Reliability strength design through inverse distributions—exponential and Weibull cases
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Weibull modulus
In this study, the Weibull distribution of five (5) hydroxyapatite (HAp) samples produced from a novel mixture of naturally derived biowastes using the sol-gel method is reported. Compressive strength values were used to estimate the Weibull parameters. The structural, morphological, and mechanical properties of the as-produced HAp materials were investigated. The results revealed the typical hexagonal crystal structure of HAp materials. Data obtained from the Weibull analysis are comparable to the data in the literature and show how an increase in porosity is related to the Weibull modulus (m). The C100 sample showed the highest Weibull modulus.
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Weibull characteristic strength and modulus were studied as a function of machining severity and orientation. The characteristic strength decreased as machining conditions were more aggressive, and longitudinally machined specimens were consistently stronger than transversely machined specimens. The Weibull modulus did not vary systematically with machining conditions, but after pooling data from sets machined under identical conditions, a low Weibull modulus and a poor fit to the Weibull distribution was obtained. The results showed that the lack of reliability of the machining process effectively decreased the Weibull modulus, a critical parameter to be used in the life prediction of machined components.
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Strength variation of alumina ceramics formed by dry-pressing of granules was examined by the number density of coarser defects. The Weibull modulus m was related to the exponent n of the power function of the number density of coarser defects by strength simulation; m = 1.82n - 1.92. The Weibull modulus was determined by a series of strength data, which were calculated by the simulated bending tests of the samples. The number density of coarser defects was given by the assumed power function. The various Weibull moduli were obtained for some kinds of series. The experimental results of the number density of coarser defects by direct observation of alumina ceramics and the Weibull modulus from strength tests were plotted well on the calculated line except a series of sample containing coarser defects with ambiguous boundaries. The relationship between the Weibull modulus and the parameter of the power function concurred with past studies; m = 2n - 2.
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In this note we give some recurrence relations satisfied by single and product moments of k-th upper record values from the exponential-Weibull lifetime distribution. Using a recurrence relation for single moments we obtain a characterization of exponential-Weibull, Weibull, exponential, Rayleigh and two parameter linear failure rate distributions.
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Weibull modulus
Exponentiated Weibull distribution
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Goodness of fit
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A Weibull space is defined by three coordinates of failure probability, effective volume and strength. The two-parameter Weibull distribution function is presented by a plane (Weibull plane) in the Weibull space. The Weibull modulus can be obtained from the gradient of the Weibull plane. Therefore, the Weibull modulus can be evaluated by the method of least squares from the strength data obtained by various tests of different effective volume specimens. The Weibull modulus estimated by this method is more accurate than that by the conventional Weibull plot method. The Weibull modulus estimated by this method is expected to be twice as accurate as that by the conventional method, particularly when the strength data are obtained from two different volumes whose ratio is more than 100.
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Exponentiated Weibull distribution
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This paper mainly introduce how to use Weibull distribution function to decide the formula of ceramic metallization.The experimental results show that the formula of metallization should be decided by average sealing strength and Weibull modulus.
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With respect to the application of the three-parameter Weibull distribution to the reliability analysis of ceramics under multi-axial stress, the following two fundamental problems are studied theoretically; (1) whether the three-parameter Weibull distribution function holds through various different stress states without contradiction, and (2) whether the shape parameter (Weibull modulus) takes the same value independent of the stress states even if the three-parameter Weibull distribution, it is shown that the same type of distributions can be derived for the uniaxial stress state and the equibiaxial stress state. However, the shape parameter takes different values depending on the stress state. The location parameter is the same independent of the stress state.
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