The validity of Weibull estimators-experimental verification

A sufficiently large number of bending tests of a recrystallized silicon carbide ceramic was performed, which gave a fundamental set of strength values. From this fundamental set, arbitrary subsets of size M were chosen by a Monte-Carlo procedure and the parameters of the two-parametric Weibull distribution were calculated by the maximum likelihood method. The dependence of the statistical distribution of the two parameters, obtained by this procedure, on the size of the subset M was investigated. It was found that the variation coefficient of the scale parameter could be well described by the equations given in the literature, whereas the variation coefficient of the Weibull modulus was much higher. It has been shown that the reason for this behaviour is that the distribution of flaws and therefore the strength of the material does not perfectly obey the Weibull statistics, for which the theoretical equations were derived. Thus, in real ceramics the numerical value for the Weibull modulus obtained from a certain number of experiments is even more indeterminate than described by the theoretical solution. This revised version was published online in November 2006 with corrections to the Cover Date.     

An improved failure‐free time estimation method

In this paper a simple order‐statistics‐based iterative procedure for estimating the failure‐free time is put forward. The method is applicable when the population lifetime follows the Weibull, C‐Weibull or Gompertz–Makeham distribution. Copyright © 1999 John Wiley & Sons, Ltd.     

Statistical models for the analysis of fracture toughness test data

The analysis of fracture toughness test data should provide distributional parameter estimates for the population of toughness predictions for use in integrity assessments. Standard statistical techniques are available which provide such information where the appropriate distributional form is normal but not where a skewed distribution (e.g. Weibull) is applicable. A procedure to analyse data for a Weibull distribution fit has been derived and is discussed. In this approach it is important to use the correct type of regression analysis. The assumptions of the statistical model, on which the analysis of multispecimen toughness data is based, are considered in the light of possible mechanisms of ductile crack growth. The implications of these mechanisms for the choice of regression analysis are discussed, and an assessment of the regression model based on the analysis of residuals is given.     

Incorporation of statistical length scale into Weibull strength theory for composites

In this paper an extension of Weibull theory by the introduction of a statistical length scale is presented. The classical Weibull strength theory is self-similar; a feature that can be illustrated by the fact that the strength dependence on structural size is a power law (a straight line on a double logarithmic graph). Therefore, the theory predicts unlimited strength for extremely small structures. In the paper, it is shown that such a behavior is a direct implication of the assumption that structural elements have independent random strengths. By the introduction of statistical dependence in the form of spatial autocorrelation, the size dependent strength becomes bounded at the small size extreme. The local random strength is phenomenologically modeled as a random field with a certain autocorrelation function. In such a model, the autocorrelation length plays the role of a statistical length scale. The focus is on small failure probabilities and the related probabilistic distributions of the strength of composites. The theoretical part is followed by applications in fiber bundle models, chains of fiber bundle models and the stochastic finite element method in the context of quasibrittle failure.     

The size effects on the mechanical behaviour of fibres

The size of a fibre affects its mechanical properties and thus is of theoretical and practical importance for studies of the rupturing process during loading of a fibrous structure. This paper investigates the overall effects of length on the mechanical behaviour of single fibres. Four types of fibres, ranging from brittle to highly extensible, were tested for their tensile properties at several different gauge lengths. Different from most previous studies where the focus has been on the gauge length effects on a single property such as fibre strength or breaking strain, this paper look comprehensively into the effects of length on all three of the most commonly studied mechanical properties, namely strength, breaking strain and initial modulus. Particular emphasis is placed on initial modulus and on the interactions between all three parameters. Influences of strain rate and fibre type on the size effects are also investigated. The effect of potential fibre slippage on experimental error is examined. An image analysis method is used to measure the real fibre elongation in comparison to the same fibre elongation obtained directly from an Instron tester. Finally, a statistical analysis is carried out using the experimental data to test the fitness of the Weibull theory to polymeric fibres. This was done as the Weibull model has been extensively utilized in examining fibre strength and breaking strain, although it is supposed to be valid only for the so-called classic fibres to which more extensible polymeric fibres do not belong. This revised version was published online in November 2006 with corrections to the Cover Date.     

Reliability of the Weibull analysis of the strength of construction materials

The breaking force during transversal loading of fibre-cement corrugated roofing sheets was measured on several test samples from a serial production. The results were statistically analyzed assuming the 2-parameter Weibull statistics. In addition, Monte Carlo statistical simulations were made by using a computerised built-in random-number generator. While smaller sample data groups, mostly containing up to 50 samples, were studied in the literature, we extended their size up to 400 samples. We showed that some trends in the evaluation of statistical parameters which hold for smaller data groups, apply well to larger data groups. In particular, we confirmed that the statistical distribution of the Weibull parameters obtained from repeated Monte Carlo simulations is log-normal. Furthermore, we considered the influence of the measurement uncertainty on the statistical parameters.     

Weibull modulus of fracture strength of toughened ceramics subjected to small-scale contacts

The Weibull modulus of toughened ceramics was evaluated. The fracture of the toughened ceramics was assumed to be initiated by surface cracks induced during small-scale contacts such as grinding and polishing and an exponential function was selected to describe the R-curve behavior. Based on a brief theoretical analysis, a numerical simulation procedure was designed to predict the fracture strength for toughened ceramics with different R-curve characteristics. The Weibull modulus of each toughened ceramic was estimated and compared with that of the un-toughened base material. It was concluded that an increase in Weibull modulus can always result from toughening. The increase in Weibull modulus was found to be related directly to the relative crack tolerance, i.e., the ratio of the initial crack size to the critical crack size. This suggests that the improvement in crack stability due to toughening is the main reason for the increased Weibull modulus.     

Effect of experimental sample size on local Weibull assessment of cleavage fracture for steel

This paper examines the sample size of the experimental datasets in calibrating the Weibull parameters for the modified three‐parameter Weibull stress framework, so as to enhance the experimental strategy for cleavage assessment of ferritic steels based on a local approach. The present work generates a large number of random and independent subsets from the ‘Euro steel’ fracture toughness database for the calibration procedure. The calibration of the Weibull parameters utilizes a subset of high‐constraint specimens and a subset of low‐constraint specimens from the Euro steel database to resolve the uniqueness issue in the calibration procedure reported in previous studies. This investigation reveals strong dependence of the calibrated Weibull modulus on both the constraint differences between the high‐constraint subset and the low‐constraint subset and the size of the selected subsets. The scale fracture toughness value, however, does not exhibit significant dependence on the constraint difference between the two subsets of specimens. The confidence level of the scale fracture toughness, nevertheless, still exhibits strong dependence on the sample size of the experimental data.     

A new method for selection of population distribution and parameter estimation

An optimization method for parameter estimation is presented with the Kolmogorov-Smirnov distance used as the objective. A step-by-step implementation procedure is given. The method is demonstrated by estimating the parameters for three-parameter Weibull distributions from three different samples (with different sample sizes). A comparison of the proposed method and the usual methods such as the least-squares method, the matching moments method and the maximum likelihood method shows that more reasonable estimates of the parameters are given by the proposed optimization method. Then, the proposed method is successfully extended to estimate the parameters for the sum of two three-parameter Weibull distributions. Based on these findings, a new procedure for selection of population distribution and parameter estimation is presented.     

Theoretical aspects of the statistical variation of strength

A modified version of Weibull's statistical theory of the strength of brittle materials is proposed, in which the expression for failure probability contains an additional term. While this term is negligible when failure originates from a flaw of relatively large size, it becomes increasingly significant as the flaw size is reduced. The resulting revised expressions for failure probability under uniform, uniaxial tension and under Hertzian indentation loading are given, and the effect of a bimodal flaw size distribution is considered in both cases. The implications with regard to the assumed invariance of Weibull statistical parameters under different experimental conditions are discussed.     

Predicting Hertzian fracture

The procedure for calculating the probability of initiation of a Hertzian cone crack as a function of indenter load and indenter radius is demonstrated and the results compared with experimental data. Such a procedure brings together the energy balance and flaw statistical explanations of Auerbach’s law. The method relies on the application of Weibull statistics in the diminishing indentation stress field. It is shown how strength parameters obtained from bending tests on bulk specimens may be used within the analysis for predicting the presence of surface flaws which lead to the initiation of a cone crack. The procedure is shown to apply to both cylindrical punch and spherical indenters. This revised version was published online in November 2006 with corrections to the Cover Date.     

Bayesian Weibull reliability estimation

The Weibull distribution is widely used as a failure model, particularly for mechanical components. This distribution is rich in shape and requires a fairly large sample size to produce accurate statistical estimators, particularly for the lower percentiles, as is usually required for a reliability analysis. In practice, sample sizes are almost always small and subjective judgement is applied, aided by a Weibull plot of the test data to determine the adequacy of the component design to meet reliability goals. The procedure is somewhat ad hoc, but apparently reasonably good results are obtained based on our experience with many past design and development programs and by comparison with actual field performance. We conjecture that the reason this procedure is successful is that test programs and methodology are standardized and quite well documented, from the standpoint of the physical test parameters. Test personnel have a wealth of experience in testing components, from one program to the next, and reliability judgements are made with regard to well-known points in the product's life. All of these factors tend to promote correct outcomes from the decision process even though sample sizes are small.

The Bayesian approach provides some structure for the application of subjective judgement to this decision process. To apply this approach, several complex decisions must be made. In this article, we have provided a structure for this decision process.     

Analysis of field failure data when modeled Weibull slope increases with time

Weibull time‐to‐fail distributions cannot be correctly estimated from field data when manufacturing populations from different vintages have different failure modes. To investigate the pitfalls of ongoing Weibull parameter estimation, two cases, based upon real events, were analyzed. First, a time‐to‐fail distribution was generated assuming the same Weibull shape parameter representing an increasing failure rate for each monthly batch or vintage of production. The shape parameter was estimated from simulated field data at regular periods as the population accumulated service time. Estimates of the shape parameter were not constant, but gradually decreased (as had occurred in a real system) with added service time. In the second case, field reliability performance was modeled to match the actual historical data for one product from a disk drive manufacturer. The actual data was proprietary and was not directly available for analysis. A production schedule was modeled with a mix of two failure characteristics. The population reaching the field in the first 12 months had a low, constant failure rate. For the second and third years of production, higher volumes were introduced that had the higher, increasing failure rates of the first case. Assessment of the mixed population at each month of calendar time resulted in an increasing Weibull shape parameter estimate at each assessment. When the two populations were separated and estimated properly, a better fit with more accurate estimates of Weibull shape parameters resulted. Copyright © 2010 John Wiley & Sons, Ltd.     

Maximum-Likelihood Estimation of the Parameters of Gamma and Weibull Populations from Complete and from Censored Samples

Iterative procedures are given for joint maximum-likelihood estimation, from complete and censored samples, of the three parameters of Gamma and of Weibull populations. For each of these populations, the likelihood function is written down, and the three maximum-likelihood equations are obtained. In each case, simultaneous solution of these three equations would yield joint maximum-likelihood estimators for the three parameters. The iterative procedures proposed to solve the equations are applicable to the most general case, in which all three parameters are unknown, and also to special cases in which any one or any two of the parameters are known. Numerical examples are worked out in which the parameters are estimated from the first m failure times in simulated life tests of n items (m ≤ n), using data drawn from Gamma and Weibull populations, each with two different values of the shape parameter.     

Reliability of a Weibull analysis using the maximum-likelihood method

We have performed extensive Monte-Carlo computer simulations of the 2-parameter Weibull statistical distribution using data groups with sizes from 5 up to 100 samples. The maximum-likelihood method was used to evaluate the resulting Weibull modulus and the scale parameter, which may be different to the input values. We confirmed some trends in the evaluation of the statistical parameters for small data groups, such as a significant biasing of the Weibull modulus. We revealed the log-normal statistical distribution of the Weibull parameters obtained from repeated Monte-Carlo simulations for several groups. We also considered the influence of the measurement uncertainty on the determination of the statistical parameters. For the experimental evidence we used bend-strength data for alumina test samples from serial production in this year. The experimental data were randomly divided into several subgroups to compare the corresponding biasing of the Weibull modulus with the Monte-Carlo results.     

Statistical analysis of compression strength of single grains of SHM powders and improvement of the strength assessment procedure

A new approach to strength assessment of diamond powders has been proposed. It is based on the statistical theory of extremes and provides the small-sampling prediction of strength properties. A comparative statistical analysis of several powder batches has been performed. The strength of grains is shown to be satisfactorily approximated by a Weibull distribution rule over the entire range of grain sizes and grades considered. The relation between the Weibull parameters and the average strength has been studied. The convergence of empirical statistics is investigated and the criterion of sufficiency of a sample size for the assessment of average compression strength of diamond grains is stated. The authors have found the possibility to reduce the number of tests, thus significantly shortening the testing duration and minimizing the anvil wear, which results in lower investigations costs. These findings provide a theoretical basis for improving the existing methods of quality inspection of SHM powders and elaborating new methods.     

A probabilistic analysis on thickness effect in fracture toughness

Fracture toughness values are often influenced by specimen thickness and they indicate generally decreasing toughness with increasing thickness. In the present paper, a probabilistic analysis has been carried out by using various kinds of toughness data in order to clarify the applicability of the weakest link model to thickness effect in fracture toughness. Moreover, a new statistical method is proposed for determining fracture toughness distribution, which is necessary for the above analysis, with taking the temperature dependency of fracture toughness into account. Thickness effect in fracture toughness is brought about by its statistical nature and the weakest link model can be applied to evaluate the thickness effect for both steel plate and its welds with heterogeneity in toughness. This thickness effect is considerably affected by Weibull shape parameter and the probability of cleavage fracture for the material. The statistical method proposed newly in this paper is sufficiently applicable and superior to the conventional method. By using this new method, Weibull parameters at a temperature of interest can be determined with much the same reliability as in the conventional method, and also Weibull parameters at lower and higher temperatures can be obtained with a certain confidence depending on the number of specimens tested.