Bayesian inference to determine elastoplastic parameter distributions

Material parameters identified by mechanical tests can vary from one specimen to another. This variability is often caused by the variability of the small-scale structures of the different specimens. Examples are geometrical variations, the variations of crystallographic orientations, and the variations of the number amount of defects present. The variability of material parameters can be described as a probability density function (PDF) as a function of the parameters. One possible way to identify such a PDF is to test numerous specimens but this entails a substantial amount of experimental efforts. In this contribution, we employ Bayes’ theorem to only test a relatively small number of specimens and use their results to infer the parameters of an initially assumed distribution. Besides avoiding an enormous amount of experimental work, a convenient result of this approach is that the distribution comes with an uncertainty in terms of the parameters. This entails that it is relatively clear how certain the obtained distribution is and how the uncertainty must be propagated, if of interest. The use of Bayesian inference results in a probability density function (PDF), a so-called posterior distribution, as a function of the quantity of interest (i.e. the parameters that describe the distributions for the material parameters). The statistical properties of the PDF, e.g. the mean parameter values, the values at which the PDF is maximum and their correlations, can be obtained by analysing the posterior distribution. In this presentation we demonstrate the approach for both elastic and elastoplastic materials which harden nonlinearly. As elastoplastic materials are characterised using several material parameters, the coupling between these parameters in the PDF will receive special attention. This coupling is incorporated via a so-called copula, which will receive special attention in this work. We will for instance present how our results change when the assumed copula is different from the one used to create the experimental measurements