulative failure probabilities of 10% and 50% increased in the order ta-C < a-C:H < a-C and ta-C < a-C < a-C:H, respectively. The variation of the failure load, represented by the Weibull slope, was minimum on ta-C and maximum on a-C:H. The rank of the anti-adhesion capacity of each DLC film with respect to the load obtained by a constant load test agreed with the rank of the failure load on each DLC film at the cumulative failure probability of 10% obtained by Weibull analysis. It was found to be possible to evaluate the anti-adhesion capacity of a DLC film under more practical conditions by combining the step loading test and Weibull analysis.

Evolutionary dynamics of the interdependent security games on complex network

With the development of information technology, the infrastructure between enterprises and the connections between businesses show complex network characteristics. The security investment made by an enterprise in a network has an impact on its neighbors, but also bears the impact of its neighbors’ security investments. Such individual interaction problems are often modeled as interdependent security games (IDS). In this paper, we study IDS models under three different attack scenarios: the total effort, the weakest link and the best shot. We use evolutionary game theory to explore the dynamics of social payoffs and the average social investments of these three models under different network topologies. The results show that under the total effort model, individuals are more inclined to increase their investments, while under the weakest link and best shot models, individuals choose to minimize their investments due to selfishness. Finally, we study the cluster effect under different network structures and find the threshold at which it exists.

A weakest link model for multiple mechanism brittle fracture — Model development and application

A multiple mechanism weakest link model for intergranular and transgranular brittle fracture is developed on the basis of experimental observations of a thermally aged low alloy steel. The model development is carried out in tandem with micro mechanical analysis of grain boundary cracking using crystal plasticity modeling of polycrystalline aggregates with the purpose to inform the weakest link model. The fracture modeling presented in this paper is carried out by using a non-local porous plastic Gurson model where the void volume fraction evolution is regularized over two separate length scales. The ductile crack growth preceding the final brittle fracture is well predicted using this type of modeling. When applied to the brittle fracture tests, the weakest link model predicts the fracture toughness distribution remarkably well, both in terms of the constraint and the size effect. Included in the study is also the analysis of a reference material.

A simplified estimation procedure for the Weibull stress parameter, m, and applications to predict the specimen geometry dependence of cleavage fracture toughness

This study describes a simplified procedure to estimate the Weibull stress parameter, m, using cleavage fracture toughness values measured from testing high constraint fracture specimens. The estimation procedure builds upon the conventional toughness scaling methodology (TSM) based on the Weibull stress coupled with the weakest link model to correct measured toughness distributions for high constraint fracture specimens with identical in-plane dimensions but varying thickness. The approach thus requires a correspondence between predictions based upon the Weibull stress methodology and the weakest link model for measured toughness values under high constraint conditions. Applications of the proposed procedure to analyze specimen geometry effects on fracture toughness values in a typical structural steel provide predictions that agree remarkably well with experimental data for shallow crack bend specimens with different thickness. These exploratory studies demonstrate the potential capability of the proposed estimation procedure which can greatly simplify engineering integrity assessments based upon the Weibull stress model.

A probabilistic Haigh diagram for notched components considering notch root plasticity due to high mean stresses

In this work, a probabilistic approach to high cycle fatigue failure probability estimation is presented, considering notched components experiencing elasto-plastic material behavior in the notch root due to high mean stresses. A parameterized weakest link model, constituted by a volume integral over the component stress field, is combined with elasto-plastic finite element simulations to account for plasticity induced stress redistributions around notches. In a calibration/prediction methodology, the considered model is applied to fatigue test results of two different aluminum alloys containing tests on eight differently notched sheet specimen types. The results show good agreement between model predictions and test data.

Predicting the fatigue performance of an additively manufactured Ti-6Al-4V component from witness coupon behavior

The relationship between the reported fatigue performance of an aerospace laser powder bed fused (L-PBF) Ti-6Al-4V component and witness coupons is examined. Traditional deterministic fatigue life models calibrated to witness coupon fatigue data are found capable of predicting the trend in median component fatigue performance, motivating deeper analysis with a probabilistic fatigue life model. A weakest link model is formulated that is applicable to heterogeneous stress states and the transition in failure mode that occurs in Ti-6Al-4V. After calibrating to the witness coupon data, the model is found capable of predicting key trends in the reported component data. After a discussion of the predicted variability and uncertainty, the model is utilized to illuminate how variations in crack initiation site density, e.g. build defects, might impact fatigue performance at the coupon and component level. By comparing to additional literature data, a wide range of sensitivity to initiation site density is illustrated, and subsequently explained via an approximate expression of the model that shows a dependence of sensitivity on the nature of the material and its initiation sites, as well as the level of inspection that is performed.

Level excursion analysis of probabilistic quasibrittle fracture

It has widely been acknowledged that no structures can be designed to be risk free, and therefore reliability analysis plays an essential role in the design of engineering structures. The recent focus has been placed on structures made of brittle heterogenous (a.k.a. quasibrittle) materials, such as ceramics, composites, concrete, rock, cold asphalt mixture, and many more at the microscale. This paper presents a level excursion model for the analysis of probabilistic failure of quasibrittle structures, in which the failure statistics is calculated as a first passage probability. The model captures both the spatial randomness of local material resistance and the random stress field induced by microstructures (e.g. randomly distributed flaws). The model represents a generalization of the classical weakest-link model at the continuum limit and it recovers the classical Weibull distribution as an asymptotic distribution function. The paper discusses two applications of the model. The model is first applied to the strength distribution of polycrystalline silicon (poly-Si) MEMS specimens. It is shown that the model agrees well with the experimentally measured strength distributions of poly-Si MEMS specimens of different sizes. The model predicts a complete size effect curve of the mean structural strength transitioning from a vanishing size effect at the small-size limit to the classical Weibull size effect at the large-size limit. The second application is concerned with the left tail of strength distribution of quasibrittle structures. By taking into account both random stress and strength fields, the model is used to investigate the origin of the power-law tail distribution of structural strength.

A calibrated weakest-link model for probabilistic assessment of LCF life considering notch size effects

This study proposes a calibrated weakest-link model for probabilistic life in low cycle fatigue (LCF) regime considering the geometric size effect and the statistical size effect. Weakest-link model is first extended to LCF regime by using damage parameter, and then calibrated based on crack propagation process. Meanwhile, experiments on standard smooth specimens and central hole plate (CHP) specimens are conducted. To improve model accuracy, parameters in the proposed model are calibrated by using Bayesian inference method. Finally, probabilistic LCF life is predicted for CHP specimens with different scales, which shows good agreement with test data.

A probabilistic Haigh diagram based on a weakest link approach

A statistical methodology for the assessment of high cycle fatigue failure probability is presented. Based on a probabilistic Haigh diagram for a unit volume in combination with a weakest link model, the approach allows failure probability quantification for arbitrarily shaped components. We demonstrate how the model can be calibrated using a maximum likelihood approach for censored data. In order to show its potential, the proposed methodology is applied on three different publicly available data sets containing tests for smooth and notched sheet specimens. In all considered cases the model predictions are in good agreement with the test data.