Bayesian analysis for determination and uncertainty assessment of strength and crack growth parameters of brittle materials

  • a Robert Bosch GmbH, Automotive Electronics, Tübinger-Str. 123, 72762 Reutlingen, Germany
  • b Robert Bosch GmbH, Corporate Sector Research and Advanced Engineering, 70465 Stuttgart, Germany
  • c German Aerospace Center (DLR), Institute of Materials Research, Linder Höhe, 51147 Cologne, Germany


To determine the strength and the crack growth parameters of brittle materials, the common approach is to first evaluate the strength and then the crack growth parameters. When the parameters are needed for different temperatures, the procedure is repeated for each temperature of interest. Since the number of test specimens is generally small, the separated analysis of strength and crack growth parameters for each tested temperature leads to large parameters uncertainty. In order to improve the accuracy of the parameters, we propose a Bayesian method that allows to combine all strength and lifetime data obtained at different temperatures and determine the distribution of all material parameters in a single analysis. The results obtained from the analysis of measured skutterudite data show that in comparison to the standard approach the presented method significantly reduces the material parameters uncertainty and therefore is well adapted for a reduced number of samples.


  • Bayesian analysis;
  • Crack growth;
  • Fatigue;
  • Strength;
  • Weibull analysis

1. Introduction

In component design, the strength and the crack growth parameters of brittle materials are relevant material properties. The strength parameters allow to forecast and avoid spontaneous failures whereas the crack growth parameters are used to predict the failure probability over the lifetime. Due to the inherent presence of statistically distributed flaws in brittle materials, these parameters are subjected to large uncertainty and required a large number of test specimens as well as statistical analysis in order to reduce their uncertainties and obtain accurate results. For the strength parameters, it is common to perform strength tests and derive the parameters from Weibull analysis. For the determination of crack growth parameters different methods can be used [1]. One of the common used approach, which we also used in this work is the Weibull analysis of lifetime data.

In many practical applications one is interested in the material properties over a wide temperature range. In such applications, the common procedure is to independently determine the material parameters for each temperature levels. This standard approach has two major drawbacks. First, it requires a large number of test specimens to get reliable results and second, it does not consider that some parameters are temperature independent according to the standard theory of fracture mechanics. For the strength analysis, pooling procedures based on regression models have been developed in order to take into account strength data from different temperature for the Weibull analysis and consequently reduce the required number of test specimens [2] ;  [3]. In the pooling approach, strength from different temperature levels are converted by a regression model to some corresponding strength at a reference temperature. This leads to loss of information since the temperature dependency of the strength cannot be exactly represented by an equation.

In this work, we present a novel method based on Bayesian networks, which allows the combined analysis of data from different temperature levels as well as the consideration of the temperature dependence of the parameters. The method does not require any kind of data transformation and can take into account strength as well lifetime parameters at the same time. Using Bayesian networks, the relationships between the material parameters and the measured data can easily be identified and based on probabilistic analysis, the distribution of all the parameters (strength and lifetime parameters) can be determined in only one step. The presented approach relies on fracture mechanic laws and on the Bayesian analysis.

In Section 2, theories about analysis of strength and lifetime data is presented. Section 3 gives a short introduction to Bayesian network. Using some measured strength data at RT and 500 °C as well as cyclic lifetime data at RT, the application of the novel Bayesian method is presented in Section 4. A comparison to the standard approach is also performed.

2. Mechanical properties of brittle material

2.1. Strength parameters

Fracture of brittle material results from large stress concentrations at microscopic flaws, which are unavoidably present in the form of pores, inclusions and precipitations. When a brittle specimen is loaded, stress singularities occur at crack tips. To describe these singularities, one defines the stress intensity factor K [1]:

View the MathML source
where the “geometry factor” Y depends on the crack size a, the geometry of crack, specimen and stress field. If the stress increases, the stress intensity factor also increases. This happens until K reaches a critical value Kc at which the specimen fails. This critical value is a material property called fracture toughness. The corresponding critical crack length is given by [4] ;  [5]:
View the MathML source
Since Kc is a material constant, the strength of a brittle component depends principally on the size of defects present in the component. The scatter of the flaws size results, therefore, in a scatter of the strength. Knowing that small flaws are not relevant for failure, the density function of flaws f(a) can be approximated by a power law:
View the MathML source

Based on this equation and on probabilistic derivation, it can be shown that the failure strength of a brittle component follows the Weibull distribution:

View the MathML source
where V0 is a normalising volume. The parameters m and σ0 denote the Weibull modulus and characteristic strength, respectively. m is a measure of the scatter of strength data and is related to the scatter of flaw size by the relation m = 2(r − 1) [1] ;  [5].