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Abstract

Advanced ceramics exhibit brittle behavior. The lack of ductility and the presence of flaws and defects of different sizes and orientations lead to scatter in failure strength. This variability depends also on the specimen size, stress distribution and stress state. The Weibull theory explains correctly this dependence [1], so the fracture strength of ceramic materials has been studied using the Weibull statistic [2][3][4][5] as recommended by ASTM Standards for reporting uniaxial strength data of these materials [6]. The Weibull statistic is also applicable to describe the scatter of the fracture toughness of steels in the ductile-brittle transition region, where failure occurs by cleavage [7][8][9][10].The two-parameter Weibull distribution function is given by:where F is the probability of rupture under an applied uniaxial tensile stress σ , m is the shape parameter or Weibull modulus and σ 0 is the scale parameter. A subject of great interest in this field is the determination of the σ value, σ p , corresponding to a predefined probability of failure p, i.e. the σ p values such thatThese values coincide with the percentile of the distribution and can be obtained as:From the estimate values (σ 0 andm) of the true values of the Weibull parameters, σ 0 and m, computed from a set of tests, an estimationσ p of σ p can be calculated by:σFor percentile points, combining Equations 3 and 4, the following relation is obtained [11]:In a previous paper, Fernández-Sáez et al. [11] pointed out that the (1 − α) confidence intervals for σ p have thewhere c 1 and c 2 fulfils the following relationships:andThese equations show the importance of the variablem ln(σ p /σ p ) in determining the percentiles of the Weibull distribution and it may be estimated from estimations of the distribution parameters m and σ 0 . The aim of this work is to select the best procedure to estimate the percentage points of this variable.Several procedures for the determination of Weibull parameters have been proposed in the literature. The most commonly used are:(1) The maximum-likelihood method, according to which the estimation of parameters should satisfy the following equation:McCool [12] showed that the maximum likelihood equation (

Kinetic Analysis of Isothermal Decomposition Process of Sodium Bicarbonate Using the Weibull Probability Function—Estimation of Density Distribution Functions of the Apparent Activation Energies

“…Namely, for every tasked operating temperature (T), we always know the values of time (t) during the isothermal experiment until the end of a given observation. Because of this fact, we can apply the linear regression method [30] for the estimation of the Weibull distribution parameters at the considered operating temperature, in the range of probability values (conversions, a(t)), which show independent behavior on E a 's. The estimates obtained from the linear regression method (LRM) may be taken as the final estimates when the data size is small.…”