Stress-dependent determination of the Weibull shape parameter for reliability prediction

A reliability prediction is essential for developing reliable products. The manufacturer should know as early as possible if the current design of its product can achieve the reliability target in the field. If crucial changes are implemented too late, considerable unforeseen costs result. Thus a first assessment of the product´s failure behavior, which describes the reliability at a given lifetime, is necessary in early development stages. For describing the failure behavior, the Weibull distribution is often used due to its flexibility. Usually, endurance-strength calculations and expert knowledge are available to derive the distribution parameters. If lifetime data from previous products is available or prototype tests have been conducted, a credible assessment is possible. However, a reoccurring problem is that existing lifetime data is not obtained at design stress of the actual development. They are rather obtained at various higher stress levels. Analyzing these data with existing models can result in inappropriate reliability predictions, especially for the failure mechanism fatigue, where the Weibull shape parameter is known to be stress-dependent. This dissertation aims to develop a methodical approach for a systematically stress-dependent determination of the Weibull shape parameter for reliability prediction considering all available sources of information. First of all, the Weibull distribution is introduced in-depth. Known influences on the shape parameter are gathered with focus on the influence of load. Existing models which partly address the stress-dependency of the shape parameter are examined. Most of these models have an underlying two parametric extreme value, log-normal or Weibull distribution with a log-linear relationship for the spread. The fatigue mechanism is elaborated regarding the material and statistical aspects. The obtained results are combined and concisely summarized: Due to the higher scatter, the higher number of cycles as well as the stronger growth rate increase of the crack initiation period - in contrast to the crack growth period -, an increasing stress leads to a higher shape parameter. Known concepts and methods, which allow for the determination of a shape parameter, are analyzed. The optimization potential is identified, which lies in a stressdependent modelling with respect to all three Weibull parameters, a differentiated field data analysis concerning actual stress and a quantified expert judgment concerning quality. Moreover, separate concepts and methods can be combined in order to increase the overall quality of the determined failure behavior. Thus, extended and new approaches are developed which address these issues. A differentiated field data approach yields a stress-dependent derivation of the Weibull shape parameter based on field data. In order to do so, simulations of the customer behavior and additional information from the customers themselves are used. As a result, linking the occurred failure with the corresponding stress-level is possible. Unknown stress-dependencies can be identified or differentiated data can be used for stressdependent analyses. Extended Weibull lifetime models are developed as a crucial part of these stressdependent analyses. Stress-dependent models comprise all three Weibull parameters to enable an adequate reliability prediction at design stress. The models are validated by means of three data sets with different stress-dependencies. A conducted simulation study highlights the wide applicability of the developed models: In most cases the new developed models are favored. Only if the failure free time is much smaller than the scale parameter, the existing models are better assessed. The more the Weibull density is right-skewed at observed stress levels, the more favorable the performance of both developed models. One of the developed models is restricted to failure mechanisms with an increasing failure rate, as fatigue, whereas the other one is unrestricted. Furthermore, a procedure is proposed based on machine learning, which empowers to quantify the probability to be an expert. This probability depends on defined attributes, e. g. work experience or publications, and is used as a measure of confidence. Finally, a confidence interval can be assigned systematically to the expert statement. The existing and developed approaches are combined in a holistic procedure, based on a Bayesian approach. Various sources of information, such as lifetime data, calculation results or expert knowledge, are taken into account and are classified in data and experience. Different sources of lifetime data are transformed to actual design stress and then integrated in the likelihood function considering the non-identical population. Algorithms for different scenarios are illustrated. The independent prior distributions of the Weibull parameters depict the available information from experience. If no data is available the procedure simplifies to a consideration of information from experience and a subsequent Monte-Carlo simulation is needed. The pragmatically holistic procedure finally leads to a systematic and comprehensive stress-dependent reliability prediction with respect to a stress-dependent Weibull shape parameter. A synthetic example substantiates the holistic procedure. The procedure is applied to various scenarios with different conditions regarding available information. The influence of parameters such as the transformation factor and expert performance is shown. This study introduces an elementary procedure in order to take into account a stressdependency on the one hand and to integrate parameter specific pre-knowledge on the other hand. New or extended approaches can be easily implemented. Further studies can focus on the determination of customer types, to ensure the adequate assignment of field data. A practical verification of the holistic procedure may be beneficial