Weakest Link Hypothesis Research Articles

Testing Weibull as a viable statistical strength distribution for Nacre

Nacre, a composite layer present in sea-shells, exhibits a remarkable combination of toughness, strength, and stiffness through its brick–mortar micro-structure, acting as a template for novel materials. Strength, one of nacre’s important properties is highly variable due to distribution of underlying material’s properties as well as various defects present in its micro-structure. Presently, researchers assume the Weakest-link hypothesis and consequently use the Weibull distribution to model this variability. However, this assumption is theoretically unproven for biological materials such as Nacre and extrapolation of the same to predict rare but catastrophic behavior would be incorrect. In this article, the suitability of Weibull distribution to account for the variability of strength in nacre is tested, using multi-scale models, developed using FEM. Through Monte Carlo based numerical experiments and Renormalization Group (RG) based arguments, it is shown that the Weakest-link hypothesis, which is commonly used to justify the use of the Weibull distribution, does not seem to hold for nacre. Micromechanics and non-local homogenization based distributions such as the one suggested by Luo and Bazant (2019) might be more appropriate for accurate extrapolation to the low probability tail.

Advances in statistical mechanics of rock masses and its engineering applications

To efficiently link the continuum mechanics for rocks with the structural statistics of rock masses, a theoretical and methodological system called the statistical mechanics of rock masses (SMRM) was developed in the past three decades. In SMRM, equivalent continuum models of stress–strain relationship, strength and failure probability for jointed rock masses were established, which were based on the geometric probability models characterising the rock mass structure. This follows the statistical physics, the continuum mechanics, the fracture mechanics and the weakest link hypothesis. A general constitutive model and complete stress–strain models under compressive and shear conditions were also developed as the derivatives of the SMRM theory. An SMRM calculation system was then developed to provide fast and precise solutions for parameter estimations of rock masses, such as full-direction rock quality designation (RQD), elastic modulus, Coulomb compressive strength, rock mass quality rating, and Poisson’s ratio and shear strength. The constitutive equations involved in SMRM were integrated into a FLAC3D based numerical module to apply for engineering rock masses. It is also capable of analysing the complete deformation of rock masses and active reinforcement of engineering rock masses. Examples of engineering applications of SMRM were presented, including a rock mass at QBT hydropower station in northwestern China, a dam slope of Zongo II hydropower station in D.R. Congo, an open-pit mine in Dexing, China, an underground powerhouse of Jinping I hydropower station in southwestern China, and a typical circular tunnel in Lanzhou-Chongqing railway, China. These applications verified the reliability of the SMRM and demonstrated its applicability to broad engineering issues associated with jointed rock masses.

A Flexible HCF Modeling Framework Leading to a Probabilistic Multiaxial Kitagawa-Takahashi Diagram

This article describes a flexible modeling framework which leads to the construction of a probabilistic, multiaxial Kitagawa-Takahashi diagram. This framework has been developed following experimental observations that clearly indicate that two independent fatigue damage mechanisms can be activated, at the same time, in metallic materials. Specifically, one damage mechanism is associated with crack initiation and the other with crack arrest. It is postulated that these damage mechanisms are more appropriately modeled using two different fatigue criteria or, more specifically, two completely different approaches to fatigue (i.e. a classical multiaxial fatigue criterion and a LEFM type criterion). Hence, the proposed modeling framework provides the possibility of combining any two suitable criteria, in a probabilistic framework based on the weakest link hypothesis and results in the continuous description of the Kitagawa diagram for any multiaxial stress state. It is shown that under certain conditions this approach is equivalent to the classical El Haddad approach to the short crack problem encountered in LEFM. However, the proposed framework is easily extended to multiaxial loading conditions. This modeling framework is demonstrated in detail via its application to multiaxial fatigue data for data taken from the literature.

A mechanistic approach to the Kitagawa–Takahashi diagram using a multiaxial probabilistic framework

The aim of this paper is to propose a flexible multiaxial modelling framework that is capable of combining two fatigue damage mechanisms so as to continuously describe the Kitagawa–Takahashi diagram. It is proposed that this diagram represents two distinct fatigue damage mechanisms: one associated with crack initiation (or microstructurally small cracks) and the other with crack propagation (or long cracks). It is further postulated that these damage mechanisms are more appropriately modelled using different fatigue criteria. A probabilistic modelling framework is proposed in which any two suitable fatigue criteria can be combined in order to simultaneously model both damage mechanisms and the transition between them. This framework is based on the weakest link hypothesis and results in a probabilistic Kitagawa–Takahashi type diagram.

Fracture Strength of Disordered Media: Universality, Interactions, and Tail Asymptotics

We study the asymptotic properties of fracture strength distributions of disordered elastic media by a combination of renormalization group, extreme value theory, and numerical simulation. We investigate the validity of the "weakest-link hypothesis" in the presence of realistic long-ranged interactions in the random fuse model. Numerical simulations indicate that the fracture strength is well-described by the Duxbury-Leath-Beale (DLB) distribution which is shown to flow asymptotically to the Gumbel distribution. We explore the relation between the extreme value distributions and the DLB-type asymptotic distributions and show that the universal extreme value forms may not be appropriate to describe the nonuniversal low-strength tail.

A micromechanical interpretation of the temperature dependence of Beremin model parameters for french RPV steel

Local approach to brittle fracture for low-alloyed steels is discussed in this paper. A bibliographical introduction intends to highlight general trends and consensual points of the topic and evokes debatable aspects. French RPV steel 16MND5 (equ. ASTM A508 Cl.3), is then used as a model material to study the influence of temperature on brittle fracture. A micromechanical modelling of brittle fracture at the elementary volume scale already used in previous work is then recalled. It involves a multiscale modelling of microstructural plasticity which has been tuned on experimental inter-phase and inter-granular stresses heterogeneities measurements. Fracture probability of the elementary volume can then be computed using a randomly attributed defect size distribution based on realistic carbides repartition. This defect distribution is then deterministically correlated to stress heterogeneities simulated within the microstructure using a weakest-link hypothesis on the elementary volume, which results in a deterministic stress to fracture. Repeating the process allows to compute Weibull parameters on the elementary volume. This tool is then used to investigate the physical mechanisms that could explain the already experimentally observed temperature dependence of Beremin’s parameter for 16MND5 steel. It is showed that, assuming that the hypothesis made in this work about cleavage micro-mechanisms are correct, effective equivalent surface energy (i.e. surface energy plus plastically dissipated energy when blunting the crack tip) for propagating a crack has to be temperature dependent to explain Beremin’s parameters temperature evolution.

Depressogenic inferential styles, negative events, and depressive symptoms in youth: An attempt to reconcile past inconsistent findings

Research examining the diathesis-stress component of the hopelessness theory (HT) in youth has provided mixed support for the theory. One explanation for inconsistent findings is the failure to consider relationships among the three inferential styles posited to serve as vulnerability factors by the theory. The weakest link hypothesis posits that an individual is as vulnerable to depression as his/her most depressogenic inferential style (DIS) makes him/her. A second explanation is that researchers have operationalized “high stress” from a nomothetic, as opposed to an idiographic, perspective. The current study tested the diathesis-stress component of the HT in 230 third-grade and 152 seventh-grade schoolchildren using both a weakest link and an idiographic approach towards analysis. Children completed measures assessing depressive symptoms and DISs. Children subsequently completed measures assessing depressive symptoms and negative events once a week for 6 weeks. Hierarchical linear modeling and time-lagged analyses indicated that a depressogenic weakest link was associated with greater elevations in depressive symptoms following elevations in stress.

The weakest link hypothesis for adaptive capacity: An empirical test

Yohe and Tol (2002. Global Environmental Change 12, 25–40) built an indexing method for vulnerability based on the hypothesis that the adaptive capacity for any system facing a vector of external stresses could be explained by the weakest of its underlying determinants—the so-called “weakest link” hypothesis. Their structure noted eight determinants, but the approach could handle any number. They quoted analogies in support of the hypothesis, but loose inference is hardly sufficient to confirm such a claim. We respond to this omission by offering an empirical investigation of its validity. We estimate a structural form designed to accommodate the full range of possible interactions across sets of underlying determinants. The perfect complement case of the pure “weakest-link” formulation lies on one extreme, and the perfect substitute case where each determinant can compensate for all others at constant rates is the other limiting case. For vulnerability to natural disasters, infant mortality and drinking water treatment, we find qualified support for a modified weakest link hypothesis: the weakest indicator plays an important role because other factors can compensate (with increasing difficulty). For life expectancy, sanitation and nutrition, we find a relationship that is close to linear—the perfect substitute case where the various determinants of adaptive capacity can compensate for each other with relative and persistent ease. Moreover, since the factors from which systems derive their adaptive capacities are different for different risks, we have identified another source of diversity in the assessment of vulnerability.

Analyse d'un mode de vieillissement dans un assemblage céramique/métal

Lors de leur cycle de fabrication, des contraintes residuelles sont generees notamment dans la partie fragile d'assemblages ceramique/metal. Ces dernieres sont responsables de l'amorcage et de l'arret des fissures generees par ce cycle, mettant en defaut les traitements probabilistes de la rupture bases sur l'hypothese du maillon le plus faible. Un modele probabiliste de fissuration est propose et permet d'apprehender de tels arrets. Une extension de ce modele considere la fiabilite des assemblages soumis a de la fatigue statique. Une methodologie d'identification des parametres aleatoires microstructuraux est proposee. La demarche est ensuite appliquee a la prevision de la duree de vie d'un composant industriel.

Long-Term Reliability of Brittle Materials: The Issue of Crack Arrest

In structures containing brittle materials, residual and/or heterogenous stresses may prevent cracks to propagate up to failure. Consequently, for such structures, crack arrest has to be accounted for and a weakest link hypothesis may not be applicable. A probabilistic crack propagation model is derived to describe instantaneous or delayed arrest phenomena. A time-dependent regime is induced by slow crack growth experienced by ceramics and glasses. A general expression is obtained in which instantaneous up to infinite propagation times can be modeled in a unified way. The results are illustrated on a case study dealing with propagation of cracks in a thin walled tube submitted to a temperature gradient through its thickness. Different types of propagation/arrest regimes can be identified.

On the probabilistic–deterministic transition involved in a fragmentation process of brittle materials

Dynamic loadings produce high stress waves leading to the fragmentation of brittle materials such as ceramics, concrete, glass and rocks. The main mechanism used to explain the change of the number of fragments with stress rate is a shielding phenomenon. However, under quasi-static loading conditions, a weakest link hypothesis may be applicable. Therefore, depending on the local strain or stress rate, different fragmentation regimes are observed. One regime corresponds to single fragmentation for which a probabilistic approach is needed. Conversely, the multiple fragmentation regime may be described by a deterministic approach. The transition between the two fragmentation regimes is discussed for high performance concrete, glass and SiC ceramics.

On crack arrest in ceramic / metal assemblies

Brittle parts of ceramic/metal assemblies are subjected to a residual stress field generated by the fabrication. During that process, cracks are initiated and the key question is whether they propagate through the whole brittle part. The use of classical probabilistic fracture models applied to the ceramic (i.e., based on a weakest link hypothesis), allow one to conclude that cracks are likely to initiate after the manufacturing process. Consequently, a crack arrest model is proposed, based on a random toughness distribution. Applied to micro-hardness experiments, the statistical parameters are identified, and the predictive capacity of the model is analyzed. The model is then used to study the reliability of ceramic/metal assemblies during the fabrication stage.

Cognitive Vulnerability to Hopelessness Depression: A Chain Is Only as Strong as Its Weakest Link

The hopelessness theory proposes that depressogenic inferential styles about the self, consequences, and causes serve as vulnerability factors to hopelessness depression. Previous research on the diathesis–stress component of the theory has examined each inferential style separately. Doing so, however, is likely to provide an inaccurate test of the theory because individuals who possess only one depressogenic inferential style but who show increases in symptoms following negative events will appear to support the theory in some analyses but to contradict the theory in others. The weakest link hypothesis posits that an individual is as vulnerable to hopelessness depression as her most depressogenic inferential style makes her. Therefore, according to this hypothesis, when testing the theory, researchers should determine each participant's degree of vulnerability using her most depressogenic inferential style. The goal of the current study was to test the diathesis–stress and symptom components of the hopelessness theory in seventh grade children using both the traditional and weakest link approaches. Seventy-nine children completed measures of depressogenic inferential styles and depressive symptoms (Time 1). Ten weeks later (Time 2), they completed measures of depressive symptoms and negative events. Neither depressogenic inferential styles about the self, consequences, nor causes interacted with negative events to predict increases in symptoms of hopelessness depression. In line with hypotheses, however, children's “weakest links” interacted with negative events to predict increases in hopelessness depression symptoms. In addition, in line with the symptom component of the theory, this interaction did not predict increases in nonhopelessness depression symptoms.

Prediction of toughness scatter in heat affected zones (HAZ) with local approach

Carbon-manganese (C-Mn) steels are widely used in welded structures. For low temperature applications, it is essential to maintain a good toughness level in the welds. In this paper, the “local approach” for cleavage fracture is used based on the weakest link hypothesis. This kind of analysis has already been applied with success to homogeneous materials and in particular to the base metal and the weld metal of structural steels. By contrast, it has never been applied to the case of heterogeneous materials such as welds. The adopted test plan included four phases of increasing complexity. First, the approach was validated in very simple configurations, where it was possible to identify without ambiguities the mechanical properties of the different zones. Then more complex welds were studied. The first part of this study included measurement of mechanical properties and local criteria of the different zones [base metal and simulated heat affected zone (HAZ)]. In the second part, a brittle zone of well controlled dimensions was produced with an electron beam. The theoretical predictions match with experimental results. The third part was a validation attempt on a simple monopass weld. Single edge notch bend (SENB) specimens with a notch in the HAZ were tested in order to measure toughness ( K, J, CTOD). Results from the first part allowed comparison of experimental results with those predicted by the local approach. Although the real configuration of the electroslag welding (ESW) was more complex than had been expected, the local approach predictions were consistent with the experimental results. The objective of the fourth part was to study a multipass weld. As in the second part, the toughness measured on SENB specimens. The toughness in multipass HAZ decreases markedly at first and then very slowly when the percentage of coarse grains intercepted by crack front increases. The predictions agree fairly well with the experimental results. This study shows that local criteria allow a realistic prediction of the fracture toughness variation not only with temperature, but also as a function of brittle zones intercepted by the crack front. The scatter is also well predicted.

Reliability analysis of brittle systems considering several random variables

A previously proposed method for the reliability analysis of brittle structures subjected to thermal shocks is extended to include the uncertainties about the magnitudes of the applied thermal shocks and the fracture strength of the brittle materials. Flaws with a given probability distribution of size are assumed to be distributed at random with a certain density per unit volume of the composed material. A probabilistic failure model in which all kinds of uncertainties can be included is presented to study the probability of crack instability of a unit volume of brittle material. The reliabilities of brittle thermal systems are evaluated based on the weakest-link hypothesis and the proposed probabilistic failure model. A numerical example is given to demonstrate the application of the present method.

Reliability analysis of brittle structures under thermal shocks

A method for the reliability analysis of brittle structures subjected to thermal shocks is presented. Flaws with a given probability distribution of size are assumed to be distributed at random with a certain density per unit volume in the structures. Criteria for crack instability are derived for brittle solids under general thermal stresses. A probabilistic failure model is presented to study the probability of crack instability for brittle solids containing cracks with uncertain crack size. The reliabilities of the brittle structures are evaluated based on the weakest-link hypothesis, which states that a structure fails when the cracks in any differential volume become unstable. A numerical example is given to demonstrate the application of the proposed method.

Failure modes of hybrid composites consisting of carbon fibre bundles dispersed in a glass fibre epoxy resin matrix

Tests have been conducted on single carbon fibres, bundles of 1000 carbon fibres impregnated with epoxy resin and on a glass/carbon hybrid composite containing similar bundles interspersed in glass/epoxy resin. The influence of the bundle dispersion on the strength of the carbon bundles is discussed and suggestions for the optimum dispersion are made. The behaviour with respect to the weakest-link hypothesis is also discussed and an explanation is proposed for the observed deviation from the theory for the bundles.

Reliability analysis of brittle structures under random loadings

A method for the reliability analysis of brittle structures subjected to random loads is proposed. The method is based on the weakest-link hypothesis and Weibull statistics for brittle materials. Initial flaws with a given expected size are assumed to be distributed at random with a certain density per unit volume. Basic concepts in random vibration theory and fracture mechanics are utilized in evaluating stress statistics, crack propagation and strength degradation. A structure fails when the stress intensity at any flaw reaches a critical value for rapid crack propagation. The failure of the structure is modeled as the first exceedance in random vibration theory. The effects of multi-vibration modes on the failure probability of the structure are included in the formulation. The evaluation of stress distribution and the computation of failure probability can be accomplished in a finite element analysis. Numerical examples on the evaluation of lifetime reliabilities of structures are given to demonstrate the feasibility of the proposed method.

Probabilistic simulation of fatigue crack growth by damage accumulation

A fatigue crack growth simulation model based on probabilistic damage accumulation is presented. The modified weakest-link hypothesis determines the reinitiation of voids ahead of the crack, which in turn determines the crack advance. From a linearly cumulated-damage standpoint, this is shown to be approximated by a statistical Miner's law. The historical damage on the material elements at the crack-tip vicinity, neglected in previous analysis, is quantified, and shown to be significant. The model includes a prediction of the fatigue threshold stress-intensity factor range, and a comparison with published crack growth rate data for three different structural metals.

On the estimation of the Weibull modulus

It is well documented that nominally identical specimens of brittle materials, e.g. ceramics, show a large variation of tensile fracture stresses and in order to use brittle materials as engineering materials the strength has to be characterized. The most widely used expression for characterization is the cumulative distribution function proposed by Weibull [ 1 ]. The Weibull function is also known to statisticians as Fisher-Tipper Type Ill distribution of smallest values or as the third asymptotic distribution of smallest extreme value [2]. The Weibull statistics is based on the the "weakest link-hypothesis" which means that the most serious flaw in the specimen will control the strength. The most serious flaw is not necessarily the largest one because its severity also depends on where it is situated. In other words, the flaw which is subjected to the highest stress intensity factor will be strength controlling. The flaws initiating fracture can conveniently be classified as intrinsic or extrinsic [3]. The intrinsic flaws are introduced during fabrication and are predominantly inclusions and voids. The extrinsic flaws are stress-induced cracks, such as surface cracks introduced during machining and microcracks resulting from large residual stresses, e.g. due to thermal contraction