Weakest link model

Whereas in the second approach of the size effects it is also assumed that fracture is controlled by defects, the strength is now considered a statistically distributed parameter rather than a physical property characterised by a single value. The statistical distribution of fibre strength is usually described by the Weibull model [22,23]. In this weakest-link model the strength distribution of a series arrangement of units of length L0 is given by... 

When discrepancies between the strength predicted by the conventional Weibull model and the experimentally obtained data occur, then multimodal Weibull distributions have been used [18, 37]. Similarly, as mentioned earlier, the Weibull/ weakest-link model can be utilized to predict the scale effect by combining the classical Weibull distribution with the weakest link theory. All these take into account the shape and scale of the distribution. Such models are reported to have shown improved accuracy of prediction including those for wool fibers [38]. 

Thus, in recent years, (8.1) and (8.4) are being used for both uniformly brittle materials as well as polymeric fibers [38]. For instance, (8.4) for WeibuU/weakest-link model was verified for wool fibers [38]. 

Because the probability of finding a flaw of a given severity increases with increase in the volume or surface of a fiber sample, the average strength tends to decrease with increase in length or diameter. Using the weakest link model, the cumulative probability of failure of a fiber can be shown to be (Hunt and McCartney, 1979) ... 

In 1951, Weibull proposed the weakest link model , and gave corresponding distribution functions on the basis of the model. The model is formulated using a chain link under symmetrical tension-stress as shown in Fig. 8.35. The chain has n links, and the whole chain would disconnect if any one link is broken under the action of the a stress. The broken strength of each link follows a uniform distribution. The critical defect or crack is the smallest crack or flaw with the least stress. In a brittle material, the critical defect or crack for a given stress distribution is the weakest link. A brittle material can be regarded as a material with n defects or cracks which are independent of each other, with the cracks under a uniform distribution of damage stress. Under the effect of an external force, the invalidation of the material... 

Paramonov, Y. Andersons, X A family of weakest link models for fiber strength distribution. Composites A Appl. Sci. Manufact. 38 (2007), pp. 1227-1233. 

Models for mechanism competition Competition is an interaction relation among two or more failure mechanisms where each mechanism contribute to a specific performance parameter and the presence of one mechanism has little or no influence on the others, as shown in Figure 2(a). Thus, the interactions among failure mechanisms can be modeled by the weakest-link model, where the failure mechanism first evolves to failure will determine the failure behavior of the component. If the failure mechanism models takes the form of equation (1) (which means, the performance parameters will vary over time), the weakest-link model can be expressed as equation (3). 

The meaning of each parameter is consistent with those in equation (4) and (5). In an actual composite ply, the three mechanisms are independent from each other and the weakest link model can be used to model the interactions among them, as shown in equation (7). 

Weibull distribution function is one of the widely used cumulative probability functions for predicting lifetime in reliability test [34]. This is because it can easily approximate the normal distribution, logarithmic normal distribution and exponential distribution functions. In addition, it is also possible to analyze data even when two or more failure modes are present at the same time. The cumulative probability F(t) of a failure system can be introduced just as Weibull distribution function based upon a weakest-link model [34], which is expressed as ... 

Size Distribution Relationships. Different models have been used to describe the size distribution of particles experiencing single and multiple fractures. A model based on fracture at the site of the weakest link and a distribution of weakest links in the system gave results that could be described as well by the Rosin-Rammler relation (56). The latter is based on the concept that fracture takes place at pre-existing flaws that are distributed randomly throughout the particle. 

A mathematical model of the hydrate reservoir typically requires several minutes to days to execute and costs typically US 10-100. Even with these low costs, unless the model is based upon extensive laboratory and field data, the model will have the weakest link of the three methods to physical reality. 

Recognizing that the continuum solvent calculations are the weakest link in pKa calculations, Ho and Coote [3] used the CPCM (with UAKS and UAFH radii), SM6, IPCM, and COSMO-RS models to predict pKa values for a common data set of neutral organic and inorganic acids. They used four different thermodynamic cycles, and in general found that the COSMO-RS, CPCM, and SM6 models worked best depending on the thermodynamic cycle used. We turn to a discussion of thermodynamic cycles in the next section. 

Turbulence closure models for the unknown terms in the flux- and variance equations for single phase reactor systems are discussed extensively by Baldyga and Bourne [5] and Fox [49] [50]. However, the application of these model closures should be treated with care as most of these parameterizations are expressed in terms of the turbulent eddy and other scalar dissipation rates (and dissipation energy spectra) which are about the weakest links in turbulence modeling. The predictive capabilities of the Re3molds averaged models... 

One can thus state that the constitutive equations for the interfacial terms are the weakest link in a multi-fluid model formulation because of considerable difEculties in terms of experimentation and modeling on a macroscopic level. The main difEculties in modeling arise from the existence of interfaces between phases and discontinuities associated with them. 

In summary, one of the weakest links in modeling reactive systems operated in bubble columns is the fluid dynamic part considering multi-phase turbulence modeling, interfacial closures, and especially the impact and descriptions of bubble size and shape distributions. For reactive systems the estimates of the contact areas and thus the interfacial mass transfer rates are likely to contain large uncertainties. 

Information about persistence is essential for the environment risk assessment of chemical substances. Persistence is needed as input for all predictive approaches, from simple leaching indexes to more complex models. Nevertheless, the availability of reliable persistence data is, at present, the weakest link in the prediction of the environmental fate of chemicals. 

Second, ABS may offer an improved control paradigm that can be implemented at the hardware level. With centralized control, infrastructure systems are vulnerable to the weakest link distribnted control can hmit, localize, and allocate risk. Some models have been proposed whereby infrastructure system agents conld antomatically reconfigure a system to heal failures (Amin, 2000). Distribnted control also enables distribnted power generation, as well as the control of mnltiple infrastructure systems. 

Weibull [54] introduced a model that used the weakest link principle, where the... 

In the sections 4 and 5 a variety of failure modes for flexible displays is presented. Knowledge of at least the dominant failure modes is required for the development of reliability models and scaling rules. These types of models, applied to brittle materials, like glass and ceramics, use weakest link statistics, in many cases Weibull statistics. 

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